### 4401

A fast, joint sparsity constraint algorithm for improved myelin water fraction mapping
Martijn Nagtegaal1, Burkhard Mädler2, Thomas Amthor3, Peter Koken3, and Mariya Doneva3

1TU Berlin, Berlin, Germany, 2Philips, Hamburg, Germany, 3Philips Research Europe, Hamburg, Germany

### Synopsis

A new method for myelin water fraction mapping from multi-echo spin echo data using a joint sparsity constraint is proposed, which is faster than previously proposed methods. This method is based on the assumption that the T2 spectrum is sparse and consists of a common small set of discrete relaxation times for all voxels. The method finds an estimation of the flip angle inhomogeneity map from the data itself, to remove the bias caused by B1 inhomogeneities. The proposed method is compared to state of the art MWF approaches in 3T brain measurements.

### Introduction

Myelin water fraction (MWF) mapping provides the means to assess myelin changes in cerebral white matter in vivo. One approach to MWF mapping is multi-component T2 analysis from multi-echo spin-echo (MESE) data, in which the T2 values below a certain threshold are assumed to belong to myelin water. The original method [1] based on the non-negative least squares(NNLS) algorithm [2] assumes non-negativity of the solution and proposes a smooth distribution to improve the noise-robustness. Later, spatial regularisation of the distribution was proposed to further improve the robustness [3].

The MESE measurement is sensitive to flip angle inhomogeneities (FAI) due to B1-variations. Methods including FAI compensation [4,5] lead to improved results, but have rather long computation times.

In this work, we propose a new, faster method for calculating the MWF including FAI-compensation using joint sparsity.

### Methods

The main premise of the proposed method is that the measured MESE signals can be represented as a linear combination of a few basis signals, corresponding to different components (e.g. myelin water, intra and extra-cellular water, free water), and that these components are the same for all voxels in the scan.

Assuming there are no flip angle inhomogeneities, the proposed method can be formulated as the following minimization problem:$$\min_{\mathbf{C}\in\mathbb{R}_{\geq 0}^{N\times J}}\lVert\mathbf{X}-\mathbf{D}\mathbf{C}\rVert^2_F+\lambda\sum_{i=1}^N\lVert \mathbf{c}^i\rVert_0$$Here $\mathbf{X}$ is a matrix containing the measured MESE signals for $J$ voxels, $\mathbf{D}$ is a matrix containing the simulated temporal signal evolutions for $N$ different T2 values, each column of $\mathbf{C}$ contains the coefficients of the $N$ components for a voxel and $\mathbf{c}^i$ denotes the ith row of $\mathbf{C}$. The parameter $\lambda$ balances the approximation error and the number of components. The joint sparsity problem without non-negativity constraints has been studied extensively in distributed compressed sensing literature [6]–[10].

To solve this problem we propose a new algorithm, the Sparsity Promoting Iterative Joint NNLS (SPIJN) algorithm. Since it is not computationally feasible to directly solve for the $\ell_0$-norm regularized problem, we use an iteratively reweighted scheme that approximates this regularization. The reweighting puts higher weight to components which are jointly used making them more likely to be selected in the next iteration. We use the weights $w_i=\lVert\mathbf{c}^i\rVert_2+\epsilon$ (here $\epsilon=10^{-4}$) ($i=1,…,N$), which is the $\ell_2$-norm of the coefficients over the FOV. The core of each iteration is formed by the NNLS algorithm [2], which is used to solve the reweighted NNLS problem $\min_{\mathbf{C}\in\mathbb{R}_{\geq 0}^{N\times~J}}\lVert\tilde{\mathbf{X}}-\tilde{\mathbf{D}}\mathbf{C}\rVert^2_F$. The following steps are performed in each iteration of the algorithm:$$\mathbf{W}_{k+1}=\operatorname{diag}(\left(\mathbf{w}\right)_{k+1}^{1/2}),\\\tilde{\mathbf{D}}_{k+1}=\begin{bmatrix}\mathbf{D}\mathbf{W}_{k+1}\\\lambda\mathbf{1}^T\end{bmatrix},\\\tilde{\mathbf{X}}=\begin{bmatrix}\mathbf{X}_{k+1}\\\mathbf{0}^T\end{bmatrix},\\\tilde{\mathbf{C}}_{k+1}=\operatorname{argmin}_{\mathbf{C}\in\mathbb{R}_{\geq0}^{N\times~J}}\lVert\tilde{\mathbf{X}}-\tilde{\mathbf{D}}\mathbf{C}\rVert^2_F,\\\mathbf{C}_{k+1}=\mathbf{W}_{k+1}\tilde{\mathbf{C}}_{k+1}.$$To account for FAI, first a large set of signal evolutions corresponding to each combination of T2 and FAI (also called a dictionary) is computed using the extended phase graph algorithm [11]. The FAI values for each voxel are determined by finding the best match between the measured signal and the dictionary of simulated signals, similar to [12]. Then, for each voxel, the matrix $D$ is selected as the submatrix containing the estimated FAI and all T2 values, and the SPIJN algorithm is performed with the selected submatrices, effectively compensating for the FA inhomogeneity. The dictionary in this work included T2 values ranging from 10ms to 5s logarithmically spaced with 140 steps and FAI of 0.6 to 1 in 100 steps.

