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Mucus flows
MATHEMATICAL
BIOSCIENCES
301
17. 301-3 13 (I 973)
Mucus Flows JOHN BLAKE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England Communicated
by Hirsh Cohen
_ ABSTRACT A model for mucus flows in the respiratory tract is considered, using a similar theory to that developed for the cilia sublayer in microorganisms [7]. An infinite array of elongated flexible bodies is used to model the ciliated epithelia of the respiratory tract from which we can obtain the average velocity fields in the mucus layer. Velocities around 1-3 cm/min. are predicted, a result which compares favourably with the observed values of mucus flow in the lungs of mammals. The flow rate is found to have a linear dependence on beat frequency and cilium length while the influence of cilia concentration is graphically illustrated. A discussion of the effects of gravity and the pulsative airflow in the diseased state (e.g., bronchitis) is given.
I. INTRODUCTION
The physiology of the respiratory tract has interested man for many centuries. Much is known about the functioning of the lung and its exchange processes by diffusion of gases in the alveoli, but little is known about the movement of mucus by cilia in the upper part of the respiratory tract, and in particular the mechanisms of this movement. The main functions of this mucus sublayer is in cleaning the inspired air of unwanted particles (e.g. aerosols, bacteria, viruses, and carcinogens in tobacco smoke) and in bringing the air to near body temperature while the relative humidity is brought close to saturation. Failure of the cilia, which leads to inefficient removal of the collected particles, may lead to disease. Obviously many air pollutants, especially tobacco smoke, have a detrimental effect on the mucus transport. Cilia are found on the outer surface of the epithelial cells which line nearly all parts of the human respiratory tract (see Sec. 2). They have a complex internal structure, consisting of a 9 + 2 fibril arrangement which causes bending in the cilium by interaction between adjacent fibrils [13]. The main function of the asymmetric beating motion of cilia is in the movement of fluids, whether it be in the lungs, microorganisms, or the reproductive tracts. The mucus layer that is secreted from the underlying 20
0 American Elsevier Publishing Company, Inc., 1973
302
JOHN
BLAKE
cells has a highly viscous (“adhesive”) quality that enables it to capture the airborn particles and to remain on the outside of the respiratory tubes around the cilia. The structure of the mucus layer will be discussed in greater detail in the Sec. 2. It should be remarked, however, that mucus has visco-elastic properties, but, to date, these are not particularly wellknown. The difficulty in finding these properties is because there is considerable doubt whether the in z;itro experiments compare favourably with the in zivo situation. There have been very few theoretical attempts to model muco-ciliary pumping; the two most important contributions coming from Barton and Raynor [2] and Ross [I 11. Barton and Raynor consider a distribution of rigid rods, modelling the cilia, over an infinite plane with the condition that the rods automatically shorten for the recovery stroke of the ciliary cycle. Using the calculations for the force exerted by a single rod in an infinite viscous fluid, they were able to calculate a shear stress at the top of the cilia sublayer for an infinite array of rods. From this, they calculated the velocity profile in the mucus layer above the cilia by assuming a variable viscosity. Their calculations agreed reasonably well with the experimental observations for volume flow rates. There are many inadequacies in this model, some of these being an inaccurate representation of the cilium’s movement, no allowance being made for the metachronal wave (the coordination wave in the array of cilia), interaction between the cilia, and the influence of the epithelial surface on the velocity profile. Ross [l l] uses an envelope approach to muco-ciliary pumping; that is, he models the beating cilia by an instantaneous waving surface which covers the tips of the cilia (for a discussion of this theory as applied to microorganisms see Blake [5] and [6]). He represents the mucus by a simple Maxwell visco-elastic model (i.e. spring and dashpot in series). Using these models for the cilia sublayer and the mucus he was able to calculate the velocity profiles, particle paths, volume flow rates, and the free surface shape of the mucus layer in terms of an asymptotic expansion of a small parameter E, which is equal to b/h where b is the amplitude of the metachronal wave and /z the depth of the mucus. However, there is some doubt about the application of the envelope model to movement of fluid by cilia which exhibits antiplectic metachronism (Blake [7]; metachronal waves move in opposite direction to normal mucus flow and effective stroke of the cilium), as occurs on the ciliated epithelia of the respiratory tract. This is because in antiplectic metachronism the cilia are relatively isolated (see Sleigh [12]) during their effective stroke, which is the main propulsive movement of the cilium, so the envelope model would be extremely doubtful in these situations. That is, we cannot replace the individuality of the ciliary beat by an envelope in antiplectic
MUCUS
303
FLOWS
metachronism. Mention should also be made of the mechanical models of Miller [B], but the number of “cilia” used in his model would not duplicate the situation in nature. In this exploratory paper, which is in a relatively new field, we set up a framework for a theoretical study of the movement of mucus in the lungs by using the theory developed for the cilia sublayer by Blake [7]. Physically the cilia sublayer was modelled by an array of flexing long slender bodies (representing the cilia) which were attached at one end to a planar surface. The flexing movements of the slender bodies coincide with ciliary movements observed in nature in the protozoa Opalina and Paramecium. The metachronal wave of COordination of the cilia is incorporated into the analysis, while the velocity field is represented by a distribution of viscous fluid singularities along the centrelines of the slender bodies. A representation for the total mean velocity field in the cilia sublayer is obtained. In Sec. 3, we apply this technique to movement of mucus by cilia in the lung. In diseased cells the concentration of cilia on the cell surface may decrease. From the viewpoint of a Newtonian fluid we investigate the effect of decreased concentration of cilia on the flow rates, while also discussing the influence of the frequency of beat and the ciliary cycle. In the healthy state of the lung, gravity and pulsative airflow effects due to inspiration and expiration of air are relatively unimportant in comparison to the mucus movement due to the cilia. However, in the unhealthy state when the mucus layer may increase tenfold and the viscosity decrease several orders of magnitude both gravity and inertial effects can become critically important (Ross [I I]). For these purely pathological situations, a brief discussion of the theory is provided in Sec. 4. Previous to both of these sections, Sec. 2 provides a relevant discussion of the physiology of the lung, which incorporates a table of the physical dimensions and parameters of both the cilia and the mucus. In the concluding section (Sec. 5), we review the principal results of this paper while indicating topics where a more thorough, or even an initial theoretical approach, should be made. 2. MUCUS TRACT
AND
CILIA
SUBLAYER
IN THE
RESPIRATORY
The lung consists of a complex branched network of tubes, terminating at the alveoli. Alveoli, where the gas exchange with the cardio-vascular systems occurs, can be found from the 17th to the 23rd generation of the lung. Figure 1 represents a schematic picture of the structure of the lung from the trachea, principal bronchus, bronchi, and bronchioles to the alveoli. Mucus covers the ciliated epithelium of the respiratory tract,
304
JOHN
BLAKE
which includes the nose, trachea, sinuses, and the proximal bronchioles. The mucus continually moves upwards towards the upper end of the trachea. The regular airflow reversals are obviously very important in contributing to the particle deposition on the surface of the mucus layer which is one of the main functions of the mucus. TRACHEA
-
18cm
diam.=lmm. 10th.
