Model-Based Simultaneous Multi-Slice (SMS) Reconstruction with Hankel Subspace Learning for Accelerated MR T1 Mapping

Abstract

Herein, we propose a novel model-based simultaneous multi-slice (SMS) reconstruction method by exploiting data-driven parameter modeling for highly accelerated T1 parameter quantification. We assume that the predefined slice-specific null space operator remains invariant along the parameter dimension. We incorporate the parameter dimension into SMS-HSL to exploit Hankel-structured and Casorati matrices. Given this consideration, the SMS signal is reformulated in k-p space as a constrained optimization problem that exploits rank deficiency for the Hankel-structured matrix and a finite-dimensional basis for a subspace containing slowly evolving signals in the parameter direction. The proposed model-based SMS reconstruction method is validated on in vivo data and compared with state-of-the-art methods with slice acceleration factors of 3 and 5, including an in-plane acceleration factor of 2. The experimental results demonstrate that the proposed method performs effective slice unfolding and signal recovery in reconstructed images and T1 maps with high precision as compared to the state-of-the-art methods.

1. Introduction

Magnetic resonance (MR) parameter mapping has been widely used for tissue characterization because different tissues have distinct parameters, such as T1 longitudinal relaxation time and T2 transverse relaxation times, and it provides insight into the mechanics of disease processes for clinical applications [1,2]. In particular, T1 parameter mapping typically involves an inversion recovery pulse coupled with echo-planar imaging (EPI) encodings, each of which is acquired with different inversion times (TI) [3]. Many T1-weighted images must be acquired during the inversion time to accurately measure the T1 values. This makes it challenging to achieve high-resolution MR parameter mapping because of the limited imaging time.

Some studies have suggested the use of parallel imaging techniques, such as SENSE and GRAPPA, to accelerate MR parameter mapping and exploit the encoding power offered by multiple coils [4,5]. In cases in which coil-sensitivity-induced spatial modulation in the phase-encoding direction may not be well localized, parallel MRI [4,5] often fails to resolve aliasing voxels, resulting in a substantial level of residual artifacts and amplified noise, even at a slightly high acceleration factor (∼3). Compressed sensing (CS), which exploits image sparsity in certain transform domains, has also demonstrated effectiveness in achieving fast acquisition and accurate reconstruction through sparse sampling [6,7]. Model-based approaches have been successfully applied to MR parameter mapping by modeling tissue signal evolution as a temporal constraint [8,9,10]. The abovementioned undersampling strategies are highly effective in increasing the speed of the EPI encoding for multi-shot imaging because of the reduction in the number of shots; however, they are relatively less time-efficient for single-shot imaging, although they mitigate the k-space signal modulation caused by T2* decay. Additionally, all the undersampling approaches result in a direct tradeoff between encoding efficiency and signal-to-noise ratio (SNR) efficiency.

Simultaneous multi-slice (SMS) MRI [11,12,13], which employs multi-band (MB) radio-frequency (RF) pulses to excite multiple slices at the same time while exploiting spatial and interchannel correlation to resolve aliased voxels in the slice direction, can potentially alleviate the problems in the undersampling strategies that are related to the low encoding efficiency in the single-shot imaging, as well as the inherent loss of signals resulting from insufficient sampling. In order to improve the SNR efficiency, some studies have combined controlled aliasing with SMS imaging to evenly distribute the aliasing between the two axes of the phase-encoding and slice directions, followed by unaliasing of the overlap voxels using parallel imaging [14]. Split slice-GRAPPA (SP-SG) is a reconstruction technique that imposes a slice leakage block constraint to effectively remove neighboring slice signals, thus allowing for improved slice separation in SMS imaging [15,16,17,18]. Recently, a SPIRiT [19]-type reconstruction method was introduced, which uses readout concatenation in the image domain to accelerate SMS encoding and enhance coil self-consistency as in SPIRiT reconstruction in an extended k-space, allowing for noise reduction (ROCK-SPIRiT) [20]. Several low-rank-promoting constraint approaches have been proposed for aliasing separation in SMS imaging [20,21,22,23,24,25,26,27,28]. The concept of virtual slices has been incorporated into SP-SG as a single-step method. In this approach, in-plane aliasing artifacts are considered additional slices that are eliminated during the reconstruction process [23]. Hankel-structured low-rank SMS reconstruction, referred to as SMS-MUSSELS [24,25], was introduced by combining the annihilating filter concept with SENSE reconstruction. Low-rank matrix modeling of local k-space neighborhoods (LORAKS) has also been applied to SMS acquisition with low-rank completion [27,28]. The hybrid-space SENSE method represents a 3D reconstruction with low-rank constraints of the SMS acquisition [29]. Furthermore, there has been a recent expansion of the SMS reconstruction method to incorporate a deep learning network [30,31].

In our previous work [32], using an SMS reconstruction method that exploits Hankel subspace learning for aliasing separation in the slice direction (SMS-HSL), we modeled an SMS signal with a Hankel matrix as a linear combination of all slices and decomposed the Hankel-structured SMS signal into slices of interest and combined complementary slices. Employing the complementary null space, the signal model is projected onto the subspace, where slices of no interest are nulled, and only a slice of interest is left.

In the present study, we extend our preliminary work to a model-based SMS-HSL framework by exploiting a data-driven parameter model for T1 mapping. First, we incorporate the parameter dimension into the SMS-HSL formulation to exploit the plural properties of both Hankel- and Casorati-structured [33] matrices. We introduce a data-driven parameter model as prior knowledge to use the similarity among different TIs and reduce the degrees of freedom of reconstruction for accurate spatial map coefficients. Thus, the proposed method estimates both spatial map coefficients and parameter image series within a single SMS optimization framework. The efficacy of this technique is validated first on the human brain. The reconstruction results of existing state-of-the art SMS algorithms such as SP-SG, SMS-HSL, SMS-MUSSELS, and ROCK-SPIRiT are also compared to demonstrate the superiority of the proposed method.

