Virtual Slice Concept for Improved Simultaneous Multi-Slice MRI Employing an Extended Leakage Constraint

Abstract

Purpose:

To develop a novel, simultaneous multi-slice (SMS) reconstruction that extends an inter-slice leakage constraint to intra-slice aliasing with a virtual slice concept for artifact reduction.

Methods:

Inter-slice leakage constraint has been used for SMS reconstruction that mitigates leakage artifacts from the adjacent slices. In this work, the leakage constraint is extended to a more general framework that includes SMS and parallel MRI as special cases by viewing intra-slice aliasing artifacts from undersampling as virtual slices while imposing data fidelity to ensure the measurement consistency. In this way, the reconstruction makes it feasible to directly estimate the individual slices from the undersampled SMS acquisition as a one-step method. The performance of the extended method is evaluated with data acquired using 2D GRE and EPI sequences.

Results:

Compared to a two-step method that performs slice unaliasing followed by inplane unaliasing, the proposed one-step method reduces aliasing artifacts by employing the extended leakage constraint while lowering the noise amplification by improving the conditioning for the inverse problem.

Conclusion:

The proposed one-step method takes advantage of virtual slices as additional encoding power for improved image quality. We successfully demonstrated that the proposed one-step method minimizes a trade-off between aliasing artifacts and amplified noises over the two-step method.

Introduction

Parallel MRI techniques such as SMASH (1), SENSE (2), and GRAPPA (3) allow for faster acquisition by skipping a fraction of phase encoding (PE) steps below the Nyquist sampling rate, and exploit complementary spatial encoding capability from multiple receiver coils for reconstruction, typically achieving 2× or 3× acceleration. Here, undersampling (R_In) is theoretically limited by the number of coils, but it is challenging to achieve high acceleration due to intrinsic $\sqrt{{\mathrm{R}}_{\text{In}}}$ signal to noise ratio (SNR) penalty resulting from the incomplete measurement.

There has recently been renewed attention in simultaneous multi-slice (SMS) MRI by acquiring linearly combined signals from all excited slices (4–6), resulting in improved SNR and imaging efficiency compared to slice-by-slice sequential acquisition. However, with increasing slice acceleration standard SMS reconstruction methods such as slice-GRAPPA (6) and SENSE/GRAPPA (7) suffer from slice leakages among simultaneously excited slices, thus yielding contrast changes and aliasing artifacts from the adjacent slices even with controlled aliasing (CAIPI) (8, 9). As an alternative, split slice-GRAPPA (SP-SG) was introduced to mitigate slice leakages by enforcing signals in other slices to zeros while keeping signals in a slice of interest during kernel calibration (10).

With superior performance in leakage block, the SP-SG has been used for various applications such as functional and diffusion MRI in conjunction with undersampling for high resolution imaging (10–12). Typically, slice separation and inplane interpolation are achieved via a two-step method by sequentially applying SP-SG and conventional GRAPPA to both slice and inplane directions, respectively. However, imperfect slice separation potentially causes error propagation to the entire k-space during inplane interpolation, in which the accuracy of the slice separation can substantially affect the subsequent reconstruction by synthesizing the signals contaminated from adjacent slices as source signals for parallel MRI.

In this work, we introduce a novel, SMS reconstruction that extends an inter-slice leakage constraint to intra-slice aliasing with a 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 the aliasing-free slice of interest while enforcing inplane aliasing and neighboring slices to zeros, respectively. Unlike the above two-step method, the proposed reconstruction is formulated as an inverse problem by imposing data fidelity as an additional constraint, which leads to a consistent solution with the acquired data avoiding signal bias. To test the feasibility of the proposed reconstruction over conventional method, numerical simulation and experimental studies were performed using retrospectively emulated and prospectively acquired SMS data with up to R=15-fold acceleration (R_Sl = 5 and R_In = 3).

Theory

Kernel Calibrations for Parallel and SMS MRI

GRAPPA has been widely used for slice separation and inplane interpolation, in which coil-by-coil reconstruction is performed by synthesizing k-space data of interest from neighboring k-space data across all coils once the synthesizing kernels are estimated during calibration. The inplane kernel ${\mathcal{G}}^{\mathrm{p}}$ is estimated by solving the inverse problem:

$$\mathcal{C}({\mathbf{x}}^{\mathrm{src}}){\mathcal{G}}^{\mathrm{p}}={\mathbf{x}}^{\mathrm{trg}}$$ [1]

where x^src is a vector of source k-space data obtained from the calibration area, $\mathcal{C}(\cdot )$ is a convolution matrix operator that stacks row-vectorized patches across all coils by sliding window through the calibration area, and x^trg is a vector of target k-space data corresponding to the convolution matrix $\mathcal{C}({\mathbf{x}}^{\mathrm{src}})$. This kernel implicitly means relative sensitivity variation between coils, which provides complementary encoding power for inplane interpolation (13, 14).

