## 2 Materials and Methods
### Projection Methods  
In the vertical projection method, each ground truth image was transformed into k-space using the Fourier transform, and the undersampled k-space lines were selected based on the DC component lines [27]. The number of parallel encoding lines was selected to be equal to the number of angles used in the radial projection. The time series in the undersampled k-space were plotted based on the ICA and CCA ROIs that were manually extracted from the ground truth images (Figure 2).  
Figure 1: Ground truth samples used in the simulation. **(a)** The regions of interest (ROIs) in the internal and common carotid arteries (ICA and CCA, respectively) were extracted from time-of-flight magnetic resonance angiography using simultaneous multislice images from five patients. **(b)** Two different pulse waveforms were applied to each ROI for simulation, resulting in a total of 10 waveforms. The time series for the left and right hemispheres of each sample were plotted, and the blue and red lines represent the time series of ICA and CCA, respectively.  
Figure 2: Workflow for the comparison of the vertical and radial projection methods. **(a)** Vertical projection method. The images acquired from simultaneous multislice (SMS) images for time-of-flight magnetic resonance angiography (TOF MRA) are Fourier transformed. Additionally, k-space is undersampled using parallel encoding based on the DC component (red line), and the undersampled k-space is reconstructed. The internal and common carotid artery (ICA and CCA) time series are then plotted using the average intensity within each ROI. The difference between the peaks of the ICA and CCA is used to calculate the pulse wave velocity (PWV). **(b)** Radial projection method. The ground truth image, i.e., the SMS image, is Radon transformed at selected projection angles to generate sinograms. The time series are created from the average signal within each ROI in these sinograms. The PWV is calculated based on the time series of the ICA and CCA obtained from the same or different angles. The yellow line indicates the projection direction.  
The radial projection method data were extracted from the sinogram derived by Radon transforming the ground truth image into a projection angle, and the time series of ICA and CCA were plotted (Figure 2). The projection angle was selected as the angle at which signal interference between blood vessels did not occur during radial projection, and isolation was assessed with the sinogram generated by Radon transforming the ROI template from 1\({}^{\circ}\) to 180\({}^{\circ}\) (Figure 3). Owing to individual variations in vascular distribution, this method allows the selection of projection angles, in which none of the ROIs overlap, ranging from a minimum of one angle to a maximum equivalent to the number of ROIs. Each vessel's time series was generated by Radon transformation at angles where they could be isolated, and the PWV calculation was performed based on the time series derived from the same or different angles.  
Finally, additional experiments were conducted to verify whether the simulation results could be applied in an actual MRI environment. Although the simulation was based on a dataset generated with a temporal resolution of 1.6 ms, achieving such a temporal resolution in an actual MRI environment is challenging. Therefore, the simulation dataset was undersampled by a factor of 10, thereby simulating images acquired at a temporal resolution of 16 ms in an actual MRI environment. This undersampled dataset was resampled by a factor of 10 to create the interpolated dataset. The differences between the simulation, undersampled, and interpolated datasets were analyzed based on the mean PWV in the patients and hemispheres, both with and without 100% Gaussian temporal noise. All processes involving vertical and radial projections were performed using MATLAB software (R2020b version, Mathworks, Natick, MA, USA).