ABSTRACT

On the epithelia of most of the respiratory tract, a dense mat of beating cilia interact with the overlying layer of airway surface liquid (ASL) to effect the removal of mucus and cellular debris from the lung. The understanding of this system, at a descriptive level, is well documented. However, the understanding of such mucociliary systems at a basic quantitative level is a fundamental field in respiratory mechanics; here we investigate one of many open topics of research in this area. After introducing techniques for the modeling mucociliary flows, we specifically consider recent measurements concerning the transport of an inert tracer within an in vitro model of the epithelial lung mucociliary system, based on human tracheobronchial epithelial cultures (1). In particular, it is found that these measurements are irreconcilable with previous modeling work. In this chap­ter we develop a simple modeling investigation of tracer transport within mucocil­iary flows to inform our understanding of this discrepancy, but must report in the negative that there is little further advancement in our understanding. This indicates that yet further modeling work is required to understand the physical basis of the experimental results, and possible research directions are suggested. 291

On the length scale of a metachronal wave, the epithelial substrate can be consid­ered as a flat plane to an excellent approximation, and one typically assumes that the cilia are evenly spread throughout the substrate in a rectangular array and beating with the same period. Taking this into account, mucociliary fluid dynam­ics constitutes an example of a thin layer, low Reynolds number flow with an oscillatory forcing that exhibits both temporal and spatial periodicity. The first step in the modeling is to represent the cilia beat and the metachronal wave. Assuming the cilia beat in a planar fashion, this is conveniently done via Fourier series with the position of the point a distance s along the (/, j) cilium of the rectangular array at time t given by:

Here, a, b are the spacings between the cilia bases in the x and y directions, with x, y representing unit Cartesian vectors across the plane of the epithelial substrate. o/27t, 2n/k give the frequency and wavelength of the metachronal wave which is, without loss of generality, taken to be in the x direction. The coefficients am( )^, bm(s) are readily determined from experimental data by least-squares fitting.The next step is to determine how the position of the beating cilia exerts a forcing on the ASL. The most convenient approximation is to assume first that the double layer stratification of the ASL can be modeled by two viscous, New­tonian fluids, one overlying the other, with the upper layer of substantially greater viscosity. This upper highly viscous layer is used as a crude representation of the mucus; the lower, less viscous, layer is reasonable representation of the peri-ciliary layer (PCL). For simplicity, we only consider the situation in which the cilia do not penetrate the mucous layer.The problem is greatly simplified by the linearity of the low Reynolds num­ber fluid dynamical equations and associated boundary conditions.* This enables us to calculate the net fluid flow by summing over the fluid flows due to the resultant force exerted by each segment, of length 8s, of each cilium on the fluid. The resultant force due to a segment, of length 8s, of the cilium (/, j) can be written in the form

where Q’j is the qth component of the vector £>i,j defined in Eq. (1), Pkq is a constant tensor, and (d ^ j/dt - uq) is the speed of the cilia relative to the speed of the surrounding fluid, moving at speed uq. This relationship arises immediately from the linearity and homogeneity of the Stokes equations on the halfplane. The calculation of Pkq can be determined using slender body techniques, by generaliz­ing the results of Gray and Hancock (5) and is essentially that of the calculation of drag coefficients for a cylinder moving in a low Reynolds number fluid flow. Such a calculation yields that

where CT is the tangential resistance coefficient with ythe ratio of the normal to tangential resistance coefficients. The relationship between the forcing and veloc­ity is given by summing over the individual solutions representing the forcing of the airway surface liquid by the segment of length 8s, a distance s along the (i,j) cilium. Thus, we are summing over (/,/), and integrating over s e [0, L], where L is the length of cilia, for individual solutions satisfying, at time t, the Stokes equation:

with the incompressibility condition V • u = 0. The variables |i, (i' are viscosities in the periciliary layer and the mucous layer, respectively; h is the height of the pericilary layer, and H is the height of the airway surface liquid (assumed con­stant). The boundary conditions are u = 0 at h = 0, z • u = 0 at z = h, z = H, with continuous velocity and normal stresses for the pericilary/mucous interface at z = h. Thus, the velocity within the periciliary layer is of the form

