FURTHER DEVELOPMENT OF SUBSPACE IMAGING TO MAGNETIC RESONANCE FINGERPRINTING: A LOW-RANK TENSOR APPROACH

Abstract

Magnetic resonance fingerprinting is a recent quantitative MRI technique that simultaneously acquires multiple tissue parameter maps (e.g., T₁, T₂, and spin density) in a single imaging experiment. In our early work, we demonstrated that the low-rank/subspace reconstruction significantly improves the accuracy of tissue parameter maps over the conventional MR fingerprinting reconstruction that utilizes simple pattern matching. In this paper, we generalize the low-rank/subspace reconstruction by introducing a multilinear low-dimensional image model (i.e., a low-rank tensor model). With this model, we further estimate the subspace associated with magnetization evolutions to simplify the image reconstruction problem. The proposed formulation results in a nonconvex optimization problem which we solve by an alternating minimization algorithm. We evaluate the performance of the proposed method with numerical experiments, and demonstrate that the proposed method improves the conventional reconstruction method and the state-of-the-art low-rank reconstruction method.

1. INTRODUCTION

Magnetic resonance imaging (MRI) is a powerful and versatile imaging modality that has revolutionized medicine and biology since its invention. Most MRI applications use contrast-weighted images that are nonlinear functions of MR tissue parameter maps (e.g., T₁, T₂, spin density), data acquisition parameters, as well as acquisition imperfections [1]. These contrast-weighted images are qualitative in nature, which have very limited capability to enable direct comparison of MRI scans from different vendors and different times. In contrast, quantitative MRI aims to measure intrinsic MR tissue parameters, which has the potential of overcoming the above limitations. Despite the great promise, quantitative MRI faces a number of key technical challenges, including long acquisition time as well as vulnerability to acquisition imperfections [2].

To address these issues, various quantitative MRI techniques have been developed over the last three decades (see [2] and references therein). As a recent paradigm of quantitative MRI, magnetic resonance (MR) fingerprinting utilizes transient-state spin dynamics and sub-Nyquist sampling to achieve quantitative MR imaging at an ultra-fast imaging speed [3]. The conventional MR fingerprinting reconstruction is based on direct pattern matching [3], which is computationally very efficient. However, this approach is sub-optimal from a statistical estimation perspective [4]. More recently, a number of advanced reconstruction approaches have been developed to improve the reconstruction accuracy, including multi-scale reconstruction [5], sparse reconstruction [6], low-rank/subspace reconstruction [7–10], and machine learning reconstruction [11, 12]. Among these methods, the low-rank/subspace reconstruction approach exploits strong spatiotemporal correlation of contrast-weighted images, which significantly improves the accuracy of reconstructed tissue parameter maps and reduces data acquisition time [7–10].

In this paper, we extend our early low-rank/subspace reconstruction method [7–9] by introducing a new multilinear low-dimensional model, i.e., low-rank tensor model [13]. With this model, we directly capture the correlation of contrast-weighted time-series images in each physical dimension, which mitigates the information loss caused by the matriciation preprocessing step in low-rank reconstruction method [14–17]. Further, we estimate the subspace associated with magnetization evolutions for the low-rank tensor model, which simplifies the resulting image reconstruction problem. The proposed formulation results in a nonconvex optimization problem, which is solved by an alternating minimization algorithm. We evaluate the performance of the proposed method with numerical experiments and representative results are shown to demonstrate the efficacy of the proposed approach. The rest of the paper is organized as follows. In Section 2, we describe the proposed approach, including the problem formulation and solution algorithm. In Section 3, we show numerical results to demonstrate the performance of the proposed approach, followed by the concluding remarks in Section 4.

2. PROPOSED APPROACH

2.1. Problem Formulation

We first describe the data model for MR fingerprinting. We denote the contrast-weighted image at the mth time point by ${\text{I}}_m\in {ℂ}^N$, and represent the sequence of contrast-weighted images by the following Casorati matrix:

$$\mathbf{\text{C}}=\left[\begin{array}{ccc}{\mathbf{\text{I}}}_1\left({\text{x}}_1\right)& \cdots & {\mathbf{\text{I}}}_M\left({\text{x}}_1\right)\\ ⋮& \ddots & ⋮\\ {\mathbf{\text{I}}}_1\left({\text{x}}_N\right)& \cdots & {\text{I}}_M\left({\text{x}}_N\right)\end{array}\right]\in {ℂ}^{N\times M}.$$