By extending a single slice to multiple slices, SP-SG takes leakage block between slices into account:

$$\left[\begin{array}{c}\hfill \mathcal{C}({\mathbf{x}}_1^{\mathrm{src}})\hfill \\ \hfill ⋮\hfill \\ \hfill \mathcal{C}({\mathbf{x}}_{\mathrm{s}}^{\mathrm{src}})\hfill \\ \hfill ⋮\hfill \\ \hfill \mathcal{C}\left({\mathbf{x}}_{{\mathrm{R}}_{\mathrm{Sl}}}^{\mathrm{src}}\right)\hfill \end{array}\right]{\mathcal{G}}_{\mathrm{s}}^{\mathrm{m}}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill ⋮\hfill \\ \hfill {\mathbf{x}}_{\mathrm{s}}^{\mathrm{trg}}\hfill \\ \hfill ⋮\hfill \\ \hfill 0\hfill \end{array}\right]$$ [2]

where ${\mathbf{x}}_{\mathrm{s}}^{\mathrm{src}}$ is a vector of source k-space data at s^th slice acquired from the calibration area, and ${\mathcal{G}}_{\mathrm{s}}^{\mathrm{m}}$ is a slice kernel for the s^th slice. This formulation emphasizes that the slice kernel is computed by passing signals in a slice of interest while enforcing signals in other slices to zeroes, thus leaving a slice of interest at the expense of SNR.

With these kernels, the individual slices can be estimated from the undersampled SMS data by sequentially solving the two forward problems for SMS and parallel MRI, in which the slice kernel is applied to separate the aliased slices followed by interpolating missing data through the inplane kernel slice-by-slice. These forward problems can be simply written as:

$$\mathbf{x}={\mathbf{G}}^{\mathrm{p}}{\mathbf{G}}^{\mathrm{m}}\mathbf{y}$$ [3]

where x is a vector of full k-space data containing all slices, y is a vector of measured SMS data, and ${\mathbf{G}}^{\mathrm{p}}=\mathrm{diag}({\mathbf{G}}_1^{\mathrm{p}},\dots ,{\mathbf{G}}_{{\mathrm{R}}_{\mathrm{Sl}}}^{\mathrm{p}})$ and ${\mathbf{G}}^{\mathrm{m}}={({\mathbf{G}}_1^{\mathrm{m}},\dots ,{\mathbf{G}}_{{\mathrm{R}}_{\mathrm{Sl}}}^{\mathrm{m}})}^{\mathrm{T}}$ are block-diagonal and block-column convolution matrices containing ${\mathcal{G}}_{\mathrm{s}}^{\mathrm{p}}$ and ${\mathcal{G}}_{\mathrm{s}}^{\mathrm{m}}$ in the appropriate locations over the whole slices, respectively.

This approach assumes that slice separation perfectly describes the undersampled data of the individual slices. However, it is challenging to accurately achieve slice separation with high accelerations, which implies that the resulting images may be susceptible to error propagation during subsequent reconstruction, thus causing crosstalk between slices as well as inplane aliasing within each slice. Therefore, it is desirable to directly estimate the individual slices from acquired data by minimizing error propagation during reconstruction.

One-step Kernel Calibration with Virtual Slice Concept

According to sampling theorem, regular undersampling in k-space reduces the FOV so that it 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 (10). Inspired by sampling theorem and leakage block, a 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:

$${\mathrm{x}}_{\mathrm{s}+\mathrm{n}\cdot {\mathrm{R}}_{\mathrm{Sl}}}(\mathrm{m}\Delta \mathbf{k})={\mathrm{x}}_{\mathrm{s}}(\mathrm{m}\Delta \mathbf{k}){\mathrm{e}}^{-i\cdot \mathrm{m}\cdot \mathrm{n}\cdot (2\pi ∕{\mathrm{R}}_{\text{In}})\cdot \Delta \mathbf{k}},\mathrm{n}=1,\dots ,{\mathrm{R}}_{\text{In}}-1$$ [4]

where m and n are 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):

