In this dissertation, I developed a boundary integral formulation to simulate a red blood cell (RBC) squeezing through a submicron slit under prescribed inlet and outlet pressures. The main application of this computational study is to investigate splenic filtrations of RBCs and the corresponding in vitro mimicking microfluidic devices, during which RBCs regularly pass through inter-endothelial slits with a width less than 1.0 micrometer. The diseased and old RBCs are damaged or destroyed in this mechanical filtration process.We first derived the boundary integral equations of a RBC immersed in a confined domain with prescribed inlet and outlet pressures. We applied a unified self-adaptive quadrature to accurately evaluate singular and nearly singular integrals, which are especially important in this fluid-structure interaction problem with strong lubrication. A multiscale model is applied to calculate forces from the RBC membrane, and it is coupled to boundary integral equations to simulate the fluid-structure interaction.After multi-step verifications and validations against analytical and experimental results, we systematically investigated the effects of pressure drop, volume-to-surface-area ratio, internal viscosity, and membrane stiffness on RBC deformation and internal stress. We found that spectrins of RBCs could be stretched by more than 2.5 times under high hydrodynamic pressure, and that the bilayer tension could be more than 500pN/（mu*m）, which might be large enough to open mechanosensitive channels but too small to rupture the bilayer. On the other hand, we found that the bilayer-cytoskeletal dissociation stress is too low to induce bilayer cytoskeleton detachment and bilayer vesiculation. A variety of cell shapes have been observed when they transmigrate the microfluidic slit, such as small tip, "Jellyfish" shape, "half-moon" shape and "whale-tail" shape. We found that two biophysical and physical conditions play a significant role for these various shapes. One is the adhesion force between the cell membrane and the channel wall. The other one is the slip condition between the cell membrane and the fluids inside the cell. In addition, we found that the surface-area-to-volume ratio has a vita effect on cell front shapes. Furthermore, we carefully examined the transit time for the cell to pass through the slit. We found that higher pressure drop and larger surface-area-to-volume ratio dramatically reduce the transit time. On the other hand, the transit time increases significantly when the viscosity of the internal fluid increases. The numerical method is also applied to another application to study the physical arresting of circulating tumor cells in the mouse brain vasculature. We predicted the mechanical microenvironment in the in vivo mouse brain vasculature. More importantly, we predicted the cell deformation and internal stress on the cell membrane under distinct pressure drops.