To enable image reconstruction from highly undersampled (k, t)-space data, we exploit strong spatiotemporal correlation of time-series images in MR fingerprinting. Rather than model the time-series images as a low-rank matrix [7–9], we use a low-rank tensor model, which in general better captures the correlation of the multi-dimensional image function in different physical dimensions. Specifically, for 2D MR fingerprinting applications, we can represent the Casorati matrix C as a three-way tensor $\mathcal{C}\in {ℂ}^{N_x\times N_y\times M}$ by separating out the two spatial dimensions for each column of C. Here N_x and N_y denote the number of encodings in each spatial dimension and note that N = N_x × N_y. Fig. 1 illustrates the low-rank tensor model of time-series images in MR fingerprinting.

Fig. 1.

Low-rank tensor model. (a) Tensor model for time-series images of MR fingerprinting. (b) Tucker decomposition of the low-rank tensor.

With the above model, the imaging equation for MR fingerprinting can be written as

$${\mathbf{\text{d}}}_c={\mathbf{\text{E}}}_c\left(\mathcal{C}\right)+{\mathbf{\text{n}}}_c,$$ (1)

where ${\mathbf{\text{d}}}_c\in {ℂ}^P$ denotes the acquired k-space data from the cth receiver coil, ${\mathbf{\text{n}}}_c\in {ℂ}^P$ denotes measurement noise, and ${\mathbf{\text{E}}}_c:{ℂ}^{N_p\times N_f\times M}\to {ℂ}^P$ denotes the imaging operator, which maps the tensor $\mathcal{C}$ to the measured k-space data d_c at the cth receiver coil. Note that E_c encompasses the Fourier encoding, sensitivity encoding, and (k, t)–space sparse sampling operations.

Invoking a low-rank tensor constraint, we formulate the time-series image reconstruction problem as follows:

$$\widehat{\mathcal{C}}=\text{arg}\underset{\mathcal{C}}{\text{min}}\sum _{c=1}^{N_c}{\Vert {\mathbf{\text{d}}}_c-{\mathbf{\text{E}}}_c\left(\mathcal{C}\right)\Vert }_2^2+R_r\left(\mathcal{C}\right),$$ (2)

where $R_r\left(\mathcal{C}\right)$ denotes the regularization functional that enforces the low-rank tensor constraint on $\mathcal{C}$. There are a number of ways of enforcing a low-rank tensor constraint, and here we use the low-rank tensor factorization based on the Tucker decomposition [13], illustrated in Fig. 1(b). Mathematically, the Tucker decomposition can be written as

$$\mathcal{C}=\mathcal{S}{\times }_1{\mathbf{\text{U}}}_1{\times }_2{\mathbf{\text{U}}}_2{\times }_3{\mathbf{\text{U}}}_3,$$ (3)

where $\mathcal{S}\in {ℂ}^{r_1\times r_2\times r_3}$ denotes the core tensor, and ${\mathbf{\text{U}}}_1\in {ℂ}^{N_x\times r_1}$, ${\mathbf{\text{U}}}_2\in {ℂ}^{N_y\times r_2}$, and ${\mathbf{\text{U}}}_3\in {ℂ}^{M\times r_3}$ are three factor matrices, and ×_i denotes the tensor i-mode product. The equivalent matrix representation to (3) can be written as

$${\mathbf{\text{C}}}_{\left(3\right)}={\mathbf{\text{U}}}_3{\mathbf{\text{S}}}_{\left(3\right)}{\left({\mathbf{\text{U}}}_2\otimes {\mathbf{\text{U}}}_1\right)}^T,$$ (4)

where C_(n) is the mode-n matricization of $\mathcal{C}$, and ⊗ denotes the Kronecker product. Similar to our early work [9], we pre-estimate U₃ from the magnetization evolutions simulated with Bloch simulations, which we denote as ${\widehat{\mathbf{\text{U}}}}_3$.

