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Structure tensor enhanced quantitative susceptibility mapping (ST-QSM)
Agnese Tamanti1, Kristian Bredies2, Marco Castellaro3, Stefan Ropele4, Berkin Bilgic5, and Christian Langkammer4

1University of Verona, Verona, Italy, 2Institute of Mathematics and Scientific Computing, University of Graz, Graz, Austria, 3Department of Information Engineering, University of Padova, Padova, Italy, 4Department of Neurology, Medical University of Graz, Graz, Austria, 5Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Harvard Medical School, MGH, Boston, MA, United States

### Synopsis

Quantitative susceptibility mapping (QSM) is an MRI technique enabling the reconstruction of a basic physical property in vivo. However, retrieving susceptibility maps from the MRI phase data requires an ill-posed inverse problem to be solved, which is often achieved using regularization approaches. In this abstract, we extend an existing QSM algorithm by incorporating weights from the linear structure tensor (ST) of the magnitude images to stabilize the regularization. The new algorithm yields improvements regarding the visual appearance and the quantitative performance of the susceptibility maps obtained.

Quantitative susceptibility mapping (QSM) enables the in vivo measurement of a basic physical property and is promising for the assessment of iron and calcium1. The Total Generalized Variation (TGV) algorithm2 solves the inverse problem to retrieve the tissue susceptibility (χ) from gradient echo phase data by regularization using a $TGV^2_\alpha$ penalty, which is a second order functional preferring piece-wise linear solutions3,4:

$$TGV^2_\alpha (\chi) = min_w \ \alpha_1 \Vert \nabla \chi - w \Vert _M + \alpha_0 \Vert \varepsilon w \Vert_M$$

Here, ∇ represents the gradient, ||.||M the Radon norm, ε the symmetrized derivative for vector fields, w the vector fields for the minimization, and α0, α1 the regularization parameters. Similarly to magnitude-stabilized QSM algorithms5,6, we extended the TGV-based QSM algorithm by incorporating weights from the linear structure tensor (ST). Because of local averaging, the ST does not only include directional information at the single point, but also includes geometric characteristics of its neighborhood7,8.

The linear structure tensor9 (ST) is the convolution of a variable Gaussian kernel (Kρ) with an orientation tensor represented by the outer product (⊗) of the gradient of the image (∇I).

$$ST = K_\rho * (\nabla I \otimes \nabla I) = K_\rho * \left( \begin{matrix} (\partial _x I)^2 & \partial _x I \partial _y I & \partial _x I \partial _z I\\ \partial _y I\partial _x I & (\partial _y I)^2 & \partial _y I \partial _z I\\ \partial _z I\partial _x I & \partial _z I \partial _y I & (\partial _z I)^2 \end{matrix} \right)$$

The weighting tensor used in the regularization process is obtained by the structure tensor modifying its eigenvalues as:

$$\lambda_{wt} = \frac{1}{1 + l\lambda_{st}^p}$$

Where λwt and λst denote the eigenvalues of the weighting and the structure tensors, respectively. For this preliminary study we choose p=4 and l so that the 10% of the modified eigenvalues in the brain were lower than a threshold of 0.3, but we will further analyze the performance of the ST varying these weighting parameters (p and l).

To determine the regularization along their correspondent eigenvectors, the weighting tensor (wT) calculated from the structure tensor was incorporated in the TGV penalty as:

$$ST\text{-}TGV^2_\alpha (\chi) = min_w \ \alpha_1 \Vert wT (\nabla \chi - w) \Vert _M + \alpha_0 \Vert \varepsilon w \Vert_M$$

QSM images were calculated with the TGV and ST-TGV algorithms based on a single transverse orientation and compared to a reference COSMOS10 reconstruction from 12 orientations from the 2016 QSM reconstruction challenge dataset with 1mm³ isotropic resolution11. Additionally, a high resolution 3D GRE dataset with 0.5mm³ was used12.

