Synopsis

Quantitative susceptibility mapping (QSM) is a meaningful MRI technique owing to its unique relation to actual physical tissue magnetic properties. The reconstruction of QSM is usually decomposed into three sub-problems which are solved independently. Here, we propose a fast reconstruction method named as fast TFI based on total field inversion. It accelerates the total field inversion by using specially selected preconditioner and the advanced solution of weighted L₀ regularization. Results from gadolinium phantom and in vivo data verified that the new method has good performance.

Introduction

Methods

The whole susceptibility distribution $$$\chi$$$ of a brain is conventionally comprised of background susceptibility $$$\chi _{H}$$$ and local susceptibility $$$\chi_{L}$$$. In this work, we estimate $$$\chi_{H}$$$ and $$$\chi_{L}$$$ jointly using the fast TFI:

$$w_{opt}=\underset{w}{\arg \min }\left \| M(F^{H}DFL_{\chi}^{-1}w-b)\right \|^{2}+\lambda\left \| W_{g}GL_{\chi}^{-1}w \right \| _{0} \; with\: G=\left [ G_{x};G_{y};G_{z}\right ]$$

where the susceptibility distribution can be calculated by $$$\chi=L_{\chi}^{-1}w_{opt}$$$, $$$M$$$ is a binary mask of the reliable phase inside ROI, $$$F$$$ is the discrete Fourier transform operator, $$$D$$$ is the dipole kernel in k-space, $$$b$$$ is the total magnetic field shift, $$$\lambda$$$ is a regularization parameter, $$$W_{g}$$$ is the weighting matrix, and $$$G$$$ is the gradient operator in r-space. When prior information concerning the susceptibility distribution $$$\chi$$$ is available, this derivation process helps to introduce a reasonable regularization operator $$$L_{\chi}^{-1}$$$, which greatly speeds up the convergence of the algorithm. In this work, $$$L_{\chi}^{-1}$$$ is constructed as follow:

$$L_{\chi }^{-1}=\left\{\begin{matrix}1,M\cap M_{R_{2}^{*}} \\ Q,otherwise\end{matrix}\right.\: M_{R_{2}^{*}}=binary(R_{2}^{*}< th)$$

where $$$Q$$$ and $$$th$$$ are constant. The matrix $$$L_{\chi}^{-1}$$$ is structured such that the difference of the matrix $$$\Gamma_{\chi }=L_{\chi}^{-1}(L_{\chi}^{-1})^{T}$$$ between voxels inside and outside ROI is roughly equivalent to the difference between the local and the background susceptibility distributions. After introducing an auxiliary variable $$$A$$$, a residual term $$$\eta$$$ and a regularization parameter $$$\mu$$$, we can obtain

$$\left\{\begin{matrix}w_{t+1}=\underset{w}{\arg \min }\left \|M(F^{H}DFL_{\chi }^{-1}w-b) \right \|_{2}^{2}+\mu \left \| A_{t}-W_{g}GL_{\chi }^{-1}w-\eta _{t} \right \|_{2}^{2}\\ A_{t+1}=\underset{A}{\arg \min }\: \mu \left \|A- W_{g}GL_{\chi }^{-1}w_{t+1}-\eta _{t} \right \|_{2}^{2}+\lambda \left \|A \right \|_{0}\\ \eta _{t+1}=W_{g}GL_{\chi }^{-1}w_{t+1}+\eta _{t}-A_{t+1}\\ \chi _{opt}=L_{\chi}^{-1}w_{opt}\end{matrix}\right.$$

Gadolinium phantom data, human healthy brain data, and pathological brain data were utilized to demonstrate the performance of the proposed method. The phase was unwrapped using a region growing algorithm. For three-step QSM methods, we removed the background field by using SHARP.

Results

Figure 1 shows the phantom QSMs reconstructed using six different methods. The linear regressions between the estimated (y) and reference (x) balloon susceptibilities were y = 0.847x+0.026 (R²=0.998) for COSMOS, y = 0.866x+0.017 (R²=0.999) for MEDI, y = 0.771x+0.011 (R²=0.996) for iLSQR, y = 0.829x+0.046 (R²=0.993) for WL₁, y = 0.909x+0.016 (R²=0.999) for TFI, and y = 0.919x+0.019 (R²=0.996) for fast TFI.

The three plane views of the QSM results from the Cornell healthy brain are displayed in Figure 2. The fast TFI method is quantitatively compared with three three-step QSM methods (MEDI¹, iLSQR², and WL₁³) and a one-step QSM method (SSQSM⁴ and TFI⁵). The COSMOS⁶ result is presented as a ground truth. The average reconstruction time was 1394 s for MEDI, 152 s for iLSQR, 88 s for WL₁, 819 s for SSQSM, 986 s for TFI, and 471 s for fast TFI. SSQSM, TFI, and fast TFI are all one-step QSM methods, but fast TFI is fastest among them. Compared to the results from the other methods, the fast TFI result reveals excellent fidelity to the COSMOS map for all three plane views. The susceptibility value from TFI in globus pallidus deviates from the ground truth. In particular, the coronal and sagittal views of the WL₁ and SSQSM results depict streaking artifacts that deteriorate the image quality.

Figure 3 shows the QSM results of a male patient with cerebral hematoma. It is obvious that the reconstructed results of all methods other than fast TFI obviously fail to restore the susceptibility distribution due to violent artifacts around the hemorrhagic area. The proposed fast TFI, benefited from the advanced solution and the prior known matrix $$$L_{\chi}^{-1}$$$, is effective in identifying the injury and successfully reconstructs the distribution.

Conclusion

Acknowledgements

References

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