The lung as a resistive tree

Abstract

This chapter gives a theoretical framework for the modeling of the respiratory tract as a resistive tree. We shall consider here the respiratory tree, or more general networks, as a collection of interconnected pipes, through which a viscous fluid is flowing according to Poiseuille’s law, so that the flux through an individual pipe depends linearly on the pressure jump between its ends (which are vertices of the network). In order to make this chapter widely accessible, we have collected all abstract considerations (which are applicable to the human lung, but also to general networks) in a final section (Section 3.3). Section 3.1 introduces Poiseuille’s law, which is the main ingredient (together with volume conservation) of the whole chapter. It simply states that when a viscous fluid flows through a pipe, the flow rate is proportional to the pressure drop. As simple as it is, this law results from the analytic solution of Partial Differential Equations, namely the Stokes system. The reader not familiar with PDE’s may simply admit Poiseuille’s law and start from Section 3.2, which introduces simple resistive networks as collections of interconnected pipes through which a viscous fluid flows. This approach reduces the parameters to pressures values at nodes and fluxes through the edges (the pipes) of the network. Dyadic trees like the respiratory tract are given a special attention, in particular the notion of resistance as a matrix is introduced. The next Section 3.3 is more abstract: it provides a general framework for resistive networks, and it defines the notion of resistance between a root and a prescribed set of nodes (which are the leafs in the case of a tree). Section 3.4 is devoted to the numerical estimation of the equivalent resistance of a tree or a network. Both deterministic and stochastic approaches are proposed. Optimality issues are investigated in Section 3.6. The chapter ends with a discussion and bibliographical notes (Section 3.7).