The proposed method was evaluated in a MESE brain dataset. The dataset was acquired on a 3T MR-scanner (Ingenia 3.0T, Philips, Best, The Netherlands) with the following parameters: 48 echoes, ΔTE=8ms,TR=1200ms,FOV=240x205x72mm3,resolution=1.25x1.25x8mm3 and 9 slices.

For comparison, MWF-maps were also estimated with two other approaches, the standard NNLS algorithm [2] with the FAI-map estimated from our approach and the B1-compensated regularized-NNLS (regNNLS) method [5].

### Results and discussion

Figs1,2 show the results for the 3T data for three slices. NNLS took 21 seconds per slice, regNNLS 22 minutes, and SPIJN 45 seconds on a simple laptop (IntelCore i5-6300U CPU @2.40GHz 2 cores, 4 threads).

The FAI-maps of both methods are similar, however, our FAI-maps show a little bit of brain structure, which is a drawback of this relatively simple approach assuming a single T2/FAI component for each voxel. In the ventricles such B1 offset has been observed before [13]. To prevent this, spatial smoothing could be applied to the FAI-map.

The NNLS MWF-maps show noise amplification, which is reduced by the regNNLS method, with the drawback of longer computation times. The SPIJN MWF-maps show better left-right-symmetry than the regNNLS maps.

### Conclusion

The proposed FAI-map approximation combined with the SPIJN algorithm using joint sparsity constraints gives results comparable to previously proposed methods with the advantage of practically feasible computation times.

### Acknowledgements

No acknowledgement found.

### References

[1] K. P. Whittall and A. L. MacKay, “Quantitative interpretation of NMR relaxation data,” J. Magn. Reson. 1969, vol. 84, no. 1, pp. 134–152, Aug. 1989.

[2] C. L. Lawson and R. J. Hanson, Solving least squares problems. SIAM, 1974.

[3] D. Hwang and Y. P. Du, “Improved myelin water quantification using spatially regularized non-negative least squares algorithm,” J. Magn. Reson. Imaging, vol. 30, no. 1, pp. 203–208, Jun. 2009. [4] D. Kumar, S. Siemonsen, C. Heesen, J. Fiehler, and J. Sedlacik, “Noise robust spatially regularized myelin water fraction mapping with the intrinsic B1-error correction based on the linearized version of the extended phase graph model,” J. Magn. Reson. Imaging, vol. 43, no. 4, pp. 800–817, Apr. 2016.

[5] T. Prasloski, B. Mädler, Q.-S. Xiang, A. MacKay, and C. Jones, “Applications of stimulated echo correction to multicomponent T2 analysis,” Magn. Reson. Med., vol. 67, no. 6, pp. 1803–1814, Jun. 2012.

[6] M. F. Duarte, S. Sarvotham, D. Baron, M. B. Wakin, and R. G. Baraniuk, “Distributed Compressed Sensing of Jointly Sparse Signals,” 2005, pp. 1537–1541.

[7] S. F. Cotter, B. D. Rao, Kjersti Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2477–2488, Jul. 2005.

[8] J. A. Tropp, A. C. Gilbert, and M. J. Strauss, “Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit,” Signal Process., vol. 86, no. 3, pp. 572–588, Mar. 2006.

[9] J. A. Tropp, “Algorithms for simultaneous sparse approximation. Part II: Convex relaxation,” Signal Process., vol. 86, no. 3, pp. 589–602, Mar. 2006.

[10] J. D. Blanchard, M. Cermak, D. Hanle, and Y. Jing, “Greedy Algorithms for Joint Sparse Recovery,” IEEE Trans. Signal Process., vol. 62, no. 7, pp. 1694–1704, Apr. 2014.

[11] J. Hennig, “Multiecho imaging sequences with low refocusing flip angles,” J. Magn. Reson. 1969, vol. 78, no. 3, pp. 397–407, Jul. 1988.

[12] N. Ben‐Eliezer, D. K. Sodickson, and K. T. Block, “Rapid and accurate T2 mapping from multi–spin-echo data using Bloch-simulation-based reconstruction,” Magn. Reson. Med., vol. 73, no. 2, pp. 809–817, Feb. 2015.

[13] W. M. Brink, P. Börnert, K. Nehrke, and A. G. Webb, “Ventricular B1+ perturbation at 7 T – real effect or measurement artifact?,” NMR Biomed., vol. 27, no. 6, pp. 617–620, Jun. 2014.

### Figures

Figure 1. The flip angle inhomogeneity (FAI) maps as determined with the B1-compensated regularized NNLS method [5] and the here proposed single component matching for multi-echo spin-echo T2 data at 3T. The FAI is caused by B1 inhomogeneities. The FAI value is the fraction between the effective FA and intended FA. A possitive B1 offset has the same effect as a negative B1 offset, therefore the range of FAI is up to 1. The obtained maps by the two methods are similar, although the proposed method shows some unexpected brain structure, especially in the ventricels.

Figure 2. The myelin water fraction (MWF) maps calculated with the NNLS algorithm [2], the B1-compensated regularized NNLS method [5] and the proposed, new SPIJN algorithm for multi-echo spin-echo T2 data at 3T. The MWF is the fraction of myelin water in a voxel compared to the total amount of water present. The proposed method gives similar results, with a better left-right symmetry and more smooth structures. Relaxation times below 40ms were considered as myelin water.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)
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