Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries

Abstract

Flow and pressure waves emanate from the heart and travel through the major arteries where they are damped, dispersed and reflected due to changes in vessel caliber, tissue properties and branch points. As a consequence, solutions to the governing equations of blood flow in the large arteries are highly dependent on the outflow boundary conditions imposed to represent the vascular bed downstream of the modeled domain. The most common outflow boundary conditions for three-dimensional simulations of blood flow are prescribed constant pressure or traction and prescribed velocity profiles. However, in many simulations, the flow distribution and pressure field in the modeled domain are unknown and cannot be prescribed at the outflow boundaries. An alternative approach is to couple the solution at the outflow boundaries of the modeled domain with lumped parameter or one-dimensional models of the downstream domain. We previously described a new approach to prescribe outflow boundary conditions for simulations of blood flow based on the Dirichlet-to-Neumann and variational multiscale methods. This approach, termed the coupled multidomain method, was successfully applied to solve the non-linear one-dimensional equations of blood flow with a variety of models of the downstream domain. This paper describes the extension of this method to three-dimensional finite element modeling of blood flow and pressure in the major arteries. Outflow boundary conditions are derived for any downstream domain where an explicit relationship of pressure as a function of flow rate or velocities can be obtained at the coupling interface. We developed this method in the context of a stabilized, semi-discrete finite element method. Flow rate and pressure distributions are shown for different boundary conditions to illustrate the dramatic influence of alternative boundary conditions on these quantities.