$$\left[\begin{array}{c}\hfill \mathcal{C}({\mathbf{x}}_1^{\text{real},\mathrm{src}})\hfill \\ \hfill ⋮\hfill \\ \hfill \mathcal{C}({\mathbf{x}}_{\mathrm{s}}^{\text{real},\mathrm{src}})\hfill \\ \hfill ⋮\hfill \\ \hfill \mathcal{C}({\mathbf{x}}_{{\mathrm{R}}_{\mathrm{Sl}}}^{\text{real},\mathrm{src}})\hfill \\ \hfill \cdots \cdots \cdots \cdots \hfill \\ \hfill \mathcal{C}({\mathbf{x}}^{\text{virtual},\mathrm{src}})\hfill \end{array}\right]{\mathcal{G}}_{\mathrm{s}}^{\mathrm{vs}}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill ⋮\hfill \\ \hfill {\mathbf{x}}_{\mathrm{s}}^{\text{real},\mathrm{trg}}\hfill \\ \hfill ⋮\hfill \\ \hfill 0\hfill \\ \hfill \cdots \cdots \hfill \\ \hfill 0\hfill \end{array}\right]$$ [5]

where $\mathcal{C}({\mathbf{x}}^{\text{virtual},\mathrm{src}})={\left[\mathcal{C}{\left({\mathbf{x}}_{{\mathrm{R}}_{\mathrm{Sl}}+1}^{\text{virtual},\mathrm{src}}\right)}^{\mathrm{T}}\cdots \mathcal{C}{\left({\mathbf{x}}_{{\mathrm{R}}_{\mathrm{In}}{\mathrm{R}}_{\mathrm{Sl}}}^{\text{virtual},\mathrm{src}}\right)}^{\mathrm{T}}\right]}^{\mathrm{T}}$, and ${\mathcal{G}}_{\mathrm{s}}^{\mathrm{vs}}$ is a virtual slice kernel for the 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 processing for SMS and parallel MRI.

Figure 1.

(a) Signal representation of the virtual slice concept including real and virtual slices. In this example, a collapsed image is acquired with 3× slice, 2× inplane, and FOV/2 CAIPI technique, yielding 3 real slices with a FOV/6 image shift and 6 virtual slices with a FOV/3 image shift, respectively. Note that the undersampled SMS data is modeled as a linear combination of both real and virtual slices in this work. (b) Illustration of how to calculate a virtual slice kernel in a matrix form for slice separation and inplane interpolation at the same time. Note that leakage block serves to eliminate aliasing offsets as well as the adjacent slices through virtual slice kernel calibration.

SMS Encoding: Data Fidelity

Based on the fact that single slice excitation can be described as a product of the RF waveform (slice profile) and phase modulation (slice position) functions, SMS excitation can be written as a linear combination of all excited slices with different phase modulation functions (15–17):

$$\begin{gathered} \widehat{\mathrm{x}}(\mathbf{k})={\int }_{\mathbf{r}}\mathrm{d}\mathbf{r}\cdot \sum _{\mathrm{s}}{\rho }_{\mathrm{s}}(\mathbf{r})\cdot {\mathrm{e}}^{i2\pi \cdot \mathrm{M}\cdot \mathrm{s}∕{\mathrm{R}}_{\mathrm{Sl}}}\cdot {\mathrm{e}}^{-i\mathbf{k}\cdot \mathbf{r}} \\ \widehat{\mathbf{x}}=\sum _{\mathrm{s}}{\mathbf{x}}_{\mathrm{s}}\cdot {\mathrm{e}}^{i2\pi \cdot \mathrm{M}\cdot \mathrm{s}∕{\mathrm{R}}_{\mathrm{Sl}}} \end{gathered}$$ [6]

where $\widehat{\mathrm{x}}(\mathbf{k})$ is the SMS data at the k-space position k in the measurement domain encoded by M (0 ≤ M ≤ R_Sl − 1) along slice direction, ρ_s (r) is a proton density of the s^th slice at the voxel position r, and x_s is the k-space data vector of the s^th slice. Here, the phase modulation terms generate different slice encoding, which enables slice separation just by performing Fourier transform along the slice direction once SMS data is fully acquired using R_S1 distinct RF pulses. As the slice Fourier encoding is applied in conjunction with undersampling mask, the acquired SMS data are simply written as:

$$\mathbf{y}={\mathbf{F}}_{\mathrm{u}}^{\mathrm{sl}}\mathbf{x}$$ [7]

where ${\mathbf{F}}_{\mathrm{u}}^{\mathrm{sl}}$ is the slice Fourier transform undersampled along slice direction. To ensure consistency between the acquired data and its estimated data, the SMS encoding in Eq. [7] is used as data fidelity under an optimization framework.

