Yongquan Ye^{1}, Xueping Li^{2}, Qinglei Zhang^{2}, Fei Zhou^{2}, Ming Li^{2}, Zhao Qing^{2}, Bing Zhang^{2}, Shuheng Zhang^{3}, Yanling Chen^{3}, and Jinguang Zong^{3}

We propose dynamically estimate, formulate and update the field components that are responsible for causing streaking artifact, as an additional regularization term for solving the QSM optimization problem. As a result, streaking artifacts arising from regions with highly disrupted local fields can be well suppressed, preventing them from spatially extending and affecting other regions of interest. The proposed method can maintain the accuracy of QSM results, and has the potential to be integrated into most QSM optimization algorithms.

Generally, regularization approaches are
solved iteratively for optimized solution. During each iteration, an
intermediate QSM solution χ_{iter} is obtained and used as input for
subsequent iteration. With χ_{iter}, the corresponding local field
map, Φ_{iter} , can be generated using the
forward dipole function D, i.e. $$$\phi _{iter}=F^{-1}DF\chi _{iter}$$$. Thus the regularization term
describing the streaking artifacts can be formulated as $$$||F^{-1}D'F\Delta\phi||_{2}^{2}$$$,
where D’ is a partial dipole kernel obtained by posing a threshold on D,
and ΔΦ is the difference between Φ_{iter} and the original local field map Φ.
The complete regularization solvers thus were:

$$$argmin_{\chi }||F^{-1}DF\chi -\phi ||_{2}^{2}+\alpha ||P\bigtriangledown \chi ||_{1}+||F^{-1}D'F\Delta \phi||_{2}^{2} $$$ [1]

We solve this minimization problem using
preconditioned conjugate gradient method^{4}.
To evaluate the effectiveness of the streaking
artifact regularization term, computer simulation and in vivo data were tested.
For simulation, a susceptibility brain model was used to generate a field map
as the input for Eq.1, which was solved with and without the artifact
regularization term. With IRB approval, 16 patients with history of brain
surgery were scanned on 3T (uMR770, UIH, Shanghai), using a 4-echo
high resolution GRE sequence. Local field map was calculated as previously
described^{5}, and QSM results with and without artifact
regularization were calculated and compared.

1. Wang, Y. and T. Liu. "Quantitative susceptibility mapping (QSM): Decoding MRI data for a tissue magnetic biomarker." Magn Reson Med, 2015. 73(1): 82-101.

2. Li, W., N. Wang, F. Yu, H. Han, W. Cao, R. Romero, B. Tantiwongkosi, T. Q. Duong and C. Liu. A method for estimating and removing streaking artifacts in quantitative susceptibility mapping. Neuroimage, 2015. 108: p. 111-22.

3. Wei H, Dibb R, Zhou Y, Sun Y, Xu J, Wang N, Liu C. Streaking artifact reduction for quantitative susceptibility mapping of sources with large dynamic range. NMR Biomed 2015;28:1294–1303

4. Bilgic, B., A. P. Fan, J. R. Polimeni, S. F. Cauley, M. Bianciardi, E. Adalsteinsson, L. L. Wald and K. Setsompop. "Fast quantitative susceptibility mapping with L1-regularization and automatic parameter selection." Magn Reson Med, 2014. 72(5): 1444-1459.

5. Yongquan Ye, Jinguang Zong, Jingyuan Lyu, and Weiguo Zhang. SWI+: A robust artifact-free SWI procedure with improved contrast. Proceedings 26th Scientific Meeting, International Society for Magnetic Resonance in Medicine, Paris, 2018(4135).

Fig.1 Computer simulation results. a) is the reference
susceptibility distribution map, which was used to generate the field map based
on which Eq.1 was solved. The resultant QSM maps with (b) and without (c)
artifact regularization term are shown, as well as their corresponding
difference maps (d & e) with the reference. When calculated with artifact
regularization term, the RMSE and SSIM of the results were 23.35 (for d) and
0.963 (for b) respectively, and when calculated without artifact regularization
term, the RMSE and SSIM were 26.35 (for e) and 0.967 (for c).

Fig.2 In vivo QSM results comparison between with and
without using artifact regularization term, on a representative patient with a
history of brain surgery. The field map and an example of the artifact
regularization term of the same image planes are also shown.