## 3 Results
### Flow Profile
\[\tau_{(x,\ y)}=\mu\ \frac{\partial v}{\partial y}\]  
While in circular channels, the shear stress is easy to calculate, the formula is more complex for rectangular channels due to the lateral walls that reduce the flow velocity. Based on the precise formula to calculate the flow rate in a rectangular channel [27], we express the shear stress according to the geometric properties of the channel (\(h\) and \(w\), as previously defined). The following equation determines the shear stress at any point of a rectangular channel:  
\[\tau_{(x,\ y)}=\mu\ \frac{Q}{S}\ \sum_{n=0}^{\infty}\frac{(-1)^{n}b\pi}{(2n+1)^{ 2}}\bigg{(}\frac{2}{\pi}\bigg{)}^{3}\frac{sin\ a\big{[}(2n+1)\frac{\pi y}{2b} \big{]}}{cos\ a\big{[}(2n+1)\frac{\pi x}{2b}\big{]}}cos\big{[}(2n+1)\frac{\pi x }{2b}\big{]}\]  
with:  
\[S=ab^{3}\frac{4}{3}\bigg{(}1-\frac{192}{\pi^{5}}\frac{b}{a}\sum_{n=0}^{\infty} \Bigg{\{}\frac{1}{(2n+1)^{5}}tan\ a\bigg{[}\frac{(2n+1)\pi a}{2b}\bigg{]} \Bigg{\}}\Bigg{)}\]  
A coordinate cross is imagined at the center of the channel, as illustrated in Figure 2c. We defined \(2a\) as the height of the channel in the direction of the \(y\)-axis, \(2b\) as the width of the channel in the direction of the \(x\)-axis, and the \(z\)-axis represents the direction of the flow. Cells seeded on the channel are located on the membrane surface: the bottom of the channel (\(y=-a\)). Microvascular endothelial cells have a very small thickness (from 0.3 to 0.5 \(\upmu\)m [28]), which can be neglected for these measurements. Likewise, the height at the center of the BMEC is greater than at the cell walls due to the presence of the nucleus, but this variability stays inferior to 0.5 \(\upmu\)m. Indeed, it is not considered as roughness and does not affect the laminarity. Firstly, we estimated the range of possible shear stresses according to the flow rates proposed by the Ibidi pump system. We used the above equations for the minimum and maximum possible flow rates (0.09 and 52.2 mL\(\cdot\)min\({}^{-1}\)) for the cells located at the center of the membrane (i.e., \(x=0\), \(y=-a\)) in our device. The results were 0.055 and 32.3 dyn\(\cdot\)cm\({}^{-2}\). This confirms that our setup can deliver a large panel of shear stresses and covers the physiological range. The calibration curve in Figure 2d shows the flow rate to set depending on the desired shear stress. For the experiments presented here, the flow rate was configured at 16.3 mL\(\cdot\)min\({}^{-1}\) to acquire a physiological pressure of 10 dyn\(\cdot\)cm\({}^{-2}\) on the cell surface seeded in our barrier device. Figure 2: Fluidic setup. (**a**) Representation of the fluidic equipment for our barrier device in well plate format. The Ibidi pump is controlled by the specific software and simultaneously drives two fluidic units with independent flow parameters. Two fluidic units connect six channels. (**b**) Picture of our barrier device in microscope slide format connected to a fluidic unit. (**c**) Illustration of the orthonormal coordinate system necessary for the flow calculations in the channel cross section. (**d**) Calibration curve of the flow rate to implement according to the intended shear stress on the cell monolayer. For 10 dyn\(\cdot\)cm\({}^{-2}\) in our barrier device, we must configure a flow rate of 16.3 mL\(\cdot\)min\({}^{-1}\). This calculation depends on the geometry of the seeding channel and parameters of the culture medium perfused. (**e**) Prediction of area of uniform shear stress along the channel width on the cell monolayer. Ibidi pump system can provide flow rates from 0.09 to 52.5 mL\(\cdot\)min\({}^{-1}\) (corresponding to a range of 0.055 to 32.3 dyn\(\cdot\)cm\({}^{-2}\)). Shear stress is distributed homogenously on 82% of the channel width. As discussed above, in a rectangular channel, the shear stress is not uniform along the channel width because of the presence of side walls. Precautions must be taken with the shear stress distribution to avoid exposing cells to an inhomogeneous shear stress field [22; 29].