=
Generation Ir\l
17th.
K
23rd
:
Ir\l
kronchioles
diam.=0.5mm.
Terminates
FIG. 1. A schematic various generations.
diagram
of the lung, showing the diameters
of the tubes at
The epithelia consists of two basic types of cell; goblet and ciliated. Goblet cells are about one quarter as abundant as the ciliated cells in the normal healthy state, but in certain conditions these goblet cells may outnumber the ciliated cells of the epithelia. As Fig. 2 illustrates, the mucus layer covering the epithelia appears to consist of two distinct regions [l 11. The lower layer consists of a “watery” liquid which has a lower viscosity than the upper mucus layer. Little is known about concentration of mucin (long chained glucoprotein) in these layers, but it seems probable that the concentration is higher in the upper layer. This could well be due to evaporation, so that the specific humidity of the inspired air is brought
MUCUS
305
FLOWS
to saturation. In some models of the ciliated epithelia, the upper layer has been assumed to be visco-elastic while the serous sublayer (which contains the cilia) consists of a Newtonian viscous fluid [ll]. Air
-
MUCUS layer
FIG. 2. Diagram illustrating the epithelial and goblet cells, cilia, and the two layers of fluid with apparently differing properties. These are called the serous fluid for the “watery” lower layer, and the upper part is the mucus layer.
Visco-elasticity occurs when flexible bodies are suspended in a Newtonian ambient fluid (see e.g. Ref. 4). The degree of visco-elasticity depends critically on the concentration of particles. Thus it is highly likely that the concentration of mucin is lower in the serous sublayer, hence the impression of being “watery.” In Table 1, information on the dimensions and distribution of the cilia and the concentration of the constituents of mucus are given. Data Cilia Length Frequency of beat Cilia spacing Type of cells Density of cilia Depth of fluid Mucus Contents:
Water
TABLE 1 on Cilia and Mucus
5-8 pm IO-20/set 0.3-0.4 I‘m ciliated, goblet 6-10/(pm)2 5-500 pm
95-97x,
Mucin
diameter wavelength h metachronal wave Metachronism average area Number/cell Average velocity of mucus
0.15-0.3 pm 20-40 pm Antiplectic 3tS40 (pm)* 200-400 5-10 mm/min
2.5- -3.O%, salts l-2%
3. CILIA INDUCED FLOW In this section we investigate the movement of mucus in the lung by cilia alone. For our model of one of the tubes in the lung, we will suppose
306
JOHN
BLAKE
it is cylindrical, of radius R, while the cilia are of length L, and the depth of fluid above the cilia is H (see Fig. 3). In the normally healthy state H/R is very small 0(10-2), being of the same order of magnitude as L/R, but in the diseased state H/R may become as large as, or even greater than, one half. In all airways of the lung L/R is always very small, so we can approximate the cilia sublayer by a planar model. We suppose that the cilia are distributed over the X,0X2 plane in a regular array with spacing a in the X1 direction and b in the X2 direction. Thus the number of cilia in each square micron is _Af = l/ab if a and b are both measured in microns (pm).
(i)
Cylindrical
Model
x3
: -----_------mu(,,,(K
(ii)
Planar
Mucus ” / Cilia
Layer StdlAy;‘r ,,\sm
Approximation
, X1
FIG. 3. Geometrical arrangement for the model of mucus flow in a cylindrical tube shown in 3(i). The planar model showing the cilia and mucus sublayers is illustrated in 3(ii).
