The stationary flow in a heterogeneous compliant vessel network

Abstract

We introduce a mathematical model of the hydrodynamic transport into systems consisting in a network of connected flexible pipes. In each pipe of the network, the flow is assumed to be steady and one-dimensional. The fluid-structure interaction is described through tube laws which relate the pipe diameter to the pressure difference across the pipe wall. We show that the resulting one-dimensional differential equation describing the flow in the pipe can be exactly integrated if one is able to estimate averages of the Reynolds number along the pipe. The differential equation is then transformed into a non linear scalar equation relating pressures at both ends of the pipe and the flow rate in the pipe. These equations are coupled throughout the network with mass conservation equations for the flow and zero pressure losses at the branching points of the network. This allows us to derive a general model for the computation of the flow into very large inhomogeneous networks consisting of several thousands of flexible pipes. This model is then applied to perform numerical simulations of the human lung airway system at exhalation. The topology of the system and the tube laws are taken from morphometric and physiological data in the literature. We find good qualitative and quantitative agreement between the simulation results and flow-volume loops measured in real patients. In particular, expiratory flow limitation which is an essential characteristic of forced expiration is found to be well reproduced by our simulations. Finally, a mathematical model of a pathology (Chronic Obstructive Pulmonary Disease) is introduced which allows us to quantitatively assess the influence of a moderate or severe alteration of the airway compliances.