Suhyung Park^{1}, Liyong Chen^{2}, Alexander Beckett^{1}, and David Feinberg^{1,2}

Simultaneous multi-slice (SMS) MRI has recently drawn attentions in its use by acquiring linearly combined signals contributed from all excited slices. In this work, we introduce a novel, SMS reconstruction that extends an inter-slice leakage constraint to intra-slice aliasing with virtual slice concept by generalizing parallel MRI as a special case, thus directly estimating the individual slices from undersampled SMS data. Motivated by the leakage block, we generate virtual slices from intra-slice aliasing signals and then penalize these virtual slices as well as real slices simultaneously by keeping only aliasing-free slice of interest while enforcing inplane aliasing and neighboring slices to zeros, respectively.

**One-step Kernel Calibration with Virtual Slice Concept:**
According to sampling theorem, regular undersampling in k-space reduces FOV that is smaller than the object, which introduces aliasing in the PE direction by a factor of FOV/R_{In}. Additionally, it is well-known that inter-slice leakage constraint is highly effective in preserving signals in a slice of interest while mitigating slice leakages. Inspired by sampling theorem and leakage block, virtual slice concept is introduced using an extended leakage constraint. To this end, assuming a collapsed image with R_{Sl}× slice and R_{In}× inplane can be represented as a sum of R_{Sl} real slices as well as (R_{In} − 1)R_{Sl} FOV shifted slices (Fig. 1a), the central concept of virtual slice is that: 1) the shifted objects from undersampling are emulated by applying phase modulation to the PE direction in k-space:

$$ \mathrm{x_{s+n·R_{Sl}}(m \Delta k)=x_s(m \Delta k)e^{-i·m·n·(2π/R_{In})\Delta k}}$$

where s, m, and n are real slice, PE line, and aliasing offset indices, respectively and 2) an extended leakage block optimization, which incorporates the aliasing offsets into the calibration as virtual slices, is built up by concatenating both real and virtual slices into (Fig. 1b):

$$ \begin{bmatrix}\mathcal{C} ( \mathrm{\mathbf{x}^{real,src}_1} )\\ \vdots \\ \mathcal{C} ( \mathrm{\mathbf{x}^{real,src}_s} ) \\\vdots \\\mathcal{C} ( \mathrm{\mathbf{x}^{real,src}_{R_{Sl}}} ) \\ ············ \\ \mathcal{C} ( \mathrm{\mathbf{x}^{virtual,src}} ) \\\end{bmatrix} \mathrm{\mathcal{G}^{vs}_s}= \begin{bmatrix}\mathbf{0} \\ \vdots \\ \mathrm{\mathbf{x}^{real, trg}_s} \\\vdots \\\mathbf{0}\\ ············ \\\mathbf{0}\\\end{bmatrix} $$

where $$$ \mathrm{ \mathcal{C} ( \mathbf{x}^{virtual,src} ) = \left [ \mathcal{C} ( \mathbf{x}^{virtual,src}_{R_{Sl}+1} )^T \cdots \; \mathcal{C} ( \mathbf{x}^{virtual,src}_{ R_{In}R_{Sl}} )^T \right ]^T}$$$, and $$$ \mathrm{\mathcal{G}^{vs}_s}$$$
a virtual slice
kernel for s^{th} slice. It is worthwhile noting that the proposed one-step calibration can be interpreted as a completely generalized and extended version of split-slice GRAPPA in that the former utilizes a aliasing offsets as well as neighboring slices to leave a slice of interest while the latter only blocks the neighboring slices, thus directly resulting in the aliasing-free images without the need of sequential process for SMS and parallel MRI.

**Optimization for Reconstruction:** We estimate the individual slice directly from the undersampled SMS data by solving the following constrained optimization:

$$ \mathrm{\underset{\mathbf{x}}{argmin}} \; \mathrm{\left \| \mathbf{G}^{vs} \mathbf{y} - \mathbf{x} \right \|_{2}^{2}} \\ \mathrm{\left \| \mathbf{y} - \mathbf{F}^{sl}_u\; \mathbf{x} \right \|_{2}^{2}<\epsilon}$$

where $$$ \mathrm{\mathbf{G}^{vs}}$$$ is a matrix containing a virtual slice kernel of
s^{th} slice in the appropriate locations, $$$ \mathbf{F}^{sl}_u$$$ is undersampled Fourier operator along slice direction, $$$ \mathbf{y} $$$ is the measured data, and ε is the noise variance that balances between the acquired and estimated data.

**Data Acquisition:** Retrospective and prospective studies were performed on both 3T and 7T scanners equipped with a 32-channel head coil. Two datasets were acquired using GRE and EPI sequences, respectively, and its imaging parameters are described in table 1.

We introduce a novel, SMS reconstruction with virtual slices that minimizes errors coming from inter-slice leakages and intra-slice aliasing simultaneously. This is accomplished by viewing aliasing offsets as virtual slices to take the importance of both sources of errors into account. By incorporating aliasing phase information, which are linearly independent with phase modulation between PE lines, into the one-step kernel calibration for virtual slice concept, the matrix conditioning becomes improved, thus resulting in a reduced g-factor compared to SP-SG+GRAPPA.

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Table 1. Imaging Parameters used in In Vivo Measurements

Fig. 1. (a) Signal representation of virtual slice concept including real and virtual slices. (b) Illustration of how to calculate virtual slice kernel in a matrix form for slice separation and inplane interpolation.

Fig. 2. (a) The nRMSEs for SP-SG+GRAPPA and virtual slice with increasing R_{In} from 1 to 3 with R_{Sl} held to 3. (b) The reconstructed images and its corresponding difference between the two methods at 3-fold acceleration (R_{Sl}=3 and R_{In}=1). (c) The reconstructed images and its corresponding error maps at 9-fold acceleration (R_{Sl}=3 and R_{In}=3).

Fig. 3. Images reconstructed using SP-SG+GRAPPA and virtual slice based on EPI data at prospective 15-fold acceleration (R_{Sl}=5 and R_{In}=3).

Fig. 4. g-factors (upper) and l-factors (bottom) calculated using SP-SG+GRAPPA and virtual slice based on EPI data at prospective 15-fold acceleration (R_{Sl}=5 and R_{In}=3).