Abstract
Quantitative Susceptibility Mapping (QSM) estimates tissue magnetic susceptibility distributions from Magnetic Resonance (MR) phase measurements by solving an ill-posed dipole inversion problem. Conventional single orientation QSM methods usually employ regularization strategies to stabilize such inversion, but may suffer from streaking artifacts or over-smoothing. Multiple orientation QSM such as calculation of susceptibility through multiple orientation sampling (COSMOS) can give well-conditioned inversion and an artifact free solution but has expensive acquisition costs. On the other hand, Convolutional Neural Networks (CNN) show great potential for medical image reconstruction, albeit often with limited interpretability. Here, we present a Learned Proximal Convolutional Neural Network (LP-CNN) for solving the ill-posed QSM dipole inversion problem in an iterative proximal gradient descent fashion. This approach combines the strengths of data-driven restoration priors and the clear interpretability of iterative solvers that can take into account the physical model of dipole convolution. During training, our LP-CNN learns an implicit regularizer via its proximal, enabling the decoupling between the forward operator and the data-driven parameters in the reconstruction algorithm. More importantly, this framework is believed to be the first deep learning QSM approach that can naturally handle an arbitrary number of phase input measurements without the need for any ad-hoc rotation or re-training. We demonstrate that the LP-CNN provides state-of-the-art reconstruction results compared to both traditional and deep learning methods while allowing for more flexibility in the reconstruction process.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In fact, the expression in (1) can also be interpreted as a Maximum a Posteriori (MAP) estimator.
- 2.
\(A^H\) denotes the conjugate transpose (or Hermitian transpose) of the matrix A.
- 3.
Any other susceptibility estimation choices can serve as an initialization as well.
- 4.
We chose an \(\ell _2\) norm for simplicity, and other choices are certainly possible, potentially providing further reconstruction improvements.
- 5.
Our code is released at https://github.com/Sulam-Group.
References
Abdul-Rahman, H., et al.: Fast three-dimensional phase-unwrapping algorithm based on sorting by reliability following a non-continuous path. In: Optical Measurement Systems for Industrial Inspection IV, vol. 5856, pp. 32–40. International Society for Optics and Photonics (2005)
Acosta-Cabronero, J., et al.: In vivo quantitative susceptibility mapping (QSM) in Alzheimer’s disease. PloS One 8(11), e81093 (2013)
Adler, J., Öktem, O.: Learned primal-dual reconstruction. IEEE Trans. Med. Imaging 37(6), 1322–1332 (2018)
Ayton, S., et al.: Cerebral quantitative susceptibility mapping predicts amyloid-\(\beta \)-related cognitive decline. Brain 140(8), 2112–2119 (2017)
Bollmann, S., et al.: DeepQSM-using deep learning to solve the dipole inversion for quantitative susceptibility mapping. NeuroImage 195, 373–383 (2019)
Bruckstein, A.M., et al.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)
Chen, W., et al.: Quantitative susceptibility mapping of multiple sclerosis lesions at various ages. Radiology 271(1), 183–192 (2014)
Chen, Y., et al.: QSMGAN: improved quantitative susceptibility mapping using 3D generative adversarial networks with increased receptive field. NeuroImage 207, 116389 (2020)
De Rochefort, L., et al.: Quantitative MR susceptibility mapping using piece-wise constant regularized inversion of the magnetic field. Magn. Reson. Med. Off. J. Int. Soc. Magn. Reson. Med. 60(4), 1003–1009 (2008)
Deistung, A., et al.: Overview of quantitative susceptibility mapping. NMR Biomed. 30(4), e3569 (2017)
Goodfellow, I., et al.: Generative adversarial nets. In: Advances in Neural Information Processing Systems, pp. 2672–2680 (2014)
Hammernik, K., et al.: Learning a variational network for reconstruction of accelerated MRI data. Magn. Reson. Med. 79(6), 3055–3071 (2018)
Jenkinson, M., Smith, S.: A global optimisation method for robust affine registration of brain images. Med. Image Anal. 5(2), 143–156 (2001)
Jung, W., Bollmann, S., Lee, J.: Overview of quantitative susceptibility mapping using deep learning: current status, challenges and opportunities. NMR Biomed. e4292 (2020). https://onlinelibrary.wiley.com/doi/abs/10.1002/nbm.4292
Jung, W., et al.: Exploring linearity of deep neural network trained QSM: QSMnet+. NeuroImage 211, 116619 (2020)
