Three-dimensional hybrid radial sampling was used to obtain self-gating signals and acquire k-space data simultaneously. As shown in Fig. 1(a), a 3D steady-state free precession (SSFP) pulse sequence was implemented with a hybrid radial k-space trajectory (Cartesian sampling along the slice encoding direction, k _z , and radial sampling in the k _x –k _y plane). All slices encoded with a specific angle (denoted as the profile) were acquired sequentially before switching to the next angle (Fig. 1(b)). Each profile included n _z slice encodings, limiting the temporal resolution to . The total number of profiles acquired and used in retrospective reconstruction was predetermined. Therefore, the profile order was determined before the scan.

Fig. 1.

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	Fig. 1. (color online) SSFP pulse sequence collects the profile with multiple slice encoding lines at a specific angle (a) and changed the profile angle after finishing the last profile (b).

The purpose of determining the optimal profile order was to obtain an approximately uniform k-space distribution for each cardiac phase. In the aforementioned hybrid radial scheme, the profile angle is related to the profile number by

(1)

In cine MRI, images are reconstructed from multiple cardiac cycles (see Fig. 2). Each cardiac cycle contains multiple profiles whose number is equal to the number of reconstructed cardiac phases. Suppose that the number of cardiac phases conforms to a normal distribution, . The profile in the j-th cardiac cycle, can be expressed as . The profile angle θ in Eq. (1) can then be replaced by

(2)

(3)

(4)

(5)

Fig. 2.

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	Fig. 2. Cine images reconstructed from multiple cardiac cycles.

Compared with a uniform distribution, the sample density along the azimuthal direction is not homogeneous, which may affect the image signal-to-noise ratio (SNR). The ratio of the non-uniform SNR to the uniform SNR is equal to the sampling efficiency (SE), which depends on the homogeneity of the sample distribution achieved by each acquisition scheme.^[21] This ratio is given by

(6)

The optimal values of the variables α and β were determined by simulation to minimize SE for a cardiac frequency range of 0.6–3 Hz, and a breath frequency range of 0.1–0.5 Hz. Ranges of α (0–2) and β (0–1) values in increments of 0.01 were evaluated. The coefficient of variation (CV) was used to evaluate the SE for different cardiac and breath frequencies.

Ten normal volunteers (eight male and two female, mean age of 32 years ± standard deviation of 7 years), without any history or symptoms of cardiac disease, were studied. Cardiac cine imaging was performed in the supine position using a 1.5-T GE HDx scanner (maximum gradient amplitude 33.0 mT/m, slew rate 120 T/m/s, Excite 14 M5 software version; GE Healthcare, Waukesha, WI, USA). All subjects provided written informed consent, and the study was approved by the local institutional ethics review board at Weill Cornell Medical College. An eight-channel cardiac phased-array coil was used for signal reception.

Two separate scans were executed in random order using the same imaging parameters. The typical imaging parameters were as follows: TR/TE=4.4/1.3 ms, flip angle is , readout bandwidth is±125 kHz, field of view is 31 cm, reconstructed image matrix is 256 × 256, slice thickness is 10 mm, and number of slice encodings is 12–14. The temporal resolution ranged from 52.8 to 61.6 ms, depending on the number of slices. The total number of profiles was 5000, and the total free-breathing 3D acquisition time was approximately 5 min.

It was impossible to ensure that the volunteers maintained the same breathing and heart rate patterns during the two continuous scans. To compare the two profile orders using the same standards, the SEs calculated using Eq. (6) were also compared for the same self-gating motion signals obtained from the above acquisitions.

The optimised nonlinear profile order was compared with the golden ratio-based profile order. A phantom was sampled using these two profile orders. The original k-space data for the phantom were selected based on self-gating signals from the ten volunteers. Twenty phantom images obtained using these two methods were compared in terms of the SNR.

Image quality metrics, including the blood SNR, myocardium-blood contrast, contrast-to-noise ratio (CNR), and image sharpness,^[8] were calculated from a mid-ventricular slice for each volunteer. The two methods were compared using a nonparametric Wilcoxon signed-rank test. All of the tests were two-sided, and was considered to indicate a significant difference. The statistical analyses were performed using statistical software (SPSS version 19; SPSS Inc., Chicago, IL, USA). Values are reported herein as means ± standard deviations.

