Comparisons between reduced order models and full 3D models for fluid-structure interaction problems in haemodynamics

When modelling the cardiovascular system, the effect of the vessel wall on the blood flow has great relevance. Arterial vessels are complex living tissues and three-dimensional specific models have been proposed to represent their behaviour. The numerical simulation of the 3D-3D Fluid-Structure Interaction (FSI) coupled problem has high computational costs in terms of required time and memory storage. Even if many possible solutions have been explored to speed up the resolution of such problem, we are far from having a 3D-3D FSI model that can be solved quickly. In 3D-3D FSI models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part. Nevertheless, in many cases, we are interested in the blood flow dynamics in compliant vessels, whereas the displacement of the domain is small and the structure dynamics is less relevant. In these situations, techniques to reduce the complexity of the problem can be used. One consists in using transpiration conditions for the fluid model as surrogate for the wall displacement, thus allowing problem's solution on a fixed domain. Another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions (Figueroa et al., 2006, Nobile and Vergara, 2008) instead of using a more realistic (but more computationally intensive) 3D elastodynamic model. Using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow. Combining the transpiration conditions with the membrane model assumption, we obtain an attractive formulation, in fact, instead of solving two different models on two moving physical domains, we solve only a Navier-Stokes system in a fixed fluid domain where the structure model is integrated as a generalized Robin condition. In this paper, we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress-strain constitutive relation adopted for the vessel wall structure. Our aim is, first, to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme, then to compare a 3D-3D FSI model and a reduced FSI one in two realistic patient-specific cases: a femoropopliteal bypass and an aorta. In particular, we are interested in comparing the wall shear stresses, in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis. More in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models. In particular, we show that, in the case of small displacements, using a 3D-3D FSI linear elastic model or the correspondent reduced order one yields many similar results.