Introduction

Standard quantitative MRI (qMRI) techniques rely on a two-step process whereby undersampled k-space data are reconstructed first, and then used in model fitting to estimate parameters of interest. Model based approaches [1,2] have been developed to incorporate mono-exponential signal models into the reconstruction, so that parameter maps can be directly estimated from undersampled data.

In this work, we propose a Non-linear Conjugate Gradient (NLCG) optimization to solve the arising optimization problem and use a scan-specific Neural Network as regularizer. Experiments show the ability of the proposed NLCG-Net to improve T1 and T2 mapping relative to subspace modeling at high accelerations, while obviating the need for an external training dataset.

Methods

·Problem Formulation
We formulate qMRI reconstruction as an optimization problem by solving the following objective function:
$$\mathop{\arg\min}\limits_{\boldsymbol{x}} \left\| \mathbf{PFCM}(\boldsymbol{x}) - \boldsymbol{y} \right\|^2 + \lambda \left\| \boldsymbol{x} - \boldsymbol{z}\right\|^2\text{(1)}$$

where

$$\boldsymbol{x} = \begin{bmatrix} \boldsymbol{M_x} \\ \boldsymbol{M_y} \\ \boldsymbol{R} \end{bmatrix}= \begin{bmatrix} \boldsymbol{M_x} \\ \boldsymbol{M_y} \\ \boldsymbol{\frac{1}{T}} \end{bmatrix}$$

Here M_x, M_y real and imaging components of transverse magnetization, R refers to R₁ or R₂ representing parameter values. y denotes acquired kspace data, z refers to regularized x and λ is regularization coefficient. PFCM are forward operators illustrated in Fig.1. M denotes the mono-exponential signal model, which has different expressions for T1 and T2 mapping; C denotes coil sensitivity maps, F denotes Fast Fourier transform, and P denotes k-space sampling mask.

To solve this, we can unroll the target function and solve it separately and iteratively. Specifically[3],
$$\boldsymbol{z_n} = \boldsymbol{R}(\boldsymbol{x_n}) (2)$$
$$\boldsymbol{x_{n+1}} = \mathop{\arg\min}\limits_{\boldsymbol{x}} \left\| \text{PFCM}(\boldsymbol{x}) - \boldsymbol{y} \right\|^2+ \lambda \left\| \boldsymbol{x} - \boldsymbol{z_n}\right\|^2 (3)$$
Where R( ) refers to regularization operation by Neural Network.

·Proposed Model
We propose NLCG-Net to solve the iterative optimization problem. The detailed architecture of NLCG-Net is presented in Fig.2. NLCG-Net directly takes k-space data as input, and unrolls the optimization into several unroll blocks. Each block has one regularization layer and one data consistency (DC), which correspond to (2) and (3). For the regularizer, we deploy a light U-Net model with only three downsample and upsample layers to restrict the parameter number and facilitate the training. In the data consistency layer, we deploy NLCG to solve (3).

·Implementation Details
NLCG-Net contains 3 Unroll blocks. A DC layer performs 20 iterations with NLCG and takes x from the last layer as iteration initialization. In the beginning, input x is acquired by performing NLCG without regularization to k-space input for 800 iterations, which can be done within 1 minute. To achieve zero-shot training, we use different sampling masks to divide data into training and validation sets, and the splitting strategy is the same as [4]. Before feeding into the network, Mx, My, and R are scaled[5] by 3 trainable parameters due to their different dynamic ranges[6]. Overall training process was performed on an NVIDIA 4090 GPU, and 300 epochs were finished within 5 hours.

·Experiments
We used a set of fully sampled spin echo multi-coil data with FOV 256 × 208 for T2 mapping. 8 echo data are deployed from TE = 23ms to 184ms, with an echo spacing of 23ms. We simulate Acceleration by creating sampling masks, with acceleration factor R = 4 and 6. Central ACS region was used with width = 24. We also validated T1 mapping on in vivo data, which has been prospectively accelerated by R = 2. 5 sets of inversion recovery TSE data with TI from 35ms to 3000ms were acquired. We furthermore simulated the R = 4 condition without utilizing ACS region data for reconstruction. For metric consideration, we used normalized root mean squared error (NRMSE).

Results

We present T2, T1, and corresponding M₀ map reconstruction results in Figs. 3-5. For comparison, we also presented T2 estimation using SENSE and Subspace. We observe that for T2 mapping, unregularized NLCG can readily achieve a good fit when considering NRMSE. However, as R increases, aliasing artifacts emerge, which were better mitigated using NLCG-Net. The proposed model could still retain the lowest NRMSE and address artifacts. For T1 mapping, it is observed that NLCG-Net has a similar performance.

Discussion and Conclusion

We proposed a model-based zero-shot self-supervised learning framework, NLCG-Net for qMRI reconstruction, which is able to reach high acceleration factors with high fidelity. Its nonlinear estimation is flexible enough for both T2 and T1 mapping, and iterative optimization problem formulation allows neural network regularization while obviating the need for an external training datasets.

Acknowledgements

No acknowledgement found.