In this talk, an overview of the results obtained in the ANR project BioFiReaDy will be presented. This interdisciplinary project aims at understanding mucociliary clearance: the natural flow of mucus in the lung, which protects bronchus from inhaled agents such as dust, pathogens or pollution particles... 3D simulations are used, as well as experimental rheology to characterize the non-Newtonian rheological behaviour of this complex biological fluid. Using suitable numerical algorithms specially developed to compute this 3D fluid-structure interaction (FSI) problem (mucus is propelled by beating cilia), it is now possible to study the biological parameters influence on the mucus flow. These algorithms are based on (i) a novative projection method which integrates the resolution of rheological shear-thinning effects, (ii) a penalization techniques coupled to a Shermann--Morrison--Woodbury approach to compute the FSI, (iii) the use of fast solvers (FFT or multigrid) and (iv) a lagrangian discretization of the convection effects. These numerical methods are particularly suitable for 3D computations since the global computational cost is linear with respect to the number of grid points, which is very interesting for a parametric study. Several results will be presented, in particular for pathological situations such as cystic fibrosis.