The relevant equations of motion for the movement of fluid are dependent on the Reynolds number, which gives an indication of the relative importance of inertial and viscous effects. The Reynolds number
MUCUS
307
FLOWS
of basic importance defined as follows
in this problem
is the cilium
Reynolds
CTLr, R, = ~ v ’
number
R,,
(1)
where CJ is the angular frequency, L the length and r. a characteristic radius of the cilium, while p is the kinematic viscosity. In the respiratory tract, R, is always very small [O(lO-“)I; so the viscous effects are more important. The equations of motion, called the Stokes flow equations at low Reynolds number, are defined as follows, vp = pv2u,
v .u
= 0,
(2)
where p is the pressure, u the velocity vector and p the dynamic viscosity (V = p/p, p the density). A problem similar to this has been discussed by Blake [7] when he models the cilia sublayer of microorganisms Opalina and Paramecium. The results obtained there can be modified to help us with a better understanding of the movement of mucus in the lung. In this model each cilium was modelled by a long slender body of length L and radius ro(s), s the distance along the cilium from its base, which is attached to an infinite plane. A regular array of these slender bodies is used to model the ciliated epithelia. The velocity field of each cilium is represented by a distribution of viscous fluid (Newtonian) singularities along the centreline of the slender body. The mean velocity field through the sublayer is obtained by summing the contributions from all the cilia of the regular array which is represented in terms of the following integral equation [7].
where U(X,) is the average velocity in the X,-direction, X, is the vertical coordinate, F1 the functional representation for the force in the X1direction, 5 specifies the movement of the cilium while w(s, t) and K(X,,
and 3:
Jw3Yr3) = ; . i
3.
0
<
5
x3
<
3
53,
3---
+
H
2
308
JOHN
BLAKE
where t3 is the vertical coordinate of the cilium’s movement and c is the metachronal wave velocity. The weight function CO(S,t) represents the influence of the metachronal wave, the positive sign for antipletic metachronism, the negative sign for sympletic metachronism for an effective stroke in the positive X,-direction. The kernel function corresponds to a Newtonian viscous fluid, but it should not be too difficult to find the kernel function for a simple visco-elastic model (e.g. Maxwell). Results obtained from this theory are applicable for the flow of mucus in the lung. The free surface condition of zero tangential stress can be satisfied at any height above the cilia sublayer because it is a constant stream and automatically satisfies this condition. Solution of this integral equation leads to the following representation for the velocity field:
u(X,)=aL+,K,.l.,~)06X, = U(L) = const.
L
where 9 represents the solution of the integral equation (see Blake, Ref. 7, for details). Thus the mucus is brought from a zero velocity (no-slip condition) at the base of the cilia to a velocity U(L) at the top of the cilia sublayer, while the mucus layer above moves with a constant velocity. Important parameters that arise in solving this integral equation are the density parameter JV, defined previously, and the nondimensional parameters
yap, CT
OL xc-_. C
(7)
The parameter y equals the ratio of the normal to tangential resistance coefficients for an elongated body in Stokes flow, while K is equivalent to the ratio of the average tip speed to the metachronal wave velocity c. Antiplectic metachronism is usually exhibited by cilia in the respiratory tract (i.e. the metachronal wave is moving in the opposite direction to the movement of the mucus). However little is known about the beat cycle of the cilia, so the best we can do is to take the beating pattern shown by a micro-organism exhibiting antiplectic metachronism. We use the data on Paramecium from Sleigh [12]. In Fig. 4 we show the beating pattern of the cilium and the induced velocity field for varying values of y, ri, andJlr. If the wavelength of the metachronal wave is kept constant, then the flow rate varies linearly with frequency. Thus if we decrease the beat frequency of the cilia we decrease the flow rate by a proportional amount. In Fig. 5 we show the variation of the flow rate against the concentration
MUCUS
309
FLOWS
CILIA
SliBLAYER
VELOCITY
PROFILE
FIG. 4. Movements of a cilium (Paramecium [12]) used in numerical calculations, the numbers indicate successive stages of the beat (equal time intervals). Some velocity profiles atN = 4 for varying values ofy and K are shown.
-!L
OL
FIG. 5. Flow rates as a function for several values of y and K.
of the cilia concentration
on the epithelial
surface
310
JOHN
BLAKE
of cilia on the epithelia. It is observed that, if some disease (e.g. asthma) causes a decrease in the cilia concentration, then there is a rather marked drop in the flow rate. There also appears to be an optimal concentration (if we considered an efficiency) because the graph asymptotes to a maximum value for the flow rate. For the frequency of beat and length of cilium in the lung, this theory predicts flow rates around 1-3 cm/min, which compares favourably with that observed naturally. We now turn to calculating the total volume flow rate Q in the cilia sublayer and the mucus layer due to ciliary activity. This can be expressed in terms of the following integral. rU*(p) dr, R s R-II-L
Q=27-r
(8)
where U”(P) = U(R - r), U(X,) being defined in Eq. (3). If we nondimensionalize velocities with respect to CTL and the volume flow rate with respect to mR3 we obtain
q’x
oTCK3
= a/?(2 - 2rr - /?)K, + 2a2K,
- 2z3K3,
where a = L/R, j3 = H/R, and K, = U(l),
U(x) dx,
K, =
(9)
t
I
xU(x)
K, = s0
s0
dx, I
which for small a and j? yields 4 N 2aPK,
+ 2x2K,.
(10)
This indicates the volume flow rate varies linearly with the depth of the mucus upper layer, so a small increase in the mucus depth, and hence volume of mucus, can be comfortably moved by the cilia. This, however, need not be the case if the mucus thickness increases dramatically, as occurs in the diseased state, when both gravity and the pulsative air flow are important. 4. INFLUENCE
OF GRAVITY
AND
PULSATIVE
AIR
FLOW
In the normal state both the influence of gravity and the pulsative airflow due to the inspiration and expiration of air are of relatively minor importance in comparison with that due to ciliary activity. However, in the diseased state when the mucus layer increases in thickness and the viscosity of the mucus decreases, both can have a detrimental effect by decreasing the volume flow rates substantially, and may in extreme cases cause flow reversal, so that the mucus flows back into the finer bronchii.
MUCUS
311
FLOWS
We can analyze the influence of gravity quite simply. profile due to the influence of gravity (Batchelor [3]) is U(X,)
= -x(x,
- 211).
