# Numerical Modelling of Erythrocyte Sticking Mechanics
## 3 Methodology  
The discrete element method (DEM) makes it possible to virtually describe the motion of various objects from very small objects (e.g., atoms) to very large objects (e.g., planets). It is important that [PERSON]'s second law is applicable and suitable for describing the motion of these objects. Different objects have different mechanical properties, which are important to identify in advance from known physical experiments. Initially, the discrete element method was used to simulate the movement of large objects (rock masses, granular soils). In this paper, DEM is used for numerical simulation of cell interaction. Translational motion (of each \(i\)-th erythrocyte) is described by [PERSON]'s second law:  
\[m_{i}\frac{d^{2}\mathbf{x}_{i}}{dt^{2}}=\mathbf{F}_{i}(t) \tag{1}\]  
where \(m_{i}\) and vector \(\mathbf{x}_{i}\) are the mass and position of the \(i\)-th erythrocyte (red blood cell). The vector \(\mathbf{F}_{i}(t)=\sum_{j}\mathbf{F}_{i,j}(t)\) corresponds to the resultant forces acting on the cell \(i\) in the event of possible interaction. Equation (1) is integrated with the 5 th-order Gear predictor-corrector scheme [31].  
During the interaction (Figure 1) of an erythrocyte, four main processes (approach, load, unload, separation) can be distinguished. Initially, the erythrocyte is close to the blood clot and interaction is possible. Figure 1 shows a model of the interaction of erythrocytes (red blood cells). The constitutive model of interaction can be described by the certain points (S-L-U-A\({}^{\prime}\)-E). When approaching (S-L), the movement of the erythrocyte corresponds to the interaction at a distance. At distances of load (L-U), unload (U-A\({}^{\prime}\)), reload-reunload (A\({}^{\prime}\)-E), the erythrocyte is in contact with the surface of the vessel. The force-displacement diagram is presented in Figure 1.  
Figure 1: Sticking process model (**a**) and initial position of the erythrocyte (**b**). If it does not have enough kinetic energy to detach, reloading begins at point A\({}^{\prime}\) and then the interaction ends at point E.  
The additional effect of cell elongation (formation of a neck in the contact zone) during unloading was not taken into account in this work. Previously, this effect (additional elongation) was considered for softer structures such as virus spikes (Jasevicius [32]). In the future, it is planned to create an additional model (for erythrocyte interaction) that will allow us to evaluate the possible elongation of cells during unloading. It should be noted, as later shown by the results of the numerical experiment, that this effect is not achieved during unloading during adhesion/sticking (the cell does not reach the value h = 0 when unloading; therefore, for h < 0, no additional elongation will be observed). This effect may be important if it is necessary to separate an attached cell from a surface (e.g., the surface of a blood vessel). As mentioned earlier, the results show that the adhesion/sticking process may not reach this effect (additional elongation). However, this effect would be important in studying the detachment of the virus from the surface (rupture of spikes at separation virus from a cell surface) when additional elongation of the spike could be observed ([PERSON] [32]).  
The interaction of the erythrocyte is described by the application of a total force:  
\[F(t)=F_{el}(t)+F_{adh}(t)+F_{dl}(t)+F_{adh,diss}(t)+F_{viscous}(t) \tag{2}\]  
where \(F_{el}(t)\) is the elastic force, \(F_{adh}(t)\) is the adhesion force, \(F_{dl}(t)\) is the electrostatic double layer force, \(F_{adh,diss}(t)\) is the adhesive-dissipative force and \(F_{viscous}(t)\) is the viscous damping force. In this work, the problem is the dynamics of the cell; therefore, both force and displacement are not controlled in time; the resulting values (force, velocity, displacement) are obtained only after integrating non-linear force-displacement equations (integrated with the 5 th Gear predictor-corrector scheme).