Kames, C., et al.: Proximal variational networks: generalizable deep networks for solving the dipole-inversion problem. In: 5th International QSM Workshop (2019)
Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
Langkammer, C., et al.: Quantitative susceptibility mapping in multiple sclerosis. Radiology 267(2), 551–559 (2013)
Langkammer, C., et al.: Quantitative susceptibility mapping: report from the 2016 reconstruction challenge. Magn. Reson. Med. 79(3), 1661–1673 (2018)
Li, W., Wu, B., Liu, C.: 5223 STI Suite: a software package for quantitative susceptibility imaging (2013)
Li, W., et al.: Susceptibility tensor imaging (STI) of the brain. NMR Biomed. 30(4), e3540 (2017)
Li, X., et al.: Magnetic susceptibility contrast variations in multiple sclerosis lesions. J. Magn. Reson. Imaging 43(2), 463–473 (2016)
Liu, C.: Susceptibility tensor imaging. Magn. Reson. Med. Off. J. Int. Soc. Magn. Reson. Med. 63(6), 1471–1477 (2010)
Liu, C., et al.: Quantitative susceptibility mapping: contrast mechanisms and clinical applications. Tomography 1(1), 3 (2015)
Liu, C., et al.: Susceptibility-weighted imaging and quantitative susceptibility mapping in the brain. J. Magn. Reson. Imaging 42(1), 23–41 (2015)
Liu, T., 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. Magn. Reson. Med. Off. J. Int. Soc. Magn. Reson. Med. 61(1), 196–204 (2009)
Liu, T., et al.: Morphology enabled dipole inversion (MEDI) from a single-angle acquisition: comparison with cosmos in human brain imaging. Magn. Reson. Med. 66(3), 777–783 (2011)
Liu, T., et al.: Nonlinear formulation of the magnetic field to source relationship for robust quantitative susceptibility mapping. Magn. Reson. Med. 69(2), 467–476 (2013)
Mardani, M., et al.: Deep generative adversarial neural networks for compressive sensing MRI. IEEE Trans. Med. Imaging 38(1), 167–179 (2018)
Osher, S., et al.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)
Özbay, P.S., et al.: A comprehensive numerical analysis of background phase correction with v-sharp. NMR Biomed. 30(4), e3550 (2017)
Parikh, N., et al.: Proximal algorithms. Found. Trends® Optim. 1(3), 127–239 (2014)
Polak, D., et al.: Nonlinear dipole inversion (NDI) enables robust quantitative susceptibility mapping (QSM). NMR Biomed. (2020). https://doi.org/10.1002/nbm.4271
Ronneberger, O., Fischer, P., Brox, T.: U-Net: convolutional networks for biomedical image segmentation. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9351, pp. 234–241. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24574-4_28
Shmueli, K., et al.: Magnetic susceptibility mapping of brain tissue in vivo using MRI phase data. Magn. Reson. Med. Off. J. Int. Soc. Magn. Reson. Med. 62(6), 1510–1522 (2009)
Smith, S.M.: Fast robust automated brain extraction. Hum. Brain Mapp. 17(3), 143–155 (2002)
Sulam, J., et al.: On multi-layer basis pursuit, efficient algorithms and convolutional neural networks. IEEE Trans. Pattern Anal. Mach. Intell. 42, 1968–1980 (2019)
Van Bergen, J., et al.: Colocalization of cerebral iron with amyloid beta in mild cognitive impairment. Sci. Rep. 6(1), 1–9 (2016)
Wang, Y., Liu, T.: Quantitative susceptibility mapping (QSM): decoding MRI data for a tissue magnetic biomarker. Magn. Reson. Med. 73(1), 82–101 (2015)
Wei, H., et al.: Streaking artifact reduction for quantitative susceptibility mapping of sources with large dynamic range. NMR Biomed. 28(10), 1294–1303 (2015)
Yoon, J., et al.: Quantitative susceptibility mapping using deep neural network: QSMnet. NeuroImage 179, 199–206 (2018)
Zagoruyko, S., Komodakis, N.: Wide residual networks. arXiv preprint arXiv:1605.07146 (2016)
Acknowledgement
The authors would like to thank Mr. Joseph Gillen, Ms. Terri Brawner, Ms. Kathleen Kahl and Ms. Ivana Kusevic for their assistance with data acquisition. This work was partly supported by NCRR and NIBIB (P41 EB015909).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
1 Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Lai, KW., Aggarwal, M., van Zijl, P., Li, X., Sulam, J. (2020). Learned Proximal Networks for Quantitative Susceptibility Mapping. In: Martel, A.L., et al. Medical Image Computing and Computer Assisted Intervention – MICCAI 2020. MICCAI 2020. Lecture Notes in Computer Science(), vol 12262. Springer, Cham. https://doi.org/10.1007/978-3-030-59713-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-59713-9_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-59712-2
Online ISBN: 978-3-030-59713-9
eBook Packages: Computer ScienceComputer Science (R0)