The velocity
(11)
Here, h = L + H and a is the angle at which the tube is inclined to the vertical, p the density of the mucus, and g the gravitational acceleration. Thus if we are standing vertically (a = 0), the flow rate at the top of the mucus layer will be
Pd12
U(h) = -__
4p
’
(12)
which indicates that it is flowing back into the lung at a rate proportional to the square of the layer thickness and inversely to the viscosity. Thus, in the diseased state, when the layer thickness increases, and the viscosity decreases damaging effects can occur. One of the more obvious solutions in chronic cases, of say, pneumonia, is to make gravity “work for you” and incline the patient downwards thus expediting the removal of mucus from the lung [see Fig. 6(d)]. Airway L+H
FIG. 6. Velocity profiles in (a, b) healthy state without gravity, (c, d) and in an unhealthy state including gravity. In all graphs Jlr = 7, and in (a), (c), and (d) y = 1.8 and K = 1.5, while in (b) y = 1.7 and K = 1.O. For (a) and (b) L = Hand in (c) and (d) H = 9L, the last two parts being at one-fifth scale. The values of the inclination angle a (in degrees) are shown on the diagram.
It is a much more complex problem to analyze the influence of the pulsative airflow on the mucus layer. It is difficult to calculate the velocity profiles in a branching system of tubes, because the profile varies considerably from the laminar Poiseuille flow of a single tube, and so far no
312
JOHN
BLAKE
complete fluid dynamic study has been made of this problem, although Pedley, Schroter and Sudlow [9, lo] have made some advances in the steady flow problem. Probably the best we can say is that the oscillatory stress developed at the free surface of the mucus by the pulsative airflow will be asymmetric in both time and space in many airways of the lung. It has been observed (Arnott, Clark, Jones [l]) that in diseases, such as chronic bronchitis where there are excess secretions, shear waves are developed on the mucus layer and two phase flow occurs. Obviously considerably more research on both the theoretical fluid dynamical and physiological sides needs to be done on this topic. In Fig. 6, graphs show some predicted velocity profiles for both healthy and unhealthy states of the lung where gravity is important. In Fig. 6(c), the influence of gravity dominates over the ciliary activity, so that the mucus flow is back into the finer tubes of the lung. In Fig. 6(d), the velocity profile is that predicted if a patient were inclined at a downward angle of 45” to the horizontal. Obviously this is a more desirable situation than that of Fig. 6(c). 5. CONCLUSIONS A self consistent theory has been developed to model mucus flow in the lung along the lines developed in a previous paper (Blake [7]). The advantage of this technique in its obvious superiority over the envelope approach in modelling the antiplectic coordination amongst the cilia of the respiratory tract. The theory presented here agrees reasonably well with the experimental observations in mammals predicting mucus flow rates around l-3 cm/min, for the frequencies and cilia lengths found in the lung. Consideration is given in the paper to the variation of flow rates against frequency, length of cilia, and cilia concentration on the epithelia, the first two obeying an increasing linear relationship, whereas the cilia concentration is more parabolic in nature (i.e., J) initially. Extension of a Newtonian model for the mucus to some visco-elastic model is envisaged, using the same techniques as applied in this present theory. The influence of gravity and the pulsative air flow are examined analytically and discussed respectively. It was seen in Sec. 4 that in certain pathological situations gravity can have a detrimental effect on the direction of mucus flows, this being best illustrated in the diagrams of Fig. 6. Pulsative airflow is a relatively new field of fluid mechanics; especially flow in branched tubes, such as occurs in the lung, and little progress has been made on its interaction with the mucus layer. Obviously an important problem to be solved is the mechanism of particle deposition on the mucus surface which is afterall one of its main functions.
MUCUS
313
FLOWS
This work was supported by a George Murray Scholarship from the University of Adelaide, C.S.I.R.O. of Australia and the A.R.G.C. The author appreciates the comments of Sir James Lighthill and Dr. E. 0. Tuck. REFERENCES 1 W. M. Arnott, S. W. Clarke, and J. G. Jones, Mass movement of gas in respiratory tubes, Chronic inflammation of the bronchi. Prog. Resp. Res. 6,2 (1971). 2 C. Barton and S. Raynor, Analytic investigations of cilia induced mucous how, Bull. Math. Biophys. 29, 419 (1967). 3 G. K. Batchelor, An Introduction to Fluid Dynamics,
Cambridge (1967). 4 G. K. Batchelor, The stress system in a suspension of force free particles, J. Fluid Mech. 41, 545 (1970). 5 J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46, 199 (1971). 6 J. R. Blake, Infinite models for ciliary propulsion,
7 J. Blake, A model for the micro-structure 55, I (1972). 8 C. E. Miller, Streamlines
induced
flow homomorphic
J. F/ltid Me&. 49,209 (1971). in ciliated organisms, J. Fluid Mech.
and particle path lines associated with a mechanically with the mammalian muco-ciliary system, Biorheol.
6, 127 (1969). 9 T. J. Pedley, R. C. Schroter, and M. F. Sudlow, Flow and pressure drop in systems of repeatedly branching tubes, J. Fluid Mech. 46,365 (1971).
10 T. J. Pedley, R. C. Schroter, and M. F. Sudlow, Energy losses and pressure drop in model of human airways, Resp. Physiol. 9, 371 (1970). 11 S. M. Ross, A Wavy Wall Analytic Model of Muco-Ciliary Pumping, Ph.D. dissertation, Johns Hopkins University (1971). 12 M. A. Sleigh, The Bioloyr of Cilia and Flagella, Pergamon, London (1962). 13 M. A. Sleigh, Cilia, Endeaoour 30, 11 (1971).
