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Novel insights on SSA-FARY: Amplitude-based respiratory binning in self-gated cardiac MRI
Sebastian Rosenzweig1,2 and Martin Uecker1,2
1Diagnostic and Interventional Radiology, University Medical Center Göttingen, Göttingen, Germany, 2Partner Site Göttingen, German Centre for Cardiovascular Research (DZHK), Göttingen, Germany

### Synopsis

Cardiac MRI is challenging because of respiratory and cardiac motion. Current clinical approaches try to bypass motion-related issues by ECG-triggering and breath-holds, which comes with several drawbacks. Alternatively, self-gating techniques can be used to determine respiratory and cardiac motion from the acquired raw-data itself. We present novel insights on the quadrature-pair self-gating signals estimated by SSA-FARY: We show that one element of each pair is certain to be in-phase with the motion it represents, as it is the result of a filtering process with a zero-phase filter. This enables the use of less respiratory bins, which decreases the computational demand.

### Introduction and Purpose

In cardiac MRI we have to account for cardiac and respiratory motion, which is usually done by ECG-triggering and breath-holds [1,2]. However, ECG-signals are error-prone when using fast-switching gradients [3] and patients might be incapable of performing breath-holds. A promising alternative is self-gated MRI, which dispenses with external devices or patient compliance by determining the respiratory and cardiac motion from the acquired data itself [4,5,6]. Recently, the SSA-FARY technique was proposed, which extracts cardiac and respiratory motion from the DC component of k-space and represents each quasi-periodic motion by quadrature pairs [7]. Here, we provide novel insights on the characteristics of these quadrature pairs: We show that one signal of each pair is generated by a zero-phase filter, for which it is in-phase with the actual physical motion and can therefore be used for amplitude-based respiratory binning. This kind of binning does not distinguish between inspiration and expiration - which in general is not ideal [7,8] - but can help to reduce the computational demand by allowing for a reduced number of bins in respiratory gating.

### Theory

SSA-FARY is based on Singular Spectrum Analysis [9] and combines three basic and easy-to-implement mathematical operations to a powerful machine-learning technique for the extraction of quasi-periodic motion and trend contained in an auto-calibration (AC-)region. In radial imaging the center of k-space of each spoke can be used as AC-region. Furthermore, the real- and imaginary part of all channels, i.e. $N_\text{c}$ coils and $N_\text{p}$ partitions, are stacked to construct an AC-region $\boldsymbol{X}$ of size[$(2\times N_\text{c} \times N_\text{p})\times N_\text{t}$], with $N_\text{t}$ the number of time-steps [7]. Note, however, that SSA-FARY is not limited to radial imaging and any appropriate AC-region can be used.

The three operations involved in SSA-FARY are (FIG_1A):

1. Zero-padding $\mathcal{Z}$
The AC-region $\boldsymbol{X}$ is symmetrically padded by $W - 1$ zeros to obtain
$\tilde{\boldsymbol{X}} = \mathcal{Z}\boldsymbol{X}$.
The odd window-size $W$ is chosen to capture ~3 seconds, which is the approximate time of a typical breathing cycle.

2. Hankelization $\mathcal{H}$
For each channel, a window of size [$1\times W$] is slided through the temporal domain and a Block-Hankel matrix $\boldsymbol{A}$ is constructed.
$\boldsymbol{A}=\mathcal{H}\tilde{\boldsymbol{X}}$.

3. Singular value decomposition SVD
The Block-Hankel matrix is decomposed using an SVD.
$\boldsymbol{A}=USV^H$

In [7,10] it was shown that the singular-vectors in $V^H$ contain $W$-sized data-adaptive filters for each channel, which act upon the zero-padded AC-region $\tilde{\boldsymbol{X}}$ to obtain the columns of $U$, also called Empirical Orthogonal Functions (EOF).