Figure 3 shows the SE CV values corresponding to various α and β values. Low CV values are inversed in brightness in Fig. 3(a). Low SE CV values for various combinations of cardiac and respiratory frequencies improve the adaptability of the sampling method. The lowest CV values for each α are shown as red dots in Fig. 3(a). The α and β value pairs had similar CV values (5% difference), as shown in Fig. 3(b). The combination of and was identified as optimal, i.e., as yielding the lowest CV of the SE.

Fig. 3.

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	Fig. 3. (color online) (a) CV of SE corresponding to different α and β values, and smaller CV values were inversed to bright for optimal viewing. The minimal CV values according to each α are indicated as red dots. CV values at red dots are plotted in panel (b).

Image reconstruction was performed for the ten subjects, for one of whom the golden ratio method had failed. Figure 4 shows images of mid-ventricular slices obtained during end-diastolic and end-systolic cardiac phases from the other volunteers. The two methods were compared to each other. A few streaking artefacts appeared in both sampling methods because of undersampling (the undersampling ratio was 2:3) and motion. Table 1 summarises the metrics of image quality for the mid-ventricular images shown in Fig. 4. Comparable image quality metrics were obtained for blood SNR, myocardium-blood CNR, myocardium contrast, and image sharpness. The differences were not statistically significant ( ).

Fig. 4.

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	Fig. 4. (a), (b) Images of mid-ventricular slices obtained with golden ratio-based profile order and (c), (d) nonlinear profile order in nine healthy volunteers.

Table 1.

Table 1.

Table 1. Metrics of image quality of the two methods . . SNR CNR Contrast Image sharpness/mm Golden ratio diastole 53.5 ± 25.3 22.0 ± 14.8 24.7 ± 6.9 0.21 ± 0.04 Nonlinear diastole 54.4 ± 28.5 22.2 ± 17.0 24.4 ± 9.9 0.24 ± 0.03 P-value 0.767 0.678 0.767 0.110 Golden ratio systole 54.4 ± 18.6 22.1 ± 10.7 24.7 ± 6.8 0.22 ± 0.05 Nonlinear systole 51.9 ± 21.0 22.2 ± 13.7 26.2 ± 11.4 0.26 ± 0.07 P-value 0.173 0.678 0.515 0.086

Table 1. Metrics of image quality of the two methods . .

Figure 5 shows a comparison of the SE for the golden ratio and nonlinear profile orders for the real self-gating signals obtained. For the same respiratory and cardiac self-gating signals, profiles from the nonlinear method had higher mean SE values (nonlinear: 0.831 ± 0.02; golden ratio: 0.80 ± 0.11), which means that the nonlinear profile order provided a more uniform distribution for the same number of profiles. As indicated by the arrows in Fig. 5, the golden ratio-based profile orders generated worse k-space distributions ( ) than the nonlinear profile orders ( ). The SE difference between the two methods was not statistically significant (p=0.60), according to the results of a two-sided Wilcoxon signed-rank test.

Fig. 5.

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	Fig. 5. (color online) Comparison of SE between golden ratio and nonlinear profile orders based on the same self-gating signals.

The SNRs determined for phantom images of golden ratio-based and nonlinear profile orders were 164.45 ± 22.81 and 176.55 ± 23.83, respectively. Figure 6 shows reconstructed phantom images and profile distributions based on the self-gating signals for a subject for whom the golden ratio-based profile order failed (as indicated by an arrow in Fig. 5). Nearly 353 profiles were used in the reconstruction, including shared neighbouring profiles. Obvious streak artefacts appeared in the golden ratio-based profile order as a result of the non-uniform distribution of the profiles. More uniform distributions were generated based on the nonlinear profile order (SE for golden ratio is 0.35; SE for nonlinear is 0.86).

Fig. 6.

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	Fig. 6. (a) Reconstructed phantom images and (b) profile distribution based on the same self-gating signals from failed subject, which is indicated by the arrow in Fig. 5.