BIOSCIENCES
301
17. 301-3 13 (I 973)
Mucus Flows JOHN BLAKE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England Communicated
by Hirsh Cohen
_ ABSTRACT A model for mucus flows in the respiratory tract is considered, using a similar theory to that developed for the cilia sublayer in microorganisms [7]. An infinite array of elongated flexible bodies is used to model the ciliated epithelia of the respiratory tract from which we can obtain the average velocity fields in the mucus layer. Velocities around 1-3 cm/min. are predicted, a result which compares favourably with the observed values of mucus flow in the lungs of mammals. The flow rate is found to have a linear dependence on beat frequency and cilium length while the influence of cilia concentration is graphically illustrated. A discussion of the effects of gravity and the pulsative airflow in the diseased state (e.g., bronchitis) is given.
I. INTRODUCTION
The physiology of the respiratory tract has interested man for many centuries. Much is known about the functioning of the lung and its exchange processes by diffusion of gases in the alveoli, but little is known about the movement of mucus by cilia in the upper part of the respiratory tract, and in particular the mechanisms of this movement. The main functions of this mucus sublayer is in cleaning the inspired air of unwanted particles (e.g. aerosols, bacteria, viruses, and carcinogens in tobacco smoke) and in bringing the air to near body temperature while the relative humidity is brought close to saturation. Failure of the cilia, which leads to inefficient removal of the collected particles, may lead to disease. Obviously many air pollutants, especially tobacco smoke, have a detrimental effect on the mucus transport. Cilia are found on the outer surface of the epithelial cells which line nearly all parts of the human respiratory tract (see Sec. 2). They have a complex internal structure, consisting of a 9 + 2 fibril arrangement which causes bending in the cilium by interaction between adjacent fibrils [13]. The main function of the asymmetric beating motion of cilia is in the movement of fluids, whether it be in the lungs, microorganisms, or the reproductive tracts. The mucus layer that is secreted from the underlying 20
0 American Elsevier Publishing Company, Inc., 1973
302
JOHN
BLAKE
cells has a highly viscous (“adhesive”) quality that enables it to capture the airborn particles and to remain on the outside of the respiratory tubes around the cilia. The structure of the mucus layer will be discussed in greater detail in the Sec. 2. It should be remarked, however, that mucus has visco-elastic properties, but, to date, these are not particularly wellknown. The difficulty in finding these properties is because there is considerable doubt whether the in z;itro experiments compare favourably with the in zivo situation. There have been very few theoretical attempts to model muco-ciliary pumping; the two most important contributions coming from Barton and Raynor [2] and Ross [I 11. Barton and Raynor consider a distribution of rigid rods, modelling the cilia, over an infinite plane with the condition that the rods automatically shorten for the recovery stroke of the ciliary cycle. Using the calculations for the force exerted by a single rod in an infinite viscous fluid, they were able to calculate a shear stress at the top of the cilia sublayer for an infinite array of rods. From this, they calculated the velocity profile in the mucus layer above the cilia by assuming a variable viscosity. Their calculations agreed reasonably well with the experimental observations for volume flow rates. There are many inadequacies in this model, some of these being an inaccurate representation of the cilium’s movement, no allowance being made for the metachronal wave (the coordination wave in the array of cilia), interaction between the cilia, and the influence of the epithelial surface on the velocity profile. Ross [l l] uses an envelope approach to muco-ciliary pumping; that is, he models the beating cilia by an instantaneous waving surface which covers the tips of the cilia (for a discussion of this theory as applied to microorganisms see Blake [5] and [6]). He represents the mucus by a simple Maxwell visco-elastic model (i.e. spring and dashpot in series). Using these models for the cilia sublayer and the mucus he was able to calculate the velocity profiles, particle paths, volume flow rates, and the free surface shape of the mucus layer in terms of an asymptotic expansion of a small parameter E, which is equal to b/h where b is the amplitude of the metachronal wave and /z the depth of the mucus. However, there is some doubt about the application of the envelope model to movement of fluid by cilia which exhibits antiplectic metachronism (Blake [7]; metachronal waves move in opposite direction to normal mucus flow and effective stroke of the cilium), as occurs on the ciliated epithelia of the respiratory tract. This is because in antiplectic metachronism the cilia are relatively isolated (see Sleigh [12]) during their effective stroke, which is the main propulsive movement of the cilium, so the envelope model would be extremely doubtful in these situations. That is, we cannot replace the individuality of the ciliary beat by an envelope in antiplectic
MUCUS
303
FLOWS
metachronism. Mention should also be made of the mechanical models of Miller [B], but the number of “cilia” used in his model would not duplicate the situation in nature. In this exploratory paper, which is in a relatively new field, we set up a framework for a theoretical study of the movement of mucus in the lungs by using the theory developed for the cilia sublayer by Blake [7]. Physically the cilia sublayer was modelled by an array of flexing long slender bodies (representing the cilia) which were attached at one end to a planar surface. The flexing movements of the slender bodies coincide with ciliary movements observed in nature in the protozoa Opalina and Paramecium. The metachronal wave of COordination of the cilia is incorporated into the analysis, while the velocity field is represented by a distribution of viscous fluid singularities along the centrelines of the slender bodies. A representation for the total mean velocity field in the cilia sublayer is obtained. In Sec. 3, we apply this technique to movement of mucus by cilia in the lung. In diseased cells the concentration of cilia on the cell surface may decrease. From the viewpoint of a Newtonian fluid we investigate the effect of decreased concentration of cilia on the flow rates, while also discussing the influence of the frequency of beat and the ciliary cycle. In the healthy state of the lung, gravity and pulsative airflow effects due to inspiration and expiration of air are relatively unimportant in comparison to the mucus movement due to the cilia. However, in the unhealthy state when the mucus layer may increase tenfold and the viscosity decrease several orders of magnitude both gravity and inertial effects can become critically important (Ross [I I]). For these purely pathological situations, a brief discussion of the theory is provided in Sec. 4. Previous to both of these sections, Sec. 2 provides a relevant discussion of the physiology of the lung, which incorporates a table of the physical dimensions and parameters of both the cilia and the mucus. In the concluding section (Sec. 5), we review the principal results of this paper while indicating topics where a more thorough, or even an initial theoretical approach, should be made. 2. MUCUS TRACT
AND
CILIA
SUBLAYER
IN THE
RESPIRATORY
The lung consists of a complex branched network of tubes, terminating at the alveoli. Alveoli, where the gas exchange with the cardio-vascular systems occurs, can be found from the 17th to the 23rd generation of the lung. Figure 1 represents a schematic picture of the structure of the lung from the trachea, principal bronchus, bronchi, and bronchioles to the alveoli. Mucus covers the ciliated epithelium of the respiratory tract,
304
JOHN
BLAKE
which includes the nose, trachea, sinuses, and the proximal bronchioles. The mucus continually moves upwards towards the upper end of the trachea. The regular airflow reversals are obviously very important in contributing to the particle deposition on the surface of the mucus layer which is one of the main functions of the mucus. TRACHEA
-
18cm
diam.=lmm. 10th.
