## 2 Materials and Methods
### Boundary Conditions  
The fluid boundary conditions simulate the internal cardiovascular system of the brain, so these should be adjusted as closely as possible to real conditions. In this case, the inlet condition is represented by a velocity profile, and the outlet condition by the blood flow pressure. Assuming that the flow is periodic over time, the Womersley velocity profile was used to obtain the physiological conditions at the artery inlet (Equation (3)),  
\[u(r,t)=\frac{2Q_{0}}{\pi a^{2}}\bigg{(}1-\frac{r^{2}}{a^{2}}\bigg{)}+Re\left\{ \sum_{n=1}^{N}\frac{Q_{n}}{\pi a^{2}}\Bigg{[}\frac{1-\frac{J_{0}\left(\beta_{n }\frac{r}{a}\right)}{I_{0}\left(\beta_{n}\right)}}{1-\frac{J_{0}\left(\beta_{ n}\right)}{I_{0}\left(\beta_{n}\right)}}\Bigg{]}e^{inovt}\right\} \tag{3}\]  
with \(\beta_{n}=\hat{r}^{3/2}\alpha_{n}=\hat{r}^{3/2}a\sqrt{\frac{n\omega}{V}}\), where \(\alpha_{n}\) is the \(n\) Womersley number, \(a\) is the inlet artery radius, \(n\) is the number of modes, and \(\omega\) is the angular frequency. In our study, there are 8 modes and an angular frequency of 7.703 s\({}^{-1}\) (the period of the cardiac cycle was 0.857 s). For this, a previously averaged blood pulse V(t) of 70 patients from Valencia's database was used, measured at the internal carotid artery using the Doppler ultrasound technique, and then the blood flow Q (t) for each case was calculated with its corresponding model input area. This averaged blood pulse is shown in Figure 3. More details about the patient's measurement and implementation can be found in [PERSON]'s work [23]. From this, at time t = 0 s the blood velocity was v(t) = 0.278 m/s. We considered this velocity value as the initial condition at the artery inlet. To obtain the pulsatile pressure conditions \(P(t)\) at the outlet, the RCR Windkessel model (Equation (4)) was used,  
\[\bigg{(}1+\frac{R_{p}}{R_{d}}\bigg{)}Q(t)+CR_{p}\frac{dQ}{dt}=\frac{P(t)}{R_{ d}}+C\frac{dP}{dt} \tag{4}\]  
where \(R_{p}\) is the proximal resistance, \(R_{d}\) is the distal resistance and \(C\) is the capacitance. These constant values were adjusted and calibrated manually. In the present work 3 different pressure conditions were considered: (1 st) a normal condition representative of a healthy patient at rest, also called \"normotensive\" and approximately between 120/80 mmHg; (2 nd) a condition representative of a grade 2 hypertension or \"hypertensive\" between 180/100 mmHg [24]; (3 rd) and a condition representative of a grade 1 isolated systolic hypertension (ISH) between 160/70 mmHg [11], which are shown in Figure 4. These hypertensive conditions should not be confused with one another since, while hypertension corresponds to high blood pressure at the entire cardiac cycle, isolated systolic hypertension implies an increase in pressure only at the systolic phase. Therefore, for the sake of clarity, we refer to the latter solely as 'ISH'. Table 3 shows the constant values for each pressure condition. Figure 4: Pressure conditions under study: Normal (Blue), Hypertension (Red) and Isolated Systolic Hypertension (ISH) (Yellow). These are used as outlet boundary conditions. Figure 3: Averaged blood flow pulse used as the inlet boundary condition. Our study prioritized the effects of the pressure conditions on blood flow dynamics. As it can be observed in the figure, we emphasized that, while the hypertensive condition was the one with the highest pressure of the entire cardiac cycle, the ISH condition has the particularity that at diastole it was the lowest pressure, while at systole it was between normal (blue) and hypertension (red), thus having a quite different shape from the other two conditions. In addition to this, the previous fluid dynamic conditions, the non-slip condition, was applied to the vessel walls of each model.