This paper presents a numerically efficient implementation of the Immersed Boundary Method (IBM), originally developed by  to simulate fluid/elastic-structure interactions. The fluid is assumed to be incompressib0le with uniform density, viscosity, while the immersed boundaries have fixed topologies with a linear elastic behavior. Based on the finite-difference method, a major numerical advantage of the IBM is the high level of uniformity of mesh and stencil, avoiding the critical interpolation processes of the cut-cell/direct methods. The difficulty of accurately simulating interaction phenomena involving moving complex geometries can be overcome by implementing large and parallel IBMcomputations on fine grids, as described in . While this paper is restricted to a two-dimensional low-Reynolds-number flow, most of the concepts introduced here should apply to three-dimensional bio-flows.We describe here the decomposition techniques applied to the IBM, in order to decrease the computational time, in the context of the parallel Matlab toolbox of . Finally, we apply the method to a blood-like suspension flow test-case.