2. Methods

2.1. Related Studies

(1) 

Simultaneous Multi-slice (SMS) Acquisition: An SMS signal is modeled by combining multiple slices:

$$\begin{array}{cc}\hfill y\left(\mathbf{k},l\right)& =\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}\sum _{n=1}^{{\mathrm{N}}_{\mathrm{r}}}{\rho }_m\left({\mathbf{r}}_{\mathrm{n}}\right)C_m({\mathbf{r}}_{\mathrm{n}},l)e^{-j\mathbf{k}·{\mathbf{r}}_{\mathrm{n}}}+n(\mathbf{k},l)\hfill \\ \hfill & =\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{x}_m(\mathbf{k},l)+n(\mathbf{k},l),\hfill \end{array}$$

(1)

where $y(\mathbf{k},l)$ is the measured aliased SMS signal in k space for the $l^{\mathrm{th}}$ coil, ${\rho }_m\left({\mathbf{r}}_{\mathrm{n}}\right)$ is the desired $n^{\mathrm{th}}$ voxel signal at the $m^{\mathrm{th}}$ slice, $C_m(\mathbf{r},l)$ is the coil sensitivity map at the $l^{\mathrm{th}}$ coil, $x_m(\mathbf{k},l)$ is the desired k-space signal at the $m^{\mathrm{th}}$ slice, and $n(\mathbf{k},l)$ is the additive noise. The coil sensitivity map ($C_m(\mathbf{r},l)$) refers to the sensitivity a particular channel to a specific location in space. To cover the desired field of view (FOV), individual receiver coils are arranged in an array, and their sensitivity maps are designed accordingly. When conducting an MR scan using multiple coils, the resulting images from each channel need to be combined to create a single image. This combination can be achieved by summing the squares of the images (sum of squares) [34,35,36]. For slice unfolding, $x_m$ is estimated using slice-specific coil sensitivity maps in either the image or k-space domain as the solution of an inverse problem in Equation (1).

(2) 

SMS-GRAPPA: To synthesize a slice of interest on the collapsed k-space data, we calculate a slice-specific SMS kernel using the collapsed and aliasing-free k-space data:

$$\begin{array}{c}\hfill {\mathcal{G}}_{\mathrm{m}}({\mathbf{x}}_1+{\mathbf{x}}_2+····+{\mathbf{x}}_{{\mathrm{N}}_{\mathrm{s}}})={\mathcal{G}}_{\mathrm{m}}\mathbf{y}={\mathbf{x}}_{\mathrm{m}},\end{array}$$

(2)

where ${\mathbf{x}}_{\mathrm{m}}$ is a vector of k-space data at the m${}^{\mathrm{th}}$ slice acquired from the calibration, ${\mathbf{y}}_{\mathrm{m}}$ is a vector of SMS k-space data emulated from the calibration, and ${\mathcal{G}}_{\mathrm{m}}$ is a matrix that generates the m${}^{\mathrm{th}}$ slice by convolving the k-space data with a series of SMS kernels. To explicitly control the cross-talk artifacts between neighboring slices, the SP-SG is introduced by enforcing signals in other slices to zero while keeping the target slice signals preserved in the k-space domain [15,16,17,18]:

$${\mathcal{G}}_m\left[\begin{array}{c}{\mathbf{x}}_1\\ ⋮\\ {\mathbf{x}}_{\mathrm{m}}\\ ⋮\\ {\mathbf{x}}_{{\mathrm{N}}_{\mathrm{s}}}\end{array}\right]=\left[\begin{array}{c}\mathbf{0}\\ ⋮\\ {\mathbf{x}}_{\mathrm{m}}\\ ⋮\\ \mathbf{0}\end{array}\right]$$

(3)

Once the SMS kernel is calculated using the inverse problem, each slice is estimated by projecting the SMS k-space data onto the subspace spanned by the SMS kernel.

(3) 

SMS-HSL: Hankel Subspace Learning for Slice Nulling: To take advantage of low-rank properties for both slice separation and data correlation in k-space, we reorganize the SMS signals as a Hankel-structured matrix, then decompose them into a slice of interest and its complementary slices:

$$\begin{array}{c}\hfill \mathcal{H}\left[\mathbf{y}\right]=\mathcal{H}\left[{\mathbf{x}}_m^c\right]+\mathcal{H}\left[{\mathbf{x}}_m\right]+\mathbf{N}\end{array}$$

(4)

where ${\mathbf{x}}_{\mathrm{m}}^{\mathrm{c}}$ is the complementary slice that combines all slices other than the ${\mathrm{m}}^{\mathrm{th}}$ slice, and $\mathbf{N}$ is the Hankel-structured noise matrix. $\mathcal{H}\left[·\right]$ is the Hankel operator that produces a Hankel-structured matrix by sliding $p_x$ × $p_y$ patches $\mathcal{H}:{\mathbb{C}}^{{\mathrm{N}}_x\times {\mathrm{N}}_y\times {\mathrm{N}}_l}\to {\mathbb{C}}^{({\mathrm{N}}_x-{\mathrm{p}}_x+1)({\mathrm{N}}_y-{\mathrm{p}}_y+1)}\times {\mathrm{p}}_x{\mathrm{p}}_y{\mathrm{N}}_l$ with ${\mathrm{N}}_x$ frequency encoders, ${\mathrm{N}}_y$ phase encoders, and ${\mathrm{N}}_l$ coils.

Conceptually, a slice of interest is estimated by projecting the aliased SMS signal onto the complementary null space:

$$\begin{array}{cc}\hfill \mathcal{H}\left[\mathbf{y}\right]{\mathcal{N}}_m^c{\mathcal{N}}_m^{c,H}& =\mathcal{H}\left[{\mathbf{x}}_m^c\right]{\mathcal{N}}_m^c{\mathcal{N}}_m^{c,H}+\mathcal{H}\left[{\mathbf{x}}_m\right]{\mathcal{N}}_m^c{\mathcal{N}}_m^{c,H}\hfill \\ \hfill & \approx \mathcal{H}\left({\mathbf{x}}_m\right){\mathcal{N}}_m^c{\mathcal{N}}_m^{c,H}\hfill \end{array}$$

(5)

The complementary null space passes a slice of interest while nulling those in the other slices. SMS-HSL further exploits the k-space correlation across all coils by imposing a low-rank prior for a slice of interest:

$$\underset{{\mathbf{x}}_{\mathrm{s}}}{\min }\frac{1}{2}{∥\left(\mathcal{H}\left[\mathbf{y}\right]-\mathcal{H}\left[{\mathbf{x}}_{\mathrm{m}}\right]\right){\mathcal{N}}_{\mathrm{m}}^{\mathrm{c}}∥}_{\mathrm{F}}^2+{\lambda }_{\mathrm{L}}{∥\mathcal{H}\left[{\mathbf{x}}_{\mathrm{m}}\right]∥}_{\ast },$$

(6)

where ${\lambda }_L$ is the regularization parameter for the Hankel low-rank prior; ${∥·∥}_{\mathbf{F}}$ is the Frobenius norm; and ${∥·∥}_{\ast }$ is the nuclear norm, which is defined as the sum of the singular values of the matrix.

