The rupture of red blood cells, known as hemolysis, is a major problem across a number of cardiovascular flows. Computational fluid dynamics is a valuable tool for evaluating the risk of hemolysis in designing new medical devices without posing a risk to patient health. Existing algorithms for hemolysis detection, however, are often empirical and lack a mechanistic underpinning linked to the actual red blood cell behavior. This work addresses this weakness by presenting and analyzing a framework to simulate individual red blood cells traversing macroscale flows that are of interest to hemolysis. The existing rheological model adopted in simulating whole blood in larger organ-scale flows produces relatively accurate results. Since this rehological model works primarily by modeling the microscale effects of the cells on the macroscopic behavior of the fluid, we employ the resolved organ-scale flow to simulate individual cells in a one-way coupled manner. The precursor to this process is computing the trajectory of the cells as Lagrangian tracers in the larger organ-scale flow, which could become prohibitively expensive in large, unstructured grids. To address this need, we present an algorithm to track these tracers and detect their binary collisions in an optimal manner. We then introduce a boundary integral method solver for simulating the motion of the red blood cell membrane. The red blood cell is treated as a thin membrane in Stokes flow situated in an infinite domain. These assumptions are made based on the microscopic size of the cells and allow for the motion of the membrane to be solved without solving for the velocity of the surrounding fluid. This fluid-mesh-free method is tested and used to validate the red blood cell behavior in a set of canonical geometries, showing good agreement with experiments and numerical benchmarks. We additionally showcase this solver in a realistic setting by employing it to study the behavior of cells in a patient anatomy receiving a Blalock-Taussig shunt. Further, we investigate the effect of cell-cell interactions on parameters of interest to hemolysis. In shear flow, we show that the effect of cell-cell interactions on individual cells can be captured by adjusting the viscosity of the fluid surrounding the cell to that of the whole blood. As long as all potential orientations of the cell are accounted for, the parameters of interest match well to first-order. By adjusting the surrounding fluid viscosity, one can perform these predictions using a single-cell simulation rather than multiple-cell simulations, thus significantly reducing the computational cost of these calculations. Finally, the solver is used to investigate the effect of turbulence on red blood cells, a phenomenon that is relatively unexplored from a cell-resolved perspective. The cells were examined in laminar and turbulent flows where the two flows have the same total shear stress on average. This study showed that the cells tend to undergo greater deformation on average in the laminar flow. However, in what is likely more relevant to hemolysis, the cells undergo a greater frequency of extreme, large-deformation events in the turbulent simulations.