Synopsis

We propose Nonlinear Dipole Inversion (NDI) for high-quality Quantitative Susceptibility Mapping (QSM) without additional regularization, while matching the RMSE of state-of-the-art regularized reconstruction techniques. In addition to avoiding over-smoothing these techniques often face, we also obviate the need for parameter selection. NDI is flexible enough to allow for reconstruction from an arbitrary number of head orientations, and outperforms COSMOS using as few as 2-direction data. This is made possible by a nonlinear forward-model that uses the magnitude as an effective prior, for which we derived a simple gradient descent update. We synergistically combine this physics-model with Variational Networks (VN) to leverage the power of deep learning in the VaNDI algorithm. VaNDI adopts this simple gradient descent rule and learns the network parameters during training, hence requires no additional parameter tuning.

Introduction

QSM solves a linear system, which relates the tissue phase $$$\phi$$$ to the unknown susceptibility $$$\chi:K\chi=\phi$$$. This ill-posed problem benefits from regularization,

$$\min_\chi||K\chi-\phi||_2^2+\lambda R(\chi)$$

This formulation assumes Gaussian noise, whereas phase noise distribution deviates from this especially in low-SNR regions (1). This has been recognized in nonlinear-MEDI (2), where a nonlinear fidelity term was utilized:

$$\min_\chi||W(e^{iK\chi}-e^{i\phi})||_2^2+\lambda ||MG\chi||_1$$

Here the magnitude $$$W$$$ serves as a noise weighting and allows derivation of a binary mask $$$M$$$ that weights the gradient computed with $$$G$$$ operator. Recently proposed FANSI algorithm (3) presents a rapid alternative to nonlinear-MEDI through parameter splitting (4,5), but comes at the cost of 3 parameters that need to be tuned. Herein, we develop a simple gradient-descent optimizer, Nonlinear Dipole Inversion (NDI), and show that magnitude weighting and nonlinear formulation act as inherent priors without need for additional regularization. NDI is simple, and can be flexibly extended to reconstruct multi-orientation data. We show that NDI matches the RMSE of FANSI without the need for parameter tuning, and without the vulnerability of over-smoothing the images. It outperforms COSMOS (6) in multi-head direction data, which is susceptible to streaking artifacts in data with small rotations. Further, we expand NDI to admit variational regularizers learned from training data. Our VaNDI outperforms NDI and FANSI, both in terms of reconstruction speed and quality.

Code/data: https://bit.ly/2RHeiF0

Theory

NDI uses gradient descent to minimize $$$f(\chi)=||W(e^{iK\chi}-e^{i\phi}||_2^2$$$, NDI uses gradient descent to minimize

$$f(\chi)=||W(\cos(K\chi)-\cos(\phi)||_2^2+||W(\sin(K\chi)-\sin(\phi))||_2^2$$

After simplifications, taking its derivative yields

$$\partial f(\chi)=2K^TW^2\sin(K\chi-\phi)$$

With this, the $$$t^{th}$$$ iteration of the reconstruction reduces to:

$$\chi^{t+1}=\chi^t-2\sum_{r=1}^NK_r^TW^2\sin(K_r\chi^t-\chi_i)$$

where we generalized the formula for multi-orientation reconstruction from $$$N$$$-directions, with $$$\phi_r$$$ and $$$K_r$$$ denoting tissue phase and dipole kernel belonging to $$$r^{th}$$$ head rotation.

Data Acquisition

Results

Multi-orientation: NDI vs COSMOS

Fig. 1 compares multi-orientation reconstructions using 2-, 3-, 4- and 5-direction data with NDI and COSMOS. COSMOS admits the closed form solution $$$\chi_{cosmos}=(\sum_rK_r^2)^{-1}\sum_rK_r^T\phi_r$$$, which is subject to artifacts if $$$\sum_rK_r^2$$$ is poorly conditioned in small rotation cases. NDI addresses this and improves reconstruction dramatically even from 2-directions.

Single-orientation: NDI vs TKD, L2 and FANSI

Fig. 2 compares 1-direction NDI against TKD (13), L2 (14) and FANSI (15), where the parameters of the latter three were tuned using the 5-direction COSMOS data. RMSEs are reported with respect to both the 5-direction COSMOS and 5-direction NDI reconstructions. NDI yields similar RMSE as FANSI, while achieving significantly sharper results, without the need for parameter tuning.

Variational NDI (VaNDI)

We developed VaNDI to further improve NDI using Variational Networks (VN) (16) by combining deep learning and nonlinear data fidelity (Fig. 3). This acts as an unrolled gradient descent with learned regularizers, where the step sizes, nonlinearities and convolutional filters are estimated during the training stage. We used an L2 loss to minimize the difference between 1-direction VaNDI and 5-direction NDI reference, with 1200 epochs, 7x7x7 kernels, and batch size of 1. Data from 8 subjects were used for training, while the 9thsubject was reserved for testing. Fig4 compares NDI (55.2% RMSE) and VaNDI (49.5% RMSE), where VaNDI further mitigated streaking artifacts and better completed the k-space.

