Self‐gated cine phase‐contrast balanced SSFP flow quantification at 0.55 T

To implement cine phase‐contrast balanced SSFP (PC‐bSSFP) for low‐field 0.55T cardiac MRI by exploiting the intrinsic flow sensitivity of the bSSFP slice‐select gradient and the in‐plane phase‐cancelation properties of radial trajectories, enabling self‐gated and referenceless PC‐bSSFP flow quantification at 0.55 T.


INTRODUCTION
2][3] While cine sequences aim to acquire functional information about the heart, phase-contrast (PC) MRI sequences [4][5][6] are used to quantify blood velocities in the chambers and great vessels.Standard PC techniques are based on spoiled gradient-echo (GRE) techniques. 6,7However, in cine 2D PC GRE, the SNR can vary significantly during the cardiac cycle as a result of the inflow effect, which may compromise visualization and quantification of slow flows.
To exploit the high SNR efficiency of bSSFP and its relative insensitivity to the inflow effect, the incorporation of flow encoding into bSSFP, termed PC-bSSFP, has been accomplished in a variety of ways.Overall et al. 8 achieved this through the addition of bipolar gradients after the readout rewinder, incorporating velocity sensitivity directly into the steady state.The addition of gradient elements, however, results in increased TR, and hence more pronounced bSSFP banding artifacts. 9By exploiting the motion sensitivity of the bSSFP slice-select gradient, an intrinsic method to encode through-plane flow was introduced by Markl et al., 10 achieving PC through inversion of the slice-select gradient in two successive experiments.This method was applied to flow imaging of CSF 11 and was expanded to in-plane flow encoding by Grinstead et al., 12 proposing to invert the Cartesian readout gradient to achieve PC along the readout direction.Depending on the desired velocity-encoding range, both methods do not impose an additional TR penalty and hence leave the bSSFP banding largely unchanged.
Up to this point, all the methods followed a traditional 2-point PC phase difference-based method involving the acquisition of two sets of images with different velocity encodings, doubling scan time relative to standard cine bSSFP imaging.To reduce scan time, Pai et al. 13 and Nielsen et al. 14 proposed a multi-echo method, combining both velocity encodings into a single yet increased TR.Another approach by Nielsen et al. 15 acquired a single PC-bSSFP image by correcting for background phase by higher-order fitting of phase of static tissue.The requirement for static background tissue means this method may be less robust in cardiac imaging, due to cardiac motion and the proximity to the lungs.The work is also closely related to prior work by Man et al., 16 who took advantage of the insensitivity of spiral readouts to in-plane flow and performed background fitting to correct the phase of a single image.In the following, we will use the term "referenceless" to refer to single-point phase velocity measurement.
Although bSSFP, and by extension PC-bSSFP, techniques are feasible at typical clinical field strengths, insufficient shimming and specific absorption rate limits can compromise their performance.5][26][27][28] At lower field strengths, relaxation times are more favorable with shorter T 1 times and reduced RF power absorption, allowing for increased flip angles.These factors, in addition to reduced field inhomogeneity, allow one to offset, in parts, the intrinsic SNR loss associated with reducing field strength.21]23 Altogether, the adaption of PC-bSSFP to lower fields is of significant interest and is the motivating factor of this work.
In this work, we synergistically combine the through-plane method of Markl et al. 10 with a tiny golden angle (TGA) radial 29 acquisition and incorporate cardiac/respiratory self-gating 30 to enable continuous radial scanning on a low-cost clinical 0.55T system.For reference, we compare these results with a PC-GRE variant [31][32][33][34] as well as those obtained with the same sequences implemented at 1.5 T. Finally, we investigate the prospect of a referenceless single-point measurement leveraging the in-plane flow-related phase cancelation of radial imaging.

