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).
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Notes
- 1.
Continuous as opposed to discrete: we aim at building a pressure field as a function over the domain occupied by the parenchyma, in some functional space which is to be defined.
- 2.
We shall investigate the possibility to embed the set of leafs or set of ends of this tree, considered as an infinite set of points, into a physical domain, but the tree itself, i.e. the collection of finite size pipes, is such that the infinite extrapolated tree would have an infinite volume.
- 3.
At least in the central part of the tract, i.e. after generation 2 or 3, and before generation 16 or 17, beyond which gas exchange occur (see Section 7.1, p. 215, for more details).
- 4.
One should not overinterpret the fact that the critical value is the same. It comes from the fact that homogeneities of the resistance and volume are opposite, and that resistances in parallel sum up in a reciprocal manner, whereas volumes sum up in a straight way. Yet, it is striking to notice that identity of critical values is still true in the 2-dimensional setting (homogeneity −2 for the Poiseuille’s resistance, and 2 for the volume), and would actually remain true in any dimension d > 3 if one would extend in a natural way Stokes equations to higher dimensions.
- 5.
We rule out the values 0 and +∞ for resistances; they could be allowed by considering that a 0 resistance between two vertices means that they are in fact identified, and an infinite resistance that they are not connected.
- 6.
Notice that this classification follows the recurrent character of the Brownian motion if ℝd (recurrent for d = 1, 2, transient otherwise).
- 7.
The fact that we restrict the L2 part of the H 1 norm to abounded domain is essential here. Indeed, in the euclidean context, the standard identification H 1 (ℝ) = H1 (ℝ) in the one-dimensional situation, with an H 1 norm defined in a standard way, means that functions in H 1 do actually vanish in some way at infinity, because the L 2 norm has to be finite. One may say that, depending on the way we define the H 1 norm, the identification between H 1 and H 0 1, which holds true in both cases, has opposite meanings.
- 8.
This object is called the projective limit of the system(ℤ/2nℤ,φ m n )-Let us say that a general understanding of this notion of projective limit is not necessary to follow this section.
- 9.
For any measurable set A ⊂ ℤ2, x ∈ℤ2, μ(x+A) = μ(A).
- 10.
The connected components of ℤ2 are singletons.
- 11.
Note that, due to the ultrametric character of ℤ2 (a ball is centered at any of its element), it contains many isometries. In particular any mapping built in a “Haar basis spirit” by interchanging the two branches of a subtree preserves distances. If one considers for example a tree such that the resistance of the left hand branch (which irrigates ℤ2) is infinite, and the resistance of the other branch (which irrigates 1 + 2ℤ2) is finite, given a function f ∈ γ0(H 1), then f ο T ∉ γ0(H 1) in general, where T interchanges both 2ℤ2 and 1 + 2ℤ2 (T (x) is x + 1 in 2ℤ2 and x − 1 in 1 + 2ℤ2).
- 12.
Note that this measure is not in general absolutely continuous with respect to Haar measure μ
- 13.
Note that \( {\zeta}_2\left(\beta \right)={\displaystyle \sum_{k=0}^{+\infty}\frac{1}{2^{\beta k}}} \), so that the product of the ζ P (β) over all prime numbers is ζ(β).
- 14.
Equivalently, it is spanned by the image by Ψ of the characteristic functions of balls of radius 1/2n. There are only 2n of them in ℤ2, which is the dimension of V n .
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© 2013 Springer-Verlag Italia
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Maury, B. (2013). The lung as an infinite tree. In: The Respiratory System in Equations. MS&A — Modeling, Simulation and Applications. Springer, Milano. https://doi.org/10.1007/978-88-470-5214-7_6
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DOI: https://doi.org/10.1007/978-88-470-5214-7_6
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