Mathematical Modeling of Mucociliary Clearance: A Mini-Review

Abstract

Mucociliary clearance is an important innate host defense of the mammalian respiratory system, as it traps foreign substances, including pollutants, pathogens, and allergens, and transports them out of the airway. The underlying mechanism of the actuation and coordination of cilia, the interplay between the cilia and mucus, and the formation of the metachronal wave have been explored extensively both experimentally and mathematically. In this mini-review, we provide a survey of the mathematical models of mucociliary clearance, from the motion of one single cilium to the emergence of the metachronal wave in a group of them, from the fundamental theoretical study to the state-of-the-art three-dimensional simulations. The mechanism of cilium actuation is discussed, together with the mathematical simplification and the implications or caveats of the results.

Keywords: cilia; clearance; mathematical modeling; mucus.

Figure 1

Schematic plots of the mucociliary system in the lung: (a) a human lung structure, (b) enlargement near the lung surface, from Blake [14] with permission.

Figure 2

(a) Cilia in the rabbit tracheal, from Sanderson and Sleight [31] with permission. (b) Flagella of a normal spermatozoon, from Oliveira et al. [32] with permission. (c) The beating pattern of a cilium can be separated into an effective stroke (left arrows) and a recovery stroke (right arrows). (d) The beating pattern of a flagellum initiates from the base and propagates to the tip as a curly wave.

Figure 3

Internal structure of a motile cilium. (a) A transmission electron microscopic image of an axoneme, with permission from Dirksen & Satir [49]. (b) A schematic drawing of the “9+2” axoneme illustrating 9 microtubule doublets connected to the central two microtubules through radial spokes. Also shown are the nexin links connecting the doublets and the dynein motor arms expending from doublets.

Figure 4

Illustration of (a) the curvature and shear waves approximating the flagellum undulating motion, with permission from Brokaw [63], (b) the tangential and normal forces (resistance forces) along the flagellum motile body, with permission from Gray [66].

Figure 5

Illustration of the stokeslet method: (a) migration of the elastic spirochete, with permission from Cortez et al. [77], and (b) bundling of flagella, with permission from Flores et al. [78].

Figure 6

(a) The cilium modeled as a finite element beam and the nearby body-fitted grid, from Mitran [85] with permission; (b) the cilium modeled as an elastic spring network, from Yang, Dillon and Fauci [87] with permission; (c) streamlines in the fluid flow induced by five cilia (rods) at a sequence of times, from Xu and Jiang [34] with permission; (d) snapshot of an large array of beating cilia, from Elgeti and Gompper [88] with permission.