THEORY
The first gradient moment created by the slice-select gradient G s is given by where t = 0 is the center of the RF pulse; t enc is the start of the slice-select rewinder gradient; G s is the gradient along the slice direction; and T E is the echo time, which in the case of bSSFP is often set to half the TR.As the slice-select gradients are inverted in 2-point PC measurement, their first moments between the acquisitions only differ in sign, and the resulting velocity-encoding strength V enc is given by where  is the gyromagnetic ratio.
Because velocity sensitivity is tied to the slice-select gradient, the achievable range of encoding strengths is dependent on the RF pulse properties and slice thickness.Specifically, the maximum achievable V enc is restricted, with decreasing time-bandwidth product or increasing slice thickness, resulting in increased maximum V enc .
Similarly, V enc can be decreased by extending the duration of the slice-select and corresponding rewinder gradients without altering the RF pulse duration, only resulting in a small increase in TR.
In the case of radial sampling with sufficiently distributed projection angles, the background phase is, on average, no longer dependent on in-plane flow.The phase accumulated due to gradients is defined as Taylor expanded the position ⃑ X(t) about an encoding time point t ′ , yielding The resulting phase is then given by By the definition of bSSFP, it follows that and for the sequences used in this work also This ensures no inter-TR cumulative phase caused by velocity or position.Similarly, by the definition of TE, one obtains Substituting Equation (8) into Equation ( 5) for encoding occurring at TE and ignoring higher-order terms results in where only the first-order term remains, as it is not compensated at TE. Expanding the dot product finally yields Defining a readout gradient G R , which is rotated by angle  for each spoke, and a slice-select gradient G s and substituting into Equation (10), yields where t ′ should be chosen to minimize contributions of higher-order terms by minimizing higher-order moments. 35This results in differing encoding times for in-plane (x, y) and through-slice (z) motion.In the case of the sequence used here, the in-plane encoding point is approximately the zero point between the readout pre-winder and readout gradient, denoted as t ∥ .The through-slice encoding point is approximately the zero point between the slice-select gradient and its subsequent rewinder, denoted as t ⊥ .Equation ( 12) can be simplified to Because TE corresponds to repetitive sampling of the k-space center, the average over all spokes is calculated as follows: Assuming a continuous approximation in the limit of N TGA → ∞, one obtains This relates directly to Equation (1), with t ⊥ = t enc .It is important to note that this only holds if there are both sufficient spokes that are sufficiently uniformly distributed on [0, 2].
Evaluating the relevant terms of Equation ( 14) on simulated TGA spoke distributions (∼120 spokes per image) gives corresponding to a residual sensitivity of less than 5% of the single spoke V enc .For the radial trajectories used in this work, this corresponds to an equivalent in-plane V enc of over 2000 cm/s.Furthermore, this is the residual sensitivity for a single cardiac bin, and the residual encoding direction for any given bin can be assumed to be random.As such, it is expected that any regularization in time during reconstruction will further reduce this sensitivity.Finally, this derivation only shows flow insensitivity for the central k-space point; however, it can also be shown that, when radial spokes are sufficiently paired, the point spread function caused by in-plane flow is negligible (Appendix A).

METHODS
A free-running radial PC-bSSFP scheme with TGA increments was implemented on a 0.55T system (MAGNETOM Free.Max; Siemens Healthineers, Erlangen, Germany) with low-performance gradients (26 mT/m amplitude, 45 T/m/s slew rate) and a 6-channel body (3 anterior, 3 posterior) array receive coil.The same sequence was implemented on a 1.5T system (Achieva; Philips Healthcare, Best, The Netherlands) and a 5-channel receive coil.
Imaging parameters are provided in Table 1.Sequence parameters were duplicated between field strengths while obeying system limits, with the exception of TR, which was shorter at 1.5 T due to increased gradient performance.A PC-GRE variant was acquired for comparison.This was achieved by inverting the readout rewinder and slice pre-winder of the PC-bSSFP sequence to increase the gradient crusher moments 32,33 in combination with a random RF phase 31 (Figure 1A).Half-way through the PC-bSSFP and PC-GRE scans, the slice-select gradients and corresponding pre-and re-winders were inverted along with inverting the sign of the frequency offset of the RF pulses (Figure 1A).No additional dummy shots or catalysts were added.The TGA increment was calculated, 29 with the seventh TGA being chosen, corresponding to an angle of 23.63 (Figure 1B).
Five healthy subjects were scanned upon written informed consent following institutional and ethics guidelines.Three scans were performed at 0.55 T on each volunteer, including breath-hold PC-bSSFP, breath-hold Sequence parameters for both 0.55 T and 1.5 T. The large difference in TR seen between field strengths is due to differences in gradient system performance.The lower table is for 0.55 T only, comparing trajectory information for free-breathing and breath-hold.PC-GRE, and free-breathing PC-bSSFP.Breath-hold PC-bSSFP was repeated at 1.5 T. Transverse slices were positioned perpendicular to the ascending aorta to measure through-plane flow.Breath-hold scans had a duration of 27 s, whereas free-breathing had a duration of 3 min.

