## 4 Discussion  
This study demonstrated that the radial projection method can resolve the problem of interference between vascular signals, which is a limitation of the vertical projection method. The results indicated that the ground truth PWV was very similar to that of the radial projection method but not to that of the vertical projection method. This also demonstrates that the radial projection method can achieve effects similar to those of full sampling in terms of PWV measurement, even with k-space undersampling. Additionally, interpolation showed that the same effects observed in the simulation could be obtained in an actual MRI environment, suggesting the potential for future in vivo applications. To evaluate the accuracy of the radial projection method, the MAE of the PWVs across different levels of temporal noise were compared with those of the ground truth and  
Figure 4: Comparison of time series and peak points among the simulation, undersampled, and interpolated datasets. The top panel shows the time series from the simulation dataset over the range of 0–4800 ms. The bottom two panels provide detailed views of the eight waveform peaks (Peak 01 to Peak 08). The black, blue, and red arrowheads indicate the peak points for the simulation, undersampled, and interpolated datasets, respectively. The interpolated dataset closely followed the simulation dataset, indicating effective reconstruction of the original signal. Although the undersampled dataset showed a similar trend to the time series of the simulation dataset, there were noticeable differences in the peak positions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Noise** & \multicolumn{3}{c}{**PWV Mean \(\pm\) SD (m/s)\({}^{\text{b}}\)**} & \multicolumn{3}{c}{\(p\)} & **Pairwise Comparisons (DSCP) \(p\)** \\
**Levels\({}^{\text{a}}\)** & **S** & **U** & **I** & **(Corrected)** & **S-U** & **S-I** & **U-I** \\ \hline
90\% & 3.02 \(\pm\) 0.80 & 1.15 \(\pm\) 0.18 & 2.92 \(\pm\) 0.36 & \textless{}0.001 \({}^{\text{a}}\) & \textless{}0.001 \({}^{\text{a}}\) & 0.645 & \textless{}0.001 \({}^{\text{a}}\) \\
100\% & 3.09 \(\pm\) 0.83 & 1.15 \(\pm\) 0.30 & 2.88 \(\pm\) 0.34 & \textless{}0.001 \({}^{\text{a}}\) & \textless{}0.001 \({}^{\text{a}}\) & 0.890 & \textless{}0.001 \({}^{\text{a}}\) \\ \hline \end{tabular}  
* indicates a statistically significant difference (\(p\) <0.05); \({}^{\text{a}}\) in the ”Noise Levels”, ”Without noise” refers to TOF MRA images acquired with SMS without temporal noise, while ”10-100%” refers to those with SMS with a temporal noise level of 10–100%. \({}^{\text{b}}\) The number of peaks used for calculating the mean and standard deviation of PWV is 8; DSCF = Dwass-Steel–Critch–Figure, I = interpolated dataset, PWV = pulse wave velocity, S = simulation dataset, SD = standard deviation, S-U = comparison between PWVs of the simulation and undersampled datasets, S-I = comparison between PWVs of the simulation and interpolated datasets, U = undersampled dataset, U-I = comparison between PWVs of the undersampled and interpolated datasets, TOF MRA = time-of-flight magnetic resonance angiography, SMS = simultaneous multislice imaging.  
\end{table} 
Table 3: \(Cont.\)vertical projection methods. The results show that the average MAE in both hemispheres using the radial projection method was much lower than that of the vertical projection method at all noise levels. Additionally, while the MAE in the radial projection method tended to increase with higher noise levels, the MAE in the vertical projection method was consistently higher, regardless of the noise level (Table 1). Furthermore, on comparing the PWVs between the ground truth and projection methods, the PWVs were not significantly different between the ground truth and radial projection method (Table 2). By contrast, the PWVs in the vertical projection method were significantly different from the ground truth PWVs in all cases, except for in the left hemisphere at the 10% and 20% noise levels.