Optimization for Reconstruction

Given the above consideration, we estimate the individual slice directly from the undersampled SMS data by solving the following constrained optimization:

$$\begin{gathered} \underset{\mathbf{x}}{\mathrm{argmin}}{\Vert {\mathbf{G}}^{\mathrm{vs}}\widehat{\mathbf{x}}-\mathbf{x}\Vert }_2^2 \\ {\Vert \mathbf{y}-{\mathbf{F}}_{\mathrm{u}}^{\mathrm{sl}}\mathbf{x}\Vert }_2^2<ϵ \end{gathered}$$ [8]

where ${\mathbf{G}}^{\mathrm{vs}}=\mathrm{diag}({\mathbf{G}}_1^{\mathrm{vs}},\dots ,{\mathbf{G}}_{{\mathrm{R}}_{\mathrm{Sl}}}^{\mathrm{vs}})$, ${\mathbf{G}}_{\mathrm{s}}^{\mathrm{vs}}$ is a matrix containing a virtual slice kernel of s^th slice in the appropriate locations and ϵ is the noise variance that balances between the acquired and estimated data. The optimization in Eq. [8] can be solved using gradient-based algorithms (18).

Here, it is important to note that the above formulation represents a more general model that covers both SMS and parallel MRI as special cases. Equation [8], if used with a single band pulse and undersampling, is equivalent to regularized SPIRiT; if used with MB pulse and without undersampling, is reduced to regularized SP-SG by including data fidelity. Another point to be noted is that the incorporation of the data fidelity in the proposed method delivers unbiased results that are still consistent with the acquired data, in which the signal bias increases with the factor of undersampling.

Methods

Data Acquisition

Numerical simulations and experimental studies were performed on both 3T and 7T scanners (Siemens Healthcare, Erlangen, Germany) equipped with a 32-channel head coil. Institutional IRB and informed consent was obtained for all subjects. Prior to actual imaging, a prescan was conducted for calibration by acquiring low frequency data from each slice. Three datasets were then acquired with imaging parameters described in detail below.

Retrospective Studies

Fully-sampled data was acquired on a 3T Trio scanner using a spoiled gradient echo (GRE) sequence. The imaging parameters were FOV=200×200 mm², slice thickness=3 mm, slice distance=30 mm, matrix size=192×192, number of slices=3, TE/TR=6.9/10.8 ms, and flip angle=15°. For simulation, the data was processed in the following way: 1) the raw data of all excited slices were linearly combined and 2) the combined k-space data was then regularly undersampled along the PE direction.

Prospective Studies

The prospective studies included 3T GRE and 7T EPI acquisitions: 1) SMS GRE images were acquired on a 3T Trio scanner with 12-fold (R_Sl=4 and R_In=3) acceleration. The imaging parameters were FOV=240×240 mm², slice thickness=3 mm, slice distance=30 mm, matrix size=192×192, TE/TR=6.9/ 10.8 ms, and flip angle=15°; 2) SMS EPI images were acquired on a 7T Magnetom scanner using 15-fold (R_Sl=5 and R_In=3) acceleration. The imaging parameters were FOV=192×192 mm², slice thickness=2 mm, slice distance=20 mm, matrix size=128×128, TE/TR=29/4220ms, EPI factor=43, and flip angle=80°. To increase the distance between aliasing voxels along slice direction, CAIPI-induced phases were applied by RF pulses and blips for GRE and EPI sequences, respectively, thus shifting the aliasing patterns of the multiple slices with respect to each other by FOV/2 in the phase encoding direction. However, the combined effect of both CAIPI-shift and inplane undersampling does not result in a FOV/2 shift between neighboring slices, leading to three alias locations for each slice, one shifted by m·FOV/2, one shifted by m·FOV/2 + FOV/3, and one shifted by m·FOV/2 + 2·FOV/3, where m is the slice number, which ends up an inter-slice shift of FOV/6 for neighboring slices. Given the consideration above, the proposed one-step calibration for real and virtual slices was performed as shown in Fig. 1.

Data Processing and Quality Metrics

All data were processed offline on a personal computer with 3.3 GHz CPU and 32GB RAM using MATLAB (Math Works Inc., Natick, MA). The size of the virtual slice kernels were set to 9 × 9, 11 × 11, and 13 × 13 (phase encoding × readout) for 9-, 12-, and 15-fold accelerations, respectively. The reconstructed multicoil images were combined using a root-sum-of-squares. The reconstruction code for the proposed virtual slice will be available as requested. As a competing method, the two-step approach, which performs slice unaliasing followed by inplane unaliasing in a sequential manner (SP-SG+GRAPPA), was implemented. The kernel sizes and regularization parameters we used for SP-SG and GRAPPA were empirically determined as optimal ones based on the parameter sweep within a reasonable range, respectively. For fair comparison, the data fidelity was imposed on the reconstruction process as with inplane undersampling.