Parameter
Retrospective cardiac binning was performed using the central k-space point as a self-gating signal (Figure 2A,B).An electrocardiogram (ECG) was recorded during scans at 1.5 T (no ECG was available on the 0.55T system) for verification of the self-gated binning.Respiratory binning along an additional dimension was performed in the same way for free-breathing scans.Cardiac and respiratory bin counts were 25 and 5, respectively, resulting in approximately equivalent spoke counts per bin for breath-hold and free-breathing scans.Before reconstruction, corrections for gradient/acquisition delay were applied by first determining shift of the maximum signal magnitude for each spoke from the k-space center.After sorting by spoke angle, a sinusoid can be fit to these shifts, 36,37 which restricts the resulting correction to a single shift for each direction (Gx, Gy).This fit is then used to correct the trajectories of all spokes.As nonuniform reconstruction is already required, this circumvents the need for any adjustments to the sampled signal itself.
Reconstruction was performed with BART 38 using the pics (parallel-imaging compressed sensing) function.Before regularized reconstruction, coil sensitivities were estimated based on the adjoint NUFFT reconstruction of the full time-averaged data set, using a direct coil (A) Sequence diagram for one TR of a radial phase-contrast balanced SSFP sequence.Gradient areas and durations are for illustration only and are not necessarily representative of actual gradient areas.Gradient-related changes for phase-contrast spoiled gradient echo are shown in green.The in-plane, "rotating" crusher moment is increased from ∼ to 2, while the crusher along the slice-select direction is based on the inversion of the slice-select prephaser.A random RF spoiling increment was also added to improve spoiling.The portion of the slice-select gradient responsible for velocity encoding is given in red, and purple for the standard (gradient moment M 1,A ) and inverted gradients (gradient moment M 1,B ), respectively.The extension of the slice-select gradient to decrease V enc is shown as dotted lines.calibration method provided by BART (caldir 39 ).All images had phase correction applied based on a linear fit of the FOV, excluding low signal regions in the lungs and background to improve visualization of higher-order phase effects.
Because the regularization method may affect magnitude and phase reconstruction differently, four regularization methods were compared: spatial L2 (L2 x ), total variation in time (TV t ), total generalized variation in time (TGV t ) and locally low rank (LLR t ).Additionally, the effect of applying a moving median filter in time 33,40 with a window size of three bins was also investigated for all regularization methods.A nonregularized, parallel-imaging reconstruction was used as a reference.The performance of the regularization methods was evaluated on (a) velocity error at peak systole in the ascending and descending aorta (AAo and DAo, respectively), (b) mean SD of magnitude in time in a background region, and (c) degree of motion smoothing in the AAo.Regions of interest for all subsequent figures and metrics are shown in Figure 3.
The first half of the 0.55T breath-hold scans, Before slice-select inversion, were taken to evaluate referenceless single-point flow estimation, corresponding to a breath-hold duration of 13 s.Background phase correction was performed using the phase of the time-averaged (unbinned) parallel-imaging reconstruction (unregularized).Polynomials up to order 4 were fit to the whole torso, excluding regions with low signal or large velocity SD in time.This was applied in PC-bSSFP scans and compared with the 2-point phase difference results.Additional scans of a static phantom bottle were performed to verify background-phase fitting performance.

