Free Interfaces at the Tips of the Cilia in the One-Dimensional Periciliary Layer

Abstract

Cilia on the surface of ciliated cells in the respiratory system are organelles that beat forward and backward to generate metachronal waves to propel mucus out of lungs. The layer that contains the cilia, coating the interior epithelial surface of the bronchi and bronchiolesis, is called the periciliary layer (PCL). With fluid nourishment, cilia can move efficiently. The fluid in this region is named the PCL fluid and is considered to be an incompressible, viscous, Newtonian fluid. We propose there to be a free boundary at the tips of cilia underlining a gas phase while the cilia are moving forward. The Brinkman equation on a macroscopic scale, in which bundles of cilia are considered rather than individuals, with the Stefan condition was used in the PCL to determine the velocity of the PCL fluid and the height/shape of the free boundary. Regarding the numerical methods, the boundary immobilization technique was applied to immobilize the moving boundaries using coordinate transformation (working with a fixed domain). A finite element method was employed to discretize the mathematical model and a finite difference approach was applied to the Stefan problem to determine the free interface. In this study, an effective stroke is assumed to start when the cilia make a 140∘ angle to the horizontal plane and the velocitiesof cilia increase until the cilia are perpendicular to the horizontal plane. Then, the velocities of the cilia decrease until the cilia make a 40∘ angle with the horizontal plane. From the numerical results, we can see that although the velocities of the cilia increase and then decrease, the free interface at the tips of the cilia continues increasing for the full forward phase. The numerical results are verified and compared with an exact solution and experimental data from the literature. Regarding the fixed boundary, the numerical results converge to the exact solution. Regarding the free interface, the numerical solutions were compared with the average height of the PCL in non-cystic fibrosis (CF) human tissues and were in excellent agreement. This research also proposes possible values of parameters in the mathematical model in order to determine the free interface. Applications of these fluid flows include animal hair, fibers and filter pads, and rice fields.