Purpose

Echo-planar time resolved imaging (EPTI) is a valuable tool for distortion-free diffusion and relaxometry imaging ^1-6.Given the large under-sampling factor in the EPTI acquisition and the intrinsic low SNR of diffusion MRI, an SNR-efficient reconstruction is vital. Subspace reconstruction is an SNR-efficient approach by reducing the number of unknowns and thus improving the conditioning of the reconstruction^2,3,6,7. In subspace reconstruction, the selection of bases is critical for reconstruction image fidelity, SNR, and computational efficiency, which can include only magnitude (“magnitude bases”) or both magnitude and phase (“complex bases”). Compared with the magnitude-only bases, complex bases can increase the accuracy of the reconstruction but at the cost of increasing number of bases, which may degrade the SNR performance and computational efficiency. A previous study has proposed to concatenate the subspace reconstructions with complex bases and magnitude bases to ensure the accuracy and SNR simultaneously ⁶. In this study, we aim to explore a new approach to ensure the accuracy and SNR of the subspace reconstruction, i.e., using magnitude-only bases together with a synergistic phase bias updating.

Methods

EPTI Acquisition
Data were acquired on a GE 3.0T Premier MRI scanner, using a 48-channel head coil and a custom diffusion prepared self-navigated EPTI sequence ⁵ (Fig. 1a). The imaging parameters were shown in Fig. 1b. A low-resolution fully sampled EPTI scan (Fig. 1c) was conducted as a calibration scan, to estimate the coil sensitivity map (S_j) and and the B₀ inhomogeneity induced phases at each TE (P_B0,k). A product single-shot EPI (ss-EPI) acquisition was conducted as a reference. In the ss-EPI acquisition, 3 extra b=0 volumes with reversed phase encoding directions were also acquired for distortion correction.
Subspace EPTI Reconstruction
The multi-TE diffusion MRI signal from EPTI acquisition can be formulated as:
$$\mathbf{d}_{i,j,k}=\mathbf{G}_{i}\mathbf{FS}_{j}\mathbf{P}_{B_{0},k}\mathbf{P}_{i}\mathbf{P}_{bias}\mathbf{m}_{k}. \left ( 1 \right )$$
where d_i,j,k is the multi-shot multi-channel diffusion k-space data at TE_k, m_k is the magnitude image at TE_k, Pbias is the phase bias term from potentially dynamic B₀ drift, eddy current and bias of the estimated motion induced phase from the navigator, P_i is the estimated motion induced phase from the navigator, P_B0,k is the B₀ inhomogeneity induced phase at different TE_k (estimated from the time-resolved calibration scan), S_j is the sensitivity map, F is Fourier transform, G is data under-sampling operator of each shot.
The underlying principle of subspace reconstruction is that the multi-TE images m_k (N_echo×1) are strongly correlated temporally and can be represented by a few subspace bases φ (N_echo×N_k) and coefficients c (N_k×1) ^2,3,6,7. Since N_k is much smaller than N_echo, the number of unknowns in the subspace reconstruction drops significantly, which improves the reconstruction conditioning.
The subspace EPTI reconstruction can be written as:
>$$\underset{\mathbf{c}}{min}\left \| \mathbf{G}_{i}\mathbf{FS}_{j}\mathbf{P}_{B_{0},k}\mathbf{P}_{i}\mathbf{\varphi }\mathbf{c} -\mathbf{d}_{i,j,k}\right \|^{2}+\lambda \left \| R\left ( \mathbf{c} \right ) \right \|_{*}. \left ( 2 \right )$$
where the first term is data consistency and the second term is locally low-rank regularization on c, λ is the regularization parameter.
If incorporating P_bias in the subspace approximation, then the subspace basesφ and coefficients c in “complex format”. Alternatively, if excluding P_bias from the subspace approximation (either ignoring P_bias or updating P_bias separately), then the subspace bases φ' and coefficients c' will be “magnitude format”.
Given Eq. 2 is bilinear in terms of P_bias and c, we will explore the alternating minimization with respect to P_bias and c' to solve the above problem. Specifically, in each iteration, we first solve the subproblem regarding c' using locally low-rank constrains (Eq. 2) ^2,3,6,7. Then the subproblem w.r.t. is

$$\underset{\mathbf{\mathbf{P}_{bias}}}{min}\left \| \mathbf{G}_{i}\mathbf{FS}_{j}\mathbf{P}_{B_{0},k}\mathbf{P}_{i}\mathbf{m' }\mathbf{P}_{bias} -\mathbf{d}_{i,j,k}\right \|^{2}. \left ( 3 \right )$$
with m' =abs(φ'c') is the magnitude of φ'c'. Eq. 3 can be solved using gradient descent ⁸ or phase cycling ^9,10.
Image Processing
The ss-EPI images were first preprocessed with FSL TOPUP¹¹ and EDDY¹² for distortion and eddy current correction. DTI metrics (FA and MD) were calculated with the b=0 and 1000 s/mm² data ¹³. Orientation dispersion index (ODI) maps were calculated with b=0, 1000 and 2500 s/mm² data with the NODDI model ¹⁴. For DTI and NODDI calculation, images at 12 adjacent TEs were averaged prior to model fitting for higher SNR purpose. For EPTI data, T2 and T2* maps were estimated using a custom Matlab script.

Results

Fig. 2 shows the FA and color-coded FA comparisons between (a) ss-EPI and (b-d) EPTI acquisition. Three EPTI results with the raw images combined from different TE ranges are shown: (b) from all TEs, (c) from TEs=83.6~93 ms, and (d) from TEs=111.8~121.2 ms. The FA results from EPTI with the new reconstruction match well with the ss-EPI reference both visually and quantitatively (histogram in Fig. 2e).
The MD maps (Fig. 3) and ODI maps (Fig. 4) are also consistent well between EPTI and the ss-EPI. Fitted T₂ and T₂^*maps from EPTI are shown in Fig. 5.

Discussion & Conclusion

Our preliminary results show that the subspace reconstruction with magnitude bases and synergistic phase bias updating can be a promising approach for high-efficacy EPTI reconstruction. Our future work includes comparing this work thoroughly with other EPTI reconstructions. We will also leverage the SNR gains from the EPTI and subspace reconstruction to evaluate more about the TE-dependent diffusion changes.

Acknowledgements

This study is funded by GE Healthcare and the Focused Ultrasound Foundation.