Self-gating accuracy
Self-gating results were compared with ECG data for scans at 1.5 T. Good agreement was found between the two methods (RMS error [RMSE] = 10.8 ms, ΔHR <0.5 bpm).
A delay between the ECG R peak and the self-gating peak was measured, with the self-gating peak occurring 150 ms after the R peak on average.

Choice of regularization
Metrics used to guide the choice of regularization type and strength are shown in Figure 4. Figure 4A,B shows underestimation of peak velocity as a function of regularization.
As regularization strength increases, velocity time curves are smoothed, resulting in underestimation of peak velocity in both the AAo and DAo.When a median filter in time is applied, peak velocities are reduced significantly, exceeding 10% error in all cases.
Figure 4C shows SD in a background region as a function of regularization.As expected, the SD in the background decreases with increasing regularization strength; however, at some point a lower limit is reached.The application of a median filter in this case further decreases SD.
In Figure 4D, spatiotemporal (x-t) plots for a slice through the AAo are shown.Little change can be seen for L2 x regularization.However, both TV t and TGV t show noticeable motion smoothing for  > 0.005, whereas (A,B) Peak velocity as a function of regularization method and strength for the ascending and descending aorta.Values are normalized to an unregularized parallel-imaging reconstruction.Dashed lines indicate the addition of a moving median filter in time after reconstruction.As regularization strength increases, peak velocity is reduced as velocity curves are smoothed out.Markers indicate three optimal choices of regularization for overall quality, magnitude, and velocity.(C) Plot of mean SD in time of the background region labeled in Figure 3. Colors, line styles, and markers correspond to plots (A) and (B).Values are normalized to an unregularized parallel-imaging reconstruction.Similar to (A) and (B), as regularization strength increases, SD decreases; however, in this case a lower limit is reached, which is dependent on regularization method and independent of median filtering.(D) X-t plots taken from the line indicated in Figure 3. Red markers indicate an example region where increasing regularization causes smoothing of features/motion.Red line indicates upper acceptable limit for motion smoothing based on visual analysis and segmented motion curves (Figure S1).LLR, locally low rank; TGV, total generalized variation; TV, total variation.motion smoothing with LLR t is noticeable for  > 0.001.Additional plots showing the effect of increasing regularization on AAo motion can be found in Figure S1.
As shown in Figure 4, the choice of regularization can strongly affect the accuracy of results.Of note, L2 x is not considered for accuracy of velocity estimation, as it does not perform sufficiently well in terms of noise and artifact reduction.For the reconstruction of velocity, LLR t performs the best in regions of flow, minimizing peak underestimation; however, it underperforms in reducing undersampling artifacts in static tissue and background when compared with TGV t or TV t .TGV t and TV t perform similarly to each other, producing smoother curves with some peak underestimation and a significant reduction in artifacts in the magnitude.
Application of a median filter in time further reduces background SD, as found in previous works. 33,40At high regularization values, median and non-median-filtered SD converges, indicating that regularization has effectively removed residual folding artifacts such that median filtering has little effect.However, this comes at the cost of peak velocity error, with significant smoothing of peak velocity.The extent of this smoothing is dependent on the number of cardiac bins used; in the case of 25 bins as used here, median filtering resulted in a minimum error of 10%, with TGV t and TV t having a baseline error of 15% in the AAo.
Spatiotemporal (x-t) plots (Figure 4D) show noticeable differences between regularization methods.Part of the pulmonary trunk (red arrows) shows significant smoothing with increased regularization.Visual evaluation of this structure can be used to determine a maximum allowable threshold for regularization for each method, indicated by the red line.This agrees with aortic motion curves extracted by segmentation, seen in Figure S1.In this metric, LLR t performs noticeably worse than TV t or TGV t , requiring a lower α to avoid motion smoothing.
Another interesting finding can be seen in the difference between the low regularization asymptotes for the AAo and DAo (Figure 4A,B).In the DAo, all regularizations lead to convergence to the same peak velocity (except for a small error for TV t ).However, for the AAo, L2 x converges to a value 2.5% higher than the other regularization methods investigated.Similar differences in convergence can be seen in the SD (Figure 4C), as well as in noticeable difference in x-t plots (Figure 4D, first column).This indicates some residual regularization processes for low , which may be specific to the regularization methods used.
Overall, TGV t performed the best, providing a balance between background phase stability, acceptable peak underestimation, and excellent magnitude reconstruction (orange marker, Figure 4).If purely magnitude images are of interest, combining a moving median filter with TGV t regularization provides the best background stability and removes any remaining flashing artifacts (yellow marker, Figure 4).Similarly, if the region of interest is well defined and has sufficient SNR, LLR gives the most accurate velocity reconstruction in regions of flow (red marker, Figure 4).For all subsequent results in this work, a combination of TGV t + median and TGV t was used for magnitude and phase, respectively.