2.2. Proposed Method

(1) 

SMS Signal Model in the k-Parameter Dimension: The signal model for the SMS with parameter dimensions can be represented by:

$$\begin{array}{cc}\hfill y\left(\mathbf{k},l,p\right)& =\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}\sum _{n=1}^{{\mathrm{N}}_{\mathrm{r}}}{\rho }_m({\mathbf{r}}_{\mathrm{n}},p)C_m({\mathbf{r}}_{\mathrm{n}},l)e^{-j\mathbf{k}·{\mathbf{r}}_{\mathrm{n}}}+n(\mathbf{k},l,p)\hfill \\ \hfill & =\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{x}_m(\mathbf{k},l,p)+n(\mathbf{k},l,p)\hfill \end{array}$$

(7)

where $y(\mathbf{k},l,p)$ is the measured aliased SMS signal in k space for the $l^{\mathrm{th}}$ coil, p is the parameter index, ${\rho }_m({\mathbf{r}}_{\mathrm{n}},p)$ is the desired $n^{\mathrm{th}}$ voxel signal at the $m^{\mathrm{th}}$ slice, $C_m({\mathbf{r}}_{\mathrm{n}},l)$ is the coil sensitivity at the $l^{\mathrm{th}}$ coil, $x_m(\mathbf{k},l,p)$ is the desired k-space signal at the $m^{\mathrm{th}}$ slice, and $n(\mathbf{k},l,p)$ is the additive noise that is modeled to have a complex white Gaussian distribution. To separate the aliased SMS signal, the complementary null space can be learned from the low-resolution reference data. It was observed in a previous study [32] that the idea of estimating a set of vectors in the noise subspace can be extended to separate overlapping slices using a mathematical model in which the null space project enforces the acquired SMS data to lie in the noise subspace only for the neighboring slice; consequently, slices of no interest are filtered out. The null space concept can be applied extensively to MR parameter mapping under the assumption that the predefined null space remains invariant along the parameter dimension. In Figure 1, the slice-specific null space shows a strong dependence on slice encoding rather than on image contrast. To avoid time-consuming calibration along the parameter dimension, the complementary null space is learned from the complementary Hankel-structured SMS signal through singular value decomposition (SVD) by taking right singular vectors corresponding to low singular values from the first TI low-resolution reference datum and projecting all parameter dimensions to separate aliasing SMS signals. Then, the null space projection effectively filters out a slice of no interest while passing through a slice of interest, regardless of the parameter dimension. An illustration is also provided in Figure S1.

$${\mathcal{H}}_p\left(\mathbf{x}\right)=\left[\begin{array}{ccc}\mathbf{x}\left({\mathbf{k}}_{0,0},1,p\right)& \cdots & \mathbf{x}\left({\mathbf{k}}_{0,0},{\mathrm{N}}_l,p\right)\\ \mathbf{x}\left({\mathbf{k}}_{0,1},1,p\right)& \cdots & \mathbf{x}\left({\mathbf{k}}_{0,1},{\mathrm{N}}_l,p\right)\\ ⋮& \ddots & ⋮\\ \mathbf{x}\left({\mathbf{k}}_{{\mathrm{N}}_x-{\mathrm{p}}_x,{\mathrm{N}}_y-{\mathrm{p}}_y},1,p\right)& \cdots & \mathbf{x}\left({\mathbf{k}}_{{\mathrm{N}}_x-{\mathrm{p}}_x,{\mathrm{N}}_y-{\mathrm{p}}_y},{\mathrm{N}}_l,p\right)\end{array}\right]$$

(8)

Given the multi-channel SMS signals in k-p space for parameter mapping, strong correlations among neighboring signals can be selected in the Hankel matrix by using sliding patches ($p_x$ × $p_y$) in k-p space over coils using a Hankel operator: ${\mathcal{H}}_t={\mathbb{C}}^{{\mathrm{N}}_x\times {\mathrm{N}}_y\times {\mathrm{N}}_l\times {\mathrm{N}}_p}\to {\mathbb{C}}^{({\mathrm{N}}_x-{\mathrm{p}}_x+1)({\mathrm{N}}_y-{\mathrm{p}}_y+1)\left({\mathrm{N}}_p\right)\times {\mathrm{p}}_x{\mathrm{p}}_y{\mathrm{N}}_l}$

$${\mathcal{H}}_t\left(\mathbf{x}\right)=\left[\begin{array}{c}{\mathcal{H}}_1\left(\mathbf{x}\right)\\ {\mathcal{H}}_2\left(\mathbf{x}\right)\\ ⋮\\ {\mathcal{H}}_{{\mathrm{N}}_{\mathrm{p}}}\left(\mathbf{x}\right)\end{array}\right]$$

(9)

Then, the SMS signal model can be represented in matrix form as:

$${\mathcal{H}}_t\left(\mathbf{y}\right)=\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{\mathcal{H}}_t\left({\mathbf{x}}_m\right)+\mathbf{N}$$

(10)

where ${\mathcal{H}}_t\left(\mathbf{y}\right)$ is the aliased Hankel matrix including all parameter dimensions, ${\mathcal{H}}_t\left({\mathbf{x}}_m\right)$ is the $m^{\mathrm{th}}$ Hankel matrix, and N is the Hankel-structured noise matrix.