=
Generation Ir\l
17th.
K
23rd
:
Ir\l
kronchioles
diam.=0.5mm.
Terminates
FIG. 1. A schematic various generations.
diagram
of the lung, showing the diameters
of the tubes at
The epithelia consists of two basic types of cell; goblet and ciliated. Goblet cells are about one quarter as abundant as the ciliated cells in the normal healthy state, but in certain conditions these goblet cells may outnumber the ciliated cells of the epithelia. As Fig. 2 illustrates, the mucus layer covering the epithelia appears to consist of two distinct regions [l 11. The lower layer consists of a “watery” liquid which has a lower viscosity than the upper mucus layer. Little is known about concentration of mucin (long chained glucoprotein) in these layers, but it seems probable that the concentration is higher in the upper layer. This could well be due to evaporation, so that the specific humidity of the inspired air is brought
MUCUS
305
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to saturation. In some models of the ciliated epithelia, the upper layer has been assumed to be visco-elastic while the serous sublayer (which contains the cilia) consists of a Newtonian viscous fluid [ll]. Air
-
MUCUS layer
FIG. 2. Diagram illustrating the epithelial and goblet cells, cilia, and the two layers of fluid with apparently differing properties. These are called the serous fluid for the “watery” lower layer, and the upper part is the mucus layer.
Visco-elasticity occurs when flexible bodies are suspended in a Newtonian ambient fluid (see e.g. Ref. 4). The degree of visco-elasticity depends critically on the concentration of particles. Thus it is highly likely that the concentration of mucin is lower in the serous sublayer, hence the impression of being “watery.” In Table 1, information on the dimensions and distribution of the cilia and the concentration of the constituents of mucus are given. Data Cilia Length Frequency of beat Cilia spacing Type of cells Density of cilia Depth of fluid Mucus Contents:
Water
TABLE 1 on Cilia and Mucus
5-8 pm IO-20/set 0.3-0.4 I‘m ciliated, goblet 6-10/(pm)2 5-500 pm
95-97x,
Mucin
diameter wavelength h metachronal wave Metachronism average area Number/cell Average velocity of mucus
0.15-0.3 pm 20-40 pm Antiplectic 3tS40 (pm)* 200-400 5-10 mm/min
2.5- -3.O%, salts l-2%
3. CILIA INDUCED FLOW In this section we investigate the movement of mucus in the lung by cilia alone. For our model of one of the tubes in the lung, we will suppose
306
JOHN
BLAKE
it is cylindrical, of radius R, while the cilia are of length L, and the depth of fluid above the cilia is H (see Fig. 3). In the normally healthy state H/R is very small 0(10-2), being of the same order of magnitude as L/R, but in the diseased state H/R may become as large as, or even greater than, one half. In all airways of the lung L/R is always very small, so we can approximate the cilia sublayer by a planar model. We suppose that the cilia are distributed over the X,0X2 plane in a regular array with spacing a in the X1 direction and b in the X2 direction. Thus the number of cilia in each square micron is _Af = l/ab if a and b are both measured in microns (pm).
(i)
Cylindrical
Model
x3
: -----_------mu(,,,(K
(ii)
Planar
Mucus ” / Cilia
Layer StdlAy;‘r ,,\sm
Approximation
, X1
FIG. 3. Geometrical arrangement for the model of mucus flow in a cylindrical tube shown in 3(i). The planar model showing the cilia and mucus sublayers is illustrated in 3(ii).
