Zhe Liu^{1,2}, Yihao Yao^{2}, and Yi Wang^{1,2}

One challenge in Quantitative Susceptibility Mapping (QSM) identified in a recent QSM workshop is zero reference. Cerebrospinal fluid (CSF) with little cellular content has been a popular choice. However, current QSM often shows inhomogeneous CSF, which may be regarded as artifacts caused by surrounding anisotropic white matter fibers in the scalar dipole inversion. We propose a regularization of minimal CSF variation for projecting out CSF inhomogeneity artifacts. Our proposed new QSM incorporates automated segmentation and regularization specific to CSF and outputs susceptibility values with automatic and uniform CSF zero reference. Accordingly, we term this novel QSM method as QSM0

The optimization problem for proposed QSM0 is:

$$\chi^*=arg\min_{\chi}\frac{1}{2}\parallel w \left( e^{if}-e^{i\left(d\star\chi\right)} \right) \parallel_2^2 + \lambda_1 \parallel M_G\triangledown\chi \parallel_1 + \lambda_2 \parallel M_{CSF}\left(\chi-\overline{\chi_{CSF}}\right) \parallel_2^2 (1)$$

with $$$\chi$$$ the susceptibility map, $$$\star$$$ the convolution with the dipole kernel $$$d$$$, $$$w$$$ the noise weighting, $$$f$$$ the local field, $$$\triangledown$$$ the gradient operator and $$$M_G$$$ the binary edge mask derived from the magnitude image ^{3}. It differs from the current QSM method MEDI ^{3 }by the addition of an L_{2}-regularization, where $$$M_{CSF}$$$ is ROI of ventricular CSF with a mean susceptibility $$$\overline{\chi_{CSF}}$$$. This additional term penalizes large susceptibility variance within ventricular CSF.

$$$M_{CSF}$$$ is determined in following automated steps: (a) Threshold $$$R_2^*$$$: $$$M_{R_2^*}=R_2^*<5s^{-1}$$$. (b) Define brain centroid: $$$\mathbf{c} =\frac{1}{N}\sum_{\mathbf{r}\subset M}\mathbf{r}$$$, where $$$M$$$ is the brain mask of $$$N$$$ voxels. (c) Define central brain region: $$$M_c=\left\{\mathbf{r}|\parallel \mathbf{r}-\mathbf{c} \parallel_2 <3\mathrm{cm}\right\}$$$. (d) Analyze connectivity: devide $$$M_c\cap M_{R_2^*}$$$ into connected components $$$M_{ci}$$$ (6-neighbor), and merge the largest 2 components: $$$M_{cCSF}=M_{c1}\cup M_{c2}$$$; Then divide $$$M_{R_2^*}$$$ into connected components $$$M_{i}$$$ (6-neighbor), and merge all components overlapping with $$$M_{cCSF}$$$: $$$M_{CSF}=\left\{\cup M_i\mid M_i\cap M_{cCSF}\neq\varnothing \right\}$$$. An example for this process is illustrated in Fig.1. The problem (1) is solved using Gauss Newton Conjugate Gradient ^{3}. In the end of the method, $$$\overline{\chi_{CSF}}$$$ is subtracted from the entire map for zero reference.

Considering white matter (WM) susceptibility anisotropy as a cause of CSF inhomogeneity, we constructed a numerical brain phantom with anisotropic susceptibility $$$\left( \chi_{13},\chi_{23},\chi_{33} \right)$$$ in WM and zero susceptibility in CSF (Fig.2). Local fields with and without anisotropic susceptibility were generated as:

$$F_1(k)=\left({\frac{1}{3}-\frac{k_z^2}{k^2}}\right)X_{33}-\frac{k_xk_z}{k^2}X_{13}-\frac{k_yk_z}{k^2}X_{23},\ \ \ \ \ \ \ \ \ \ \ \ F_2(k)=\left({\frac{1}{3}-\frac{k_z^2}{k^2}}\right)X_{33}\ \ \ \ (2)$$

QSMs were reconstructed from these two fields using MEDI ($$$\lambda_1=0.001$$$) and QSM0 ($$$\lambda_1=0.001,\lambda_2=0.1$$$) and compared to $$$\chi_{33}$$$

MEDI and QSM0 were then applied to human brains of 8 patients with Multiple Sclerosis (MS). The *in vivo* data were acquired at 3T with 20° flip angle, 24cm FOV, $$$0.5\times0.5\times2\mathrm{mm^3}$$$ resolution and 4.8msec echo spacing. Background field was estimated and removed using PDF ^{4}. For quantitative comparison, susceptibility was measured for each lesion, relative to normal appearing white matter (NAWM) at the contralateral mirror site.

In simulation (Fig.2), both MEDI and QSM0 recovered $$$\chi_{33}$$$ from field $$$F_2$$$ within similar root-mean-square-error (RMSE) of 25.4% and 25.8%, respectively. However, for field $$$F_1$$$ containing anisotropic contribution, QSM0 had a smaller RMSE (74.2%) than that of MEDI (77.3%), and QSM0 reduced artifacts in deep brain compared with MEDI (Fig.2, red arrows).

For all 8 subjects, compared to MEDI, QSM0 achieved artifact suppression (Fig.3) similar to simulation, a 5-fold reduction of the susceptibility standard deviation within ventricle CSF (Fig.4), and MS lesion-to-NAWM contrasts with a strong correlation to MEDI $$$\left(k=1.11, R=0.93\right)$$$. The scatter and Bland-Altman plots (Fig.5) of MS lesion susceptibility measurements from all subjects showed a small 1.3ppb bias and a narrow [-8.7,11ppb] limits of agreement between MEDI and QSM0.

1. Wang, Y, et al. Quantitative susceptibility mapping (QSM): decoding MRI data for a tissue magnetic biomarker. MRM. 2015; 73(1): 82-101.

2. Straub, S, et al. Suitable reference tissues for quantitative susceptibility mapping of the brain. MRM. 2016. DOI 10.1002/mrm.26369.

3. Liu, T, et al. Nonlinear formulation of the magnetic field to source relationship for robust quantitative susceptibility mapping. MRM. 2013. 69(2): 467-476.

4. Liu, T, et al. A novel background field removal method for MRI using projection onto dipole fields (PDF). NMR in BioMed. 2011. 24(9): 1129-1136.

Figure 1. Illustration of automated ventricle CSF segmentation. A raw binary mask M_{R2*} is generated by thresholding R_{2}^{*} map, then used to extract ventricular CSF (M_{CSF}) using a connectivity analysis.

Figure 2. Flowchart for numeric simulation. With field generated by anisotropic model, QSM0 achieves lower RMSE with respect to *χ*_{}_{33} than that of MEDI. In particular, QSM0 suppresses the hypo-intense error close to the substantia nigra and subthalamic nucleus (red arrows).

Figure 3. In vivo brain QSM of a MS patient using MEDI (left column) and QSM0 (right column). Hypo-intensity is suppressed using QSM0 (indicated by red arrows).

Figure 4. Standard deviation of susceptibility (in ppb) within ventricular CSF.

Figure 5. Scatter (left) and Bland-Altman (right) plots of QSM measurements of lesion relative to NAWM.