Yuxin Hu^{1}, Qiyuan Tian^{2}, Grant Yang^{1}, Jennifer A McNab^{3}, Bruce Daniel^{3,4}, and Brian Hargreaves^{1,3,4}

We developed a nonlinear model with simultaneous phase and magnitude updates for iterative multi-shot DWI reconstruction. In addition, locally low-rank regularization along the diffusion encoding direction was included in the proposed model to utilize angular correlation for DTI reconstruction. In-vivo high-resolution and high b-value images have been acquired to validate the proposed method and the proposed method significantly reduces image noise.

Introduction

Theory

*Locally low-rank regularization for DTI *

Typically, 30 or even more directions are acquired in DTI to achieve high SNR, which is far more than the minimum required 6 directions, thus some redundancy exists between different directions. Fig. 1a shows how we construct the spatial-direction locally low-rank matrix given the magnitude images of all directions.

*Nonlinear model *

To utilize the angular correlation, motion-induced phase variations between directions and shots need to be considered. The complex MR image can be written in terms of magnitude and phase explicitly, and then the k-space signal for direction d and shot s can be represented as the following,

$$ y_{d,s} = EFS(m_d \cdot e^{j\theta_{d,s}}) + n $$

where E, F and S represent sampling operator, Fourier transform and sensitivity encoding operator, respectively, $$$m_d$$$ represents the magnitude image after diffusion encoding along direction d, and $$$e^{j\theta_{d,s}}$$$ represents the phase of direction d, shot s, which may comes from B0 inhomogeneity, bulk motion and other sources, and n denotes noise.

We developed the following general model for joint DTI reconstruction,

$$\min_{m,\theta} \sum_{d = 1}^{ND}\sum_{s = 1 }^{NS}\frac{1}{2}\left \| E_{d,s}FS (m_{d} \cdot e^{j\theta_{d,s}}) - y_{d,s} \right \|_{2}^{2} + \lambda_{1}g_m(m) $$

where $$$ND$$$ is the number of diffusion encoding directions, and $$$NS$$$ is the number of shots. The first term in the cost function is about the forward model. The second term is a regularization term to utilize the correlation between images from different directions. In this work, we used LLR regularization as described aboe.

We use alternating minimization with respect to the magnitude and phase separately to solve the above problem^{[7]}. The phase of each shot and each direction can be updated independently, and the proximal gradient method is used to update magnitude. Fig. 1b shows the pipeline of the reconstruction using SENSE reconstruction as initialization^{[8]}.

*Experiment design and data acquisition *

With IRB approval, data were acquired in four volunteers on a 3T GE Signa Premier scanner using a 2D DW EPI sequence and a 48-channel head coil. Three experiments were designed to demonstrate the feasibility of the proposed method (experiment 1), test the performance of the proposed method on high-resolution data (experiment 2), and high-b-value data (experiment 3), respectively. The scan parameters were shown in Table 1.

*Image reconstruction and processing *

Images were reconstructed with both the product MUSE^{[9]} and the proposed method (sensitivity maps from non-diffusion-weighted data using ESPIRiT^{[10]}). Diffusion data were processed using FSL's eddy function^{[11]}. DTI and BEDPOSTX models were fitted using FSL's dtifit and bedpostx functions.

Results and discussion

Figure 1 shows the results with 1.2mm isotropic resolution from product MUSE and the proposed method. The FA encoded V1 map from the proposed method is consistent with that from MUSE, which demonstrates the feasibility of the proposed method. The decreased noise level in the proposed method benefits from the fact all directions are reconstructed jointly, and a LLR regularization term is used to utilize their correlation.

This improvement is more significant when the number of encoding directions and resolution are increased (Fig. 4). Figure 4 shows a representative slice reconstructed by these two methods under different resolutions. The DWIs reconstructed by MUSE (Fig. 4 rows b, d and f, columns i-iii) are very noisy, which results in unusable V1 maps (Fig. 4 rows b, d and f, column iv). While the proposed method shows markedly reduced noise level, and the reconstructed DWIs and V1 maps are clearly visually better (Fig. 4 rows a, c and e). The proposed method shows similar improvements for high b-value images as shown in Fig. 5 with resolution 1mm isotropic and b-value 2000.

Thank Frank Ong for sharing the code.

Research support from R01-EB009055, P41-EB015891 and GE Healthcare.

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Figure 1. a) Construction of spatial-direction locally
low-rank matrix from images of all diffusion-encoding directions. b)
Reconstruction pipeline of the proposed method for multi-shot DTI data with
SENSE as initialization.

Table 1. Acquisition parameters for three experiments.

Figure 2. 3 out of 30 directions 4-shot DW
images (columns i-iii) and corresponding FA encoded V1 maps (column iv)
reconstructed by MUSE (row b) and the proposed method (row a) from experiment 1
(resolution = 1.2 mm iso., b-value = 1000 $$$s/{mm}^{2}$$$).

Figure 3. 3 out of 75 directions 4-shot DW images (columns i-iii) and corresponding FA encoded V1 maps (column iv) reconstructed by MUSE (rows b, d and f) and the proposed method (rows a, c and e) from experiment 2 (resolution = 0.9/0.8/0.7 mm iso., b-value = 1000 $$$s/{mm}^{2}$$$). The windows of V1 maps are tuned separately for better visual quality (0-1 for MUSE, and 0-0.8 for proposed method). The proposed method shows significantly reduced noise level.

Figure 4. 3 out of 75 directions 4-shot high b-value DW
images (columns i-iii) and corresponding f1 encoded Dyads1 maps (column iv)
reconstructed by MUSE (rows b) and the proposed method (rows a) from experiment
3 (resolution =1 mm iso., b-value = 2000 $$$s/{mm}^{2}$$$).