We observe that contrast dynamics slowly vary according to different TIs on the time-invariant background signals, while tissues of interest exist even in the different slices, although their spatial locations differ from one another. The former means that the contrast dynamics of a single-slice image can be represented on the low-dimensional subspace across different TIs while sharing the same spatial basis functions. The latter implies that all slices share similar temporal basis functions, as evidenced by similar contrast dynamics across slices. Figure 2A shows the T1 weighted multi-contrast images that describe a mono-exponential relaxation process. Using principal component analysis, we can express the desired T1-weighted multi-contrast images as the product of spatial coefficient maps and parameter basis functions. For example, the first three principal components are selected as the parameter basis functions computed from each slice and SMS, as shown in Figure 2B, in which the singular values drop quite rapidly by decomposing the information into a significant signal and noise-like subspaces. Interestingly, the parameter basis functions of SMS acquisition are considerably similar to those of slice-by-slice acquisition. Because such a parameter subspace is complementary to the redundancy between slices, its use might provide additional gain by reducing the degrees of freedom in the spatial coefficient maps with predefined parameter basis functions. Accordingly, the Casorati matrix can be represented as a product of the two subspaces:

$${\mathbf{X}}_m=\left[\begin{array}{ccc}{\mathbf{x}}_m\left({\mathbf{k}}_1,1,p_1\right)& \cdots & {\mathbf{x}}_m\left({\mathbf{k}}_1,1,p_{{\mathrm{N}}_{\mathrm{p}}}\right)\\ ⋮& \ddots & ⋮\\ {\mathbf{x}}_m\left({\mathbf{k}}_{{\mathrm{N}}_{\mathrm{r}}},1,p_1\right)& \cdots & {\mathbf{x}}_m\left({\mathbf{k}}_{{\mathrm{N}}_{\mathrm{r}}},1,p_{{\mathrm{N}}_{\mathrm{p}}}\right)\\ {\mathbf{x}}_m\left({\mathbf{k}}_1,2,p_1\right)& \cdots & {\mathbf{x}}_m\left({\mathbf{k}}_1,2,p_{{\mathrm{N}}_{\mathrm{p}}}\right)\\ ⋮& \ddots & ⋮\\ {\mathbf{x}}_m\left({\mathbf{k}}_{{\mathrm{N}}_{\mathrm{r}}},2,p_1\right)& \cdots & {\mathbf{x}}_m\left({\mathbf{k}}_{{\mathrm{N}}_{\mathrm{r}}},2,p_{{\mathrm{N}}_{\mathrm{p}}}\right)\\ ⋮& \ddots & ⋮\\ {\mathbf{x}}_m\left({\mathbf{k}}_{{\mathrm{N}}_{\mathrm{r}}},{\mathrm{N}}_l,p_1\right)& \cdots & {\mathbf{x}}_m\left({\mathbf{k}}_{{\mathrm{N}}_{\mathrm{r}}},{\mathrm{N}}_l,p_{{\mathrm{N}}_{\mathrm{p}}}\right)\end{array}\right]\sim {\mathbf{U}}_m\mathbf{V}$$

(11)

where ${\mathbf{U}}_m\in {\mathbb{C}}^{\left({\mathrm{N}}_r{\mathrm{N}}_l\right)\times {\mathrm{N}}_R}$ is the spatial coefficient, $\mathbf{V}\in {\mathbb{C}}^{{\mathrm{N}}_R\times {\mathrm{N}}_p}$ is the temporal basis, and ${\mathrm{N}}_R$ is the number of principal components. Given the SMS signal model that considers signal evolution in the parameter dimension, we can rewrite Equation (11) as:

$${\mathcal{H}}_t\left(\mathbf{y}\right)=\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{\mathcal{H}}_t\left({\mathcal{A}}_m\left({\mathbf{U}}_m\mathbf{V}\right)\right)+\mathbf{N}$$

(12)

where ${\mathcal{A}}_m$ is the operator for ${\mathcal{A}}_m={\mathbb{C}}^{\left({\mathrm{N}}_r{\mathrm{N}}_l\right)\times {\mathrm{N}}_p}\to {\mathbb{C}}^{{\mathrm{N}}_x\times {\mathrm{N}}_y\times {\mathrm{N}}_l\times {\mathrm{N}}_p}$, and N is the Hankel-structured noise matrix. It is assumed that acquired SMS data along the parameter dimension are followed by an exponential increase in T1 relation times. We hypothesize that this increase can be delineated by linear combination of the temporal basis found by performing singular value decomposition (SVD) of the measured SMS data (Figure S2).

(2) 

Aliasing Separation: Given the considerations mentioned above, we incorporate null space projection (${\mathcal{N}}_m^c$) and a low-rank prior (${\sum }_{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{∥{\mathcal{H}}_t\left({\mathcal{A}}_m\left({\mathbf{U}}_m\mathbf{V}\right)\right)∥}_{\ast }$) into the single optimization problem with data fidelity. The proposed method is summarized in (Algorithm 1):

$$\begin{array}{c}\hfill \widehat{\mathbf{U}}=\underset{\mathbf{U}\mathrm{in}\{{\mathbf{U}}_1,\cdots ,{\mathbf{U}}_{{\mathrm{N}}_{\mathrm{s}}}\}}{\arg \min }\frac{1}{2}{∥{\mathcal{H}}_t\left(\mathbf{y}\right)-\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{\mathcal{H}}_t\left({\mathcal{A}}_m\left({\mathbf{U}}_m\mathbf{V}\right)\right)∥}_{\mathrm{F}}^2\\ \hfill +\frac{1}{2}\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{∥\left({\mathcal{H}}_t\left(\mathbf{y}\right)-{\mathcal{H}}_t\left({\mathcal{A}}_m\left({\mathbf{U}}_m\mathbf{V}\right)\right)\right){\mathcal{N}}_m^c∥}_{\mathrm{F}}^2\\ \hfill +{\lambda }_L\sum _{m=1}^{{\mathrm{N}}_{\mathrm{s}}}{∥{\mathcal{H}}_t\left({\mathcal{A}}_m\left({\mathbf{U}}_m\mathbf{V}\right)\right)∥}_{\ast }\end{array}$$

(13)

where the first term is data fidelity, the second term is the reconstruction fidelity, the third term (${\left|\right|.\left|\right|}_{\ast }$) is the low-rank prior, and ${\lambda }_L$ is the regularization parameter. The null-space projection errors in the second term are independent of slices, and the low rankness of ${\mathcal{H}}_t$ is independent of slices. Therefore, we can use projection onto convex sets (POCS) to ensure the data fidelity of every iteration.

