On simple well-balanced semi-implicit and explicit numerical methods for blood flow in networks of elastic vessels with applications to FFR prediction

Abstract

This paper proposes a semi-implicit 2D and a fully explicit 1D finite volume scheme for the simulation of blood flow in axially symmetric compliant vessels characterized by variable mechanical and geometrical parameters. The computational efficiency of the two methods are compared using patient-specific simulations designed to predict the haemodynamic impact of partial vessel occlusion in coronary trees.

The first method is a staggered semi-implicit one and solves a two-dimensional blood flow model with moving boundaries, derived from the Navier–Stokes equations in an axially symmetric geometry, by splitting it into two subsystems: one containing the nonlinear convective terms and a second subsystem for the pressure-related terms. An explicit approach is used for the nonlinear convective terms, while the pressure subsystem is treated implicitly. This leads to a CFL-type time step restriction which depends only on the bulk velocity of the flow and not on the speed of the pressure waves. The scheme is by construction well balanced for flow at rest and variable material parameters.

The second method is a novel fully explicit collocated path-free path-conservative finite volume scheme for simulating one-dimensional blood flow in networks of elastic vessels. The method is exactly well-balanced for flow at rest and general material parameters.

Both methodologies are then coupled to a simple 3D approach for the treatment of junctions where each junction is represented by a 3D cell and the Euler equations are employed to approximate the velocity and pressure unknowns. Thanks to a multidimensional numerical flux which takes into account the elementary information of the junction geometry, namely the normal vectors and areas of the incident vessels, the schemes are able to correctly capture the reflected waves, taking into account the effect of the different incident angles of the vessels at a junction.

The proposed methodologies are first validated using classical computational fluid dynamics benchmark tests and then applied to solve the flow dynamics in a network of multiple elastic arteries. In addition, to demonstrate the ability of the proposed methods to deal with a real clinical context, we study hemodynamics in patients affected by stable coronary artery disease, the pathological condition that occurs when an abnormal narrowing of the vessel wall is present. The capability of both methods to predict the Fractional Flow Reserve (FFR) index is shown and the results are compared with in vivo measurements and numerical estimates obtained with a 3D flow solver for a large number of patients.