The idea of the research project is to provide new fast and effective mathematically justified numerical strategy for multiscale modeling of the bloodstream in the network of vessels and for cardiac valves, based on the hybrid dimension models, reducing dimension to one in the main part of the blood circulation system and keeping three-dimensional zooms in small neighborhoods of „specific events“ such as bifurcations, clot formation zones, malformations, stents, etc. This approach will allow to reduce considerably the computational resources of the solvers and accelerate computations, which is especially important in the case of the necessity of an urgent surgery intervention. The novelty of this approach is related to its multiscale nature: starting from a detailed three-dimensional viscous  flow model at microscopic scale (Stokes, Navier-Stokes equations or non-Newtonian fluid equations with rigid or elastic wall boundary conditions), studying an asymptotic behavior of a solution of this complicated mathematical problem (here the small parameter is the ratio of the thickness and the length of vessels) this approach allows to derive rigorously justified partially reduced models keeping very high accuracy of approximation. The objective of the project is to develop multiscale mathematical and numerical tools (multiscale models and new effective mathematically justified methods) for the blood circulation, to improve the existing model of the mitral valve surgery and to create an interdisciplinary team at Vilnius University with strong international leadership in multiscale mathematical modeling with biomedical applications. This team will possess and continue to develop the effective numerical tools and software for modeling of several cardiac surgery manipulations. The tools and the software will be used as a supplementary source of information in preparation of surgery operations.