Accelerating multi-echo water-fat MRI with a joint locally low-rank and spatial sparsity-promoting reconstruction

Abstract

Objectives

Our aim was to demonstrate the benefits of using locally low-rank (LLR) regularization for the compressed sensing reconstruction of highly-accelerated quantitative water-fat MRI, and to validate fat fraction (FF) and ${R_2^*}$ relaxation against reference parallel imaging in the abdomen.

Materials and methods

Reconstructions using spatial sparsity regularization (SSR) were compared to reconstructions with LLR and the combination of both (LLR+SSR) for up to seven fold accelerated 3-D bipolar multi-echo GRE imaging. For ten volunteers, the agreement with the reference was assessed in FF and ${R_2^*}$ maps.

Results

LLR regularization showed superior noise and artifact suppression compared to reconstructions using SSR. Remaining residual artifacts were further reduced in combination with SSR. Correlation with the reference was excellent for FF with $R^2$ = 0.99 (all methods) and good for ${R_2^*}$ with $R^2$ = [0.93, 0.96, 0.95] for SSR, LLR and LLR+SSR. The linear regression gave slope and bias (%) of (0.99, 0.50), (1.01, 0.19) and (1.01, 0.10), and the hepatic FF/${R_2^*}$ standard deviation was 3.5%/12.1 s$^{-1}$, 1.9%/6.4 s$^{-1}$ and 1.8%/6.3 s$^{-1}$ for SSR, LLR and LLR+SSR, indicating the least bias and highest SNR for LLR+SSR.

Conclusion

A novel reconstruction using both spatial and spectral regularization allows obtaining accurate FF and ${R_2^*}$ maps for prospectively highly accelerated acquisitions.

Appendix: Update terms and solutions of proximal operators

Update terms of the SB algorithm enforce the coupling with penalty terms:

$$\begin{aligned}& \varvec{b}^{(k+1)}_b= {} \varvec{b}^{(k)}_b + \varvec{x}^{(k+1)} - \varvec{d}^{(k+1)}_b, \nonumber \\ & \varvec{b}^{(k+1)}_x= \varvec{b}^{(k)}_x + \nabla _x \varvec{x}^{(k+1)} - \varvec{d}^{(k+1)}_x, \nonumber \\ &\varvec{b}^{(k+1)}_y= \varvec{b}^{(k)}_y + \nabla _y \varvec{x}^{(k+1)} - \varvec{d}^{(k+1)}_y, \nonumber \\ &\varvec{b}^{(k+1)}_w= \varvec{b}^{(k)}_w + \varvec{W} \varvec{x}^{(k+1)} - \varvec{d}^{(k+1)}_w. \end{aligned}$$

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The proximal operator for the TV formulation is realized via generalized shrinkage as

$$\begin{aligned} \varvec{d}_x^{(k+1)}= & {} \left( \varvec{s}^{(k)} - \dfrac{\mu _d}{\lambda _d} \right) _{+} \dfrac{\nabla _x \varvec{x}^{(k+1)} + \varvec{b}_x^{(k)} }{\varvec{s}^{(k)}}, \end{aligned}$$

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$$\begin{aligned} \varvec{d}_y^{(k+1)}= & {} \left( \varvec{s}^{(k)} - \dfrac{\mu _d}{\lambda _d} \right) _{+} \dfrac{\nabla _y \varvec{x}^{(k+1)} + \varvec{b}_y^{(k)} }{\varvec{s}^{(k)}}, \end{aligned}$$

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with

$$\begin{aligned} \varvec{s}^{(k)} = \sqrt{ \left( \nabla _x \varvec{x}^{(k+1)} + \varvec{b}_x^{(k)} \right) ^2 + \left( \nabla _y \varvec{x}^{(k+1)} + \varvec{b}_y^{(k)} \right) ^2 } , \end{aligned}$$

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where $(\varvec{a})_+$ performs component-wise ${\text {max}}\left( 0,\varvec{a}_j \right)$, while soft-thresholding of wavelet coefficients is used for DWTs [29]:

$$\begin{aligned} \varvec{d}_w^{(k+1)} = {\text {soft}} \left( \varvec{W} \varvec{x}^{(k+1)} + \varvec{b}_w^{(k)}, \dfrac{\mu _w}{\lambda _w} \right) . \end{aligned}$$

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The LLR proximal functional can be solved separately in the case of non-overlap** blocks by singular value soft-thresholding and placing the result back with the adjoint operator $B^{\dagger }$ [28, 33],

$$\begin{aligned} \varvec{d}_b^{(k+1)}= & {} \sum _{p \in \Omega } B^{\dagger }_p \left( \text {SVT} \left( B_p \left( \varvec{x}^{(k+1)} \right) ,\dfrac{\mu _b}{\lambda _B} \right) \right) , \nonumber \\ \text {using SVT}(\varvec{X},\theta )= & {} \varvec{U} {\text {diag}}\left( (\varvec{\sigma } - \theta )_+ \right) \varvec{V}^T. \end{aligned}$$

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