Analysis of lumped parameter models for blood flow simulations and their relation with 1D models

Abstract

Abstract. This paper provides new results of consistence and convergence of the lumped parameters (ODE models) toward one-dimensional (hyperbolic or parabolic) models for blood flow. Indeed, lumped parameter models (exploiting the electric circuit analogy for the circulatory system) are shown to discretize continuous 1D models at first order in space. We derive the complete set of equations useful for the blood flow networks, new schemes for electric circuit analogy, the stability criteria that guarantee the convergence, and the energy estimates of the limit 1D equations.Mathematics Subject Classification. 35L50, 35M20, 47H10, 65L05, 76Z05.

Methods of Blood Flow Modelling

“…More appropriate outflow boundary conditions are studied in [100]: selected parts of small arteries and microcirculation are modelled as structured trees whose root impedance can be evaluated from linearization of the governing equations. Many authors perform time-domain coupling of the 1D blood flow models with electric circuits (lumped parameters) 0D models [4,16,36,73,80,87,92]. The 0D models provide correct boundary conditions for global blood flow models reproducing specific reflections from natural microcirculatory bed.…”

A 1D pipe finite element with rigid and deformable walls

“…is a dimensionless energy parameter often used in pipe flow (Formaggia et al 2001, Sherwin et al 2003a, Milisic and Quarteroni 2004, Sherwin et al 2003b, Sochi 2013 …”

Spectral models for 1D blood flow simulations

“…This hydraulic network is a closed loop and shows a different behavior at various locations. For instance, the wave propagation in the arteria tree is of greater influence than in the capillary bed where the flow is almost steady [1]. The mathematical and numerical modeling of the human cardiovascular system has become a topic of great interest in the recent years (see e.g.…”

“…[7], [8], [9]). For instance, the Windkessel and similar lumped models are often used to represent blood flow and pressure in the arterial system [1]. These lumped models can be derived from electrical circuit analogies where current I represents arterial blood flow and voltage V represents arterial pressure.…”

“…Over the last years, one-dimensional models of human arterial system have been intensively investigated [1], [14], [9], [2]. In [1], it has been shown that, for the particular case of blood flow, lumped networks can be regarded as first order discretization of one-dimensional linear systems.…”