$$\begin{array}{c}\hfill \widehat{{\mathbf{U}}_s}=\underset{{\mathbf{U}}_s}{\mathrm{argmin}}\frac{1}{2}{∥\left({\mathcal{H}}_t\left(\mathbf{y}\right)-{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_s\mathbf{V}\right)\right)\right){\mathcal{N}}_s^c∥}_{\mathrm{F}}^2\\ \hfill +{\lambda }_L{∥{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_s\mathbf{V}\right)\right)∥}_{\ast }\end{array}$$

(14)

The above equation is reformulated by introducing new auxiliary variables for a low-rank prior under the variable-splitting algorithm [37,38,39]:

$$\begin{array}{c}\hfill \{{\widehat{\mathbf{U}}}_s,{\widehat{\mathbf{Q}}}_s\}=\underset{{\mathbf{U}}_s,{\mathbf{Q}}_s}{\arg \min }\frac{1}{2}{∥\left({\mathcal{H}}_t\left(\mathbf{y}\right)-{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_{\mathrm{s}}\mathbf{V}\right)\right)\right){\mathcal{N}}_s^c∥}_{\mathrm{F}}^2\\ \hfill +{\lambda }_L{∥{\mathbf{Q}}_s∥}_{\ast }+\frac{\beta }{2}{∥{\mathbf{Q}}_s-{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_s\mathbf{V}\right)\right)∥}_{\mathrm{F}}^2\end{array}$$

(15)

where ${\mathbf{U}}_s$ is the desired signal (${\mathbf{U}}_s\in {\mathbb{C}}^{\left({\mathrm{N}}_r{\mathrm{N}}_l\right)\times {\mathrm{N}}_R}$), and ${\mathbf{Q}}_s$ is the auxiliary matrix for the slice-specific Hankel-matrix. The unknown variables $({\mathbf{U}}_s,{\mathbf{Q}}_s)$ can be recovered by solving the following minimization problem using the alternating direction method (ADM) [37,38,39]. The first step is to minimize with respect to $\mathbf{Q}$, which is estimated using a singular-value soft thresholding method:

$$\begin{array}{cc}\hfill {\widehat{\mathbf{Q}}}_s^{n+1}& =\underset{{\mathbf{Q}}_s}{\arg \min }{\lambda }_L{∥{\mathbf{Q}}_s∥}_{\ast }+\frac{\beta }{2}{∥{\mathbf{Q}}_s-{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_s\mathbf{V}\right)\right)∥}_{\mathrm{F}}^2\hfill \\ \hfill & ={\mathcal{S}}_{{\lambda }_L/\beta }\left({\mathcal{H}}_t\left({\mathcal{A}}_s\left({\widehat{\mathbf{U}}}_{\mathrm{s}}^n\mathbf{V}\right)\right)\right)\hfill \end{array}$$

(16)

where ${\mathcal{S}}_{{\lambda }_L/\beta }\left(·\right)$ is the singular value shrinkage function that is defined by:

$$\begin{array}{c}\hfill {\mathcal{S}}_{{\lambda }_L/\beta }\left(·\right)=\left\{\begin{array}{cc}\hfill & \left|·\right|-{\lambda }_L/\beta ,\left|·\right|\ge {\lambda }_L/\beta \hfill \\ \hfill & 0,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

(17)

The second step is to minimize with respect to ${\mathbf{U}}_s$, which is reduced to a least squares approximation problem by:

$$\begin{array}{c}\hfill {\widehat{\mathbf{U}}}_s^{n+1}=\underset{{\mathbf{U}}_s}{\mathrm{argmin}}\frac{1}{2}{∥\left({\mathcal{H}}_t\left(\mathbf{y}\right)-{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_{\mathrm{s}}\mathbf{V}\right)\right)\right){\mathcal{N}}_s^c∥}_{\mathrm{F}}^2\\ \hfill +\frac{\beta }{2}{∥{\mathbf{Q}}_s-{\mathcal{H}}_t\left({\mathcal{A}}_s\left({\mathbf{U}}_s\mathbf{V}\right)\right)∥}_{\mathrm{F}}^2\\ \hfill {\mathcal{A}}^{\ast }{\mathcal{H}}_t^{\ast }\left(\left[{\mathcal{H}}_t\left(\mathbf{y}\right){\mathcal{N}}_{\mathrm{s}}^{\mathrm{c}}{{\mathcal{N}}_s^c}^{\ast }+\beta \left({\widehat{\mathbf{Q}}}^n+1\right)\mathbf{V}\right]{\left[{\mathcal{N}}_s^c{{\mathcal{N}}_s^c}^{\ast }+\beta \mathbf{I}\right]}^{-1}\right){\mathbf{V}}^H\end{array}$$

(18)

Q and U estimations are performed for all simultaneously excited slices. Then data fidelity is enforced, convergence is checked as follows.

$$\underset{s=1\cdots {\mathrm{N}}_{\mathrm{s}}}{\max }\left\{\frac{{∥{\widehat{\mathbf{U}}}_s^{n+1}-{\widehat{\mathbf{U}}}_s^n∥}_{\mathrm{F}}^2}{{∥{\widehat{\mathbf{U}}}_s^n∥}_{\mathrm{F}}^2}\right\}\le ϵ\mathrm{or}n>{\mathrm{N}}_{\max }$$

where $ϵ$ is the predetermined tolerance, and ${\mathrm{N}}_{max}$ is the maximum number of iterations.