The relevant equations of motion for the movement of fluid are dependent on the Reynolds number, which gives an indication of the relative importance of inertial and viscous effects. The Reynolds number
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of basic importance defined as follows
in this problem
is the cilium
Reynolds
CTLr, R, = ~ v ’
number
R,,
(1)
where CJ is the angular frequency, L the length and r. a characteristic radius of the cilium, while p is the kinematic viscosity. In the respiratory tract, R, is always very small [O(lO-“)I; so the viscous effects are more important. The equations of motion, called the Stokes flow equations at low Reynolds number, are defined as follows, vp = pv2u,
v .u
= 0,
(2)
where p is the pressure, u the velocity vector and p the dynamic viscosity (V = p/p, p the density). A problem similar to this has been discussed by Blake [7] when he models the cilia sublayer of microorganisms Opalina and Paramecium. The results obtained there can be modified to help us with a better understanding of the movement of mucus in the lung. In this model each cilium was modelled by a long slender body of length L and radius ro(s), s the distance along the cilium from its base, which is attached to an infinite plane. A regular array of these slender bodies is used to model the ciliated epithelia. The velocity field of each cilium is represented by a distribution of viscous fluid (Newtonian) singularities along the centreline of the slender body. The mean velocity field through the sublayer is obtained by summing the contributions from all the cilia of the regular array which is represented in terms of the following integral equation [7].
where U(X,) is the average velocity in the X,-direction, X, is the vertical coordinate, F1 the functional representation for the force in the X1direction, 5 specifies the movement of the cilium while w(s, t) and K(X,,
and 3:
Jw3Yr3) = ; . i
3.
0
<
5
x3
<
3
53,
3---
+
H
2
308
JOHN
BLAKE
where t3 is the vertical coordinate of the cilium’s movement and c is the metachronal wave velocity. The weight function CO(S,t) represents the influence of the metachronal wave, the positive sign for antipletic metachronism, the negative sign for sympletic metachronism for an effective stroke in the positive X,-direction. The kernel function corresponds to a Newtonian viscous fluid, but it should not be too difficult to find the kernel function for a simple visco-elastic model (e.g. Maxwell). Results obtained from this theory are applicable for the flow of mucus in the lung. The free surface condition of zero tangential stress can be satisfied at any height above the cilia sublayer because it is a constant stream and automatically satisfies this condition. Solution of this integral equation leads to the following representation for the velocity field:
u(X,)=aL+,K,.l.,~)06X, = U(L) = const.
L
where 9 represents the solution of the integral equation (see Blake, Ref. 7, for details). Thus the mucus is brought from a zero velocity (no-slip condition) at the base of the cilia to a velocity U(L) at the top of the cilia sublayer, while the mucus layer above moves with a constant velocity. Important parameters that arise in solving this integral equation are the density parameter JV, defined previously, and the nondimensional parameters
yap, CT
OL xc-_. C
(7)
The parameter y equals the ratio of the normal to tangential resistance coefficients for an elongated body in Stokes flow, while K is equivalent to the ratio of the average tip speed to the metachronal wave velocity c. Antiplectic metachronism is usually exhibited by cilia in the respiratory tract (i.e. the metachronal wave is moving in the opposite direction to the movement of the mucus). However little is known about the beat cycle of the cilia, so the best we can do is to take the beating pattern shown by a micro-organism exhibiting antiplectic metachronism. We use the data on Paramecium from Sleigh [12]. In Fig. 4 we show the beating pattern of the cilium and the induced velocity field for varying values of y, ri, andJlr. If the wavelength of the metachronal wave is kept constant, then the flow rate varies linearly with frequency. Thus if we decrease the beat frequency of the cilia we decrease the flow rate by a proportional amount. In Fig. 5 we show the variation of the flow rate against the concentration
MUCUS
309
FLOWS
CILIA
SliBLAYER
VELOCITY
PROFILE
FIG. 4. Movements of a cilium (Paramecium [12]) used in numerical calculations, the numbers indicate successive stages of the beat (equal time intervals). Some velocity profiles atN = 4 for varying values ofy and K are shown.
-!L
OL
FIG. 5. Flow rates as a function for several values of y and K.
of the cilia concentration
on the epithelial
surface
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JOHN
BLAKE
of cilia on the epithelia. It is observed that, if some disease (e.g. asthma) causes a decrease in the cilia concentration, then there is a rather marked drop in the flow rate. There also appears to be an optimal concentration (if we considered an efficiency) because the graph asymptotes to a maximum value for the flow rate. For the frequency of beat and length of cilium in the lung, this theory predicts flow rates around 1-3 cm/min, which compares favourably with that observed naturally. We now turn to calculating the total volume flow rate Q in the cilia sublayer and the mucus layer due to ciliary activity. This can be expressed in terms of the following integral. rU*(p) dr, R s R-II-L
Q=27-r
(8)
where U”(P) = U(R - r), U(X,) being defined in Eq. (3). If we nondimensionalize velocities with respect to CTL and the volume flow rate with respect to mR3 we obtain
q’x
oTCK3
= a/?(2 - 2rr - /?)K, + 2a2K,
- 2z3K3,
where a = L/R, j3 = H/R, and K, = U(l),
U(x) dx,
K, =
(9)
t
I
xU(x)
K, = s0
s0
dx, I
which for small a and j? yields 4 N 2aPK,
+ 2x2K,.
(10)
This indicates the volume flow rate varies linearly with the depth of the mucus upper layer, so a small increase in the mucus depth, and hence volume of mucus, can be comfortably moved by the cilia. This, however, need not be the case if the mucus thickness increases dramatically, as occurs in the diseased state, when both gravity and the pulsative air flow are important. 4. INFLUENCE
OF GRAVITY
AND
PULSATIVE
AIR
FLOW
In the normal state both the influence of gravity and the pulsative airflow due to the inspiration and expiration of air are of relatively minor importance in comparison with that due to ciliary activity. However, in the diseased state when the mucus layer increases in thickness and the viscosity of the mucus decreases, both can have a detrimental effect by decreasing the volume flow rates substantially, and may in extreme cases cause flow reversal, so that the mucus flows back into the finer bronchii.
MUCUS
311
FLOWS
We can analyze the influence of gravity quite simply. profile due to the influence of gravity (Batchelor [3]) is U(X,)
= -x(x,
- 211).
