Synopsis

Keywords: Data Processing, Quantitative Imaging, phase-cycled bSSFP, fat fraction

A novel cluster-based sparse total-variation minimization (CSTV) dictionary matching algorithm for multi-compartment parameter mapping applied to a phase-cycled balanced steady state free precession (PC-bSSFP) signal profile is presented. A dictionary encoding off-resonance frequency and relaxation time ratio is matched to the measured profiles with the proposed method to recover the underlying fat fraction, proton density, and water-fat-separated images simultaneously. The performance of the proposed method is tested and compared with a Laplacian regularized non-negative least squares dictionary matching method in phantom and in vivo experiments.

Introduction

Signal profiles obtained with phase-cycled (PC) balanced Stead-State Free Precession (bSSFP) acquisitions are typically known to be $$$T_1⁄T_2$$$ dependent with a pass-band and a and stop-band based on the underlying magnetic field inhomogeneity (B0) and tissue chemical shift¹. Profile asymmetries have been observed in the presence of multiple compartments². Recently, Signal Profile Asymmetries for Robust multi-Compartment Quantification (SPARCQ)³ was proposed to decompose the measured PC-bSSFP profiles into a weighted sum of dictionary entries which differ in relaxation time ratio and off-resonance frequency, enabling fat-water separation and fat fraction (FF) quantification. SPARCQ uses a non-negative least-squares fit with a Laplacian-penalization regularizer that operates on the dictionary, treating each voxel independently, therefore the inherent sparsity of a possible solution and information from neighboring voxels are not used. The current study aims to develop a cluster-based total-variation minimization (CSTV) dictionary matching scheme that uses the sparsity in dictionary entries to reconstruct a measured PC-bSSFP profile while simultaneously minimizing the variation of weights among the voxels that are members of the same cluster.

Methods

Theory: The proposed CSTV formulation problem can be stated as follows:
$$\left \|\begin{bmatrix}\phi\\\bar\lambda^T\end{bmatrix}W_c-\begin{bmatrix}Y_c\\\bar0\end{bmatrix}\right\|_2^2+\gamma\left\|GW_c^T\right\|_2^2\ subject\ to\ (W_c)_{i,j}\geq0$$
Where $$$\phi$$$ represents dictionary entries with Bloch-simulated PC-bSSFP profiles, $$$\bar{\lambda}$$$ is a constant vector with value $$$\lambda$$$, $$$W_c=[w_1w_2,...]$$$ are the concatenated weights of cluster $$$c$$$, $$$Y_c$$$ is the concatenated measured PC-bSSFP profiles of cluster $$$c$$$, $$$\gamma$$$ is a regularization parameter and $$$G$$$ is the finite difference operator. By augmenting $$$\phi$$$ with $$$\bar\lambda^T$$$ and $$$Y_c$$$ with $$$\bar 0$$$, considering the non-negativity constraint on weights, sparsity on the solution is enforced⁴.By minimizing $$$||GW_c^T ||_2^2$$$ , weights of voxels belonging to the same cluster are forced to be similar. The constrained optimization problem was solved with Alternating Direction Method of Multipliers (ADMM)⁵. $$$W_c$$$ update is based on a solution of the formation AX+XB=C by a series of matrix factorizations⁶.
Clustering is based on a dot product fit where voxels that are fitted to the same off-resonance frequency interval are assigned to the same cluster. 10 clusters were used in the current study. After their computation in each voxel, the weights are projected onto the off-resonance frequency dimension of the dictionary. Considering the water being on-resonance and the periodicity of the PC-bSFFP profile with $$$1/TR$$$, projected weights around $$$0$$$ and $$$1/TR\ Hz$$$ are assigned to water and remaining weights are assigned to fat. Proton density (PD), fat fraction (FF) and water fraction (WF) is then calculated as follows:$$PD=\sum{w},FF=\frac{\sum{w^f}}{PD},WF=1-PD$$ Where $$$w^f$$$ denotes the weights that are assigned to fat (Figure 1).
MRI Experiments: PC-bSSFP and multi-echo gradient-recalled echo (MEGRE) acquisitions were performed in a phantom with known fat fractions at 3T (MAGNETOM Prisma, Siemens Healthcare, Erlangen, Germany) with acquisition parameters given in Table 1. The same acquisitions were performed in knees of 2 human volunteers.
Data Processing: B0 and FF maps along with the fat and water images were recovered from MEGRE measurements with a graph-cut based method⁷. A bone marrow fat model was used for in vivo data⁸, and the measured³ fat spectrum was used for phantom data.
To be independent of B0, the measured PC-bSSFP profiles are shifted according to the recovered B0. Dictionary matching is conducted on the processed PC-bSSFP profiles with both SPARCQ, and CSTV and PD, FF, and WF maps are recovered as described³.
Data Analysis: Algorithm performance was determined in vitro by calculating the mean and standard deviation of FF maps in each region of interest (ROI). Root mean squared errors (RMSE) in each ROI is with respect to the reference FF values are calculated. Linear regression and $$$R^2$$$ values for different methods are reported. In vivo results are compared visually.

Results and Discussion

In phantom studies, SPARCQ and CSTV yield similar PD, water and fat images. SPARCQ, CSTV and MEGRE derived fat fractions match the reference with a maximum RMSE value of $$$3.1%$$$ (Figure 3-d). Linear regression of obtained FF maps yield $$$0.9979$$$, $$$0.9978$$$ and $$$0.9977$$$ $$$R^2$$$ values for CSTV, SPARCQ and MEGRE, respectively. The $$$0%$$$ fat fraction vial is blended into the background with CSTV (Figure 2, red-circle), which suggests the efficiency of the total variation minimization, corroborated by its RMSE of $$$0.02$$$ (Figure 3-d). CSTV method increases RMSE performance compared with original SPARCQ in phantoms (Figure 3-d), showing the value of sparsity and minimal-variation assumptions, with a drawback of an increased number of regularization parameters.
In vivo, the water images obtained with SPARCQ and CSTV appear less homogenous compared with those obtained with MEGRE (Figure 4). This is likely related to a bias in fat signal because of the chosen flip angle. SPARCQ FF map in axial slice and CSTV FF map in sagittal slice yield underestimated fractions (Figure 4), which suggests a requirement for different regularization parameter selection for different orientations and/or slices. In MEGRE-based fat images, bone and joint region appears darker, which suggests a signal loss, to which PC-bSSFP-based fat images are more robust (Figure 4).

Conclusion

Dictionary matching approach to the PC-bSSFP profiles offer a robust fat fraction estimation independent of a spectral fat model. Promoting sparsity while enforcing intra-cluster minimized variation increases the performance of dictionary matching to the PC-bSSFP profiles by means of fat fraction estimation.

Acknowledgements

This study was supported by the Swiss National Science Foundation (grant number PCEFP2_194296)