Figure 1 shows the color coded ST utilized for stabilization and figure 2 the resulting QSM images obtained with TGV and ST-TGV algorithms as well as the 12-orientations COSMOS reconstruction. ST-weighting improves the delineation of anatomical structures and additionally, yielded less underestimation of susceptibility, which is a common issue of many QSM algorithms. These improvements were also observed in the 0.5mm³ isotropic high resolution QSM images shown in figure 3

The proposed ST-QSM method retrieves susceptibility maps using a regularization approach where the TGV penalty is stabilized by structure tensor weights derived from the magnitude images. In particular, this ST prior information yielded more homogeneous susceptibilities in white matter and ventricles while iron-rich deep gray matter structures showed less underestimation of susceptibilities. Although ST-TGV showed generally higher susceptibilities in the basal ganglia (Figures 2 and 3), more systematic work including broad variation of the TGV and ST parameters is required to analyze and understand the impact regarding susceptibility underestimation

### Acknowledgements

No acknowledgement found.

### References

1. Wang Y, Liu T. Quantitative susceptibility mapping (qsm): decoding MRI data for a tissue magnetic biomarker. Magnetic resonance in medicine, 2015, 73(1):82101.

2. Langkammer C, Bredies K, et al. Fast quantitative susceptibility mapping using 3D EPI and total generalized variation. Neuroimage, 2015;111:622–630.

3. Bredies K. Recovering Piecewise Smooth Multichannel Images by Minimization of Convex Functionals with Total Generalized Variation Penalty , pages 4477. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014.

4. Bredies K, Kunisch K, Pock T. Total generalized variation. SIAM Journal on Imaging Sciences, 2010;3(3):492526.

5. Liu T, Liu J, et al. Morphology enabled dipole inversion (MEDI) from a single-angle acquisition: comparison with COSMOS in human brain imaging. Magn. Reson. Med. 2011;66:777–783.

6. Kee Y, Cho J, et al. Coherence enhancement in quantitative susceptibility mapping by means of anisotropic weighting in morphology enabled dipole inversion. Magnetic Resonance in Medicine. 2017

7. Estellers V, Soatto S, Bresson X. Adaptive regularization with the structure tensor. IEEE Transactions on Image Processing, 2015; 24(6):17771790.

8. Brox T, Rein van den Boomgaard et al. Adaptive Structure Tensors and their Applications, pages 1747. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.

9. Jähne B. Spatio-temporal image processing: theory and scientic applications, volume 751. Springer Science & Business Media, 1993

10. Liu T, Spincemaille P, et al. Calculation of susceptibility through multiple orientation sampling (cosmos): a method for conditioning the inverse problem from measured magnetic field map to susceptibility source image in mri. Magnetic Resonance in Medicine, 2009; 61(1):196204.

11. Langkammer C, Schweser F, et al. Quantitative susceptibility mapping: Report from the 2016 reconstruction challenge. Magnetic Resonance in Medicine; 2017

12. Bilgic B, Xie L, et al. Rapid multi-orientation quantitative susceptibility mapping. Neuroimage. 2016;125:1131-1141.

### Figures

Fig. 1: Structure tensor (first row) obtained from the magnitude images (second row) of the QSM ‘2016 challenge dataset in axial (left), coronal (central) and sagittal (right) sections. The colors of the structure tensor correspond to the direction of its principal eigenvector (red, green and blue for the x,y and z axis, respectively), while the pixel intensity is determined by the principal eigenvalues of the weighting tensor.

Fig. 2: Susceptibility maps reconstructed from the QSM 2016 challenge data: (a,A) the COSMOS reference reconstructed from 12 head orientations; (b,B) the TGV and (c,C) the proposed ST-TGV methods. The first row shows central axial sections, the second row shows the magnifications of the basal ganglia region. Stabilization with the structure tensor (ST) reduces underestimation of the susceptibility. All susceptibility maps are scaled from -0.10 to 0.14 ppm.

Fig.3: Comparison of the TGV (left) and the proposed ST-TGV QSM reconstructions (right). ST-TGV shows more homogeneous appearance of the white matter (AB) and the ventricles (CD). Generally, iron-rich deep gray matter such as the red nucleus and the substantia nigra (EF) and the basal ganglia (GH) are better delineated and show higher susceptibilities. The susceptibilities are scaled between -0.10 and 0.14ppm.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
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