The lung as an infinite tree

Abstract

We investigate in this chapter the possibility to elaborate a mathematical model of the lung as an infinite resistive tree. The approach is an extension of Section 3.3 (p. 74) to the case of infinite networks. In the infinite setting, the notions of boundary and functions on this boundary have to be designed with care. We give some general properties of networks regarding those questions, in particular the possibility to prove trace theorems, and we detail the case of an infinite dyadic tree. We propose (in Section 6.2.1) a framework based on the ring of dyadic integers, which makes it possible to quantify the regularity of trace functions in a very natural way, by means of an adapted Fourier transform. We investigate in Section 6.3 the possibility to embed the tree (its ends) into a physical domain. This approach raises difficult regularity issues: if a trace function has some regularity with respect to the tree, what about its embedded counterpart? We shall see that, under some assumptions on the way the tree is plugged into the domain, some regularity of the embedded field holds. The approach proposed here is quite academic, since its core (the infinite tree) has an abstract nature which may seem quite unrelated to the actual respiratory tract. Yet, it gives a better understanding of the notion of pressure field within the parenchyma, and it allows to build continuous mechanical models of the overall lung, while respecting the very nature of dissipative phenomena in the tree (Sections 6.4 and 6.5).