Edith Franziska Baader1, Thomas Jochmann2, Jens Haueisen2, Robert Zivadinov1,3, and Ferdinand Schweser1,3
1Buffalo Neuroimaging Analysis Center, Department of Neurology at the Jacobs School of Medicine and Biomedical Sciences, University at Buffalo, The State University of New York, Buffalo, NY, United States, 2Department of Computer Science and Automation, Technische Universität Ilmenau, Ilmenau, Germany, 3Center for Biomedical Imaging, Clinical and Translational Science Institute, University at Buffalo, The State University of New York, Buffalo, NY, United States
Synopsis
Quantitative
susceptibility mapping (QSM) is increasingly being used to study the
brain iron homeostasis and white matter pathology. However, all QSM
algorithms that are currently used in the clinical setting use a
physical model that neglects the well-established anisotropic
magnetic susceptibility of myelin. In this work, we demonstrate that
an extended U-Net allows solving a Vector QSM model that accounts for
field perturbations caused by off-diagonal tensor elements. The
proposed Deep Vector QSM yielded improved estimates of
χ33 compared to conventional QSM.
Introduction
Quantitative
susceptibility mapping (QSM)1 is increasingly being used
to study the brain iron homeostasis and white matter pathology. All
clinically-used QSM algorithms calculate a single (scalar)
susceptibility value $$$\chi$$$
per voxel, assuming that tissue susceptibility is isotropic ($$$d$$$:
unit dipole;
$$$\Delta B$$$:
field perturbation; $$$\star$$$: convolution):2
$$\Delta B= \chi \star d \text{. (1)}$$
The
well-established anisotropic magnetic susceptibility of myelin3
renders this assumption invalid in the white matter. Using the rank-2
susceptibility tensor and assuming $$$B=B_0 \cdot (0,0,1)^T$$$,
the field perturbation is given by the three tensor elements $$$\chi_{13}$$$,
$$$\chi_{23}$$$,
and
$$$\chi_{33}$$$:4
$$\Delta B= \chi_{13} \star d_{13} + \chi_{23} \star d_{23} + \chi_{33} \star d_{33} \text{. (2)}$$
Here,
$$$d_{33}=d$$$,
and $$$d_{13}$$$ and
$$$d_{23}$$$ are the specific unit response functions for the off-diagonal tensor
elements
$$$\chi_{13}$$$ and $$$\chi_{23}$$$,
respectively. The 2016 QSM Reconstruction Challenge demonstrated that
ignoring the additional terms introduces artifacts in susceptibility
maps.4 While Susceptibility Tensor Imaging (STI)5
can measure the complete tensor, its requirement of at least six head
orientations renders the technique clinically infeasible.
Liu
et al.6 have proposed solving Eq. (2) with an algorithm
similar to the MEDI approach7. In this work, we
demonstrate that an extended U-Net allows solving Eq. (2). The
proposed Deep Vector QSM yielded improved estimates of
$$$\chi_{33}$$$ compared to conventional QSM.Methods
Network
architecture and training: We extended Jochmann et al.’s
generalized U-Net architecture8,9 by a third decoder arm
(Fig. 1) and trained the network (Tensorflow v2.0;
NVidia-GeForce-2080Ti) using synthetic maps of
$$$\chi_{13}$$$,
$$$\chi_{23}$$$ and
$$$\chi_{33}$$$ at the output nodes and field maps simulated according to Eq. (2) at
the input nodes. We used 2400 training and 600 validation data sets
as described previously (96x96x96).8 To increase the
robustness of the network to noise and different ranges of
susceptibility values, we added Gaussian noise to the field maps and
included scaling-based data augmentation.
Conventional
QSM: For comparison of the proposed technique (Eq. 2) with
conventional DeepQSM10 (Eq. 1), we trained a U-Net with
only $$$\chi_{33}$$$ and field maps simulated according to Eq. (1).
Evaluation:
We applied both techniques to the 2016 QSM Reconstruction Challenge
dataset4 and compared results to the provided
gold-standard STI dataset.Results
Application of Deep Vector QSM
to the validation data demonstrated disentanglement of tensor
elements that had a measurable effect on the field map (Fig. 2).
Inaccuracies occured when an object was hardly visible in the magnetic
field and when objects had more than one non-zero tensor component
at the same time. The prediction of DeepQSM included objects that
belonged to the $$$\chi_{13}$$$ and
$$$\chi_{23}$$$
source
maps. The $$$\chi_{33}$$$ maps of Deep Vector QSM and STI were highly
similar and did not show apparent artifacts (Fig. 3). Reconstruction
quality was substantially improved compared to the map obtained with
DeepQSM (U1-Net) from the same field-map (Fig. 4). The $$$\chi_{13}$$$ and
$$$\chi_{23}$$$ maps demonstrated shared features
using Deep Vector QSM or STI (Fig. 3-arrows).Discussion
We demonstrated the
feasibility of solving the Vector QSM problem (Eq. 2) using an extended U-Net architecture. While the cross-diagonal tensor terms
showed only few anatomical features in both VectorQSM and STI, the
extended physical model improved the $$$\chi_{33}$$$ map compared to
conventional QSM. The disentanglement of all three tensor elements
from a single field map is likely enabled by the different
mathematical structure of the three convolution kernels in Eq. (2).
Future work will focus on improved synthetic training data to further
improve the reconstruction quality in vivo.Conclusion
Deep neural networks
enable the solution of more realistic physical models for
quantitative phase MRI. The improved estimation of tissue
susceptibility in the white matter will bring us one step closer to a
reliable application of QSM in the white matter.Acknowledgements
Research reported in
this publication was funded by the German Academic Exchange Agency
(DAAD) Rise Worldwide program and the National Center for Advancing
Translational Sciences of the National Institutes of Health under
Award Number UL1TR001412. The content is solely the responsibility of
the authors and does not necessarily represent the official views of
the NIH of DAAD.References
1. Reichenbach JR et
al. Quantitative Susceptibility Mapping: Concepts and Applications.
Clin Neuroradiol 2015, 25(S2):225–30.
2. Schweser et al.
Foundations of MRI phase imaging and processing for Quantitative
Susceptibility Mapping (QSM). Z Med Phys 2016, 26(1):6–34.
3. Li W et al.
Susceptibility Tensor Imaging (STI) of the Brain: Review of
Susceptibility Tensor Imaging of the Brain. NMR Biomed 2017,
30(4):e3540.
4. Langkammer C et
al. Quantitative susceptibility mapping: Report from the 2016
reconstruction challenge. Magn Reson Med 2018. 79(3): 1661–73.
5. Liu, C.
Susceptibility tensor imaging. Magn Reson Med 2010,
63(6):1471-1477.
6. Liu, T et al.
Vector Model for Quantitative Susceptibility Mapping (Vector QSM).
ISMRM 2015, p928.
7. Liu J. Morphology
enabled dipole inversion for quantitative susceptibility mapping
using structural consistency between the magnitude image and the
susceptibility map. Neuroimage. 2012;59(3):2560–2568.
8. Jochmann T et al.
U2-Net for DEEPOLE QUASAR–A Physics-Informed Deep Convolutional
Neural Network that Disentangles MRI Phase Contrast Mechanisms. ISMRM
2019, p320.
9. Ronneberger O,
Fischer P, Brox T. U-Net: Convolutional Networks for Biomedical Image
Segmentation. arXiv:1505.04597 [cs] 2015.
10. Bollmann et al.
DeepQSM - Using Deep Learning to Solve the Dipole Inversion for
Quantitative Susceptibility Mapping. NeuroImage 2019,
195:373-383.