Jonas Walheim^{1} and Sebastian Kozerke^{1}

We present a low-rank + sparse reconstruction method which resolves respiratory motion in 4D flow magnetic resonance imaging as a low-rank signal component. Respiratory motion resolved 4D flow MRI data is reconstructed and compared to the total variation based XD-GRASP method and a standard parallel imaging acquisition protocol. Good agreement of the reconstructed results with the reference shows that a low-rank model is effective in resolving respiratory motion in 4D flow magnetic resonance imaging.

Respiratory motion resolved 4D flow
magnetic resonance imaging^{1} promises increased scan efficiency and valuable
physiological information in various congenital diseases such as the Fontan
circulation^{2}.

Respiratory motion-resolved MR
imaging frameworks^{3,4} acquire data continuously throughout the respiratory cycle with
subsequent binning of the data into discrete respiratory motion states. In
reconstruction, redundancies between different motion states are exploited to
improve reconstruction accuracy. Typically, correlation among neighboring respiratory
motion states is exploited by penalizing total variation (TV) along the
respiratory motion dimension in an iterative image reconstruction approach^{4,5}. However, penalizing finite differences on a relatively
coarse resolution can lead to underestimation of velocities in 4D flow MRI^{6}.

The objective of the present study is
to propose 5D Flow MRI based on resolving respiratory motion as a low-rank
component in a low-rank + sparse (L+S)^{7} reconstruction and compare it to XD-GRASP and a standard
respiratory-gated parallel imaging protocol.

Using the insight that 4D
flow data can be well represented as the sum of a low-rank component and a sparse
component in the Hadamard transform domain^{8}, the
approach is extended here to include the respiratory motion dimension. The
corresponding minimization problem reads as follows:

$$\underset{L,S}{\operatorname{argmin}} ||\Omega\mathcal{F} \mathcal{S}(L+S)-y||_2^2 +\lambda_L||L||_*+\lambda_S||\Psi S||_1, \> s.t. X=L+S $$Where $$$\Omega,\mathcal{F},\mathcal{S}$$$ denote the undersampling operator, Fourier transform and sensitivity maps, respectively. $$$X\in \mathbf{C}^{N_x \times N_{hp} \times N_{ms} \times N_{k_v} }$$$ denotes the image tensor with spatial locations, heart phases, respiratory motion states and velocity encodings. $$$L$$$ and $$$S$$$ refer to the low-rank and sparse component, respectively. For the low-rank regularization, $$$L$$$ is reordered as a Casorati matrix $$$L\in \mathbf{C}^{N_x \times N_{hp} N_{ms} N_{k_v} }$$$. $$$\Psi$$$ is the Hadamard matrix [+1 +1 +1 +1;+1 +1 -1 -1; +1 -1 +1 -1;+1 -1 -1 +1], i.e. the differences between the different velocity encodings are used as sparse representation. An illustration of the L+S decomposition with Hadamard sparsity is provided in Figure 1.

*Data acquisition
*

4D Flow data in the aortic
arch of a healthy volunteer was acquired on a 3T Philips Ingenia system
(Philips Healthcare, Best, the Netherlands) using a Cartesian four-point
phase-contrast gradient-echo sequence with uniform venc of 220 cm/s, a spatial
resolution of 2.5x2.5x2.5 mm^{3} and 16 cardiac phases. The measurement was performed with a 28 channel
receiver coil which was compressed to 5 virtual channels^{9}.

The acquisition process is
illustrated in Figure 2. The measured data were binned into 4 different respiratory
motion states with equal acceleration factors of approximately 7 per bin. For
comparison, a standard 4D Flow MRI protocol using SENSE^{10} with an acceleration
factor of 2 and using a 5 mm navigator window was obtained.