Breath-hold PC-bSSFP
Images at both peak systole and diastole for PC-bSSFP (0.55 T, 1.5 T) and PC-GRE are shown in Figure 5, in which SNR is scaled to the 1.5T results, with 0.55T PC-bSSFP being multiplied by 2× and 0.55T PC-GRE by 3× to match color scales.Both 0.55T images show better uniformity, whereas 1.5T images have better SNR.
Overall, there is agreement of velocity and flow values between breath-hold PC-bSSFP and PC-GRE at 0.55 T (RMSE v = 5.8 cm/s, RMSE Q = 23 mL/s), with average velocity and flow overestimated by 1.8 cm/s and 7.5 mL/s by PC-bSSFP.Velocity at peak systole was overestimated by 6 cm/s by PC-bSSFP on average relative to PC-GRE.PC-bSSFP at 0.55 T and 1.5 T have more mismatch (RMSE v = 9.7 cm/s, RMSE Q = 24 mL/s), with 1.5 T overestimating average velocity and flow compared with 0.55 T (4 cm/s, 3 mL/s).Total flow between 0.55 T and 1.5 T showed good agreement in the AAo (90 vs. 87 mL) and DAo (48 vs. 45 mL).A correlation plot comparing PC-bSSFP with PC-GRE at 0.55 T is shown in Figure 6A, revealing good agreement as well as the previously mentioned velocity overestimation of PC-bSSFP at higher velocities.

Free-breathing PC-bSSFP
Figure 7 compares breath-hold and free-breathing PC-bSSFP.Images show peak systole and diastole, with excellent similarity, whereas curves show velocity, VNR, and SNR for both AAo and DAo.Again, there is excellent similarity in the inset regions when comparing breath-hold with free-breathing acquisitions.Free-breathing velocities in the AAo and DAo at 0.55 T showed good agreement with breath-hold (RMSE v = 5.7 cm/s, RMSE Q = 27 mL/s), with a small overall velocity error (−0.5 cm/s) and flow error (0.05 mL/s).Free-breathing SNR in the AAo was slightly reduced compared with breath-hold (76 ± 9 vs. 90 ± 11), with average VNR being similarly affected (23 vs. 28).A correlation plot based on these results is shown in Figure 6B.An additional comparison video is shown in Video S1.

F I G U R E 5
Images at peak systole and diastole for phase-contrast balanced SSFP (PC-bSSFP) at 0.55 T/1.5 T and phase-contrast spoiled gradient-echo (PC-GRE) at 0.55 T. Magnitude images are in units of SNR, with PC-bSSFP at 0.55 T scaled by 2× and PC-GRE scaled by 3×, to better match PC-bSSFP at 1.5 T. PC-bSSFP at 0.55 T has superior uniformity and lung vasculature detail; however, 1.5T PC-bSSFP is slightly sharper.Note that 1.5T PC-bSSFP results had some issues with coil sensitivity, causing the lack of uniformity.From the background in the velocity images, PC-GRE appears to have the least streaking, whereas 1.5T PC-bSSFP has the most.Overall, there is good agreement in flow structure and strength among the three scans.