The velocity
(11)
Here, h = L + H and a is the angle at which the tube is inclined to the vertical, p the density of the mucus, and g the gravitational acceleration. Thus if we are standing vertically (a = 0), the flow rate at the top of the mucus layer will be
Pd12
U(h) = -__
4p
’
(12)
which indicates that it is flowing back into the lung at a rate proportional to the square of the layer thickness and inversely to the viscosity. Thus, in the diseased state, when the layer thickness increases, and the viscosity decreases damaging effects can occur. One of the more obvious solutions in chronic cases, of say, pneumonia, is to make gravity “work for you” and incline the patient downwards thus expediting the removal of mucus from the lung [see Fig. 6(d)]. Airway L+H
FIG. 6. Velocity profiles in (a, b) healthy state without gravity, (c, d) and in an unhealthy state including gravity. In all graphs Jlr = 7, and in (a), (c), and (d) y = 1.8 and K = 1.5, while in (b) y = 1.7 and K = 1.O. For (a) and (b) L = Hand in (c) and (d) H = 9L, the last two parts being at one-fifth scale. The values of the inclination angle a (in degrees) are shown on the diagram.
It is a much more complex problem to analyze the influence of the pulsative airflow on the mucus layer. It is difficult to calculate the velocity profiles in a branching system of tubes, because the profile varies considerably from the laminar Poiseuille flow of a single tube, and so far no
312
JOHN
BLAKE
complete fluid dynamic study has been made of this problem, although Pedley, Schroter and Sudlow [9, lo] have made some advances in the steady flow problem. Probably the best we can say is that the oscillatory stress developed at the free surface of the mucus by the pulsative airflow will be asymmetric in both time and space in many airways of the lung. It has been observed (Arnott, Clark, Jones [l]) that in diseases, such as chronic bronchitis where there are excess secretions, shear waves are developed on the mucus layer and two phase flow occurs. Obviously considerably more research on both the theoretical fluid dynamical and physiological sides needs to be done on this topic. In Fig. 6, graphs show some predicted velocity profiles for both healthy and unhealthy states of the lung where gravity is important. In Fig. 6(c), the influence of gravity dominates over the ciliary activity, so that the mucus flow is back into the finer tubes of the lung. In Fig. 6(d), the velocity profile is that predicted if a patient were inclined at a downward angle of 45” to the horizontal. Obviously this is a more desirable situation than that of Fig. 6(c). 5. CONCLUSIONS A self consistent theory has been developed to model mucus flow in the lung along the lines developed in a previous paper (Blake [7]). The advantage of this technique in its obvious superiority over the envelope approach in modelling the antiplectic coordination amongst the cilia of the respiratory tract. The theory presented here agrees reasonably well with the experimental observations in mammals predicting mucus flow rates around l-3 cm/min, for the frequencies and cilia lengths found in the lung. Consideration is given in the paper to the variation of flow rates against frequency, length of cilia, and cilia concentration on the epithelia, the first two obeying an increasing linear relationship, whereas the cilia concentration is more parabolic in nature (i.e., J) initially. Extension of a Newtonian model for the mucus to some visco-elastic model is envisaged, using the same techniques as applied in this present theory. The influence of gravity and the pulsative air flow are examined analytically and discussed respectively. It was seen in Sec. 4 that in certain pathological situations gravity can have a detrimental effect on the direction of mucus flows, this being best illustrated in the diagrams of Fig. 6. Pulsative airflow is a relatively new field of fluid mechanics; especially flow in branched tubes, such as occurs in the lung, and little progress has been made on its interaction with the mucus layer. Obviously an important problem to be solved is the mechanism of particle deposition on the mucus surface which is afterall one of its main functions.
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This work was supported by a George Murray Scholarship from the University of Adelaide, C.S.I.R.O. of Australia and the A.R.G.C. The author appreciates the comments of Sir James Lighthill and Dr. E. 0. Tuck. REFERENCES 1 W. M. Arnott, S. W. Clarke, and J. G. Jones, Mass movement of gas in respiratory tubes, Chronic inflammation of the bronchi. Prog. Resp. Res. 6,2 (1971). 2 C. Barton and S. Raynor, Analytic investigations of cilia induced mucous how, Bull. Math. Biophys. 29, 419 (1967). 3 G. K. Batchelor, An Introduction to Fluid Dynamics,
Cambridge (1967). 4 G. K. Batchelor, The stress system in a suspension of force free particles, J. Fluid Mech. 41, 545 (1970). 5 J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46, 199 (1971). 6 J. R. Blake, Infinite models for ciliary propulsion,
7 J. Blake, A model for the micro-structure 55, I (1972). 8 C. E. Miller, Streamlines
induced
flow homomorphic
J. F/ltid Me&. 49,209 (1971). in ciliated organisms, J. Fluid Mech.
and particle path lines associated with a mechanically with the mammalian muco-ciliary system, Biorheol.
6, 127 (1969). 9 T. J. Pedley, R. C. Schroter, and M. F. Sudlow, Flow and pressure drop in systems of repeatedly branching tubes, J. Fluid Mech. 46,365 (1971).
10 T. J. Pedley, R. C. Schroter, and M. F. Sudlow, Energy losses and pressure drop in model of human airways, Resp. Physiol. 9, 371 (1970). 11 S. M. Ross, A Wavy Wall Analytic Model of Muco-Ciliary Pumping, Ph.D. dissertation, Johns Hopkins University (1971). 12 M. A. Sleigh, The Bioloyr of Cilia and Flagella, Pergamon, London (1962). 13 M. A. Sleigh, Cilia, Endeaoour 30, 11 (1971).