With the above low-rank tensor model, the problem formulation (2) can be rewritten as:

$$\begin{gathered} \underset{\mathcal{C},{\mathbf{\text{U}}}_1,{\mathbf{\text{U}}}_2,{\mathbf{\text{S}}}_{\left(3\right)}}{\text{min}}\sum _{c=1}^{N_c}{\Vert {\mathbf{\text{d}}}_c-{\mathbf{\text{E}}}_c\left(\mathcal{C}\right)\Vert }_2^2 \\ +\lambda {\Vert \mathcal{C}-{\mathbf{\text{fold}}}_3\left({\widehat{\mathbf{\text{U}}}}_3{\mathbf{\text{S}}}_{\left(3\right)}{\left({\mathbf{\text{U}}}_2\otimes {\mathbf{\text{U}}}_1\right)}^T\right)\Vert }_F^2, \end{gathered}$$ (5)

where λ denotes the regularization parameter, and the operator fold_n converts the nth mode matrix representation into the tensor representation.

2.2. Solution Algorithm

The proposed formulation (5) is a non-convex optimization problem, which we solve by an alternating minimization algorithm. For simplicity of notation, we denote the cost function in (5) as $f\left({\mathbf{\text{U}}}_1,{\mathbf{\text{U}}}_2,{\mathbf{\text{S}}}_{\left(3\right)},\mathcal{C}\right)$. The proposed algorithm is as follows. At the (k + 1)th iteration, we update the optimization variables as follows:

$${\mathbf{\text{U}}}_1^{k+1}=\text{arg}\underset{\mathbf{\text{U}}}{\text{min}}\hspace{0.17em}f\left({\mathbf{\text{U}}}_1,{\mathbf{\text{U}}}_2^k,{\mathbf{\text{S}}}_{\left(3\right)}^k,{\mathcal{C}}^k\right),$$ (6)

$${\mathbf{\text{U}}}_2^{k+1}=\text{arg}\underset{\mathbf{\text{U}}}{\text{min}}\hspace{0.17em}f\left({\mathbf{\text{U}}}_1^{k+1},{\mathbf{\text{U}}}_2,{\mathbf{\text{S}}}_{\left(3\right)}^k,{\mathcal{C}}^k\right),$$ (7)

$${\mathbf{\text{S}}}_{\left(3\right)}^{k+1}=\text{arg}\underset{\mathbf{\text{S}}}{\text{min}}\hspace{0.17em}f\left({\mathbf{\text{U}}}_1^{k+1},{\mathbf{\text{U}}}_2^{k+1},{\mathbf{\text{S}}}_{\left(3\right)},{\mathcal{C}}^k\right),$$ (8)

$${\mathcal{C}}^{k+1}=\text{arg}\underset{\mathcal{C}}{\text{min}}\hspace{0.17em}f\left({\mathbf{\text{U}}}_1^{k+1},{\mathbf{\text{U}}}_2^{k+1},{\mathbf{\text{S}}}_{\left(3\right)}^{k+1},\mathcal{C}\right).$$ (9)

Here we update the optimization variables in (6)–(8) as follows:

$${\mathbf{\text{U}}}_1^{k+1}={\mathbf{\text{C}}}_{\left(1\right)}^k{\left[{\widehat{\mathbf{\text{U}}}}_3^T\otimes {\left({\mathbf{\text{U}}}_2^k\right)}^T\right]}^{†}{\left({\mathbf{\text{S}}}_{\left(1\right)}^k\right)}^{†},$$

$${\mathbf{\text{U}}}_2^{k+1}={\mathbf{\text{C}}}_{\left(2\right)}^k{\left[{\widehat{\mathbf{\text{U}}}}_3^T\otimes {\left({\mathbf{\text{U}}}_1^{k+1}\right)}^T\right]}^{†}{\left({\mathbf{\text{S}}}_{\left(2\right)}^k\right)}^{†},$$

$${\mathbf{\text{S}}}_{\left(3\right)}^{k+1}={\left({\widehat{\mathbf{\text{U}}}}_3^k\right)}^{†}{\mathbf{\text{C}}}_{\left(3\right)}^k{\left[{\left({\mathbf{\text{U}}}_2^{k+1}\right)}^T\otimes {\left({\mathbf{\text{U}}}_1^{k+1}\right)}^T\right]}^{†}.$$

where † denotes the Moore-Penrose pseudo-inverse. For the subproblem (9), we solve the following large-scale least-squares problem, i.e.,