Referenceless velocity estimation
Figure 8 shows the results of varying order polynomial fits on the background phase of a static phantom bottle.
As polynomial fit order is increased, the residual background phase in the center of the phantom is reduced; however, some phase remains at the edges of the bottle.Phase correction results using a third-order polynomial fit to the entire torso, excluding regions with low signal or large velocity SD in time, as well as curves for single-point flow estimation on a single subject are shown in Figure 9. Phase correction significantly reduces offsets within the heart, although it does not manage to remove background phase in the outer edges of the torso, with a similar phase pattern to the phantom bottle results in Figure 8. Velocity curves show excellent agreement, with Pixel-wise correlation plots extracted from the ascending and descending aorta.(A) Breath-hold phase-contrast balanced SSFP (PC-bSSFP) compared with breath-hold phase-contrast spoiled gradient-echo (PC-GRE) at 0.55 T. PC-bSSFP tends to slightly overestimate high velocities in the ascending aorta (positive large velocities) and underestimate low velocities when compared with PC-GRE.(B) breath-hold PC-bSSFP to free-breathing PC-bSSFP at 0.55 T. On average, the agreement is very good.Some small clusters off badly correlated pixels are present (red markers); however, these are from only 1 volunteer.

F I G U R E 7
Comparison of breath-hold (BH) to free-breathing (FB) phase-contrast balanced SSFP (PC-bSSFP) for systole (A) and diastole (B) with approximately equivalent spoke counts per bin.Insets in velocity images have additional masking to remove low signal (lung).Structurally, free-breathing results are nearly identical to breath-hold, with only minor smoothing in some flow regions.(C-E) Flow rate, velocity-to-noise ratio (VNR), and SNR in the ascending and descending aorta, corresponding to the regions of interest in Figure 2, with free-breathing shown as dashed lines.In general, there is excellent agreement between free-breathing and breath-hold, with the only significant difference being a small SNR decrease for free-breathing (E).

F I G U R E 8
(A) Uncorrected background phase results for a static phantom bottle at 0.55 T. Corrections are applied with polynomials of increasing order from zeroth (B) to fourth (F).Mean and SD for a 6-cm region at the center of the phantom (black circles) are also given, showing reduction in both mean and SD with polynomial order.At third order, the criteria of Gatehouse et al. 42 is satisfied.Residual background (BG) phase can still be seen at third order due to static field inhomogeneity.a significant reduction in velocity error compared with uncorrected results (uncorrected error = 14.2 ± 15 cm/s, corrected error = −1.8± 5.2 cm/s).
SNR was decreased by a factor of approximately √ 2 when compared with phase difference methods (SNR = 111), resulting in SNR = 76.1 ± 7.6.

DISCUSSION
Free-running cine PC-bSSFP has been successfully implemented, enabling referenceless flow quantification at 0.55 T during breath-hold and free-breathing.Good accuracy of PC-bSSFP velocity estimation along with overall improved SNR when compared with PC-GRE could be demonstrated.Free-breathing results showed excellent agreement with breath-hold results, with comparable velocity curves and image quality.Referenceless single-point imaging was shown to be feasible, producing good agreement in velocity relative to 2-point phase-difference methods.

