A note on mucus shear rates

Respiration

PhJ:?ioh?g_v (1973) 17. 394399;

A NOTE

Abstract.

Some order is brought

due to ciliary

activity.

sublayer, Mucus

while in the other rheologists

rheological

should

properties

ON MUCUS

into calculations

In the cilia sublayer

of the cilium beat, are theoretically regions

North-Holland

obtained.

Publishing

SHEAR

Cotnpun~~, Amstc~rdml

RATES

of the shear rates found in the mucus layers of the lung

shear rates of the order of 27~ f set-‘, The maximum

shear rates are found

of the mucus layer shear rates less than

now be able to carry

out more appropriate

where f is frequency in the “watery”

10 set-’

experiments

cilia

can be expected. in determining

the

of mucus.

Cilia sublayer

Mucus

Metachronism

Shear rates

Considerable confusion appears in the literature concerning the shear rates experienced by the mucus which lines the outer surface of the “tubes” in the upper part of the respiratory tract. Some papers (e.g. Dulfano, Adler and Philippoff, 1971) consider shear rates of 0 (1 set- ‘) in cases of chronic bronchitis while others (e.g. Reid, 1970) take shear rates up to 2000 set- ‘. Obviously there needs to be some clarification of the shear rates we might expect to find in the in uivo situation. In this paper calculations are made on the time averaged shear rates for two differing ciliary movements corresponding to observations made on protozoa Opalina and Paramecium (Sleigh, 1962). The theoretical analysis obtaining the mean velocity profiles (in a Newtonian fluid) is discussed in a previous paper (Blake, 1972a). Although the movements of the cilia lining the lung will be different to both of these we can still get some idea of the order of magnitude of the shear rates from these calculations. This is because we might expect the motion of cilia in the lung to lie somewhere in between the two movements (see fig. l), (a) corresponding to Opalina and (b) to Paramecium. The coordination wave amongst the cilia in (a) is in the same direction as the effective stroke (called symplectic metachronism). This means that the cilia are always close together in all stages of their beat because during the effective stroke the preceding cilium is in the process of completing its stroke while the next cilium is just

394

MUCUS

SHEAR

395

RATES

(b)

(a) Fig. 1. Models of the ciliary beat from (a) Opalina and (b) Par~~e~j~~.

The effective stroke in {a) is from S--9

in (b) from 1-3 (from Blake, 1972a). about

to start its movement. In the recovery stroke they are nearly parallel to the epithelial surface so they are naturally close together. In (b) the coordination wave moves in the opposite direction to the effective stroke (called antiplectic metachronism) and it is believed (Ross, 1971) that this type of coordination is found in the lung. During the effective stroke the cilia beat in relative isolation because the preceding and following cilia have just completed or are awaiting their effective strokes respectively. This means there is considerable more freedom of movement in antiplectic metachronism. General observations, from the fluid dynamical angle, can be made about both the movement of cilia and the resulting mean velocity profile. The ciliary beating cycle consists of two distinct movements, one being termed the effective stroke when the cilium moves in the same direction as the fluid and the other being the recovery stroke in the opposite direction to the general fluid movement. In the effective stroke, the cilium beats like a rigid rotating arm sticking straight out of the wall (cell surface). Physically it does this for two fluid dynamical reasons. The first is because when an elongated body, such as a cilium, moves perpendicular to its body axis it exerts a force on the fluid nearly twice as great as it would when moving tangentially at the same velocity. Secondly the further each point of the cilium is away from the cell surface the greater the volume of fluid it strongly influences. This, of course, is achieved when it is sticking nearly perpendicularly out of the wall. Conversely during the recovery stroke the cilium is moving tangentially, hence a smaller force, and also closer to the wall so it is influencing a much smaller volume of fluid. A fuller mathematical discussion of these points can be found in a previous paper (Blake, 1972a). Added to this, we have the fact that the cilium moves faster in the effective stroke than the recovery so the force is larger by comparison.

