Li Guo^{1}, Yihao Guo^{1}, Yingjie Mei^{1,2}, Jijing Guan^{1}, Wufan Chen^{1}, and Yanqiu Feng^{1}

MEDI reduces streaking artifacts in QSMs by minimizing total variation in smooth regions in the susceptibility map. However, MEDI still contains artifacts near image edges because this method does not impose any constraint on voxels near edges. We aim to improve the reconstruction of quantitative susceptibility map from MR phase data by introducing morphology-adaptive TV regularization which imposes the TV constraint on the whole susceptibility map but with different weights in smooth and non-smooth regions. The performance of the proposed method is demonstrated in both simulation and in vivo data sets.

The reconstruction of quantitative susceptibility map using morphology-adaptive TV regularization can be formulated as follows:

$$$\min_{\chi}\parallel W\left(F^{-1}DF\chi-\phi\right)\parallel_2^2+\lambda_{1}\parallel M\triangledown \chi\parallel_{1}+\lambda_{2}\parallel \left(1-M\right)\triangledown \chi\parallel_{1}$$$ (1)

where χ is the susceptibility map, $$$\phi$$$ the local tissue phase, $$$F$$$ the Fourier transform operator, $$$D$$$ the dipole kernel in k-space, $$$W$$$ the noise weighting, $$$\parallel \star\parallel_2^2$$$ the L2 norm, $$$\triangledown$$$ the gradient operator, $$$M$$$ the binary mask of smooth regions in the magnitude image, $$$\lambda_{1}$$$ and $$$\lambda_{2}$$$ regularization parameters, $$$\parallel\star\parallel_{1}$$$ the L1 norm, and $$$\parallel\triangledown\chi\parallel_{1}$$$ denotes the L1 norm of gradient, i.e., TV. The second term picks a solution with smooth regions matching those of magnitude images, and thus can effectively suppress streaking artifacts in these smooth regions. The third term imposes the piecewise constant constraint on reconstructed susceptibility map in non-smooth regions to reduce quantification errors at these edge voxels. Eq. (1) provides a spatially adaptive regularization for the QSM inversion. The regularization parameter $$$\lambda_{2}$$$ is generally set to be smaller than $$$\lambda_{1}$$$, and this means less smooth constraint on voxels near edges. When parameter $$$\lambda_{2}$$$ equals to zero, the proposed method reduces to MEDI. When $$$\lambda_{2}$$$ equals to $$$\lambda_{1}$$$, Eq. (1) is a conventional TV constrained QSM inversion without prior morphological information from magnitude images.

To evaluate the performance of the proposed method, the simulation and in vivo data sets from the online QSM repository [4] were reconstructed using the proposed method and MEDI. The associated reference standard images are also provided by the online QSM repository. The parameters used in MEDI were set to the optimal values suggested by previous work [5]. The parameters used in the proposed method were as follows: $$$\lambda_{1}=$$$0.003 and $$$\lambda_{2}=$$$0.00001 for numerical simulation data, and $$$\lambda_{1}=$$$0.003 and $$$\lambda_{2}=$$$ 0.0003 for in vivo data.

Fig.
1 QSM reconstruction of numerical brain data set. Fist row: reference standard with COSMOS. Second row: susceptibility maps by MEDI and corresponding
error maps. Third row: susceptibility maps by the proposed method and
corresponding error maps. An enlarged portion of the images are displayed in
the red boxes.

Fig.
2 QSM reconstruction of an in vivo data set. Fist row: reference standard with
COSMOS. Second row: susceptibility maps by MEDI. Third row: susceptibility maps by
the proposed method. An enlarged portion of the images are displayed in the red
boxes.

Table 1 Quantitative comparison of MEDI
and the proposed method.