Algorithm 1: Model-based SMS Reconstruction of MR T1 Mapping

1.  Task: Find $\mathbf{U}$ minimizing Equation (13) 2.  Initialization:  Iteration index: n = 1  Predefined tolerance: $ϵ$ = 1 × ${10}^{-4}$  Regularization parameters: ${\lambda }_L$ and $\beta$  Predefined temporal basis: $\mathbf{V}$ computed by performing SVD on the acquired SMS data (Figure S2)  Complementary null space: ${\mathcal{N}}_m^c$ computed by performing SVD right singular vectors (Figure 1A) 3.  Algorithm:  Step 1: Update ${\widehat{\mathbf{U}}}_s$, ${\widehat{\mathbf{Q}}}_s$ as Equations ((14)–(17))  Step 2: Increase iteration index n  Step 3: Stop if $n>{\mathrm{N}}_{\max }$ or $\left\{\frac{{∥{\widehat{\mathbf{U}}}_s^{n+1}-{\widehat{\mathbf{U}}}_s^n∥}_{\mathrm{F}}^2}{{∥{\widehat{\mathbf{U}}}_s^n∥}_{\mathrm{F}}^2}\right\}\le ϵ$ 4.  Out: ${\mathbf{U}}_s^{n+1}\mathbf{V}\to \mathbf{X}$

3. Experiments

Experimental studies were performed to evaluate the performance of the proposed reconstruction method using retrospective SMS data was acquired via conventional inversion recovery EPI. To compare image reconstruction algorithms, undersampled data were reconstructed with SP-SG, SMS-MUSSELS, and the proposed method. All reconstruction algorithms were implemented offline using Matlab software (R2017b; MathWorks Inc., Natick, MA, USA). All processes of the proposed algorithm were implemented in the k-space domain.

(1) 

IR-GE-EPI: Fully sampled brain k-space data were collected by a 3T GE Discovery scanner equipped with a 32-channel receiver head coil using a 2D IR-GE-EPI sequence. The following parameters were used for IR-GE-EPI: FOV = $240\times 240{\mathrm{mm}}^2$, matrix size = $120\times 120$, slice thickness = 2 mm, number of slices = 25, TR/TE = 30,000 ms/40 ms, and 15 IR scans (TI = 50, 250, 450, 650, 850, 1050, 1250, 1450, 1650, 1850, 2050, 2250, 2450, 2650, and 2850 ms). For algorithm validation, our experiments used four slices (3rd, 10th, 17th, and 24th slices) from this dataset with an SMS factor of 4.

(2) 

IR-SE-EPI: Fully sampled brain k-space data were collected by a 3T Prisma equipped with a 20-channel receiver head coil using a 2D IR-SE-EPI sequence. The following parameters were used for IR-SE-EPI: FOV = $220\times 220{\mathrm{mm}}^2$, matrix size = $128\times 128$, slice thickness = 3 mm, number of slices = 30, TR/TE = 20,000 ms/27 ms, 16 IR scans (TI = 34, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1400, 1600, 1800, and 2000 ms), and partial Fourier (PF) = 62.5%. To compare the slice unaliasing performance of the current state-of-the-art techniques listed in the Introduction, our experiments used five slices (1st, 7th, 13th, 19th, and 25th slices) from this dataset with ab SMS factor of 5.

(3) 

Slice-Shuffled SMS IR-SE-EPI: Prospective brain k-space data were collected by a 3T GE Discovery scanner equipped with a 32-channel receiver head coil using a slice-shuffled SMS IR-SE-EPI sequence [40]. The following parameters were used for slice-shuffled SMS IR-EPI: FOV = $240\times 240{\mathrm{mm}}^2$, matrix size = $120\times 120$, slice thickness = 2 mm, number of slices = 75, TR/TE = 3000 ms/25 ms, 13 IR scans (TI = 50 + $n$ × TR/N ms, with n = 0, 1, …, 24), specific CAIPI phase shift = FOV/3, in-plane acceleration = 2, and SMS factor = 5.

All datasets were processed offline on a personal computer with 3.3 GHz CPU and 64 GB RAM using MATLAB (The Math Works, Natick, MA, USA). For retrospective SMS experiments, we applied a CAIPI-induced phase shift to the fully sampled k-space data of each slice, then linearly combined all the excited slices. To unalias the collapsed slices, we compared model-based SMS-SHL with SP-SG and SMS-HSL using codes available at https://doi.org/10.5281/zenodo.5149780 and https://misl-skku.github.io/software.html, respectively. The reconstructed images were visualized after combining the multichannel images using the root sum of squares, yielding a single image at each TI. To estimate the T1 values, a nonlinear least-squares fitting method was performed on a pixel-by-pixel basis. Experimental datasets and reconstruction parameters are shown in table in the Supplementary Materials.

For quantitative evaluation, the normalized root mean square error (nRMSE) was used as $\frac{1}{\mathbf{max}\left({\rho }_s\right)}\sqrt{\frac{1}{\mathrm{N}}{\sum }_{r=1}^{\mathrm{N}}{({\rho }_s-{\widehat{\rho }}_s)}^2}$, where ${\rho }_s$ and ${\widehat{\rho }}_s$ are the reference and estimated images, respectively, and $\mathrm{N}$ is the total number of pixels. To clearly visualize the differences between the reconstruction methods, we calculated the relative image error maps using $\frac{\left|{\rho }_s-\widehat{\rho }\right|}{\left|{\rho }_s\right|}\times 100\%$ and the T1 error maps using $\frac{\left|T_1-\widehat{T_1}\right|}{\left|T_1\right|}\times 100\%$ [6,41].

SMS-MUSSELS reconstruction was performed with a $7\times 7$ kernel size, and Fista iterations (number of iterations = 100, predefined tolerance = 0.3) with a regularization factor of $\lambda$ = 1.25. Regularized ROCK-SPIRiT was implemented with a $7\times 7$ kernel size and the Tikhonov regularization parameter set to ${10}^{-2}$ for calibration. The regularization parameters for reconstruction were set to default values, and conjugate-gradient (CG) and outer-loop iterations were 5 and 20, respectively. In SP-SG reconstruction, the two-step approach, which involved slice unaliasing followed by in-plane reconstruction, was implemented (SP-SG + GRAPPA). For a fair comparison, a reconstruction kernel of SP-SG, which produces the smallest nRMSE, was manually determined by changing the kernel size from $3\times 3$ to $7\times 7$ and set to $5\times 5$. Each kernel was estimated using 32 phase-encoding lines in the central region of k space by sweeping Tikhonov regularization parameters from $5.0\times {10}^{-1}$ to $5.0\times {10}^{-6}$ and set to $5.0\times {10}^{-4}$.

