# Seminário Matemática Aplicada e Análise Numérica

### On the development of a numerical model for the simulation of air flow in the human airways

The main motivation for this study is the air flow in the human respiratory system, although similar problems are also common in other areas of biomedical, environmental or industrial fluid mechanics. The detailed experimental studies of respiratory system in humans and animals are very challenging and even impossible in many cases due to various medical, technical or ethical reasons. This leads to the development of more and more realistic mathematical and numerical models of the flow in airways including the complex geometry of the problem, but also various fluid- and bio-mechanics features. The main difficulties are not just in the geometrical complexity of the computational domain with several levels of branching, but also in the need to prescribe mathematically suitable, but yet sufficiently realistic boundary conditions for the computational model. This leads to a complex multiscale problem, whose solution requires large amount of complicated and time-consuming numerical calculations.

In this work we are considering simplified simulations in a two-dimensional rigid channel coupled with a one-dimensional extended flow model derived from a 3D fluid-structure interaction (FSI) model under certain conditions. For this purpose we built a simple test code employing an immersed boundary method and a finite difference discretization. At this stage the air flow in human airways is considered as incompressible, described by the Navier-Stokes equations. This simple code was developed with the aim of testing and improving boundary conditions using reduced order models. The incompressible model will later be replaced by a compressible one, to be able to evaluate the impact of intensive pressure changes in human airways while using realistic, patient specific airways geometry. The main idea is to use different dimensional models, 3D(2D), 1D and 0D, with different levels of complexity and accuracy and to couple them into a single working model.

In the present talk, first results of the 2D-1D coupled toy model will be presented, focusing on the main features of the computational setup, coupling strategy and parameter sensitivity. In addition, some long term outlook of the more complex 3D-1D(-0D) model will be discussed.

Acknowledgment: Center for Computational and Stochastic Mathematics - CEMAT (UIDP/04621/2022 IST-ID).