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Mucus transport in the lung
Malhl Compur. ModeNing, Printed in Great Britain
Vol. I I,
MUCUS TRANSPORT
pp.797-800.
IN
0895.7 I77/88 $3.00 + 0.00 Pergamon Press plc
1988
THE LUNG
Manju Agarwal Department of Applied Science, Institute of Eng. and Technology, Aliganj Extension P.O., Post Bag-l, Lucknow J.B. Shukla Department of Mathematics, Canada T6G 2Gl
University
of Alberta,
Edmonton,
Alberta,
Abstract. In this Paper, the mucus transport in the lung due to metachronal wave motion of the cilia and longitudinal wave motion of the epithelium caused by coughinq is studied by taking into account the effect It is shown that as the of shear force at the mucus air interface. parameters characterisinh these factors increase, the mucus flow rate increases. By taking into account the variation of viscosity across the mucus layer (two layer model), it is also pointed out that the flow rate decreases as the viscosity of mucus layer in contact with the air mucus interface increases. Keywords.
Mucus; metachronal
wave; viscosity. Generally the viscosity of the mucus blanket chanqes appreciably with depth due to air flowing past the blanket and temperature gradient across it increasing the viscosity towards the air mucus interface. In the following analysis, since the mucus blanket layer is of an order of magnitude thicker than the length of the cilia, the fluid transport in the serous layer is assumed to be negligible.
INTRODUCTIDY Cilia are small hairlike projections on the epithelial cells lined all along the bronchial respiratory tract, the inner lininq of which is covered by mucus blanket. The mucus secreted from the underline cells, is a hiohlv viscous fluid, the main function of which.is‘to clear the inhaled air of unwanted particles and to bring it to near body temperatures. In recent decades the mechanism of cilia motion and mucus transport have been studied by many investigators [Miller, 1966, 1967, 1969; Barton and Raynor, 1967; Blake 1971, 1973, 1975; Lardner and Shack, 1972; Blake and !a!inet,19801.
The non-dimensional equations governing the slow motion of mucus layer, under long wave in the moving frame can length approximation, be written as follows;
(1)
It may be noted here that thounh the mucus transport in tubles due to metachronal wave along a ciliated surface (peristaltic wave) has been studied TLardner and Shack,. 19721. no attempt has been made to apply this con&t in the study of motion of mucus blanket due to cilia in the lung. In view of the above, we study in this paper the transport of mucus in the lung by taking into account the followino factors: 1.
2.
3.
?JLp=0
As the movement of mucus is assumed to be qoverned by metachronal (peristaltic) wave due to cilia motion, the longitudinal wave due to muscular activitv of the epithelium bv couqhinq and expiration/inspiration of the air-in the _ lunq, the following boundary conditions in nondimensional form are prescribed,
Effect of metachronal wave alonq the ciliated surface by representing it with peristaltic wave. Effect of shear stress on the mucus air interface due to air motion (insoiration , and expiration). Effect of motion of eoithelial surface due to muscular activity caused by coughing. MATHEMATICAL
(3)
ar
w = w. sin Z-1 x - 1
MODEL
u = WI r=hS
The physical situation of the movement in the lung is idealised by a circular tube geometry, the inner wall of which is ciliated (see Fig.1) It is noted that there exists two distinct fluid layers, the scrous layer consisting of watery liquid with much lower viscosity than the mucus blanket lying on the tip of cilia.
at
r=h
at
r=h
2
- dw p dr = ~~ sin 2~ x
u=Wir=h$
797
s
(4)
798
Proc.
6th Ini.
Conf
on Mathematical
Modeilinn
hs = 1 + cs sin 2~ x h=
a +
E
1 (hz-h2)(Wo I3 = I
(5)
sin ~II x
Equations (l)-(5) have been written by using the following dimensionless parameters and variables [see Figure 1 for symbols]. r=g
%
“=!L c,
u=u’,
,
ES
0
J5
i=“,
c1
%
ij =
S
S
and
~=~
S
aS
where X,R and x,r are dimensionless stationary and moving coordinates corresponding t;,d;m;ns;onal coordinates (XI,R')_ and 1s dlmenslonless time; n(r) is viscosi;y function; ns is viscosity of the
stress and velocity.
is interface
~~ sin 2nx c
is the peristaltic
being
the flux
q = (q' 1 m:C
in the moving
system)
which Ap
=
(6)
I1
;
q
is given
'f2 f 01
to the waves
can be written
.
