Geometrical multiscale modeling is a strategy advocated in computational hemodynamics for representing in a single numerical model dynamics that involve different space scales. This approach is particularly useful to describe complex vascular networks and has been applied to the study of cerebral vasculature, where a one-dimensional (1D) description of the circle of Willis, relying on the one-dimensional Euler equations, has been coupled to a fully three-dimensional (3D) model of a carotid artery, based on the solution of the incompressible Navier-Stokes equations.
Even if vascular compliance is often not relevant to the meaningfulness of 3D results (e.g. in large arteries), it is crucial in the multiscale model, since it is the driving mechanism of pressure wave propagation. Unfortunately, 3D simulations in compliant domains still demand computational costs significantly higher than the rigid case. Appropriate matching conditions between the two models have been devised to gather the effects of the compliance at the interfaces and to obtain reliable results still solving a 3D problem on rigid vessels.
More precisely, we introduce a lumped parameter model at the interface, in the form of a RCL network, giving a simplified representation of the compliance of the 3D vessel in the multiscale model. For simple cases, e.g. a cylindrical pipe, numerical results are promising, showing that the multiscale model can both capture the correct wave propagation (in comparison with a fully 1D model) and compute the local 3D flow. In more complex situations, like the circle of Willis, results compare well with a fully 1D model, however a mathematically sound fine tuning of the parameters is required.
We point out that this approach can be easily extended, for instance to the analysis of the coronary artery bypass, the 3D model representing the grafted and the host arteries, and the coronary circulation being described by 1D models.
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