Quantitative susceptibility mapping (QSM) is very useful for obtaining biological information such as iron content or blood oxygen saturation. Determining a susceptibility distribution is an ill-conditioned inverse problem caused by the zero-cone region of a dipole function in k-space. To solve the problem, calculation of susceptibility through multiple-orientation sampling (COSMOS) was proposed [1], but it is clinically impractical to carry out on human subjects in scan time and with the standard receiver coil size. We have proposed a QSM reconstruction of a single orientation sampling method that combines an iterative least-square minimization and adaptive edge-preserving filtering [2]. The method has less variation in estimated magnetic susceptibility with respect to the value of the reconstruction parameters compared with regularization methods [3,4]. In this study, we demonstrate a qualitative and quantitative comparison between susceptibility calculated by the proposed method and COSMOS in healthy volunteers.

Algorithm Figure 1 shows an illustration of the proposed QSM reconstruction method. The method consists of three steps: (I) iterative least-square minimization [steps (a) - (d), (l)], (II) adaptive edge-preserving filtering to the susceptibility map in the minimization process [steps (e), (f)], and (III) weighted addition of the susceptibility map in k-space before and after filtering [steps (g) - (k)]. The cost function in the iterative least square minimization is defined by the equation: $$$e(\chi _i)=||W(C\chi _i-\delta)||_2^2$$$, where $$$C$$$ denotes a matrix representing the convolution with the dipole kernel, $$$\delta$$$ denotes the measured local field, and $$$W$$$ denotes the weighting matrix. In step (e), the adaptive edge-preserving filter is defined by the equation: $$$ \chi_{smooth}(r)=\mu(r)+\frac{\sigma(r)^2-\nu(r)^2}{\sigma(r)^2}{(\chi_{i}(r)-\mu(r)}) $$$ , where r denotes the coordinates of a voxel, and $$$\mu$$$ and $$$\sigma^2$$$ denote the average and variance of the local window, i.e., 3 × 3 × 3, in the susceptibility map, respectively. $$$\nu^2$$$ is a parameter corresponding to the noise variance that is calculated by the local variance of the difference of susceptibility before and after filtering. In steps (g) - (k), the weighting matrices $$$G$$$ and $$$G_k$$$ are defined by the equations: $$$ G(k)=\begin{cases}1 & |D(k)|\geq a_{th}\\|D(k)|/a_{th} & |D(k)| <a_{th} \end{cases} $$$ , $$$ G_k(k)=1-G(k) $$$ , where $$$k$$$ denotes the coordinates in k-space, $$$a_{th}$$$ denotes the boundary threshold, and $$$D(k)$$$ denotes a dipole function in k-space $$$\left(D(k)=\frac{1}{3}-\frac{k_z^2}{k_x^2+k_y^2+k_z^2}\right)$$$. In this study, we used $$$a_{th}$$$ = 0.8 in reference to the previous study [2].

Human study Three healthy volunteers were recruited (male, 24 - 29 years old) to be studied on a 3T MR imaging scanner (TRILLIUM OVAL, Hitachi Ltd., Japan) with 15-channel head coil. A 3D RF-spoiled gradient-echo sequence was used, that is, TR/TE = 37/33 ms, FA = 15°, FOV: 240 × 240 × 144 mm, matrix: 200 × 200 × 120 (zero filled to 480 × 480 × 240), and voxel size of scan: 1.2³ mm³. For the first scan, the head was not rotated. To apply COSMOS, we obtained two additional images for which the main magnetic field was changed by rotating the head position (tilted to the right and the back). For quantitative evaluation of susceptibility between the proposed method and COSMOS, we performed a region of interest (ROI)-based and voxel-based quantitative comparison. For the ROI-based analysis, we selected the straight sinus, right-and-left of globus pallidus, putamen, caudate nucleus, substantia nigra, and red nucleus as the ROIs.

Figures 2 show a susceptibility map estimated by using the proposed method [(a) - (c)] and COSMOS [(d) - (f)] for the three volunteers. Both methods had similar susceptibility map quality, and the brain structures of the susceptibility map generated by the proposed method were preserved, as well as those of COSMOS. Figures 3 show the results of the ROI-based [(a), (b)] and voxel-based [(c), (d)] analysis. Figures 3 (a) - (c) show good agreement between the proposed method and COSMOS. The voxel-based Bland-Altman analysis in figure 3 (d) shows that the variations in a slice of the susceptibility between both methods were slightly larger. It is assumed that the number of scans is different for both methods. We demonstrated that the proposed method could generate a high-quality susceptibility map by combining an iterative least-square minimization and adaptive edge-preserving filtering. The results from human brain experiments showed good agreement with COSMOS. The proposed method is useful for generating a high-quality susceptibility map of the human brain.