$$\begin{gathered} \left[\sum _{c=1}^{N_c}{\mathbf{\text{E}}}_c^{*}{\mathbf{\text{E}}}_c+\lambda \mathcal{I}\right]{\mathcal{C}}^{k+1}=\sum _{c=1}^{N_c}{\mathbf{\text{E}}}_c^{*}\left({\mathbf{\text{d}}}_{\text{c}}\right)+ \\ \lambda \hspace{0.17em}{\mathbf{\text{fold}}}_3\left[{\widehat{\mathbf{\text{U}}}}_3{\mathbf{\text{S}}}_{\left(3\right)}^{k+1}{\left({\mathbf{\text{U}}}_2^{k+1}\otimes {\mathbf{\text{U}}}_1^{k+1}\right)}^T\right], \end{gathered}$$

with an iterative algorithm (e.g., the pre-conditioned conjugate-gradient algorithm), where $\mathcal{I}$ is an identity operator, and ${\mathbf{\text{E}}}_c^{*}$ is the adjoint operator of E_c.

Given that (5) is a nonconvex optimization problem, the optimal solution generally depends on the initialization. Here we initialize $\mathcal{C}$ with the low-rank reconstruction [9], and initialize U₁, U₂, and S₍₃₎ using the higher-order singular value decomposition of $\mathcal{C}$ [13]. This initialization consistently produces good reconstruction results, although other initializations might potentially provide even better performance.

3. RESULTS

In this section, we show representative results from numerical simulations to demonstrate the performance of the proposed method. Here we used a numerical brain phantom which was created using the underlying ground truth T₁, T₂, and spin density maps from the Brainweb database [18]. We performed Bloch simulations to generate the contrast-weighted time-series images using the inversion recovery fast imaging with balanced steady-state precession (IR-bSSFP) sequence [3]. We performed highly-undersampled spiral acquisition to acquire k-space data. At each time point, we acquired one spiral interleave, and a set of fully-sampled data consists of 48 spiral interleaves [19]. We added noise to measured k-space data such that SNR = 20 dB. We reconstructed the contrast-weighted images using the conventional MR fingerprinting reconstruction [3], the low-rank reconstruction [9], and the proposed low-rank tensor reconstruction, from which we estimated the tissue parameter maps. Here we chose r = (210; 180; 6) for the proposed low-rank tensor reconstruction, while we chose the rank r = 6 for the low-rank reconstruction. Moreover, we empirically selected the regularization parameters λ for the proposed method to optimize its performance. For both the low-rank reconstruction and the proposed method, we estimated the subspace associated with the magnetization evolution from the dictionary simulated using the Bloch equation.

We first evaluated the performance of the above three methods at the acquisition length M = 250. Fig. 2 shows the reconstructed T₁ and T₂ maps as well as the associated error maps. As can be seen, the proposed method outperforms the conventional MR fingerprinting reconstruction and the low-rank reconstruction, illustrating the benefit of incorporating a low-rank tensor model into image reconstruction. Note that the improvement is larger for T₂ map than T₁ map. Next, we evaluated the performance of the three methods with respect to different acquisition lengths. Fig. 3 shows the reconstruction error versus different acquisition lengths, which further illustrates the performance of the proposed approach.

Fig. 2.

Reconstructed T₁ and T₂ maps and associated error maps at the acquisition length M = 250 and SNR = 20 dB. (a) T₁ reconstruction and relative error maps. (b) T₂ reconstruction and relative error maps. As can be seen, the proposed low-rank tensor reconstruction outperforms the conventional MR fingerprinting reconstruction and the state-of-the-art low-rank reconstruction. Note that the improvement for T₂ is larger than that for T₁.

Fig. 3.

Reconstruction accuracy versus acquisition length for (a) T₁ and (b) T₂ reconstructions. Note that the results here are consistent with the observations in Fig. 2.

4. CONCLUSION

This paper presents a new low-rank tensor based reconstruction method for MR fingerprinting, which exploits strong spatiotemporal correlation of contrast-weighted time-series images. An alternating minimization algorithm was developed to solve the resulting optimization problem. The proposed method provides improved accuracy compared to the conventional MR fingerprinting reconstruction and the low-rank reconstruction. The effectiveness of the proposed method was illustrated with numerical experiments.

Low-rank tensor models are often very effective to represent high-dimensional imaging data. In this proof-of-concept study, we investigate its efficacy for 2D MR fingerprinting applications. In the future work, it would be very interesting to extend the proposed method for volumetric quantitative imaging applications (e.g., simultaneous multislice [20] or 3D imaging applications [21]). Moreover, the proposed method could potentially integrate with the optimized quantitative imaging acquisitions [22, 23], which may further improve the accuracy of tissue parameter maps.