Shuhui Cai^{1}, Li Zhang^{1}, Congbo Cai^{1}, and Zhong Chen^{1}

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 L_{0}
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 (R^{2}=0.998) for COSMOS, y = 0.866x+0.017 (R^{2}=0.999) for MEDI, y = 0.771x+0.011 (R^{2}=0.996)
for iLSQR, y = 0.829x+0.046 (R^{2}=0.993) for WL_{1}, y = 0.909x+0.016 (R^{2}=0.999) for TFI, and y = 0.919x+0.019 (R^{2}=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 (MEDI^{1}, iLSQR^{2}, and WL_{1}^{3})
and a one-step QSM method (SSQSM^{4} and TFI^{5}). The COSMOS^{6}
result is presented as a ground truth. The average
reconstruction time was 1394 s for MEDI, 152 s for iLSQR, 88 s for WL_{1},
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 WL_{1} 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.

1. Liu J, Liu T, de Rochefort L, et al. Morphology enabled dipole inversion for quantitative susceptibility mapping using structural consistency between the magnitude image and the susceptibility map. Neuroimage, 2012;59(3):2560-2568.

2. Li W, Wang N, Yu F, et al. A method for estimating and removing streaking artifacts in quantitative susceptibility mapping. Neuroimage, 2015;108:111-122.

3. Bilgic B, Fan AP, Polimeni JR, et al. Fast
quantitative susceptibility mapping with L_{1}-regularization and
automatic parameter selection. Magn Reson Med, 2013;72(5):1444-1459.

4. Chatnuntawech I, McDaniel P, Cauley SF, et al. Single-step quantitative susceptibility mapping with variational penalties. NMR Biomed, 2017;30(4):e3570.

5. Liu Z, Kee Y, Zhou D, et al. Preconditioned total field inversion (TFI) method for quantitative susceptibility mapping. Magn Reson Med, 2016;78(1):303-315.

6. Liu T, Spincemaille P, de Rochefort L, et al. Calculation of susceptibility through multiple orientation sampling (COSMOS): a method for conditioning the Inverse problem from measured magnetic field map to susceptibility source image in MRI. Magn Reson Med, 2009;61(1):196-204.

Figure 1 QSM results of the gadolinium phantom reconstructed with
COSMOS (A), MEDI (B), iLSQR (C), WL_{1} (D), TFI (E) and fast TFI (F) methods.

Figure 2 Coronal, sagittal, and axial views of the QSM results of
the Cornell healthy brain reconstructed with COSMOS (A), MEDI (B), iLSQR (C),
WL_{1} (D), SSQSM (E), TFI (F), and fast TFI (G) methods.

Figure 3 Results
of the patient with cerebral hematoma. (A) Amplitude image,
(B) R_{2}^{* }map; (C) total field map; (D-H) QSM results
reconstructed with MEDI (D), iLSQR (E), WL_{1} (F), SSQSM (G), TFI (H),
and fast TFI (I) methods.