To evaluate the effect of image quality on aliasing artifacts and SNR, we used the following metrics: 1) nRMSE: The root-mean-square-error (nRMSE) was calculated as:

$$\mathrm{nRMSE}=\frac{1}{\max ({\rho }_{\mathrm{s}})}\cdot \sqrt{\frac{1}{\mathrm{N}}\sum _{\mathrm{r}=1}^{\mathrm{N}}{({\rho }_{\mathrm{s}}-{\widehat{\rho }}_{\mathrm{s}})}^2}$$ [9]

where ρ_s and ${\widehat{\rho }}_s$ are the reference and its estimated images, and N is the total number of pixels. 2) ℓ-factor: To assess signal leakage from the other slices, the leakage factors were produced by applying the reconstruction kernels to the calibration data across all excited slices (10, 22), in which the reconstruction kernels just pass the signals in a slice of interest while nulling those contributed from all other slices. 3) g-factor: For noise evaluation, geometry factors were calculated using the reconstruction kernels of each method as in standard GRAPPA (23).

Results

Retrospective Studies

Reconstruction Validation

The effect of the data fidelity on image quality is demonstrated in Fig. 2. The nRMSE was plotted versus the penalty parameter λ with its corresponding images at 6-fold acceleration (R_Sl = 3 and R_In = 2). In the absence of data fidelity, nRMSE is much higher due to signal dropouts by simultaneously limiting the influence of aliasing offsets and slice leakages (Fig. 2b). By incorporating the data fidelity into virtual slice reconstruction, nRMSE decreases gradually with increasing λ and becomes minimal for the penalty value λ ≈ 0.8, implying that data fidelity is primarily responsible for signal bias (Fig, 2c). Then, nRMSE rises rapidly for high penalty values by introducing aliasing artifacts at the expense of SNR (Fig. 2d).

Figure 2.

(a) The nRMSE for virtual slice is plotted versus data fidelity penalty λ. (b)-(d) The reconstructed images for penalty values λ=0, 0.8, and 5, respectively. Note that the nRMSE becomes minimal at λ ≈=0.8, thus providing a trade-off between SNR and accuracy.

Performance Comparisons using GRE data

Figure 3a shows the variation of nRMSE in both SP-SG+GRAPPA and virtual slice with increasing R_In from 1 to 3 while setting R_Sl to 3. The nRMSE values consistently rises as a function of R_In, and the nRMSE difference between the two methods also increases with increasing R_In. Without inplane undersampling (R_In = 1), neither SP-SG nor virtual slice show obvious visual artifacts, thereby exhibiting minor differences between the two methods outside the brain area as depicted in Fig. 3b (blue arrows). At 9-fold acceleration (R_Sl = 3 and R_In = 3), the SP-SG+GRAPPA suffers from inplane aliasing and noise amplification in the middle of FOV (white arrows), whereas the image quality are improved with virtual slice, respectively, as shown in Fig. 3c. The calculated g-factor and ℓ-factors were displayed in Supporting Fig. S1.

Figure 3.

(a) The nRMSEs for SP-SG+GRAPPA and virtual slice are plotted versus inplane undersampling 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 using the two methods at 9-fold acceleration (R_Sl = 3 and R_In = 3). Note that the proposed virtual slice converge to SP-SG with marginal differences at R_In = 1 (blue arrow), while showing overall improvement with increasing R_In in terms of aliasing artifacts and noises compared to SP-SG+GRAPPA (white arrows).

Prospective Studies

Performance Comparisons using GRE data

Figure 4a demonstrates the resulting images using SP-SG+GRAPPA and virtual slice based on undersampled SMS GRE data at 12-fold acceleration (R_Sl = 4 and R_In = 3). The SP-SG+GRAPPA demonstrates aliasing artifacts originating from both slice (white arrow) and inplane (yellow arrow) directions, while the proposed virtual slice mitigates these issues, although aliasing artifacts are slightly visible in the lower part of the second slice. The corresponding g-factors and ℓ-factors are displayed in Supporting Fig. S2.

Figure 4.

(a) Images reconstructed using SP-SG+GRAPPA and virtual slice based on GRE data at prospective 12-fold acceleration (R_Sl = 4 and R_In = 3) and (b) Images reconstructed using SP-SG+GRAPPA and virtual slice based on EPI data at 15-fold acceleration (R_Sl = 5 and R_In = 3). Note that the proposed virtual slice tends to suppress aliasing artifacts originating from both slice (white arrow) and inplane (yellow arrow) directions, respectively, as compared to SP-SG+GRAPPA.