Velocity profiks From the previous discussion of the ciliary cycle we can make observations about the velocity profiles in the cilia sublayer (i.e. region containing the cilia). We define the velocity u(x, t) by (1)

u(x,t)=U(x)+ul(x,t),

II’=0

396

J. BLAKE

where u(x, t) is the velocity parallel to the wall, which has been separated into the time averaged part U(x) and the oscillatory component ui (x, t) where x is the vertical distance above the wall and t is time. If we suppose L is the length of the cilium then OdxdL. From the general

movement

of the cilium we would anticipate

lui) to be consider-

ably less than JIJI in the region x > L/2 because here the fluid only “feels” the effective stroke of all the cilia. Conversely in the region x -=zL/2 the fluid is under the influence of both effective and recovery strokes so Juij and IIJI are comparable in magnitude but both are very much less than U(L), the mean velocity at the top of the cilia sublayer. We should also remember that the mucus sticks to the cell surface (no-slip condition) so ~(0, t) is equal to zero. Some mean velocity profiles (U(x)) are shown in fig. 2 for varying values of parameters y, K and n defined as follows (Blake, 1972a) (2)

y=$,

r(=-

fSL C

T

and n which represents the cilia density on the cell surface (usually n=49/(pm)‘). C, and CT are the normal and tangential resistance (drag) coefficients for an elongated body at low Reynolds number (Gray and Hancock, 1955) respectively and C is the metachronal wave speed. An estimate of the oscillatory component u1 (x, t) is marked on fig. 2. The following general comments can be made about the mean velocity profile. The magnitude of the mean velocity profile is very small, in comparison to the velocity at the top of the cilia sublayer, in the region less than 0.2L in model (a) and less than 0.5 L in (b). In (a) from 0.2 L to 0.5 L there is a rapid increase in velocity (corresponding to maximum shear rates) while in (b) the rapid increase is from 0.5 L to 0.8 L. From 0.8 L-L both velocity profiles flatten out and are nearly constant. The calculations for the movements due to (b) predict weak backflow (“reflux”) in the region below L/2. This is probably not the case for the cilia in the respiratory

L

Fig. 2. Velocity profiles

for differing

values of y, K and n. (1) and (2) are for the ciliary movements

(a), (3) (4) and (5) are for (b) while (6) is a hypothetical velocity profile for the cilia sublayer an estimate of the oscillatory components.

due to

in lung, (---) is

MUCUS

SHEAR

397

RATES

tract, but it is interesting to note that if in some diseased state the ciliary cycle were altered considerably then backflow near the base could result. In fig. 2, we have drawn a velocity profile that might be expected in the cilia sublayers of the respiratory tract; the mean velocity being small in the region Q-O.2 L, a rapid increase from 0.2 L-O.6 L and then flattening out from 0.6 L-L with the velocity of movement of the upper layer being the same as that at the top of the cilia sublayer (Blake, 1972b). These calculations were carried out assuming a Newtonian viscous fluid at very low Reynolds number with a zero stress condition on the exterior boundary. Variations from all three of these constraints could alter the profile, but it is doubtful if any would to a marked extent. The region in which the cilia beat is often called the serous sublayer (Ross, 1971) which is thought to contain a fluid with near Newtonian properties. Shear rates

From the calculations for the mean velocity profile it is quite easy to obtain the mean shear rates, these being illustrated in fig. 3 for both models (a) and (b). In (a) the maximum shear rates is 0.9 D set- ’ which occurs in the range 0.3 L-OS L while in (b) they range from 2-3 CTset- ’ which occurs at 0.7 L.

+$set-1)

(a)

I

-l-O\

Fig. 3. Shear rates corresponding dimensionalised

to the numbering

on fig. 2. The shear rates and cilium length are non-

with respect to o set-

’ and L(pm) respectively.

