# In Silico Investigation of the RBC Velocity Fluctuations in Ex Vivo Capillaries
## 4 Results and Discussion
Simulation visualizations are shared in the Supplementary Materials for each capillary (S1, S2, S3, S4) with three different ht. This investigation is conducted in four stages. First, we present the data of the RBCs averaged in each 5 \(\upmu\)m interval throughout the capillary, along with their average values and deviation. Second, temporal variations during the simulation time are incorporated into the spatial average evaluation. After revealing the spatial variations, as a third step, we individually calculate the temporal average of each RBC data and study the transport variation between the RBCs. Finally, in the fourth step, RDW is introduced in cases of high hematocrit and compared with non-RDW RBC transport in capillaries. The hematocrit values achieved with these settings in all capillaries and their equivalent regularized tubes are shown in Figure 6a,b with respect to \(I_{cc}\) and ht. We observe the lowest ht in the S4 capillary, which has the highest average hydraulic diameter and the lowest \(I_{cc}\), while the highest \(I_{cc}\) is observed with the S2 capillary. As a general trend, ht is almost always higher in tubes except the S2 capillary, which is the most uniform capillary. Although their confinement ratio is relatively the same, the \(I_{cc}\) values of the S1 and S2 capillaries are 0.91 and 0.93, respectively, and it can clearly be observed that the deviation from the tube is higher in the S1 capillary. Figure 6c shows the averaged velocities (in each 5 \(\upmu\)m) of the red blood cells across the capillaries and the tubes, and Table 3 shows the difference of the average velocities between the capillaries and their corresponding tubes. The tube results show nearly uniform behavior, whereas in the capillaries the velocity of the RBC varies with respect to the position due to changing \(I_{cc}\). Since an average velocity value is imposed at the inlet boundary condition and RBCs tend to stay near the center of the channels, it is expected to \begin{table}
\begin{tabular}{l l l l l} \hline
**Capillary** & **S1** & **S2** & **S3** & **S4** \\ \hline Average hydraulic diameter (\(\upmu\)m) & 4.89 & 5.22 & 6.77 & 6.9 \\ \hline Length (\(\upmu\)m) & 46.9 & 48 & 55 & 49.6 \\ \hline Average Inlet velocity (mm/s) & 0.75 & 0.86 & 1.44 & 1.5 \\ \hline Flow rate (\(\upmu\)m\({}^{3}\)/s) & 13,992 & 21,365 & 53,266 & 56,156 \\ \hline \(I_{cc}\) & 0.91 & 0.93 & 0.84 & 0.67 \\ \hline Capillary Number & 0.07 & 0.08 & 0.13 & 0.14 \\ \hline \end{tabular}
\end{table} 
Table 2: Flow–capillary properties. see a higher RBC velocity than the average inlet velocity in these channels. Table 3 shows the highest velocity difference in each capillary calculated 5 \(\upmu\)m apart. We can clearly observe an inversely proportional relation with \(I_{cc}\), and it has a minimum of 18% in the S2 capillary (\(I_{cc}=0.91\)) and 57% in the S4 capillary (\(I_{cc}=0.67\)). The velocity difference increases with an increase in ht in each capillary, but it is negligible compared to the effect of \(I_{cc}\) on the velocity difference. Therefore, it can be clearly stated that the \(I_{cc}\) change is more crucial than the ht change for the 8-22 ht range. In regular tubes, the velocity difference is at most 1.5% for all cases due to the constant diameter and \(I_{cc}=1\). This indicates that even a small deviation from the ideal channel can cause spatial heterogeneity in RBC velocity, and for more complex capillaries, such as S4, this difference can reach 57%. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Capillary** & \(I_{cc}\) **(\%)** & **Ht Low** & **Ht Mid** & **Ht High** \\ \hline S4 & 69 & 55.5 & 56.88 & 57.27 \\ \hline S3 & 84 & 33.23 & 33.38 & 36.72 \\ \hline S2 & 93 & 24.99 & 25.15 & 26.33 \\ \hline S1 & 91 & 18.22 & 18.41 & 18.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average RBC velocity difference (%) in each 5 \(\upmu\)m interval for different ht levels (low, mid, and high). Figure 6: (**a**) Ht with respect to \(I_{cc}\) for each capillary and the equivalent tube.