Performance Comparisons using EPI data

Figure 4b shows results based on undersampled SMS EPI data at 15-fold acceleration (R_Sl = 5 and R_In = 3). Similar to the previous GRE results, the proposed virtual slice consistently outperforms SP-SG+GRAPPA in suppressing structured aliasing artifacts (white arrows), while partially mitigates noise amplification due to a large kernel size of 13 × 13 comparable to those of SP-SG+GRAPPA particularly in the first three slices. The corresponding g-factors and ℓ-factors are displayed in Supporting Fig. S3.

Discussion

We introduce a novel, SMS reconstruction with virtual slices that minimizes errors 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, with the benefits of the virtual slice concept being demonstrated through several human experiments. 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 (Fig. 3–5). Imposing data fidelity on reconstruction enables the unbiased results even in the presence of undersampling (Fig. 2).

Figure 5.

Reconstructed images, image differences, and their corresponding correlation matrices using GRAPPA and virtual slice with 4-fold acceleration (R_Sl = 1 and R_In = 4). Note that the incorporation of phase modulated aliasing information into calibration step suppresses aliasing artifacts without apparent noise enhancement resuiting from the improved conditioning of the correlation matrix at the cost of image blurring.

The proposed virtual slice concept is a simple extension of SP-SG, where aliasing offsets are concatenated along the slice direction as virtual slices, thus enforcing the reconstructed image to reside in a subspace spanned by the aliasing-free image. This work demonstrates that there exist virtual slice kernels that yield the full-FOV k-space for all slices without loss of generality. This means that the proposed virtual slice allows for handling parallel MRI as well as SMS MRI under the unified framework by inheriting the great flexibility of SP-SG that can control weighting parameters between inplane aliasing and slice leakages.

Since coil sensitivities vary smoothly in the image domain and thereby are band-limited in k-space, the kernel size in GRAPPA and slice GRAPPA is typically set to a reasonably small number such as 4 × 5 and 5 × 5. As the kernel size is not large enough to capture the complexity of the relationship between neighboring signals with the large number of calibration data in the presence of noises (# of kernel coefficients << # of equations), the calibration matrix becomes overfitted, showing the following trend: As the model complexity increases, the RMSE decreases, reaches a minimum, then increases (19–21). From this perspective, the proposed virtual slice kernel is more susceptible to noise because it is overly restrictive on both real and virtual slices, which requires more complex model with a larger kernel size. In the GRE experiment with 12-fold acceleration (R_Sl = 4 and R_In = 3), supporting Fig. S4 shows the reconstructed images and calculated g-factors using virtual slice concept with increasing kernel sizes from 5×5 to 13×13. As expected, the proposed method substantially increases g-factors with a 5×5 kernel size resulting from overfitting. As the kernel size is increased from 9×9 to 13×13, the g-factor reaches its minimum noise and then slightly increased noises due to excessive model complexity. In the EPI experiment with 15-fold acceleration (R_Sl = 5 and R_In = 3), the impact of a large kernel size can be appreciated by the amplified noise as shown in supporting Fig. S4. To address the suboptimal model accuracy, the proposed method, which increases the number of equations, can be combined with either virtual coil concept (24, 25) or nonlinear kernel regression model (26) by increasing the number of of kernel coefficients.

In this work, the noise performance strongly depends on the conditioning for the inversion of the effective calibration matrix including real and virtual slices ${\mathcal{C}}^{\mathrm{vs}}=[\mathcal{C}({\mathbf{x}}^{\text{real},\mathrm{src}});\mathcal{C}({\mathbf{x}}^{\text{real},\mathrm{src}})]$, which implies that the minimization of g-factor can be achieved by enforcing off-diagonal elements of the correlation matrix ${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}$ to zeros while keeping on-diagonal elements of ${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}$ similar to each other, thereby becoming better conditioned for the inverse problem (27). With the incorporation of phase modulated aliasing information into calibration process, the correlation matrix ${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}$ results in lower off-diagonal elements by 1 + e^{i·r·2π/R_In} (r=1, ⋯ , R_In − 1) while evenly increasing all the diagonal elements by 2, which means that the columns of the correlation matrix tend to be less correlated as compared to conventional GRAPPA, thus resulting in improved matrix conditioning for the inverse problem. Figure 5 shows the reconstructed images, image differences, and their corresponding correlation matrices using GRAPPA and virtual slice with 4-fold acceleration. As expected, GRAPPA typically yields a trade-off between artifacts and noise amplification depending on regularization degree. Particularly, the reconstructed image is susceptible to noises for weak regularization due to the ill-conditioning of the correlation matrix in the presence of significant off-diagonal elements. Interestingly, the proposed virtual slice introduces different signal behaviors with varying regularization degrees as compared to GRAPPA. Increasing the regularization degree in virtual slice tends to oversmooth the reconstructed images with slightly amplified noises, resulting in significant blurring artifacts around the boundaries of skull. As the regularization becomes weak, the proposed virtual slice is still effective in suppressing noises and artifacts due to the improved conditioning of the correlation matrix by forcing off-diagonal elements to be small, but some minor blurring artifacts can still be observed in the image difference. Given the consideration above, the proposed virtual slice shows superior performance in suppressing noise and aliasing artifacts over GRAPPA reconstruction at the cost of blurring artifacts. On the other hand, GRAPPA lowers overall artifact level in this example with optimal kernel size and regularization parameter by minimizing a trade-off between noises and aliasing artifacts, but the image quality is relatively susceptible to reconstruction parameters.

