# Finite Element Analysis on Wall Fluid Shear Stress on Cells under Oscillatory Flow
## 4 Discussion  
In a gradient FSS field under unidirectional flow, RAW264.7 osteoclast precursors do not move along the direction of fluid flow but instead migrate toward the low FSS region [7]. This striking phenomenon may be related to the polarized distribution of local wall FSS on the cell surface that may further activate the calcium ion channels on the cell membrane and finally regulate the cell migration (Figure 5). This study mainly aimed to investigate the effects of oscillatory flow frequency, cell spacing, and global FSS level on the ARV of local wall FSS on the cell surface through fluid-solid coupled FE simulation. RAW264.7 osteoclast precursors migrate toward the low FSS region at a low FSS level of 0.1 Pa, but move along the flow direction at a high FSS level 2.0 Pa under a gradient FSS of 0.2 Pa/mm in unidirectional flow [7]. Therefore, the gradient of oscillatory flow adopted in this study was 0.2 Pa/mm, and the two FSS levels were 0.1 and 2.0 Pa. Three frequencies of oscillatory flow including 0.25, 1, and 4 Hz were chosen to mimic the different movement modes of the human body. With the increase in oscillatory frequency, the fluctuating amplitude of ARV on the central cell surface also increased, during which, a zero or negative ARV indicates a reduction in the directional migration of RAW264.7 osteoclast precursors toward the low FSS [23; 24; 25]. At a high FSS level, the fluctuation of ARV intensified. These results indicate that a high level of FSS or a large oscillatory frequency may reduce the directional migration of RAW264.7 osteoclast precursors. Reynolds number and Womersley number are the two physical parameters necessary to characterize an incompressible oscillating fluid flow [26]. The Reynolds number in the present work was 0.5, indicating that the viscous forces are predominant, i.e., the fluid flow is laminar. The Womersley number (\(\alpha\)) is the ratio between the oscillating flow frequency with respect to viscous effects, i.e., \(\alpha=\delta(\omega/\
u)^{0.5}\) (where \(\delta\), \(\omega\) and \(\
u\) are the channel half height, the angular frequency defined as \(\omega=2\pi\)f, and the kinematic viscosity of the fluid) [27]. Therefore, when substituting the parameters of the modified plate flow chamber in this study into the above expression, we obtained \(\alpha=\) 1.0, 2.0 and 4.0 for 0.25, 1 and 4 Hz loading frequencies. In these cases, viscous forces tend to dominate the flow [26]. Therefore, laminar fluid flow was adopted in the fluid-solid coupling simulation in this study. The effect of different cell spacings on the ARV of wall FSS on the central cell surface was also investigated to identify whether the separation distance of osteoclasts is relevant  
Figure 5: Schematic of the mechanism of gradient FSS-induced directional migration of osteoclast precursors. The global gradient FSS first induces the polarization distribution of local wall FSS on cell surface, then activates the mechanosensitive ion channels in the cell membrane and produces the polarized distribution of cytosolic calcium concentration, and finally leads to the directional migration of osteoclast precursors toward the low FSS region. to their migration under fluid flow. The results showed that the ARV on the central cell surface is inversely proportional to the cell spacing; that is, ARV increases when the cell spacing is small. Most of the ARV values are positive, and their fluctuation becomes relatively weak when the cells approach each other. A positive ARV reveals the polarized distribution of local wall FSS that may finally induce the directional migration of RAW264.7 osteoclast precursors to the low FSS region. These results indicate that the crowded osteoclast precursors have a higher tendency to migrate than the separate ones under the fluid flow with gradient FSS. A positive or negative ARV reveals the polarization distribution of local wall FSS on the cell surface and may be used to explain the orientation of cell migration.