# Numerical Modelling of Erythrocyte Sticking Mechanics
## 3 Methodology
### Unload, Contact  
Unloading begins after reaching the maximum displacement of the erythrocyte under load. The general expression for the total force is:  
\[F(t)=F_{el}(t)+F_{L,adh}+F_{adh,diss}(t)+F_{dl}(t)+F_{el}(t)+F_{drag}(t)+F_{viscous }(t) \tag{5}\]where \(F_{el}(t)\) is the elastic force (Hertz model), \(F_{L_{,}~{}adh}(t)\) is the adhesion force at point L (Figure 1), \(F_{adh,diss}(t)\) is the dissipative adhesion force, \(F_{dl}(t)\) is the electrostatic double layer force, \(F_{drag}(t)\) is the drag force, and \(F_{viscous}(t)\) is the viscous damping force. A description of each force (unload) is presented by [PERSON] [31].  
Energy dissipation due to adhesion. Energy dissipation \(W_{adh,diss}\) due to the change in influence of adhesion can be described theoretically ([PERSON] [31]):  
\[W_{adh,diss}=k_{H}\cdot\left|F_{L,adh}\cdot a_{F=0}\right| \tag{6}\]  
where \(F_{L,adh}\) is the adhesion force at point L (Figure 1), \(a_{F=0}\) is the initial interaction distance at which the equilibrium of molecular attraction and repulsion potentials is assumed; it is considered as the minimum molecular center separation. The coefficient \(k_{H}\) gives ability to consider the different amount of energy dissipation. In this simulation, this coefficient is equal to \(k_{H}=1\). The dissipative adhesive force is described by the following equation:  
\[F_{adh,diss}(t)=-\frac{2W_{adh,diss}}{h_{U}+a_{F=0}}.\overline{h}(t) \tag{7}\]  
where \(\overline{h}(t)\) is the normal strain and \(\overline{h}(t)=\frac{h_{U}-h(t)}{h_{U}}\) and depends on the maximum overlap (displacement) \(h_{U}\).  
Energy dissipation due to viscous damping. To simulate the dissipative behavior of an erythrocyte during the sticking process, a viscous damping model is taken into account. During dissipative oscillations, the kinetic energy of the erythrocyte is dissipated into heat. The viscous model is described based of the Tsuji model ([PERSON] et al. [33], [PERSON] and [PERSON] [34]):  
\[F_{viscous}(t)=\alpha_{d}\sqrt{m_{eff}K_{el}(t)}.\overline{h}(t)\cdot\left(- \dot{h}(t)\right) \tag{8}\]  
where \(\alpha_{d}\) is the damping coefficient and \(\dot{h}(t)\) is the displacement rate (velocity) [34]. The stiffness \(K_{el}(t)\) of the erythrocyte is taken into account (Hertz model) according to the equation:  
\[K_{el}(t)=\frac{2}{3}E_{eff}\sqrt{R_{eff}h(t)} \tag{9}\]  
where \(E_{eff}\) is the effective modulus of elasticity of the erythrocyte and blood clot, while \(R_{eff}\) represents the effective radius.  
When an erythrocyte interacts with the boundary, the effective mass \(m_{eff}\) is equal to the mass \(m_{eff}=m_{i}\) of the erythrocyte, since the boundary mass \(m_{j}\) (blood clot) is considered to be significantly greater than the mass of the erythrocyte \(m_{j}\gg m_{i}\).  
In the general case, the effective mass \(m_{eff}\) of interacting bodies is determined by the expression:  
\[m_{eff}=\left(\frac{1}{m_{i}}+\frac{1}{m_{j}}\right)^{-1}. \tag{10}\]  
The following process is the reload. This takes into account the decrease in the influence of viscous damping during unloading. The effect of attraction due to adhesion in contact is considered to be time dependent.