Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids. In biofluidic applications, including the interaction between blood flow and cardiovascular tissue, the coupling between the fluid and structure is highly nonlinear because the density of the structure (tissue) and the density of the fluid (blood) are roughly the same. In such problems, geometric nonlinearities of the fluid-structure interface and significant exchange in the energy between the moving fluid and structure play important roles in the physical and mathematical description of the underlying biological problem. The problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with different mechanical characteristics. No mathematical results exist so far that analyze existence of solutions to fluid-structure interaction problems in which the structure is composed of several different layers. In this talk we summarize the main difficulties in studying the underlying problem, and present a computational scheme based on which the existence of a weak solution to this class of FSI problems was obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of the fluid-structure interface with mass regularizes evolution of the FSI solution. This means that in our large (muscular) arteries, the inner-most layer of arterial walls, which consists of an elastic lamiae covered with endothelial cells, smooths out the propagation of the pressure wave in the cardiovasuclar system. All theoretical results will be illustrated with numerical examples.  This is a joint work with Boris Muha (University of Zagreb, Croatia), and with Martina Bukac (Notre Dame).