# In Silico Investigation of the RBC Velocity Fluctuations in Ex Vivo Capillaries
## 4 Results and Discussion  
Each simulation runs for 1 s of physical time, revealing the spatiotemporal RBC velocity profiles in four different capillaries with varying levels of luminal irregularities and their straight tube equivalents. Three different inlet ht levels are investigated in each capillary, since ht could be a major determinant of velocity characteristics. The RDW was also investigated to represent the actual biological situation since red blood cells actually have heterogeneity in their diameters. The diameter of the standard RBCs is selected as 5.8 \(\upmu\)m to represent the mouse RBCs, since the capillary geometers employed are acquired from a mouse (ex vivo). RBC data are collected only when the RBC moves across the original length of the capillaries; i.e., if the RBC does not move across the capillary by the end time, that data is discarded. The coefficient of variation (CoV) (\(CoV=V_{STD}/V_{\text{arg}}\)) [52] is also presented to reveal the velocity heterogeneity in different configurations, where STD is the standard deviation and avg is the average. Due to the unique shape of the capillaries, the evaluation of the results using only the hydraulic diameter and the deviation is not sufficient to explain the complex data. Hence, here we define a capillary circularity index (\(l_{cc}\)) to quantify the irregularity of the capillaries. Equation (6) shows the definition, where \(\alpha\) is the aspect ratio, \(\chi=D_{RBC}/D\) is the confinement ratio, \(D_{ah}\) is the average hydraulic diameter of the geometry, \(D\) is the hydraulic diameter, and n is the number of cross-section slices representing the capillary. When the cross-section is not circular due to the \(\alpha\) and \(\chi\) terms and when there is a diameter variation due to the final term, \(l_{cc}\) will start to reduce. Here, the term \(\alpha\) is responsible for capturing the porosity when used together with the final term, and the term \(\chi\) is responsible for capturing the confinement. In a perfect cylinder the final term will be zero with \(l_{cc}=1\), and any number less than unity means that the geometry is not a straight cylinder, and the velocity across the geometry will deviate more as \(l_{cc}\) decreases. For example, in a converging-diverging nozzle-type confined geometry, which is widely used in in vivo RBC studies, \(\alpha\) and the final term will always be one, but \(\chi\) will increase in the confinement  
Figure 5: (**left**) Schematic drawing of the slit validation and the initial orientation of the RBCs as a, b, c. (**right**) Side-by-side RBC deformation comparison with the the results by [PERSON] et al. [48] at three different times with three different initial orientations as (**a–c**) (Darker RBCs are the current results.). region, so \(I_{cc}\) will reduce. Another example is a rectangular channel, where \(\alpha\) and \(\chi\) are equal to unity, but the final term will increase due to the average hydraulic diameter, so \(I_{cc}\) will reduce. It is important to note that, due to the inclusion of the term \(D_{avg}\), this index is intended to compare a given geometry with its ideal tubular equivalent and is not suitable for comparing different capillaries with each other. The average hydraulic diameter, length, average input velocity \(I_{cc}\), and the number of capillaries of each capillary are given in Table 2. Here, the capillary number is calculated in equivalent tubes with the average inlet velocity. \[I_{CC}=1-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{\alpha_{i}}\frac{1}{\chi_{i}}\frac{ \left|D_{i}-D_{ah}\right|}{D_{ah}} \tag{6}\]  
Three different ht levels were used in the feeding section to achieve low, medium, and high hematocrit levels for each capillary. The orientation of the RBCs and the number of RBCs are equivalent in each hematocrit level to compare the RBC transport differences in each capillary and their equivalent tubes while feeding the RBCs almost in the same manner.