In terms of computation, the complexity of the proposed method is dominated by calculating correlation matrices for each slice, $\mathcal{C}({\mathbf{x}}_{\mathrm{s}})$, including real and virtual slices, resulting in a computational complexity $\mathcal{O}({({\mathrm{N}}_{\mathrm{s}}\cdot {\mathrm{N}}_{\mathrm{p}})}^2\cdot {\mathrm{N}}_{\mathrm{k}})$, where N_k and N_p are the patch size of the kernel vectorized across coils and the number of patches stacked over the k-space of the calibration, respectively, which takes as long as the number of slice kernels. Furthermore, the implementation in the image domain leads to $\mathcal{O}({\mathrm{N}}_{\mathrm{c}}^2\cdot {\mathrm{N}}_{\mathrm{v}})$ complexity per iteration in the gradient descent-based method, where N_c and N_v are the number of coils and image size, respectively. The computation becomes demanding particularly when a large kernel size is applied with large number of coils at the expense of kernel accuracy. To achieve clinically reasonable reconstruction time, it is noted that the proposed algorithm can be efficiently parallelized using a synergetic combination of multi-core CPUs with GPUs (28, 29), in which the calibration, which requires high computational cost, is performed in parallel on multi-core CPUs for each slice while the reconstruction, which requires data synthesis and Fourier transform that are independent of processing order, is implemented on GPUs with the large number of cores. The computation time can be further accelerated by using coil compression techniques.

In conclusion, a proposed generalized SMS reconstruction based on virtual slice concept has been presented. In vivo experiments demonstrate that the proposed method achieves better image quality in the presence of undersampling with slice acceleration. Though the aliasing artifacts from aliasing offsets and slice leakages are suppressed at the cost of SNR, the proposed method still provides overall SNR gain compared to conventional two-step method.

Supplementary Material

Supporting Information

Supporting Figure S1. g-factors (upper) and ℓ-factors (bottom) calculated using SP-SG+GRAPPA and virtual slice based on GRE at retrospective 9-fold acceleration (R_Sl = 3 and R_In = 3).

Supporting Figure S2. g-factors (upper) and ℓ-factors (bottom) calculated using SP-SG+GRAPPA and virtual slice based on GRE data at prospective 12-fold acceleration (R_Sl = 4 and R_In = 3).

Supporting Figure S3. 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).

Supporting Figure S4. The effect of virtual slice kernel sizes (5×5, 9×9, and 13×13) on reconstructed images and calculated g-factros based on GRE data at prospective 12-fold acceleration (R_Sl = 4 and R_In = 3). With a small kernel size, the g-factors result in higher noise amplification due to overfitting. With increasing kernel sizes, the g-factors reaches its minimum noise and then slightly increases noises due to excessive model complexity.

Appendix

To provide an intuitive comprehension of SNR improvement, a simple example is demonstrated by omitting both real neighboring slices and virtual neighboring aliasing offsets here while taking just aliasing offsets of a slice of interest into account, thereby limiting virtual slice concept to parallel imaging. With inplane undersampling (R_In=2), the effective calibration matrix for real and virtual slices is constructed by concatenating an aliasing offset shifted by FOV/2 into the original calibration matrix:

$$\left[\begin{array}{c}\hfill {\mathbf{x}}^{\text{real},\mathrm{trg}}\hfill \\ \hfill 0\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \mathcal{C}({\mathbf{x}}^{\text{real},\mathrm{src}})\hfill \\ \hfill \mathcal{C}({\mathbf{x}}^{\text{virtual},\mathrm{src}})\hfill \end{array}\right]{\mathcal{G}}^{\mathrm{vs}}=\left[\begin{array}{c}\hfill {\mathcal{C}}^{\text{real}}\hfill \\ \hfill {\mathcal{C}}^{\text{virtual}}\hfill \end{array}\right]{\mathcal{G}}^{\mathrm{vs}}={\mathcal{C}}^{\mathrm{vs}}{\mathcal{G}}^{\mathrm{vs}}$$ [A.1]