For the model-based SMS-HSL, we used a $6\times 6$ window size in the Hankel-structured matrix while employing five principal components of the parameter basis functions in the Casorati matrix. The cutoff singular value for null-space selection was set to a maximum of 0.05. The effect of the regularization parameters in Equation (16) on the image reconstruction was investigated by calculating nRMSE between the reference and the estimated proposed model-based SMS-HSL with varying ${\lambda }_L$ and $\beta$ values. Reconstruction errors were lowest with ${\lambda }_L=1.0\times {10}^{-6}$ and $\beta =1.0\times {10}^{-13}$ and gradually increased as ${\lambda }_L$ and $\beta$ deviated from the optimal values. ${\lambda }_L$ and $\beta$ were set to $1.0\times {10}^{-6}$ and $1.0\times {10}^{-13}$, respectively. Figure S3 shows the sensitivity of the proposed method to the regularization parameters (${\lambda }_L$ and $\beta$) and convergence behaviors with patch size. For SMS-HSL without a magnitude prior, all reconstruction parameters were identical to those of the model-based SMS-HSL.

The calibration region consisted of 32 fully sampled phase-encoding lines in the central k space of the first TI data for retrospective studies. In a prospective study, we acquired fully sampled single-band EPI data for calibration. Representative results from the above datasets validate the effectiveness of the proposed method.

4. Results

4.1. Image Reconstruction: Stepwise Validation

To validate the model-based SMS-HSL, we tested the stepwise performance by generating the following four sets of images with an SMS factor of 4: (1) reference, (2) null space only, (3) null space + low rank, and (4) null space + low rank + parameter subspace model. Figure 3 shows the reconstructed images (TI = 2050 ms) and their corresponding T1 maps. The null-space-only reconstruction creates signal shading by reducing tissue conspicuity, particularly in the aliased location between neighboring slices. Accordingly, the T1 maps obtained using the null-space-only method are corrupted by noise-like artifacts resulting from ill-conditioned problems. Imposing a low-rank prior on the Hankel-structured matrix effectively increases the signal intensity in the collapsed area, leading to improved T1 maps.

However, it exhibits aliasing artifacts in the low-SNR area, as indicated by the red arrows, although not clearly observed in the corresponding T1 maps. In contrast, by imposing a parameter subspace on the Casorati matrix with the Hankel low rank, the proposed method achieves the best reconstruction results by removing the aliasing artifacts, as indicated by the white arrows, yielding more accurate T1 values comparable to those of the reference.

4.2. Retrospective Studies: SMS Simulation

We compared our model-based SMS-HSL with SP-SG and SMS-HSL without magnitude priors with an SMS factor of 5.

Figure 4A,B illustrate the reconstructed images and their corresponding error maps at TI = 34 ms. SP-SG is severely contaminated with noise, particularly in the central region of the brain, owing to the restriction of the interslice leakage constraint. Despite substantially mitigating noise amplification, SMS-HSL without a magnitude prior suffers from blurring and a lack of sharpness. By imposing parameter basis functions, our proposed method suppresses noise while introducing slight image blurring. Note that the complementary properties derived from the Casorati and Hankel matrices are less vulnerable to the aliasing and noise observed in SP-SG and SMS-HSL without magnitude priors. Consistent with the results shown in the error maps, our proposed method presents the lowest nRMSE for all slices.

Figure 5 shows a single exemplary slice at four TIs, which provide different T1 contrasts for subsequent T1 mapping. As indicated by the red arrows, both SP-SG and SMS-HSL without magnitude priors exhibit severe noise and aliasing artifacts in the late TIs because of the SNR penalty around a signal null point for gray- and white-matter tissues, whereas the results from the model-based SMS-HSL are consistent with those of the reference.

Figure 6A,B show the corresponding T1 values and voxel-wise error maps estimated using different TIs. Because of the high-noise g factor, the T1 map obtained by SP-SG is corrupted around the cerebrospinal fluid (CSF), whereas the edge structures of the T1 maps obtained by SMS-HSL without a magnitude prior tend to be oversmoothed around the boundaries of the brain. In contrast, the T1 values estimated by the proposed method agree well with those estimated by the reference, yielding more accurate T1 values. This trend is pronounced in the magnified images of the fourth slice, as shown in Figure 6C. Table S4 confirms the excellent T1 accuracy and precision for all reconstructed slices compared to other methods.

Such an improvement can also be observed in the T1 signal evolution across multiple TIs of representative voxels in the image. Figure 7A,B show the signal intensity curves along the parameter dimension in the white- and gray-matter tissues. The specific regions of interest (ROIs) were manually selected from each tissue within the yellow circle (A: white matter; B: gray matter) by averaging pixel values within a 3 × 3 window. For white matter, SP-SG and SMS-HSL without magnitude priors showed some bias from the actual T1 curve at the late TIs in the opposite direction. For gray matter, SP-SG deviates the most from the reference, while SMS-HSL without a magnitude prior is relatively close to the reference, although there are signal drops in the middle of the T1 curve. In contrast, the proposed method recovers a more accurate signal curve with reduced noise and errors for both white- and gray-matter tissues, especially at the late TIs. Figure 7C shows the corresponding histograms of the T1 values for the central slice presented in Figure 6. We observe that the T1 histogram of SP-SG is significantly lower than those of SMS-HSL without a magnitude prior and model-based approaches around 1000 ms because of the presence of white matter, in addition to exhibiting the highest peak around 3000 ms, demonstrating increased noise amplification in the center of the FOV. However, SMS-HSL without a magnitude prior and model-based approaches agree well with the reference results, indicating that null-space-based slice separation is performed in a stable manner, despite the different reconstruction qualities of SMS-HSL without a magnitude prior and model-based approaches.

Figure 8 shows a comparison between SMS-MUSSELS and model-based SMS-HSL reconstructions with an SMS factor of 3SMS-MUSSEL exhibits structure-aliasing artifacts with noise amplification, whereas model-based SMS-HSL can address these issues in the reconstructed images and T1 maps. There are two reasons for this result. For slice separation, the model-based SMS-HSL further generalizes multiple null-space vectors by flexibly controlling the signal and null-space vectors, unlike SMS-MUSSELS, which uses a main eigenvector for sensitivity estimation [42]. Additionally, the higher performance of the model-based SMS-HSL suggests that the estimation of the spatial coefficient maps in the Casorati matrix can complement the low rankness in the Hankel-structured matrix by reducing the number of degrees of freedom during signal recovery.