$U_t^k=\frac{1}{\lambda_k}\sum_{c=1}^{2 \cdot N_\text{c} \cdot N_\text{p}}\sum_{j=1}^{W}\tilde{\boldsymbol{X}}_c^{t+j}V_{cj}^k$

Like this, each oscillatory signal contained in the AC-region yields a pair of EOFs, which represent the desired gating signal. These EOFs are in quadrature and thus can be thought of as generalized sine-cosine pairs. Harris and Yuan [10] showed for the single-channel case, that for sufficiently large time-series, i.e. $N_\text{t}>>W$, one EOF of each quadrature pair is generated by a symmetric filter and one by a skew-symmetric filter, respectively (FIG_1B). In our experience, this also holds true for the multi-channel case. The consequence of this finding is remarkable:

The EOF generated by the symmetric (zero-phase) filter will always be in sync with the actual motion, as a zero-phase filter does not induce any phase-shift.

This property of the quadrature pairs allows for two different ways of sorting the acquired data according to their respiratory and cardiac phase:

1. Amplitude-binning (FIG_1B)
Amplitude-binning sorts the data acoording to the amplitude of the respective gating signal. For SSA-FARY, the in-phase EOF must be utilized, which can be detected by analysing the symmetry of the corresponding filters. Note, that amplitude-binning can be used for respiratory motion [11,12] but is not applicable for cardiac motion, as contraction and relaxation of the heart must be distinguished.

2. Phase-binning (FIG_1C)
Phase-binning resolves the entire cycle of a quasi-periodic motion by sorting the data according to the angle defined by the quadrature pair and can be used for both respiratory and cardiac gating [7].

### Methods

We revisit the self-gating and reconstruction of the 1min radial 3-slice SMS-FLASH acquisition (TE/TR=1.79/2.90ms) described in [7]. Contrary to the publication, which utilizes SSA-FARY phase-binnig for both respiratory (9 bins) and cardiac motion (25 bins), we here use SSA-FARY amplitude-binning to sort the respiratory motion into 5 bins only. Multi-dimensional image reconstruction using BART [13] is performed as described in [7]. RING is used for gradient-delay correction [14].

### Results

The background of FIG_2A shows the diaphragm motion extracted from a real-time reconstruction [15,16]. On top, the respiratory EOF-pair is plotted. For each EOF, FIG_2B shows 4 representative filters. Note, that the symmetric filters (green) correspond to the in-phase EOF in FIG_2A, which therefore was used for amplitude-binning.

FIG_3 shows the cardiac cycle for all three slices resolved into 5 amplitude-binned respiratory states. The different depths of breathing can be appreciated considering the white reference lines.

### Discussion and Conclusion

SSA-FARY can be used for robust respiratory and cardiac self-gating. To reduce the number of required bins for respiratory motion, amplitude-binning can be used by selecting the appropriate in-phase EOF of the quadrature pair. This approach was demonstrated on a radial SMS-FLASH acquisition and futhermore successfully tested in multiple other experiments (not shown).

### Acknowledgements

This work was supported by the German Centre for Cardiovascular Research (DZHK) and funded by the German Research Foundation (DFG) under Grant UE 189/1-1 and in part by NIH under grant U24EB029240.

### References

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[2] Y. L. Liu et al. "A monitoring, feedback, and triggering system for reproducible breath-hold MR imaging" Magn. Reson. Med. 1993; 30:507–511.

[3] R. Rokey et al., "Monitoring of acutely III patients during nuclear magnetic resonance imaging: Use of a time-varying filter electrocardiographic gating device to reduce gradient artifacts", Magn. Reson.Med., vol. 6, no. 2, pp. 240-245, Feb. 1988.

[4] A. C. Larson, R. D. White, G. Laub, E. R. McVeigh, D. Li and O. P. Simonetti, "Self-gated cardiac cine MRI", Magn. Reson. Med., vol. 51, no.1, pp. 93-102, 2004.