where Up is the pth component of the ASL flow field, denoted U in vector form below. G]pq(x, QJ(s, t)) is the (/?, q) component of the Green’s function matrix,i.e., the pth component of the solution of Eq. (2), denoted u, for the special case where F = xq, with xq denoting the qth Cartesian unit vector. The solution in the mucous layer is basically the same, though the Green’s function matrix is slightly different, due to the different viscosity and boundary conditions associ­ated with the mucous layer. With this formalism, one can thus, in theory, calculate the velocity profile caused by the beating cilia. In practice, one typically calcu­lates a time-averaged Green’s function matrix, which takes a much simpler form than Gpq(x, t); this is the basis of numerous papers modeling mucociliaryfluid flow, which therefore calculate time-averaged properties of the flow only (2-4). A typical time averaged result for a mucociliary flow is displayed in Figure

1. Further details concerning such calculations may be found in Refs. 2-4. The key points to note about such results is that, first, there is essentially no time-averaged fluid flow within micrometers of the epithelial substrate, though oscilla­tory flow may occur at substantially smaller distances from the epithelial sub­strate. Second, from the assumption that the metachronal wave is planar and beats in the x direction, one can additionally deduce that the resulting fluid flow has no y dependence, which is a useful simplification below. EXPERIMENTAL INVESTIGATIONS OF TRACER TRANSPORT

In a series of experiments, Matsui et al. (1) used conventional and confocal mi­croscopy to investigate the transport of fluorescent microspheres and photoacti-vated fluorescent dyes within well-differentiated human tracheobronchial epithe­lial cell cultures exhibiting spontaneous, radial mucociliary transport. These studies resulted in a number of key observations, including: The samples used were selected for the observation that the velocity of the fluorescent microspheres labeling the mucous layer exhibited a velocity profile with purely azimuthal motion of magnitude vazi = co r, where r is the radial distance from a center of zero motion, and co is constant.Tracer transport was investigated at points midway along the radius of mu­

cous rotation. Up to the limits of resolution, it was observed that the PCL was transported at approximately the same rate as the mucus (40 pms-1) or, to be more precise, that the transport of tracer indicated that the PCL was transported at approximately the same rate as the mucus.Removing the mucous layer reduced PCL transport by —80%. Clearly, such observations seem to refute the results from previous modeling; Figure 1 clearly shows that the periciliary layer is (on time averaging) stagnant. There are accordingly a number of interesting questions. The most important one, and the one we consider here, is: Does the tracer transport faithfully represent the fluid flow?This requires careful consideration of whether tracer diffusion obscures the true motion of the periciliary layer, especially given the fact the fluid flow will have oscillatory components both parallel and perpendicular to the epithelial substrate. By definition, such flow contributions are not captured in the time-averaged calculations presented in the literature. Furthermore, the resultant mix­ing from the oscillatory components of the flow entails that the tracer is dispersed and mixed more than one would expect from considering just the time-averaged flow. The key question is therefore whether this additional mixing could explain the similar tracer transport rates observed in the mucous and periciliary layers, or whether one has to include additional mechanisms within the modeling to explain the experimental results. MODELING TRACER TRANSPORT

To address the key question posed above we must model the transport of a tracer within a mucociliary fluid flow, denoted by U(x, z, 0 = (u(x, z, 0, v(x, z, t)) below, where t represents time, and x, z are the coordinates perpendicular and parallel to the epithelial substrate, respectively. The tracer is dispersed by a com­bination of diffusion and convection (with neither mechanism completely domi­nating tracer transport). Consequently, the equation describing the concentration of tracer particles is

where D(z), U(x, z, t) are the diffusion and velocity profiles, respectively. The boundary conditions for this equation are that no tracer is lost through the epithe­lial substrate or through the air/mucous layer interface, and that c —> 0 as x —> ±oo. The initial conditions are plotted in Figure 2. The diffusion profile, D(z), is plotted in Figure 3; it can be seen to decrease extremely rapidly at the boundary between the PCL and the mucus.