Using the effective calibration data, the proposed virtual slice kernel is given by:

$${\mathcal{G}}^{\mathrm{vs}}={({\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}})}^{-1}{\mathcal{C}}^{\mathrm{vs},\mathrm{H}}\left[\begin{array}{c}\hfill {\mathbf{x}}^{\text{real},\mathrm{trg}}\hfill \\ \hfill 0\hfill \end{array}\right]$$ [A.2]

Here, the noise performance strongly depends on the matrix conditioning of ${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}$, which can be represented with phase modulation as:

$${\mathcal{C}}^{\text{real}}=\left[\begin{array}{cc}\hfill {\mathbf{c}}_{11}\hfill & \hfill {\mathbf{c}}_{12}\hfill \end{array}\right],{\mathcal{C}}^{\text{virtual}}=\left[\begin{array}{cc}\hfill {\mathbf{c}}_{11}\hfill & \hfill {\mathbf{c}}_{12}{e}^{i\pi }\hfill \end{array}\right]$$ [A.3]

$${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}={\mathcal{C}}^{\text{real},\mathrm{H}}{\mathcal{C}}^{\text{real}}+{\mathcal{C}}^{\text{virtual},\mathrm{H}}{\mathcal{C}}^{\text{virtual}}$$ [A.4]

$${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}=\left[\begin{array}{cc}\hfill 2{\mathbf{c}}_{11}^{\mathrm{H}}{\mathbf{c}}_{11}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2{\mathbf{c}}_{12}^{\mathrm{H}}{\mathbf{c}}_{12}\hfill \end{array}\right]$$ [A.5]

By generalizing the correlation matrix (R_In= n), Eq. [A.5] can be written as:

$${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}=\left[\begin{array}{ccc}\hfill 2{\mathbf{c}}_{11}^{\mathrm{H}}{\mathbf{c}}_{11}\hfill & \hfill \cdots \hfill & \hfill {\sum }_{\mathrm{m}=0,1}{\mathbf{c}}_{11}^{\mathrm{H}}{\mathbf{c}}_{1\mathrm{n}}{\mathrm{e}}^{\mathrm{i}\cdot \mathrm{m}\cdot (\mathrm{n}-1)\cdot 2\pi ∕{\mathrm{R}}_{\text{In}}}\hfill \\ \hfill {\sum }_{\mathrm{m}=0,1}{\mathbf{c}}_{12}^{\mathrm{H}}{\mathbf{c}}_{11}{\mathrm{e}}^{\mathrm{i}\cdot \mathrm{m}\cdot 1\cdot 2\pi ∕{\mathrm{R}}_{\text{In}}}\hfill & \hfill \cdots \hfill & \hfill {\sum }_{\mathrm{m}=0,1}{\mathbf{c}}_{12}^{\mathrm{H}}{\mathbf{c}}_{1\mathrm{n}}{\mathrm{e}}^{\mathrm{i}\cdot \mathrm{m}\cdot (\mathrm{n}-2)\cdot 2\pi ∕{\mathrm{R}}_{\text{In}}}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\sum }_{\mathrm{m}=0,1}{\mathbf{c}}_{1\mathrm{n}}^{\mathrm{H}}{\mathbf{c}}_{11}{\mathrm{e}}^{\mathrm{i}\cdot \mathrm{m}\cdot (\mathrm{n}-1)\cdot 2\pi ∕{\mathrm{R}}_{\text{In}}}\hfill & \hfill \cdots \hfill & \hfill 2{\mathbf{c}}_{1\mathrm{n}}^{\mathrm{H}}{\mathbf{c}}_{1\mathrm{n}}\hfill \end{array}\right]$$ [A.6]

Note that with modulation terms the correlation matrix ${\mathcal{C}}^{\mathrm{vs},\mathrm{H}}{\mathcal{C}}^{\mathrm{vs}}$ results in lower off-diagonal elements by 1 + e^{(i·r·2π/R_In)} (r=1 ~ n − 1) while increasing diagonal elements by 2, respectively, thus resulting in improved matrix conditioning for the inverse problem.