Shouchang Guo^{1}, Douglas C. Noll^{2}, and Jeffrey A. Fessler^{1}

Oscillating steady state (OSS) imaging is a new fMRI acquisition method that substantially improves SNR by exploiting a large and oscillating signal. However, the oscillation nature of the signal leads to an increased number of acquisitions. To improve the temporal resolution and address the nonlinearity of the OSS signal, we propose a novel dictionary-based regularization method for OSS reconstruction to reconstruct dramatically undersampled (e.g. R = 12) data. The proposed method leads to better image quality than CG-SENSE and does not require any temporal filtering like low-rank methods, therefore the undersampling directly leads to an improved fMRI temporal resolution. The high SNR advantage of OSS is also well preserved.

OSS fMRI data has two time dimensions due to the acquisition pattern.The oscillation dimension is called "fast time" ($$$t_f$$$), and the dimension corresponding to the fMRI time course is called "slow time" ($$$t_s$$$). We model the fast time dimension with parameters $$$m_0,\ T_2$$$, and $$$\Delta f$$$ in a voxel-by-voxel manner, where $$$m_0$$$ captures the signal magnitude, $$$T_2$$$ accounts for the tissue properties, and $$$\Delta f$$$ denotes off-resonance frequency due to $$$B_0$$$ field inhomogeneity. We exclude $$$T_1$$$ because the dominant effect is an approximate scaling on the apparent $$$m_0$$$. (See Fig. 2)

The proposed reconstruction problem that incorporates the OSS signal model is:

$$\hat{\mathbf{X}}_s =\underset{\mathbf{X}_s \in \mathbb{C}^{N_x \times N_x \times N_f}}{\operatorname{argmin}} \ \frac{1}{2}\lVert\mathcal{A}(\mathbf{X})-\mathbf{y}_s \rVert_2^2+ \beta \sum_{x,y,t_s} R\left(\mathbf{X}\left[x,y,:\right]\right), \quad s = 1,\ldots,N_s,$$

$$R(\mathbf{v}) = \underset{m_0,T_2,\Delta f}{\operatorname{min}}\ \lVert \mathbf{v}-m_0\Phi (T_2,\Delta f)\rVert_2^2,$$

where $$$\mathbf{X}_s \in \mathbb{C}^{N_x \times N_y \times N_f}$$$ represents the images at the $$$s$$$th of $$$N_s$$$ time points to be recovered from limited k-space measurements $$$\mathbf{y}_s$$$, $$$\mathcal{A}(\mathbf{\cdot})$$$ is a linear operator presenting the MRI physicsconsisting of the non-uniform Fourier transform and coil sensitivities,and $$$\mathbf{v} \in \mathbb{C}^{N_f}$$$ is a vector of fast-time image values for one particular voxel. The fast-time signal model is denoted $$$\Phi(T_2,\Delta f) \in \mathbb{C}^{N_f}$$$. The regularization parameter $$$\beta$$$ was selected based on the condition number of the system matrix. We alternate between updating the image $$$\mathbf{X}$$$ and the dictionary based regularizer. The minimization of the parametric prior is a nonlinear estimation problem, and we solve it using the variable-projection method ^{3} and a signal dictionary. Specifically, we construct the dictionary with signals generated from a discretized range of parameters $$$T_2,\Delta_f$$$ using Bloch simulations, and estimate the 3 parameters through grid search. The $$$\mathbf{X}_s$$$ update is a quadratic least-squares problem solved by the conjugate gradient method and is easily parallelized across temporal frames. We sampled the data with randomized variable-density (VD) spirals and pseudo-random golden-angle rotations between consecutive interleaves and frames. The retrospective undersamplinggives an acceleration factor of 6 compared to fully sampled uniform-density spirals. For prospective undersampling, we acquired only 8% of the fully measurements. The data were collected on a 3T GE MR750 scanner with FOV = 22 cm, matrix size = 168, slice thickness = 2.5 mm. The OSS parameters are TR = 15 ms, nc = 10, TE = min, $$$t_f = 10$$$ and flip angle = 10$$$^\circ$$$. The human functional task is flashing-checkerboard visual stimulus with left and right stimulus.

1. Zhao, Bo, et al. "Maximum likelihood reconstruction for magnetic resonance fingerprinting." IEEE transactions on medical imaging 35.8 (2016): 1812-1823.

2. Assländer, Jakob, et al. "Low rank alternating direction method of multipliers reconstruction for MR fingerprinting." Magnetic resonance in medicine 79.1 (2018): 83-96.

3. Golub, Gene, and Victor Pereyra. "Separable nonlinear least squares: the variable projection method and its applications." Inverse problems 19.2 (2003): R1.