Jingu Lee^{1,2}, Hyeong-geol Shin^{1}, Hyunsung Eun^{1}, Dongmyung Shin^{1}, and Jongho Lee^{1}

In this study, the recently proposed magnetic susceptibility source separation method, which separates the paramagnetic susceptibility source from the diamagnetic susceptibility source, was validated using Monte-Carlo simulation and phantom experiment. The results demonstrate that the method successfully separates the paramagnetic and diamagnetic susceptibility sources in both simulation and experiment.

**Purpose**

**
[Reconstruction algorithm] **The
positive and negative susceptibility maps ( and ) were reconstructed by solving the following equation
(Fig. 1): $$argmin_{\chi_{pos}(r),\chi_{neg}(r)}\mid({R_2}'(r)+i2\pi f(r))-D_m\cdot(\mid\chi_{pos}(r)\mid+\mid\chi_{neg}(r)\mid)-i2\pi D_p(r)\otimes(\chi_{pos}(r)+\chi_{neg}(r))\mid_2+g(\chi_{pos}(r),\chi_{neg}(r))$$

where $$$D_m$$$ is a proportionality constant
explaining for the effect size of susceptibility on R_{2}’, and $$$D_p$$$ is a dipole pattern^{2}. The $$$g(\chi_{pos}(r),\chi_{neg}(r))$$$ is an optional regularization term^{3}.

**
[Monte-Carlo simulation]
**The nine simulation segments were designed to
consider three susceptibility compositions (positive only, negative only, and half
and half) and three source concentrations (|0.0125|, |0.025|, and |0.0375| ppm)
(Fig. 1a-b). Each segment contained 9×9×9
voxels with a 100
µm size. Within a voxel, 1000 protons were randomly distributed by Gaussian
diffusion model (1 µm^{2}/ms).
The susceptibility sources (diameter = 1 µm, susceptibility = |520| ppm)
were randomly located within the central voxel.
The voxel-averaged magnitude and
phase were calculated from the sum of proton signals (T_{2} = 100 ms). The
R_{2}* decays and frequency shift were estimated using curve fitting. R_{2}’
was calculated as R_{2}* - R_{2}. All the simulation was
repeated 100 times and averaged.
The
resulting data were processed using Eq. 1, generating the positive and negative
susceptibility maps. $$$D_m$$$ was calculated from the slope of
the simulated R_{2}’ values w.r.t. the assigned susceptibility values.

**[phantom experiment]
**For phantom manufacturing, a cylindrical container (diameter =
120 mm) was constructed with nine small cylinders (diameter = 7 mm) (Fig. 2a).
The medium was filled with 1.5% agarose gel. The nine small cylinders were
filled with positive and/or negative susceptibility sources in agarose.
For the positive and negative susceptibility sources, iron oxide (Fe_{3}O_{4};
0.25, 0.50, 0.75 mg/ml; first row) and calcium carbonate (CaCO_{3};
12.5, 25.0, and 37.5 mg/ml; second row) were used. The two sources were mixed for the third row with the same concentrations as the first two rows. For R_{2}*
and frequency shift, multi-echo gradient-echo were acquired at 3T: FOV = 192×192×120 mm^{3}, voxel size = 1×1×2 mm^{3},
TR = 60 ms, and TE = 2.6:4.9:27.1 ms. For R_{2}
estimation, multi-echo spin-echo data were acquired: FOV = 192×192 mm^{2},
voxel size = 1×1 mm^{2}, slice thickness = 2 mm, number of slices = 40,
TR = 4000 ms, and TE = 9:9:90 ms.
The data were processed for R_{2}’ and
frequency shift^{1}, and the source separation algorithm was applied to
generate the positive and negative susceptibility maps. For $$$D_m$$$,
the slope of R_{2}’ plot w.r.t. the absolute susceptibility values was calculated from the positive and negative
only data.

**Results**

**[Monte-Carlo simulation**] The
simulation setup is visualized in Fig. 2a-b. The simulated frequency shift
maps show dipole patterns (Fig. 2c) whereas R_{2}’ shows localized
signal change (Fig. 2d). The linear regression between R_{2}’ and
susceptibility yields $$$D_m$$$ of 247 Hz/ppm (Fig. 2e). When the susceptibility source separation method is applied, the
positive and negative susceptibility maps are reconstructed with high fidelity
(R^{2} = 0.99 with the assigned maps; Fig. 2f-g). The results shown in the bar graph reveal consistent
measurements (Fig. 2h).

**[phantom experiment]** The phantom experimental results further corroborate our method. The phantom setup
is shown in Figs. 3a-b. The frequency shift map shows dipole patterns around the
susceptibility sources (Fig. 3c). On the other hand, R_{2}’ is dependent on the absolute sum of the susceptibility as shown
in Fig. 3d. When the R_{2}’ values of the single sources are plotted w.r.t. the
susceptibility (Fig. 3e; blue dots: Fe_{3}O_{4}
and red dots: CaCO_{3}), linear dependence is observed for both sources with the average $$$D_m$$$ of 279 Hz/ppm. When the
sources are separated (Fig. 3f-g), they clearly show positive and negative susceptibility contributions exclusively. In
the third row, the mixed sources are successfully separated, revealing the
similar positive and negative susceptibility concentrations with
the first and second rows (Fig. 3h). These results demonstrate that our method
successfully separated positive and negative susceptibility sources with high
accuracy.

[1] Lee, J., Nam, Y., Choi, J.Y., Shin, H.G., Hwang, T., Lee, J., 2017. Separating positive and negative susceptibility sources in QSM. Proc Intl Soc Mag Reson Med 25, 0751.[2] Salomir, R., de Senneville, B.D., Moonen, C.T.W., 2003. A fast calculation method for magnetic field inhomogeneity due to an arbitrary distribution of bulk susceptibility. Concepts in Magnetic Resonance Part B: Magnetic Resonance Engineering 19B, 26-34.[3] Liu, T., Liu, J., de Rochefort, L., Spincemaille, P., Khalidov, I., Ledoux, J.R., Wang, Y., 2011. Morphology enabled dipole inversion (MEDI) from a single-angle acquisition: comparison with COSMOS in human brain imaging. Magn Reson Med 66, 777-783.