Wall shear stress distribution in a compliant airway tree

In the lung, the walls of the bronchi are protected by a mucus layer that is constantly renewed and motioned toward the oesopharyngeal region. Its role is to capture the inhaled particles and to remove them from the lung. The mucus layer is submitted to a shear stress by the air flow in the bronchi. The conditions under which it induces a displacement of the mucus, such as during cough, are still not clearly established. Nevertheless, the air--mucus interaction justifies several common chest physiotherapy technics used to help the draining of the mucus in prevalent diseases such as asthma, bronchiolitis or cystic fibrosis. Hence, understanding how the wall shear stress distributes in the lung could bring interesting highlights on these therapies. We develop a mathematical model to study the distribution of the wall shear stress induced by an outgoing air flow in an airway tree. This model accounts for the main physical processes that determines the wall shear stresses, more particularly the compliance of the airways and the air inertia. We show that the wall shear stress exhibits a maximum within the tree whose amplitude and location depend on the amount of air flow and on the "tissue" pressure surrounding the airways. Our analysis suggests that a tuning of the air flow and tissue pressure might allow to control, at least partially, the air--mucus interaction in the lung.