Sugil Kim^{1,2}, Suhyung Park^{2}, and Jaeseok Park^{2}

MR parameter mapping has been potentially of great value in diagnosing pathological diseases, but is difficult to be translated to clinical applications due to prohibitively long imaging time. It was recently shown in [1-4] that simultaneous multi-slice (SMS) imaging is highly efficient in reducing imaging time while well maintaining SNR. In this work, we propose a novel, model-based SMS reconstruction approach with Hankel subspace learning (Model-based SMS-HSL) for highly accelerated MR parameter mapping under the hypothesis that the null space in the spatial dimension, which filters out slices of no interest, is time-invariant in the parameter dimension while the dimension of temporal basis, which is found from signal evolution models, is limited.

MR parameter mapping has been potentially of great value in diagnosing pathological diseases, but is difficult to be translated to clinical applications due to prohibitively long imaging time. It was recently shown in [1-4] that simultaneous multi-slice (SMS) imaging is highly efficient in reducing imaging time while well maintaining SNR. In this work, we propose a novel, model-based SMS reconstruction approach with Hankel subspace learning (Model-based SMS-HSL) for highly accelerated MR parameter mapping under the hypothesis that the null space in the spatial dimension, which filters out slices of no interest, is time-invariant in the parameter dimension while the dimension of temporal basis, which is found from signal evolution models, is limited. With increasing SMS factors up to 5, the proposed, model-based SMS-HSL outperforms conventional split-slice-GRAPPA in accurately estimating T1 relaxation times with relatively low artifacts and noise, potentially enabling whole-brain T1 mapping within 64sec.

**
SMS Signal Model in
k-Parameter Dimension: **Since
SMS signals in k-space can be represented by linear superposition of all
excited slices, MR signals in k-parameter dimension is denoted in a Casorati
matrix form using the Hankel operator by:

\begin{equation} \begin{split} \mathcal{H}\mathrm{\left ( \textbf{d} \right)}= {\sum_{s=1}^\mathrm{N_s}}\mathcal{H} {\left ( vec\textbf({X}_s \right )) } +\textbf{N},\;\;\;\textbf{X}_s=\begin{bmatrix} \rho_{s}(k_{1},t_{1}) & \cdots & \rho_{s}(k_{1},t_{M}) \\\vdots & \ddots & \vdots \\ \rho_{s}(k_{n},t_{1})&\cdots & \rho_{s}(k_{1},t_{M}) \end{bmatrix}\in \mathbb{C}^{\mathrm{N} \times \mathrm{M}}\end{split} \end{equation}

where, **H**(·) is the Hankel operator, **d
**is the measured SMS signals in k-space, **X _{s}** is the
desired k-space,

**Model-Based SMS-HSL
Reconstruction for T1 Mapping: **It is assumed that the signal evolution of **Xs** along the parameter dimension follows
an exponential increase by T1 relaxation times. Although the temporal signal
evolution is non-linear, we hypothesize that it can be delineate by linear
superposition of the temporal basis that is found by performing singular value
decomposition (SVD) of the model-based simulation data. It is noted that the temporal signal variation can be
synthesized using only a few principal eigenvectors in **V _{t}**.

$$\begin{equation} \begin{split} \label{eq:contSMSSPASE} \hat{\textbf{U}}_\mathrm{s} = \mathrm{arg} \; \underset{\textbf{U}_\mathrm{s}}{\mathrm{min}} \; \| \mathcal{H}\mathrm{\left ( \textbf{d} \right)} - {\sum_{s=1}^\mathrm{N_s}}\mathcal{H} {\left ( vec(\textbf{F}\textbf{U}_\mathrm{s}\textbf{V}_\mathrm{t} \right )) } \| + \frac{\mathrm{\lambda_L} }{2} \left \| \left ( \mathcal{H}\mathrm{\left ( \textbf{d} \right )}-\mathcal{H}\mathrm{\left ( \textbf{F}\textbf{U}_\mathrm{s}\textbf{V}_\mathrm{t} \right )} \right ) \mathrm{\mathcal{N}^c_s} \right \|_{\mathrm{F}}^{2} + \mathrm{\lambda_L} \left \| \mathcal{H}\mathrm{\left ( \textbf{F}\textbf{U}_\mathrm{s}\textbf{V}_\mathrm{t} \right )} \right \|_* \end{split} \end{equation}$$

where **U _{s}** is
the spatial coefficient matrix,

**Simulations and Experiments:** To evaluate the feasibility of the proposed model-based
SMS-HSL in highly accelerated T1 mapping, multiple sets of data were acquired
using multi-point inversion recovery 2D EPI pulse sequence. Imaging parameters
were: TR/TE=20000ms/27ms, FOV=220x220 mm2,
matrix size=128x128, slice thickness=3mm, 16 inversion times with 34, 100, 200,
300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1400, 1600, 1800, 2000ms, and
partial Fourier factor=5/8, CAIPI factors=1/5 FOV shift. With increasing SMS
factors, all images were reconstructed using conventional SP-SG and model-based
SMS-HSL for comparison, and parametric T1 maps were correspondingly found
voxel-wise by nonlinear fitting.

ICT&Future Planning (2016M3C7A1913844)

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4. Suhyung Park, Jaeseok Park. SMS-HSL: Simultaneous Multi-Slice Aliasing Separation Exploiting Hankel Subspace Learning. ISMRM 24th Annual Meeting & Exhibition, Singapore. ,0610