4.3. Prospective Studies: Experimental Validation

Figure 9 shows the resulting images obtained using SP-SG, ROCK-SPIRiT, SMS-HSL without a magnitude prior, and the proposed model-based SMS-HSL method on undersampled SMS data at 10-fold acceleration. The SP-SG-reconstructed images show significant noise amplification in the third slices and severe aliasing artifacts in the fourth slices. The regularized ROCK-SPIRiT method reduced the noises compared to SP-SG while maintaining the aliasing artifacts. SMS-HSL without a magnitude prior substantially reduced the noise but did not eliminate aliasing artifacts. Note that compared with other methods, the proposed model-based method exhibits better image quality, reducing aliasing artifacts, as indicated by the red arrows.

5. Discussion and Conclusions

In this work, we describe a novel SMS T1-mapping technique that combines Hankel subspace learning with a parameter model for slice separation. In contrast to previously proposed SMS reconstruction methods that only take advantage of slice-specific encoding information, the proposed method incorporates a parameter basis directly from the aliased slices into the SMS-HSL reconstruction framework to enable improved slice separation and more accurate parameter fitting under the assumption that the parameter variations for all inversion times are invariant across slices. The shared parameter subspace provides prior information in addition to the existing slice-specific null-space constraint, outperforming the state-of-the-art methods and increasing accelerations with good accuracy and precision in image reconstruction and T1 mapping.

The proposed model-based SMS-HSL uses parameter redundancy across slices to estimate the parameter subspace of the Casorati matrix, which results in a simplified problem that can be solved efficiently by reducing the number of degrees of freedom. The parameter subspace is estimated from the observed aliased slice under the assumption that all slices share the same parameter subspace. As shown in [32], conventional SMS-HSL suffers from severe contrast loss without a magnitude prior during slice separation because of the sharing of null-space information between slices. In contrast, the parameter subspace is complementary to the slice-specific null space and can improve the SMS-HSL in reconstruction and T1 mapping. Additionally, the proposed framework can be extended to jointly estimate the spatial and parameter bases for each slice using a power factorization algorithm [43,44], in which ${\mathbf{U}}_m$ and $\mathbf{V}$ are estimated in an alternating manner.

In the case of acceleration along both the slice- and phase-encoding directions, the proposed method sequentially reconstructs unaliased slices, followed by in-plane GRAPPA reconstruction. The proposed algorithm, which exploits rank deficiency using a Hankel-structured matrix and temporal basis priors, can be applied to data obtained with an SMS acceleration only and in combination with in-plane undersampling. This is evidenced by Figure 9, where among the methods examined for image reconstruction from prospectively undersampled data with both SMS and in-plane accelerations, the proposed model-based technique presents relatively higher performance in aliasing separation and noise suppression.

The proposed reconstruction method was implemented using an ADM algorithm that typically requires 30 iterations to converge, although its computation time depends on the data dimension. For each iteration, the algorithm involves two matrix inversions and SVD calculations for each variable update. The computational complexity depends on several factors, including the number of parameter dimensions, the patch size, the number of coils, the number of stacked patches, the effective rank, and the number of simultaneously excited slices:

$$O\left(N_n{N}_s{N}_p\left\{{\left(p_x{p}_y{N}_l\right)}^3+r^3+{(N_x-p_x+1)}^3{(N_y-p_y+1)}^3\right\}\right)$$

(19)

where $N_n$ is the total number of iterations, $N_s$ is the number of slices, $N_p$ is the number of parameters, $p_x$ × $p_y$ is the patche size, r is the rank, and $N_l$ is the number of coils. To address the computational challenges associated with an increasing number of coils and larger patch size in the Hankel operation, it becomes impractical to perform singular-value decomposition at each iteration while minimizing the nuclear norm in Equation (16). In order to overcome this computational problem, we employed a matrix factorization method [38,39], implementing a conjugate gradient algorithm for matrix inversion in Equation (17) and the precomputed inversion of $({\mathcal{N}}_s^c{{\mathcal{N}}_s^c}^{\ast }+\beta \mathbf{I})$ in Equation (17) [43,44]. The computational complexity of our method per iteration is slightly higher than that of SMS-HSL in the presence of the parameter basis model. In our reconstruction, the computation times of SMS-HSL and our method were 15 min and 28 min, respectively, for all T1-weighted images. However, our codes were implemented in MATLAB and were not optimized for speed. Accordingly, we believe that there are several approaches to mitigate the computational burden, such as channel compression in the preprocessing step and element-wise multiplication in the image space rather than convolution in k space for the null space projection in Equation (17), making our reconstruction framework more practical.

The proposed model-based SMS-HSL estimates the slice-specific null space from a separate calibration scan. It potentially yields signal inconsistencies between the calibrated and “true” null spaces in the presence of either subject motions or magnetic field drifts, leading to quantitative errors with cross-talk artifacts between the neighboring slices. To mitigate potential mismatches, we believe that the calibration process can be incorporated into our proposed framework by controlling the EPI blips without loss of generality. Similar to regular undersampling of the dynamic parallel MRI with shifts in the k-space grids across time [45], the sampling pattern is temporally interleaved with different slice encodings across TIs to simultaneously provide complementary slice encoding and k-space whole coverage, and inverse Fourier transform is performed along the slice direction for null-space estimation. Accordingly, integrated calibration in echo-planar trajectories can further help address this limitation.

In conclusion, this work introduces a model-based SMS-HSL by exploiting a data-driven parameter model for T1 quantification, utilizing the combined properties of Hankel- and Casorati-structured matrices. To resolve the SMS reconstruction, we employ a complementary null space based on a time-invariant slice-specific null-space operator. This null-space operator preserves the slice of interest while nulling the signal from the other slices. Furthermore, the proposed framework can be extended to jointly estimate the spatial and parameter bases for each slice in Equation (11), resulting in a reduction in the degrees of freedom during the reconstruction process. Thus, we incorporate null-space projection and low-rank priors into a single SMS framework in Equation (13). The experimental results show that even when subjected to high SMS acceleration, the proposed method consistently outperforms SP-SG, SMS-MUSSELS, and ROCK-SPIRiT. We successfully demonstrated the effectiveness of accelerated SMS T1 quantification for aliasing separation.