Breath-hold PC-bSSFP
The observed overestimation of PC-bSSFP in comparison to PC-GRE is in agreement with previous findings 12 and can be seen from Figure 6A to come into play mostly for higher velocities.This may be caused by additional phase accumulation associated with acceleration (and higher-order motion) due to the pulsatile nature of the flow.As shown in Eqs. ( 6) and ( 7), this sequence is zeroth-moment and first-moment compensated over TR; however, higher-order moments may be nonzero over TR.
In particular, the second moment is nonzero both over TR and at TE, and as such, phase is accumulated over multiple repetitions, possibly contributing to acceleration-related errors.Specifically, positive acceleration leading up to peak flow may accumulate some additional phase, contributing to the overestimation.Figure 6A also demonstrates a small underestimation of velocity by PC-bSSFP at low flow rates.This can be more prominently seen if including the pulmonary trunk in the region of interest (Figure S2), indicating that another reason for the underestimation may be in-plane steady flow.
The direct comparison between 0.55T and 1.5T PC-bSSFP is more challenging due to differences in coils and slice/volunteer position.Difference in positioning in particular increases possible errors in region of interest-based velocity estimation.Taking this into consideration, the average error of 4 cm/s and RMSE v of 9.7 cm/s can be considered a reasonably good agreement between field strengths.
The SNR difference between PC-bSSFP and PC-GRE at 0.55 T was significant, with an increase of 1.8× from PC-GRE to PC-bSSFP on average.Furthermore, during high flow, PC-bSSFP exhibits nonlinear, inflow-related SNR changes, including both increase and decrease, 41 whereas PC-GRE shows an SNR increase.However, even in the worst-case situations, PC-bSSFP still exceeds the SNR of PC-GRE.

F I G U R E 9
Results of single-point flow correction using time-averaged background phase.Before correction, significant phase offset in the heart is present in single-point data (A) both at systole and diastole, resulting in an offset of velocity results (E) and velocity errors on the order of 15 cm/s (F).(B) Reconstructing time-averaged data with parallel imaging results in a very similar, smooth phase structure.(C) Background phase after correcting with a third-order polynomial fit, showing a small amount of remaining higher-order phase terms and significantly reduced phase offset.Results from applying this third-order correction are shown in (D), with corrected velocity curves (G) showing good agreement with the phase difference method, and average errors reduced to less than 3 cm/s (< 2% V enc ) (H).AAo, ascending aorta.
Comparing SNR between 0.55 T and 1.5 T, we found factors of 2× for both the adjoint and binned reconstructions.Based on acquisition bandwidth and field strength, a factor of 2.2× is expected when going from 0.55 T to 1.5 T. This estimate does not account for more favorable relaxation times associated with decreasing field strength, as well as the increased flip angle used for the 0.55T scans.Therefore, these factors would increase the SNR at 0.55 T and may explain the decrease in difference; as such, our observed differences can be considered reasonable.
Finally, it should be emphasized that this work was performed on a 0.55T system with relatively low-performance gradients.Higher gradient strength and slew rate would allow reduced deadtimes (gradient ramps, nonsampling time), improving SNR through longer sampling durations or increased spoke counts (through reduced TR).

Free-breathing PC-bSSFP
Free-breathing reconstruction produced similar results to breath-holds, with excellent agreement of velocity curves, minimal additional blurring of structures, and a minor reduction in SNR.Correlation between breath-hold and free-breathing was found to be very good; however, some small clusters of poorly correlated data can be seen (Figure 6B).These originate from one subject due to misregistration of free-breathing and breath-hold respiratory states, which may be correctable by manually shifting respiratory bin positions such that one of the bins better lines up with the breath-hold state.