Ultra-high resolution NDI at 7T

Multi-orientation NDI was also evaluated at 0.5 mm³ isotropic resolution at 7T and Wave-CAIPI (17)(R=15x, 5:13min/orientation), where a 32-ch coil [18] was used for high-quality imaging. This, however, limits the achievable head rotations (0, 7, 13 degrees) which makes reconstruction difficult. NDI still provides high-quality reconstruction as in Fig. 5.

Conclusion

Acknowledgements

References

[1] H. Gudbjartsson and S. Patz, “The Rician distribution of noisy MRI data.,” Magn. Reson. Med., vol. 34, no. 6, pp. 910–4, Dec. 1995.

[2] T. Liu, C. Wisnieff, M. Lou, W. Chen, P. Spincemaille, and Y. Wang, “Nonlinear formulation of the magnetic field to source relationship for robust quantitative susceptibility mapping,” Magn. Reson. Med., vol. 69, no. 2, pp. 467–476, Feb. 2013.

[3] C. Milovic, B. Bilgic, B. Zhao, J. Acosta-Cabronero, and C. Tejos, “Fast nonlinear susceptibility inversion with variational regularization,” Magn. Reson. Med., vol. 80, no. 2, pp. 814–821, Aug. 2018.

[4] T. Goldstein and S. Osher, “The Split Bregman Method for L1-Regularized Problems,”SIAM J. Imaging Sci., vol. 2, no. 2, pp. 323–343, Jan. 2009.

[5] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Found. Trends® Mach. Learn., vol. 3, no. 1, pp. 1–122, 2010.

[6] T. Liu, P. Spincemaille, L. de Rochefort, B. Kressler, and Y. Wang, “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., vol. 61, no. 1, pp. 196–204, Jan. 2009.

[7] J. Yoon, E. Gong, I. Chatnuntawech, B. Bilgic, J. Lee, W. Jung, J. Ko, H. Jung, K. Setsompop, G. Zaharchuk, E. Y. Kim, J. Pauly, and J. Lee, “Quantitative susceptibility mapping using deep neural network: QSMnet,” Neuroimage, vol. 179, pp. 199–206, Oct. 2018.

[8] M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V. Jellus, J. Wang, B. Kiefer, and A. Haase, “Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA),” Magn. Reson. Med., vol. 47, no. 6, pp. 1202–1210, 2002.

[9] S. M. Smith, “Fast robust automated brain extraction,” Hum. Brain Mapp., vol. 17, no. 3, pp. 143–155, Nov. 2002.

[10] M. Jenkinson, P. R. Bannister, M. Brady, and S. M. Smith, “Improved Optimization for the Robust and Accurate Linear Registration and Motion Correction of Brain Images,”Neuroimage, vol. 17, no. 2, pp. 825–841, Oct. 2002.

[11] W. Li, B. Wu, and C. Liu, “Quantitative susceptibility mapping of human brain reflects spatial variation in tissue composition,” Neuroimage, vol. 55, no. 4, pp. 1645–1656, 2011.

[12] F. Schweser, K. Sommer, A. Deistung, and J. R. Reichenbach, “Quantitative susceptibility mapping for investigating subtle susceptibility variations in the human brain.,” Neuroimage, vol. 62, no. 3, pp. 2083–100, Sep. 2012.

[13] K. Shmueli, J. A. de Zwart, P. van Gelderen, T.-Q. Li, S. J. Dodd, and J. H. Duyn, “Magnetic susceptibility mapping of brain tissue in vivo using MRI phase data,”Magn. Reson. Med., vol. 62, no. 6, pp. 1510–1522, Dec. 2009.

[14] B. Bilgic, I. Chatnuntawech, A. P. Fan, K. Setsompop, S. F. Cauley, L. L. Wald, and E. Adalsteinsson, “Fast image reconstruction with L2-regularization.,” J. Magn. Reson. Imaging, vol. 00, pp. 1–11, Nov. 2013.

[15] C. Milovic, B. Bilgic, B. Zhao, J. Acosta-cabronero, and C. Tejos, “A Fast Algorithm for Nonlinear QSM Reconstruction,” 2017.

[16] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, and F. Knoll, “Learning a variational network for reconstruction of accelerated MRI data,” Magn. Reson. Med., vol. 79, no. 6, pp. 3055–3071, 2018.

[17] B. Bilgic, B. A. Gagoski, S. F. Cauley, A. P. Fan, J. R. Polimeni, P. E. Grant, L. L. Wald, and K. Setsompop, “Wave-CAIPI for highly accelerated 3D imaging,” Magn. Reson. Med., vol. 73, no. 6, pp. 2152–2162, 2015.

[18] B. Keil, C. Triantafyllou, M. Hamm, and L. L. Wald, “Design Optimization of a 32-Channel Head Coil at 7T,” Proc Intl Soc Mag Reson Med, vol. 18, no. 6. p. 1493, 2010.