Multiscale Modelling of the Respiratory Track : a Theoretical Framework

Abstract. We propose here a decomposition of the respiratory tree into three stages which correspond to different mechanical models. The resulting system is described by the Navier-Stokes equation coupled with an ODE (a simple spring model) representing the motion of the diaphragm muscl. We prove that this problem has at least one solution locally in time for any data and, in the special case where the external forces are equal to zero and if the initial conditions are small enough, that the solution exists globally in time. Note moreover that, in the case where the spring stiffness is equal to zero, we obtain an existence result globally in time provided, once again, that the data are small enough. The behaviour of the global model is illustrated by two-dimensional simulations.