A new index for characterizing micro-bead motion in a flow induced by ciliary beating: Part I, experimental analysis

Abstract

Mucociliary clearance is one of the major lines of defense of the respiratory system. The mucus layer coating the pulmonary airways is moved along and out of the lung by the activity of motile cilia, thus expelling the particles trapped in it. Here we compare ex vivo measurements of a Newtonian flow induced by cilia beating (using micro-beads as tracers) and a mathematical model of this fluid flow, presented in greater detail in a second companion article. Samples of nasal epithelial cells placed in water are recorded by high-speed video-microscopy and ciliary beat pattern is inferred. Automatic tracking of micro-beads, used as markers of the flow generated by cilia motion, enables us also to assess the velocity profile as a function of the distance above the cilia. This profile is shown to be essentially parabolic. The obtained experimental data are used to feed a 2D mathematical and numerical model of the coupling between cilia, fluid, and micro-bead motion. From the model and the experimental measurements, the shear stress exerted by the cilia is deduced. Finally, this shear stress, which can easily be measured in the clinical setting, is proposed as a new index for characterizing the efficiency of ciliary beating.

Fig 1. Scheme of experimental setup.

Fig 2. Determining the location of the ciliated edges.

The operator places 5 points (blue crosses) which are the ends of 4 line segments (red dashed lines) defining the location of the cilia wall.

Fig 3. Example of wavelength determination.

(Left) 104 line segments are placed along the cilia wall, perpendicular to it. (Right) The mean grey level of each segment oscillates in time. The corresponding phase each segment is plotted as a function of the curvilinear abscissa of the center of each line. A linear regression is performed on the phase-abscissa relationship, the slope of this regression directly giving the metachronal wavelength (blue dashed line).

Fig 4. Measuring the cilia density.

(Left) Two different regions are delimited on the real system, the green rectangle corresponds to the cilia region while the red ellipse corresponds to the background. (Right) Schematic representation of the same system: black rectangles represent the cilia, e_c is the cilium diameter and d_c is the typical distance between two cilia.

Fig 5. Schematic representation of the stroke of an individual cilium and the envelope model.

(Left) The trajectory of the stroke cycle is assumed to follow an elliptic motion. (Right) Representation of the envelope model covering the cilia layer and the propagation of the metachronal wave. (Inspired by Velez-Cordero et al. [28]).

Fig 6. Schematic elliptic motion of an individual ciliary tip.

Fig 7. Vertical velocity component versus horizontal velocity component.

Each square corresponds to one of the 195 micro-beads measured. Red, green and purple dashed lines correspond respectively to |u_y| = |u_x|/10, |u_y| = |u_x|/4 and |u_y| = |u_x|/2.

Fig 8. Micro-bead velocities versus distance to the ciliated edge.

(Left) Examples of mean velocity obtained in 25 micro-beads in 3 ciliated edges (blue, green and red). Each square corresponds to one micro-bead. Horizontal and vertical error bars display for each bead the standard deviation of the distance to cilia and velocity, respectively. Dashed lines are the parabolic regression on each ciliated edges. (Right) The 195 measured micro-beads are presented into 2 groups of equal sizes, according to their distance to the ciliated edge. Bars exhibit a significant velocity difference (p < 0.05). Error bars represented the standard error of the mean (SEM).

Fig 9. Effective porosity parameter ϕ.

(Left) The effective slip length ϕ, obtained in each one of the 24 ciliated edges by fitting the measured micro-bead velocity with a parabola, plotted against cilia density. (Right) Effective slip length ϕ fitted from micro-bead measurements plotted against the effective slip length computed from the cilia density using Eq 8. The red line corresponds to the linear regression.

Fig 10. Comparison between measured and simulated micro-bead velocities.

(Left) Experimental velocities against simulated velocities in the 195 measured microbeads. The red line corresponds to y = x. (Right) Bland-Altman plot. The blue dashed line corresponds to the mean value while the red dashed lines correspond to the mean value ± 1.96 standard deviation.