We adress a depth study about reconstruction methods for blood flows from medical data. We set the procedure as an interpolation problem of the data using physical information about the underlying phenomena. The abstract framework for the variational time-dependent approach we use is discussed in [1], and it is nothing but a continuation of the previos work [2]. The novelties of the presented strategies are: 1) a greedy strategy for the transducer location, 2) realistic 4D-flow images generated using sub-sampled k-space data, 3) viscosity reconstruction with background dynamics using power-law, Carreau and Cross models, 4) the use of a bias correction method for the measurements using an offline built model of the observation process [3]. We explore and verify the feasibility of the reconstruction method with several test-cases, for both, full 3D state estimation (of velocity, pressure and viscosity), and related quantities, such as wall shear stress, pressure drops and vorticity fields.