[5] A. C. Larson et al., "Preliminary investigation of respiratory self-gating for free-breathing segmented cine MRI", Magn. Reson. Med., vol. 53, no. 1, pp. 159-168, Jan. 2005.

[6] S. Uribe et al., "Whole-heart cine MRI using real-time respiratory self-gating", Magn. Reson. Med., vol. 57, no. 3, pp. 606-613, Mar. 2007.

[7] S. Rosenzweig, et al. "Cardiac and Respiratory Self-Gating in Radial MRI Using an Adapted Singular Spectrum Analysis (SSA-FARY)", IEEE Trans. Med. Imag., vol. 39, no. 10, pp. 3029-3041, Oct. 2020.

[8] K. Nehrke et al., “Free-breathing cardiac mr imaging: study of implications of respiratory motion—initial results,” Radiology, vol. 220, no. 3, pp. 810–815, 2001.

[9] R. Vautard and M. Ghil, “Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series.” Phys. D, Nonlinear Phenomena, vol. 35, no. 3,pp. 395–424, 1989.

[10] T. Harris and H. Yuan, “Filtering and frequency interpretations of singular spectrum analysis”, Physica D, vol. 239, no. 20-22, pp. 1958–1967, 2010.

[11] L. Feng et al., “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.

[12] J. Paul et al., “High-resolution respiratory self-gated golden angle cardiac MRI: comparison of self-gating methods in combination with k-t SPARSE SENSE”, Magn. Reson. Med., vol. 73, no. 1, pp. 292–298, 2015.

[13] M. Uecker et al. "Berkeley advanced reconstruction toolbox", In Proc. Intl. Soc. Mag. Reson. Med., vol. 23, p. 2486, 2015.

[14] S. Rosenzweig et al.“Simple auto-calibrated gradient delay estimation from few spokes using radial intersections (RING)”, Magn. Reson. Med., vol. 81, no. 3, pp. 1898–1906, 2019.

[15] M. Uecker et al. “Nonlinear inverse reconstruction for real-time MRI of the human heart using undersampled radial FLASH”, Magn. Reson. Med., vol. 63, no. 6, p. 1456–1462, 2010.

[16] S. Rosenzweig et al. “Simultaneous Multi-Slice Real-Time Imaging with Radial Multi-Band FLASH and Nonlinear Inverse Reconstruction”, in Proc. Intl. Soc. Mag. Reson.Med., vol. 24, p. 0518, 2017.

### Figures

FIG_1 SSA-FARY self-gating. A) Zero-padding, Hankelization and SVD. Each oscillation contained in the AC-region is reflected by an EOF-pair in the columns of $U$. B) The EOF which is in-phase with the actual physical motion can be used for amplitude-binning of respiratory motion. This EOF is determined by considering the symmetry of the corresponding filters given by $V^H$: The symmetric filters (green) correspond to the in-phase EOF. C) For cardiac motion amplitude-binning must not be used, so the data is binned according to the phase defined by the quadrature pair [7].

FIG_2 Respiratory gating with SSA-FARY. A) The background shows the diaphragm-motion extracted from a real-time reconstruction of the entire time-series. On top, EOF 3 and EOF 4, i.e. the third and fourth column of the SVD-Matrix $U$, are plotted. B) For each EOF, four representative filters extracted from the$V^H$-matrix are depicted. Note, that the symmetric filters (green) correspond to EOF 4, which is in-phase with the diaphragm and which can therefore be used for amplitude-binning.

FIG_3 SSA-FARY-gated compressed sensing recontruction of the heart. All three SMS-slices are depicted. The respiratory motion is resolved into 5 bins using SSA-FARY amplitud-binning. The white lines serve as reference to appreciate the different respiratory states. Bin 1: End-expiration, Bin 5: End-inspiration. The slightly blurred appearance of the fifth bin is a result of the limited number of spokes which were aquired in the typically shorter end-inspiration state.

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)
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