We propose to present in parallel two mathematical models for the lungs, obtained by pursuing opposite strategies:
1) OD model based on a single-pipe equivalence for the bronchial tree, single box for the alveoli, and single spring for the thoracic cage;
2) Infinite dimensional model: the number of generation is sent to infinity, so that the bronchial tree is identified to an infinite dyadic tree, which is embedded onto a continuous elastic medium.
The first strategy, which has given rise to several models in the last decades,
gives a fairly good representation of normal breathing and some forced maneuvers, and it can be used to study fine properties of the respiration process.
In particular we shall present how, by defining the efficiency of the lungs as depending on the quantity of diffused oxygen in the blood, it gives some insight in the possible role of the smooth muscle in the overall respiration process.
The second one raises some delicate mathematical issues related to
the proper definition of a pressure field on the collection of alveola, and to the way the set of leafs is actually embedded onto the respiratory organ.