Zhe Wu^{1}, Hongjian He^{1}, Yi Sun^{2}, and Jianhui Zhong^{1,3}

The simultaneous estimation of images and coil sensitivities using joint nonlinear inversion (JNLINV) has been shown to be effective for

The joint estimation of images and coil sensitivities from undersampled k-space data using nonlinear inversion is recently proposed and demonstrates that no calibration is required for multi-echo and multi-contrast parallel imaging acquisition. This joint nonlinear inversion (JNLINV) method [1], derived from the previously proposed NLINV approach [2], takes advantage of the fact that coil sensitivities are independent of echoes or contrasts and achieves one-sixth RMSE of that with NLINV. This calibrationless method has been successfully applied for accelerating many protocols with a negligible RMSE, including multi-echo, phase-cycled bSSFP sequences, and some multi-contrast protocols [1].

This study proposes a grouped JNLINV (gJNLINV) method that achieves about a 3-fold reduction in reconstruction time with a similar RMSE as JNLINV for the multi-echo data with a long echo train. We also demonstrate the application of this method for myelin water imaging using multi-echo GRE data.

Under-sampled data $$$y$$$ from the images $$$\rho$$$ with coil sensitivities $$$c$$$ and k-space sampling pattern $$$P$$$ are represented by $$$y=PF(c\rho)$$$, where $$$F$$$ is the Fourier transform. It is simplified as a nonlinear operation $$$y=N(x)$$$ in the JNLINV method. The unknown $$$x$$$ contains both images and coil sensitivities: $$$x=[\rho,c]$$$, and can be computed by solving the update steps $$$dx=[d\rho,dc]$$$ using a linear approximation through the Jacobian $$$D$$$ (or the Fréchet derivative) [2] at the $$$n$$$th update step of $$$x$$$ with coil terms and an $$$l_{2,1}$$$ norm of image gradients as the penalty terms:

$$min_{dx}||DN(x^n)dx+N(x^n)-y||_2^2+\alpha^n||W(c^n+dc)||_2^2+\beta^n||\nabla(\rho^n+d\rho)||_{2,1}\qquad\qquad(1)$$

where $$$n$$$ is the index of the current update step, $$$W$$$ is a low-pass filter for smoothing the coil sensitivities, $$$\nabla$$$ is a spatial gradient operator, and $$$\alpha$$$, $$$\beta$$$ are coefficients of penalties and are reduced 2-fold in each outer iteration of the $$$dx$$$ updates.

For the under-sampled multi-echo data with a large echo train (e.g., 32 echoes or more), the direct application of the standard JNLINV is time consuming due to the substantial number of unknowns. We propose the gJNLINV to accelerate the reconstruction of all the echoes:

(i) Apply JNLINV to the first few echoes for coil sensitivities;

(ii) Divide the remaining echoes into groups which are processed using parallel computation to save reconstruction time. The coil sensitivities obtained from step (i) are subsequently applied in the operation $$$N$$$, and the coil terms are removed from both $$$x$$$ and Eq. (1), i.e., $$$x=[\rho]$$$, and the objective function for the $$$dx$$$ optimization becomes:

$$min_{dx}||DN(x^n)dx+N(x^n)-y||_2^2+\beta^n||\nabla(\rho^n+d\rho)||_{2,1}$$

The multi-echo GRE data were acquired on a 3T MAGNETOM Prisma (Siemens Healthcare, Erlangen, Germany) using 16-head elements of a 20-channel head-neck coil. The sequence parameters were as follows: 32 echoes, TE = 2.7 : 1.5 : 49.2 ms, TR = 62 ms, voxel size = 0.94 × 0.94 × 2 mm^{3}, and acquisition matrix = 256 × 256 × 64.

Reconstruction was performed on a server (Xeon E5-2680, 256 GB memory) with MATLAB R2014a. All 32 echo images were evenly divided into eight echo groups (i.e., four echoes in each group) and were manually 2 × 2 under-sampled. Echoes 1 to 4 were processed using JNLINV for coil sensitivities, and the remaining echo groups were reconstructed using the gJNLINV through parallel computing. $$$\alpha^0=0.05$$$, $$$\beta^0=0.1$$$.

Myelin water fraction (MWF) maps were calculated from all the reconstructed images with 32 TEs. The data was fitted to a three-pool model [3]:

$$s(TE)=A_{my}e^{-TE/T_{2,my}^*}+A_{ma}e^{-TE/T_{2,ma}^*}+A_{mx}e^{-TE/T_{2,mx}^*}$$

where $$$A$$$ is the signal amplitude of each component ($$$my$$$: myelin, $$$ma$$$: axon, $$$mx$$$: mixed pool). The MWF was calculated as $$$MWF=A_{my}/(A_{my}+A_{ma}+A_{mx})$$$

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5. Prasloski T, Rauscher A, MacKay AL, Hodgson M, Vavasour IM, Laule C, Madler B. Rapid whole cerebrum myelin water imaging using a 3D GRASE sequence. Neuroimage. 2012;63:533–539.

Fig.
1. Reconstructed 2 × 2 accelerated 32-echo mGRE images and coil sensitivities
using standard JNLINV simultaneously with the data from all 32 echoes. The long
computation time is prohibitive for most clinical applications. Note the
discontinuities in the coil maps (low intensities indicated by the yellow arrows),
demonstrating the impaired fidelity of coil estimation.

Fig. 2. Reconstructed 2 × 2 accelerated 32-echo mGRE images and coil sensitivities using the proposed gJNLINV. The coil maps are estimated from the first four echoes, which are among the highest SNRs of all the echoes, and the remainder of the echoes were estimated using these coil sensitivities. This avoids the discontinuities of the coil sensitivities demonstrated in Fig. 1.

Fig. 3. Myelin water fraction maps calculated from data with full acquisition, proposed grouped JNLINV, and standard JNLINV. The results are similar: the difference maps between the two JNLINV methods and the full acquisition demonstrate mostly a noise-like pattern.