# In Silico Investigation of the RBC Velocity Fluctuations in Ex Vivo Capillaries
## 2 Physical and Mathematical Description of the Problem
### RBC Modeling  
RBCs are elastic objects, and in this study we assume that they are composed of discrete elements that form a thin elastic membrane immersed in the surrounding incompressible fluid. We use the FEM-based Zavodszky material model [30] to represent the RBCs and the immersed boundary method (IBM) [31; 32] for fluid-solid coupling. The forces acting on the RBC are \(F_{link}\), \(F_{bend}\), \(F_{area}\), and \(F_{volume}\). We assume that during small deformations, all forces change linearly and are independent of each other. Equation (1) shows the link force, which acts between surface points and is responsible for the response to the stretching and compression of the spectrum network. Here, \(p\) is a biological value, specifically the spectrin link length, taken as 7.5 pN [33]. The relative extension is given by \(dL=(L_{i}-L_{0})/L_{0}\), where \(L_{0}\) is the initial distance between the vertices. Equation (2) shows the bending force, which acts between neighboring faces and is responsible for the bending response of the RBC membrane. This response is due to the \(\sim\)29 nm thickness of the spectrin network [34], where \(d\theta=(\theta_{i}-\theta_{0})/\theta_{0}\), and \(\theta_{0}\) is the initial angle between the faces. Equation (3) shows the area force, which represents the membrane's response to stretching and compression at the surface area of each element. Here, \(dA=(A_{i}-A_{0})/A_{0}\), where \(A_{0}\) is the initial area of the triangle. Equation (4) shows the volume force, a global force that enforces a pseudoincompressibility to the RBC. Here, \(dV=(V_{i}-V_{0})/V_{0}\), and \(V_{0}\) is the initial volume of the RBC. This force ensures that the volume is conserved at a specified \(\tau_{vol}\). The model parameters are provided in Table 1, yielding a Young's modulus (\(E_{s}\)) of 27.82 \(\upmu\)N/m, a shear modulus (\(K_{s}\)) of 10.72 \(\upmu\)N/m, and a Poisson ratio of 0.29. These continuum mechanical properties align with the experimental measurement range of previous studies [35; 36; 37]. It should be noted that these model parameters are dependent on the Lagrangian resolution. The shared values are for a 0.5 \(\upmu\)m spacing. When the spacing changes, constants are modified to yield the same macroscopic material properties (e.g., shear moduli and \begin{table}
\begin{tabular}{l l l l} \hline \hline \(F_{link}\) & 15 & \(\tau_{link}\) & 3 \\ \hline \(F_{bend}\) & 80 & \(\tau_{bend}\) & \(\tau/6\) \\ \hline \(F_{area}\) & 5 & \(\tau_{area}\) & 0.3 \\ \hline \(F_{vol}\) & 20 & \(\tau_{vol}\) & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RBC material model constants. Poisson ratio). For example, in the main simulations, due to the refined RBC membrane mesh, \(\Delta s/\Delta x\approx 0.74\), where \(s\) is the Lagrangian particle distance of the RBC. This also applies to the RDW cases, where the diameters of the smaller and larger RBCs are 5.5 and 6.1 \(\upmu\)m, respectively. The Foppl-von Karman number, given in Equation (5), can be used to characterize RBC deflection. It is generally taken as \(\sim\)400 [38], although higher values [39; 40; 41; 42] or lower values [43; 44] are also commonly observed. \[F_{link} =-\frac{\kappa_{link}dL}{p}[1+\frac{1}{\tau_{link}^{2}-dL^{2}}] \tag{1}\] \[F_{bend} =-\frac{\kappa_{bend}d\theta}{L_{0}}[1+\frac{1}{\tau_{bend}^{2}-d \theta^{2}}]\] (2) \[F_{area} =-\frac{\kappa_{area}dA}{A_{0}}[1+\frac{1}{\tau_{area}^{2}-dA^{2}}]\] (3) \[F_{volume} =-\frac{\kappa_{volume}dV}{V_{0}}[1+\frac{1}{\tau_{volume}^{2}- dV^{2}}]\] (4) \[1/\Gamma=K_{B}/K_{s}r^{2} \tag{5}\]  
Currently, this material model has been shown to be stable in the low/high shear regions [30]. To our knowledge, there is no publication that uses this model in capillaries.