Agnese Tamanti^{1}, Kristian Bredies^{2}, Marco Castellaro^{3}, Stefan Ropele^{4}, Berkin Bilgic^{5}, and Christian Langkammer^{4}

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.

**Introduction**

Quantitative
susceptibility mapping (QSM) enables the in vivo measurement of a
basic physical property and is promising for the assessment of iron
and calcium^{1}.
The Total Generalized Variation (TGV) algorithm^{2}
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 solutions^{3,4}:

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

Here,
∇
represents the gradient, ||.||_{M}
the Radon norm, ε
the symmetrized derivative for vector fields, w
the vector fields for the minimization, and α_{0}, α_{1}
the
regularization parameters.
Similarly
to magnitude-stabilized QSM algorithms^{5,6},
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 neighborhood^{7,8}.

**Methods**

The linear
structure tensor^{9}
(ST) is the convolution of a variable Gaussian kernel (*K _{ρ}*)
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
λ_{wt}
and
λ_{st}
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 COSMOS^{10}
reconstruction from 12 orientations from the 2016 QSM reconstruction
challenge dataset with 1mm³ isotropic resolution^{11}.
Additionally, a high resolution 3D GRE dataset with 0.5mm³ was
used^{12}.

**Results**

**Discussion and Conclusion**

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.

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.