Sebastian Rosenzweig^{1,2} and Martin Uecker^{1,2}

^{1}Diagnostic and Interventional Radiology, University Medical Center Göttingen, Göttingen, Germany, ^{2}Partner Site Göttingen, German Centre for Cardiovascular Research (DZHK), Göttingen, Germany

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.

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

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.

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.

$$$\boldsymbol{A}=\mathcal{H}\tilde{\boldsymbol{X}}$$$.

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:

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

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.

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].

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.

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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.