*Data reconstruction *

Coil
sensitivity maps were calculated from a separate calibration scan using ESPIRiT^{11}. The
SENSE scan was reconstructed with Tikhonov regularization and using the same sensitivity maps as used in the CS reconstructions.
XD-GRASP and
SENSE reconstructions were performed using the Berkeley Advanced image Reconstruction
Toolbox^{12}. L+S reconstruction was
implemented in Python 2.7.12. The regularization parameters were automatically determined
based on the minimum deviation of velocities in the ascending aorta relative to
the standard SENSE reconstruction. Concomitant field correction was applied to
the signal phase according to^{13} and eddy currents were
corrected for with a linear model fitted to stationary tissue^{14}.

*Data analysis*

Results were compared using GTFlow (Gyrotools LLC, Zurich, Switzerland). Velocities were calculated for a plane in the ascending aorta. Velocity magnitude images and streamline visualizations were assessed qualitatively.

A framework for 5D Flow MRI was implemented with L+S and XD-GRASP and its feasibility tested in-vivo. Both methods show good agreement in terms of velocity profiles and maps with the standard 4D Flow MRI protocol, however permit 3.5x faster acquisition and resolve the respiratory dimension in addition. Further in-vivo experiments are warranted to test for significant differences between L+S and XD-GRASP.

In
this work, similar flow fields were reconstructed in inspiration and expiration
which can be expected for aortic flow. Future investigations will be aimed at
studying venous flow and its dependency on the respiratory state^{15}. To this end, the
acquisition framework has to be extended to include a multi-venc encoding
scheme^{16}, in order to provide
sufficient velocity-to-noise ratio for low velocities.

[1] J. Y. Cheng, T. Zhang, M. T. Alley, M. Uecker, M. Lustig, J. M. Pauly, and S. S. Vasanawala, “Comprehensive multi-dimensional MRI for the simultaneous assessment of cardiopulmonary anatomy and physiology,” Sci. Rep., vol. 7, no. 1, p. 5330, 2017.

[2] M. Gewillig, “The Fontan circulation,” Heart, vol. 91, no. 6, pp. 839–846, 2005.

[3] A. Sigfridsson, L. Wigström, J. P. E. Kvitting, and H. Knutsson, “k-t2 BLAST: Exploiting spatiotemporal structure in simultaneously cardiac and respiratory time-resolved volumetric imaging,” Magn. Reson. Med., vol. 58, no. 5, pp. 922–930, 2007.

[4] L. Feng, L. Axel, H. Chandarana, K. T. Block, D. K. Sodickson, and R. Otazo, “XD-GRASP: Golden-angle radial MRI with reconstruction of extra motion-state dimensions using compressed sensing,” Magn. Reson. Med., vol. 75, no. 2, pp. 775–788, 2016.

[5] S. S. Cheng, Joseph Y., Zhang, Tao, Alley, Marcus T., Lustig, Michael, Pauly, John M., Vasanawala, “Ultra-High-Dimensional Flow Imaging (N-D Flow),” in ISMRM 2016, 2016.

[6] C. Santelli, M. Loecher, J. Busch, O. Wieben, T. Schaeffter, and S. Kozerke, “Accelerating 4D Flow MRI by Exploiting Vector Field Divergence Regularization.”

[7] D. K. Sodickson, R. Otazo, and E. Candes, “Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,” vol. 1136, pp. 1125–1136, 2015.

[8] G. Valvano, N. Martini, A. Huber, C. Santelli, C. Binter, D. Chiappino, L. Landini, and S. Kozerke, “Accelerating 4D flow MRI by exploiting low-rank matrix structure and hadamard sparsity,” Magnetic Resonance in Medicine, no. September, pp. 1–12, 2016.

[9] T. Zhang, J. M. Pauly, S. S. Vasanawala, and M. Lustig, “Coil compression for accelerated imaging with Cartesian sampling,” Magn. Reson. Med., vol. 69, no. 2, pp. 571–582, 2013.

[10] K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, “SENSE: Sensitivity encoding for fast MRI,” Magn. Reson. Med., vol. 42, no. 5, pp. 952–962, 1999.

