A 2-adic approach of the human respiratory tree

TL;DR

This paper introduces a 2-adic mathematical framework for modeling the human bronchial tree, enabling precise analysis of pressure fields and ventilation using Sobolev spaces and Fourier transforms on the 2-adic integers.

Contribution

It develops a novel 2-adic approach to analyze pressure and flux in the human respiratory tree, connecting Sobolev space theory with biological modeling.

Findings

Established Sobolev space structure on the infinite resistive tree

Derived trace theorems similar to classical Sobolev spaces

Explicitly formulated the ventilation operator as a Riesz kernel convolution

Abstract

We propose here a general framework to address the question of trace operators on a dyadic tree. This work is motivated by the modeling of the human bronchial tree which, thanks to its regularity, can be extrapolated in a natural way to an infinite resistive tree. The space of pressure fields at bifurcation nodes of this infinite tree can be endowed with a Sobolev space structure, with a semi-norm which measures the instantaneous rate of dissipated energy. We aim at describing the behaviour of finite energy pressure fields near the end. The core of the present approach is an identification of the set of ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can be defined in a very natural way by means of Fourier transform, which allows us to establish precised trace theorems which are formally quite similar to those in standard Sobolev spaces, with a Sobolev regularity which depends on the growth rate of resistances, i.e. on geometrical properties of the tree. Furthermore, we exhibit an explicit expression of the "ventilation operator", which maps pressure fields at the end of the tree onto fluxes, in the form of a convolution by a Riesz kernel based on the 2-adic distance.