## 2 Methods
### CFD Simulations  
For a homogeneous incompressible Newtonian fluid with a low Reynolds number, viscosity is considered unchanged at a stable temperature, and continuum flow is governed by the Navier-Stokes equation,  
\[\
abla\cdot\
u=0 \tag{4}\]  
\[\frac{\partial\
u}{\partial t}+(\
u\cdot\
abla)v=f-\frac{1}{\rho}\
abla p+v \
abla^{2}v \tag{5}\]  
where \(\frac{\partial\
u}{\partial t}\) is unsteady acceleration, \((\
u\cdot\
abla)v\) is convective acceleration, and \(f\), \(\frac{1}{\rho}\
abla p\), and \(v\
abla^{2}\
u\) represent the effect of body force, pressure gradient, and viscosity. When the Reynolds number is very small, inertial effects are negligible compared to viscous effects, so Equation (5) could be rewritten as the following,  
\[\
abla p=f+\mu\
abla^{2}\
u \tag{6}\]  
which is called the Stokes equation. Cell movement studies in biomechanics could be considered as Stokes flow because of their small dimension and slow velocity. Corresponding to the experimental observations, a blood vessel was simplified as a 3D channel with a dimension of 1000 \(\upmu\)m (length) \(\times\) 700 \(\upmu\)m (width) \(\times\) 550 \(\upmu\)m (height). Flow was fully developed to Poiseuille flow under a low Reynolds number condition. The melanoma cell was modeled as an ellipsoid, while the PMN was modeled as a teardrop by the following equation,  
\[\frac{y^{2}}{b^{2}}+\frac{z^{2}}{a^{2}}=\left(1-\frac{x_{2}}{a^{2}}\right) \left(1+\frac{\beta x}{a}\right)^{2} \tag{7}\]  
where constant \(a\) is one half of cell length, and unknown constants (\(b\), \(c\), \(\beta\)) are determined as [PERSON] et al. previously described [30]. The pre-deformed shape of the cells is determined by the measurements in Tables 1 and 2. In order to save time, a region of 300 \(\upmu\)m (length) \(\times\) 150 \(\upmu\)m (width) \(\times\) 150 \(\upmu\)m (height) in the central bottom flow was actually calculated. A PMN was fixed in the center of the bottom wall, and the upper stream profile was simplified to linear distribution. The boundary conditions were set to be velocity inlet and pressure outlet. The side walls and the top wall were treated as velocity boundaries, while the bottom wall was considered a fixed wall. Commercial software ANSYS-FLUENT was used to solve the problem with the single-precision solver and SIMPLE algorithm. Since quasi-steady state is an important characteristic of Stokes flow, the instant flow field is determined only by the boundary conditions at these time periods and is not affected by the movement history of particles and other boundaries. During the process of \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{3}{*}{**Conditions**} & \multicolumn{6}{c}{**PMN**} & \multicolumn{6}{c}{**TC**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Low Shear**} & \multicolumn{2}{c}{**High Shear**} & \multicolumn{2}{c}{**Low Shear**} & \multicolumn{2}{c}{**High Shear**} \\ \cline{2-9}  & **L (\(\upmu\)m)** & **H (\(\upmu\)m)** & **L (\(\upmu\)m)** & **H (\(\upmu\)m)** & **L (\(\upmu\)m)** & **H (\(\upmu\)m)** & **L (\(\upmu\)m)** & **H (\(\upmu\)m)** \\ \hline isolated & 8.8 & 5.2 & 11.6 & 4.9 & 15.3 & 14.48 & 15.15 & 11.4 \\ approach & 9.3 & 5.55 & 10.3 & 4.9 & 12.3 & 9.3 & 12.2 & 9.9 \\ collision & 8.25 & 4.8 & 6.6 & 6.6 & 14.25 & 13.65 & 14.7 & 12.15 \\ transient & 8.7 & 8.1 & 7.8 & 6.7 & 13.35 & 13.05 & 14.6 & 13.5 \\ aggregate & 7.65 & 6.3 & 7.05 & 5.85 & 15.3 & 12.0 & 12.3 & 10.05 \\ detach & 9.3 & 4.65 & 10.95 & 5.7 & 14.25 & 11.85 & 15.45 & 11.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cell dimensions measured in sFg solution.