[11] M. Uecker, P. Lai, M. J. Murphy, P. Virtue, M. Elad, J. M. Pauly, S. S. Vasanawala, and M. Lustig, “ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA,” Magn. Reson. Med., vol. 71, no. 3, pp. 990–1001, 2014. [12] F. Ong, M. Uecker, U. Tariq, A. Hsiao, M. T. Alley, S. S. Vasanawala, and M. Lustig, “Berkeley advanced reconstruction toolbox,” Magn. Reson. Med., vol. 73, no. 2, pp. 828–842, 2015.

[13] M. A. Bernstein, X. J. Zhou, J. A. Polzin, K. F. King, A. Ganin, N. J. Pelc, and G. H. Glover, “Concomitant gradient terms in phase contrast MR: Analysis and correction,” Magn. Reson. Med., vol. 39, no. 2, pp. 300–308, 1998.

[14] P. G. Walker, G. B. Cranney, M. B. Scheidegger, G. Waseleski, G. M. Pohost, and A. P. Yoganathan, “Semiautomated method for noise reduction and background phase error correction in MR phase velocity data,” J. Magn. Reson. Imaging, vol. 3, no. 3, pp. 521–530, 1993.

[15] F. K. Nakhjavan, W. H. Palmer, and M. Mcgregor, “Influence of respiration on venous return in pulmonary emphysema,” Circulation, vol. 33, no. 1, pp. 8–16, 1966.

[16] C. Binter, V. Knobloch, R. Manka, A. Sigfridsson, and S. Kozerke, “Bayesian multipoint velocity encoding for concurrent flow and turbulence mapping,” Magn. Reson. Med., vol. 69, no. 5, pp. 1337–1345, 2013.

[17] J. Y. Cheng, K. Hanneman, T. Zhang, M. T. Alley, P. Lai, J. I. Tamir, M. Uecker, J. M. Pauly, M. Lustig, and S. S. Vasanawala, “Comprehensive motion-compensated highly accelerated 4D flow MRI with ferumoxytol enhancement for pediatric congenital heart disease,” J. Magn. Reson. Imaging, vol. 43, no. 6, pp. 1355–1368, 2016.

[18] S. Wundrak, J. Paul, J. Ulrici, E. Hell, and V. Rasche, “A small surrogate for the golden angle in time-resolved radial MRI based on generalized fibonacci sequences,” IEEE Trans. Med. Imaging, vol. 34, no. 6, pp. 1262–1269, 2015. [19] T. Zhang, J. Y. Cheng, Y. Chen, D. G. Nishimura, J. M. Pauly, and S. S. Vasanawala, “Robust self-navigated body MRI using dense coil arrays,” Magn. Reson. Med., vol. 76, no. 1, pp. 197–205, 2016.

Figure 1: Illustration of the low-rank+sparse decomposition of 4D flow data.
Signals are separated into a low-rank component with little variation over the
different signal dynamics, and a sparse component which captures the rapid
variation in the aorta.

Figure 2: Acquisition scheme. Data are
continuously acquired with a Cartesian golden angle scheme^{17,18}. Respiratory
motion is derived from the k0 profile, which is repeatedly acquired.
The data are then retrospectively assigned to different motion states. The
respiration curve of the undersampled scans was extracted from the profiles
through the k-space center, using a PCA based approach combined with coil
clustering^{19}. The resulting sampling
patterns fulfill the incoherency criterion required for a CS reconstruction.

Figure 3: Reconstruction results in
expiration and inspiration for XD-GRASP and L+S compared to a SENSE scan in
expiration. In expiration, both methods show good agreement with the SENSE
scan. In inspiration, both methods show similar results, but no reference
measurement was available.

Figure 4:
Comparison of flow profiles in the ascending aorta (AA) and velocity components
in systole in right-left(RL), anterior-posterior(AP) and foot-head(FH)
directions. All methods show good agreement with the SENSE scan. XD-GRASP shows
a tendency to produce increased artifacts as seen for velocities encoded in RL
direction (marked by arrows).

Figure 5: Qualitative comparison of streamline
visualizations during systole in expiration and inspiration for L+S, XD-GRASP
relative to standard SENSE.