Referenceless velocity estimation
Evaluation of polynomial order for background phase correction in a static phantom as shown in Figure 8 demonstrates a general reduction in error with increasing fit order.To satisfy the criteria of Gatehouse et al., 42 a background phase error of 0.6 cm/s or lower is required, which is satisfied in both mean and SD by a third-order polynomial (Figure 8E).This is in agreement with the findings of Nielsen et al., 15 who found that third order was sufficient to correct local background phase.Velocity results based on single-point flow estimation with third-order polynomial correction produced flow curves with excellent agreement to the two-point phase-difference method.The time-averaged background phase showed minimal structure in regions of flow, with smoothly varying phase (except for low signal lung regions), seen in Figure 9B.After correction, some residual higher-order phase terms remained; however, the magnitude of these offsets was generally less than 2.5 cm/s.Additionally, similar residual phase patterns can be seen at the edges of the static phantom (Figure 8E), indicating that these phase terms are likely due to static field inhomogeneity, rather than being a result of PC-bSSFP.Increasing the fit order (as seen in Figure 8F) or moving to a different fit method such as circular harmonics may help remove these terms; however, this would require significantly more refinement and would likely be less generalizable.
Due to removal of the difference operation, the V enc of these acquisitions doubles to 300 cm/s.As such, the resulting average velocity error of −1.8 ± 5.2 cm/s corresponds to less than 1% of V enc .Furthermore, by reducing the single-point V enc to match that of the difference method (150 cm/s), VNR can be improved; however, this comes at the cost of longer TR, as the slice-select gradient needs to be extended to achieve the required M 1 .Additionally, the magnitude of the background phase would likely remain unchanged, and as such, the relative effect of any residual background phase after correction would increase, increasing from 2% of V enc to 4% of V enc in this case.
In general, the magnitude of the errors when fitting the torso is comparable to the previous finding of Nielsen et al., 15 reporting RMSEs ranging from 0.1 to 0.2 radians for third-order fitting, which would correspond to 9.5-19 cm/s for a V enc of 300 cm/s, compared with RMSE v = 5.8 cm/s in our work.The single-point images shown here still provide sufficient SNR/VNR, enabling the acquisition in clinically acceptable breath-hold duration of 13 s.It is important to note that, while this method still requires further investigation, it is nonetheless feasible on a low-field system.Specifically, different slice positions and orientations would be required to properly evaluate the quality of the time-averaged phase-correction image and the robustness of the correction.Furthermore, application of this method to free-breathing images in slice orientations with more noticeable breathing motion would be needed to properly evaluate its effectiveness in free-breathing acquisitions.

CONCLUSION
Self-gated radial PC-bSSFP incorporating through-plane gradient inversion is feasible on low-cost, low-field 0.55T systems, producing high-quality cine images while permitting simultaneous cardiac flow measurements during free-breathing and without the need for ECG gating.By exploiting in-plane phase-cancelation properties of radial trajectories and dedicated phase correction, referenceless PC-bSSFP flow measurements are feasible with scan times identical to those of standard cine bSSFP imaging.
(B) Example seventh tiny golden angle radial k-space sampling.Acq, acquisition; GRE, spoiled gradient echo.F I G U R E 2 (A) Process for extracting respiratory and cardiac navigator signals from k-space center (K0 Signal).Bandpass and lowpass filters are tuned based on the frequency peak seen in the spectra.(B) Graphical depiction of binning based on navigator signal.Each heartbeat is divided evenly into 25 intervals and readout spokes are subsequently grouped into 25 bins based on these intervals.Heartbeats with erroneously long or short durations are rejected.FFT, fast Fourier transform.

F
I G U R E 3 Magnitude (left) and phase (right) images showing regions of interest for subsequent figures and statistics.Ascending aorta (AAo) and descending aorta (DAo) are segmented to quantify both velocity and flow information.
dependence, and clinical implications.J Magn Reson Imaging.2004;20:697-705.doi:10.1002/jmri.2016342.Gatehouse P, Rolf M, Graves M, et al.Flow measurement by cardiovascular magnetic resonance: a multi-centre multi-vendor study of background phase offset errors that can compromise the accuracy of derived regurgitant or shunt flow measurements.J Cardiovasc Magn Reson.2010;12:1-8.doi:10.1186/1532-429X-12-5than 1% of the DC signal.As such, the effect of these higher frequency components can be ignored.Assume rectangular gradients and given prephaser time T p , half readout time T ′ , perturbed flat time , perturb time  t , readout gradient strength G, and prephaser gradient strength G p , M 1,R (TE +  t ) = − G p T 2