Integrated Multi-Model Description of the Human Lungs

The respiratory system realizes the transfer of oxygen from the outside air to the alveolar membrane, through which it diffuses into the blood. As pure diffusion is far from being sufficient to realize that transfer, most of it is of advective type, and this advection is triggered by inflation-deflation cycles of the parenchyma. The mechanical part of the lungs can then be seen as a tree-like domain (conducting airways) embedded in an elastic medium. The flow in the upper part is inertial (incompressible Navier-Stokes equations), whereas inertia can be neglected for deeper branches (Stokes equations), which allows to use Poiseuille’s law for each branch, and consequently Darcy like equations on the corresponding subtrees.

We address here the delicate issues in terms of theory, numerics, and modeling, raised by the coupling of those models (Navier-Stokes, Darcy equations on a network, elasticity equations).

Keywords: Alveolar membrane, Darcy equations on a network, elasticity equations, elastic medium, incompressible Navier-Stokes equations, inflation-deflation cycle, oxygen transfer, parenchyma, respiratory system, tree-like domain, ventilation process.