The diagrams of fig. 3 indicate a rather complementary note for the two models (a) and (b) in terms of the shear rate. In (a) the larger shear rates are found in the region less than L/2 while in (bj they are found in the region above L/2. This is because of the difference in the cycle of beat of the cilia in each model. In (a) the cilia are closer together than (b) so they have less space in which to move freely (i.e. in Opalina n= 10 while in Paramecium n=f-i although in the mathematical analysis we can vary n

J. BLAKE

398

as we please !). Thus there is not the same distinction strokes and so the velocities structure is different.

between the effective and recovery

are less (in (a) 0.2 oL, in (b) 0.5 (TL) and the shear rate

There appears to be considerable confusion in the literature about shear rates in the mucus layer. Basically four different mechanisms can induce the shearing in the mucus ; these being the cilia, gravity, pulsative airflow due to inspiration and expiration of air and the visco-elasticity of the mucus, although coughing (peristalsis) will induce some shearing as well. Before discussing each of these separately, a brief discussion of the structure of the mucus layer is needed. It is observed (?) to consist of a lower “watery” layer (Newtonian) containing the cilia, while the upper part is very viscous (viscoelastic). This increased viscosity at the free surface could be due to an increase in concentration of mucin, due to evaporation, or a change in structure of the molecule. In a Newtonian fluid the shear stress due to the cilia is confined to the cilia sublayer, that is, between the cilia and the epithelial surface. Because this is dimensionally relatively small 0 (5 pm) and the velocities equivalent to those observed on the surface the shear rates will be large in this region. One would expect the cilia to dominate the shear rates in the cilia sublayer as they are the “driving mechanism” for the upward flow of mucus. If some other mechanism was dominating the ciliary movement in this layer something would be drastically wrong! The other three mechanisms will principally be of importance in the upper layer of the mucus, although viscoelasticity may influence the velocity field in the cilia sublayer. Gravity is generally unimportant‘ except perhaps in the diseased state (Blake, 1972b) when the depth of mucus increases dramatically which may cause reversal of the flow back into the fine; bronchii of the lungs. In the normal state the shear rates due to gravity will be up to 1 set- ’ while in the diseased state it may increase to 20 set- ‘. The influence of the pulsative air flow is counteracted by the increase in viscosity of the mucus at the free surface, so its influence is unimportant in the normal state. The influence of viscoelasticity is the unknown quantity at this stage because very little reliable information is available on its physical characteristics (Ross, 1971). The purpose of this paper is to give the experimentalists some idea of the shear rates involved so that more suitable experiments

can be carried

out on mucus

to find its visco-elastic

properties.

Conclusions The main purpose of this paper was to calculate the shear rate found in the cilia sublayer of organisms, which we can correlate with that due to the ciliated epithelia in the lung. It was found that shear rates of O(o set- ‘), CJ= radian frequency = lOO150 set- ‘, are found in the cilia sublayer. Above the sublayer, shear rates of less than 10 set- ’ would be expected. Acknowledgements The author acknowledges the support of a George Murray scholarship from the University of Adelaide and CSIRO of Australia. He also thanks Dr. R. C. Schroter for suggesting and discussing this topic with him.

MUCUS

SHEAR

399

RATES

List of symbols used u(x, t) :general mucus velocity held U(x) : time averaged velocity u’ (x, t) : oscillatory velocity L f

: length of cilium : frequency of cilium beat

cr

: radian frequency of cilium, = 2 nf : vertical distance above wall : time : normal resistance coefficient : tangential resistance coefficient : nondimensional parameter = C,/ Cr : metachronal wave speed

X

t CN CT Y C K n

: nondimensional frequency parameter : cilia density on epithelial surface

= cr L/c

References Blake, J. (1972a). A model for the microstructure Blake, J. (1972b). Mucus

in ciliated

J. Fluid. Me&

organisms.

flows (Submitted

to Mathematical

Biosciences).

Dulfano. M. J., K. Adler and W. Philippoff Resp. Dis. 104: 88898.

(1971). Sputum

visco-elasticity

Gray, J. and G. J. Hancock Reid, L. (1970). Chronic

(1955). The propulsion

bronchitis

-. A report

Ross, S. (1971). A wavy wall analytical University. Sleigh, M. A. (1962). The Biology

of sea-urchin

on mucus research.

model of muco-ciliary

of Cilia and Flagella.

spermatozoa.

bronchitis.

Am. Rec.

J. Exp. Biol. 32: 802-814.

Proc. Roy. Instn. Gt. Br. 43 : 4388463.

pumping.

London,

in chronic

55: l-23.

Ph. D. dissertation,

Pergamon.

Johns Hopkins