Changyu Sun^{1}, Yang Yang^{2}, Craig H. Meyer^{1}, Xiaoying Cai^{1}, Michael Salerno^{1,2,3}, Daniel S. Weller^{1,4}, and Frederick H. Epstein^{1,3}

Simultaneous multislice (SMS) imaging provides through-plane acceleration. While current reconstruction methods for non-Cartesian imaging (and also for Cartesian imaging) utilize either in-plane or through-plane coil information, we reasoned that a slice-SPIRiT model could utilize both in-plane and through-plane kernel calibration information, and potentially outperform methods like conjugate-gradient SENSE (CG-SENSE). We developed a slice-SPIRiT method and compared it to CG-SENSE for spiral cardiac cine imaging. Slice leakage artifacts using slice-SPIRiT were 52.9% lower than using CG-SENSE in phantoms, and the artifact power of slice-SPIRiT was 24.2% less than CG-SENSE in five volunteers. Slice-SPIRiT is a promising method for spiral SMS imaging.

We reasoned that a slice-SPIRiT model could utilize both in-plane and through-plane kernel calibration information, and outperform methods like CG-SENSE, which make use of only in-plane coil sensitivity. The proposed slice-SPIRiT reconstruction is illustrated in Figure 1 and the proposed slice-SPIRiT model can be expressed in Equation 1 as follows:

$$\underset{\min}{arg\min}\lVert \left( \sum_{z=1}^{NS}{P_z\cdot D\left ( m_z \right)} \right) -y \rVert ^2 + \lambda _1 \lVert \left (G-I \right) m \rVert ^2+ \lambda _2\lVert m \rVert ^2,$$

where $$${NS}$$$ is the number of MB slices, $$$z$$$ is the slice number,$$$P_z$$$ is the CAIPRINHA phase modulation matrix for
the $$$z^{th}$$$ slice, the $$$D$$$ operator performs the Fast Fourier transform $$$(FFT)$$$ and inverse-gridding^{6} of the
Cartesian images, $$$m_z$$$,
to the spiral k-space,
$$$m_z$$$ is the multicoil image of the $$$z^{th}$$$ slice, $$$y$$$ is the acquired MB spiral data, $$$\lambda _1$$$ is the weight for the in-plane^{7}
calibration consistency, $$$G= \left (\begin{matrix} G_1& \cdots& 0\\\vdots& \ddots& \vdots\\0& \cdots& G_{NS}\\\end{matrix}\right) $$$ is the operator of concatenated in-plane SPIRiT^{7}
kernels, $$$G_z$$$,
for the $$$NS$$$ slices, $$$I= \left (\begin{matrix} I_1& \cdots& 0\\\vdots& \ddots& \vdots\\0& \cdots& I_{NS}\\\end{matrix}\right) $$$ is the concatenated unit matrices, $$$m=\left( \begin{array}{c}m_1\\ m_2\\ \vdots\\ m_{NS}\\ \end{array}\right)$$$ is the matrix of concatenated images for the $$$NS$$$ slices, and $$$\lambda _2$$$ is the weight for the Tikhonov regularization in the
image domain.

In
addition, if we define the operator $$$H=\sum_{z=1}^{NS}{P_z\cdot D}$$$
, then the conjugate of $$$H$$$ is $$$H^*$$$,
and $$$H^*$$$ is the key operator to calculate the
gradient $$$\nabla _m \left( \lVert H\left( m_z \right) -y \rVert ^2 \right) =2H^*\left( H\left( m_z \right) -y \right)$$$.
However, the
dimension of the gradient doesn't match the dimension of the concatenated
separated slices, $$$m$$$. To solve this problem, we use an approximation
for $$$H^*$$$, namely $$$H^*=D_{z}^{*}P_{z}^{*}$$$, where
$$$D_{z}^{*}=IFFT\left( SSG_z\cdot C \right)$$$,
$$$C$$$ is the gridding operator^{6}, and $$$SSG_z$$$ is the slice-separating kernel as used in the
split-slice GRAPPA method^{2}.
In this way, the data consistency term
in Eq. 1 utilizes the through-plane GRAPPA kernel and enforces joint estimation
of the separated slices. LSQR^{8} is used as the conjugate gradient solver
for this model (denoted as CG in Fig.1a).

Illustration of the
proposed slice-SPIRiT reconstruction for SMS CAIPIRINHA spiral images. Slice-SPIRiT utilizes through-plane and
in-plane calibration consistency, and data consistency. (a): the conjugate gradient (CG) algorithm of
slice-SPIRiT for the reconstruction of spiral SMS images. b: The operator
is
based on gridding and slice-GRAPPA kernels.

Phantom studies using
spiral cine gradient-echo MRI (MB=3) show that reconstruction using slice-SPIRiT
(d-f) reduces slice leakage artifact compared to CG-SENSE (a-c). The reference
standard single-band images at matched locations were reconstructed by NUFFT^{6} (g-i).

Comparison of slice-SPIRiT
and CG-SENSE for image recovery applied to spiral cine gradient echo MRI (MB=3)
of three short-axis slices in a human volunteer. For CG-SENSE (a-c), multiple
slice-leakage artifacts are observed (red arrows). The artifacts are reduced
using slice-SPiRIT (d-f). SB images at matched locations are presented as
reference standards (g-i).