Synopsis

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.

Introduction

Respiratory motion resolved 4D flow magnetic resonance imaging¹ promises increased scan efficiency and valuable physiological information in various congenital diseases such as the Fontan circulation².

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 MRI⁶.

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.

Methods

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 domain⁸, 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 mm³ and 16 cardiac phases. The measurement was performed with a 28 channel receiver coil which was compressed to 5 virtual channels⁹.

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 SENSE¹⁰ 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 ESPIRiT¹¹. 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 Toolbox¹². 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 to¹³ and eddy currents were corrected for with a linear model fitted to stationary tissue¹⁴.

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.

Results

Discussion

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 state¹⁵. To this end, the acquisition framework has to be extended to include a multi-venc encoding scheme¹⁶, in order to provide sufficient velocity-to-noise ratio for low velocities.

Acknowledgements

References

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