Simultaneous Multi-Slice Spiral-CEST Encoding with Hankel Subspace Learning: ultrafast whole-brain z-spectrum acquisition

Suhyung Park^{1}, Sugil Kim^{1,2}, and Jaeseok Park^{3}

**1) SMS-HSL Reconstruction:** The proposed SMS scheme models the SMS Hankel-structured matrix as the linear
superposition of Hankel matrices of all excited slices $$$\bf{N_s}$$$ :

$$\bf{\mathcal{H}\left ( y\right ) = \sum_{m=1}^{N_s} \mathcal{H}\left ( x_m \right ) + N}$$

where $$$\bf{\mathcal{H}\left (\cdot\right )}$$$ is the Hankel
operator, $$$\bf{y}$$$ is the measured SMS signals in k-space, $$$\bf{x_m}$$$ is the
desired k-space at the m^{th} slice, and $$$\bf{N}$$$ is the additive noises. To selectively
estimate the slice of interest while nulling the other slices, Hankel composite
matrix is constructed by combining all slices other than a slice of interest $$$\bf{\mathcal{H}\left ( x_s^c \right ) = \sum_{m=1}^{N_s} \mathcal{H}\left ( x_m \right ) - \mathcal{H}\left ( x_s \right )}$$$, and the
complementary null space $$$\bf{\mathcal{N}_s^c}$$$ is then learned
taking right singular vectors corresponding to small singular values below a
certain value following SVD (Fig. 1). By
projecting $$$\bf{\mathcal{H}\left ( y \right )}$$$ onto the
subspace spanned by $$$\bf{\mathcal{N}_s^c}$$$, the slice of interest $$$\bf{\mathcal{H}\left ( x_s \right )}$$$ can be
well-separated from $$$\bf{\mathcal{H}\left ( y \right )}$$$ suppressing most of the contribution
of $$$\bf{\mathcal{H}\left ( x_s^c \right )}$$$ while holding the signal components of $$$\bf{\mathcal{H}\left ( x_s \right )}$$$ intact:

$$\bf{\mathcal{H}\left ( y \right )\mathcal{N}_s^c=\mathcal{H}\left ( x_s \right )\mathcal{N}_s^c}$$

Exploiting the facts that $$$\bf{\mathcal{H}\left ( x_s^c \right )}$$$ has a non-empty null space and $$$\bf{\mathcal{H}\left ( x_s \right )}$$$ is highly rank-deficient (Fig. 1), the reconstruction is posed as an optimization problem by simultaneously imposing null projection and low-rank prior with data consistency:

$$\bf{\hat{x}_s = \underset{x_s}{min} \ \ \frac{1}{2}\left \| \mathcal{H}\left ( y\right ) - \sum_{m=1}^{N_s} \mathcal{H}\left ( x_m \right ) \right \|_F^2 + \frac{\lambda_N}{2}\left \| \left ( \mathcal{H}\left ( y \right ) - \mathcal{H}\left ( x_s \right ) \right ) \mathcal{N}_s^c \right \|_ F^2 + \lambda_L \left \| \mathcal{H}\left ( x_s \right ) \right \|_*}$$

where $$$\bf{\lambda_N}$$$ and $$$\bf{\lambda_L}$$$ are regularization parameters. The complementary null spaces are estimated from CEST reference scan, and the optimization is solved using variable splitting method by alternating between well-defined sub-problems.

**2) Experimental Evaluation:** To
test the feasibility of the SMS-HSL in accelerating CEST imaging, we performed the
SMS acquisition on a 2D whole-brain spiral CEST datasets with RF-segmented
uneven irradiation. The CEST sequence consists of two saturation pulses, each
of which lasted 3sec and 0.5sec, respectively. All images were acquired with
a spatial matrix 64x64 using a single-shot spiral trajectory through 32receiver coils, and 21different offset were performed varying from -5 to 5ppm
with a frequency interval of 0.5ppm. The total scan time for acquiring
whole-brain with 30slices was 7min 50sec, and the reference scan was
performed without saturation pulse for CEST normalization and SMS calibration. The proposed SMS-HSL reconstruction was compared with Split slice-GRAPPA (SP-SG) as a competing method.

1. Ward KM, Aletras AH, Balaban RS. A new class of contrast agents for MRI based on chemical exchange using saturation transfer. J Magn Res 2000;143(1):79–87.

2.van Zijl P, Yadav N. Chemical exchange saturation transfer (CEST): what is in a name and what isn't? Magn Reson Med 2011; 65: 927–948.

3. Uecker M, Lai P, Murphy MJ, Virtue P, Elad M, Pauly JM, Vasanawala SS, Lustig M. ESPIRiT-an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA. Magn Reson Med 2014;71:990–1001.

4. Zhang J, Liu C, Moseley ME. Parallel reconstruction using null operations. Magn Reson Med 2011;66:1241–1253.

5. Sun PZ, Cheung JS, Wang E, Benner T, Sorensen AG. Fast multislice pH-weighted chemical exchange saturation transfer (CEST) MRI with unevenly segmented RF irradiation. Magn Reson Med 2011; 65(2): 588–594.**Figure 3.** Comparison of z-spectrums for SP-SG and SMS-HSL reconstruction in the region of white (red circle) and grey (blue circle) matter in (a). Note that the proposed SMS-HSL exhibits high fidelity to the reference over the entire range of saturation offsets while SP-SG gradually deviates from the reference particularly around ±(1.5~5) ppm.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

0297