( hl
hlI
r 2 hs
hi
hl = a
at
r = h
and
r = hs,
as follows.
= al + el sin 2~ x
(15)
is the dimensionless where a =al/as. al 1 mean radius of the central interface and is the dimensionless amplitude of e1 = b,/as
2
(1 - $2
0 = (1 - a*)[(1 +$)
- 2(1+WoCs) 2+ 3CE
(8)
(9) sin 2n x dx
h (r
for
(16) +F T
11 - (1 - $)%
30
I2 = 1
for
the wave at this interface. After using Equation (14) in Equation (8) to Equation (12) we have from Equation (13), the expression for 0 as follows.
where
I1 =
>1
S
on integration and taking p(O) - p(1) = 0, we get I2+'3
(8).
(14)
hl
2r w dr
O I1
(13)
where the viscosity of the layer in contact Assuming the with the air is considered large. form of the waves at the interface r = hl, is
1;s
q=r
C2)
time
us
by
q =
+ ; ($
by Equation
1
uS
wave
h - =. sin 2n x / ' $ dr r n flux
is given
U1 ;=__=;
similar
The dimensionless
q
q + (1 -a*)
;=--1Us -
h r ' J211 (r-$idr
- 2,
- hr)dr
In such a case,
shear
Integrating equation (1) and using corresponding boundary conditions (4) for W, we get the dimensionless velocity
W = (W. sin 2n x-l)
($
It is noted here that the expression for flow rate in the fixed coordinate system given by Equation (13) is valid for any general radial To study the effect of variation of viscosity. viscosity variation more simply and analytically we consider the case of Step-wise Variation of Viscosity as shown in Figure 1.
mucus layer in contact with the serous layer; W. sin 2~ x is longitudinal velocity of the epithelium;
+-
The time averaged flux ij for a complete period T = x/c is given by Q
where
cus
b =$
(12)
c ’
'a Tos =-,h=~,a,-$
TO
+ r3 - 2h2r)dr
(11)
w=!c
c ’
w
C
($
fp.lis
w=!c
’
fl
IhS h 1 2C
=
x
, t=Ct’ x
x=x-t
dx 0
and
,R=R',x'=x,~=X, S
fl
sin 2~ x-l)
(l- Ef)" 2+ 3$
ES
F3 = -2a3[l Log Ul
;
-
kO!J
alI
(17)
(10)
+ &$l
-k)
+ 1 - $1
Proc. 6th Int. Conf. on Muthemutical
It is noted
i, = 1, in Equation
that for
we get the constant viscosity when ~~ = 0, W. = 0 Equation
799
Modelling
(16)
case. Further (16) reduces to
0.30 r
the flow flux for peristaltic transport in a circular tube as obtained in Shukla et al. (1980). DISCUSSION
AND CONCLUSIONS
To see the effect of various parameters on the mucus flow rate Equation (16) is calculated and plotted in Figures (Z)-(4).
0
= 5.0
/L ,
0.12
It is noted from these figures that for fixed the time average flow rate 0 inTo, a, al, creases
as
effects
of variation
which
E
S
increases. of
it is noted that
In Figure
Q
(2) the
0
is shown from
w.
increases
as
increases. Similarly we see from Figure that 0 increases as ~~ increases.
0.02
0.06
0.04
0.10
0.06
-
6s
(-w,) (3) FIG. 2.
The effect of the viscosity of the mucus layer in contact with the air on the flow rates is shown in Figures (4). It is observed from this figure that 0 decreases as ; increases. From the Equation (16) it is further noted for > 0, the increase or decrease of (I with
Variation
of
0
with
~~
and
Wo.
0.30
ES
wO
depends
upon whether
it is negative
or
positive. t
0.24
1 -
0.16
-
0.12
-
8
?‘, U’ 0.06
a
= 0.75
cl,
I
0.95
AP=
-
0
wo = -5.0 jL,=50 I
0
1
0.02
004
1
1
0.06
0.06
1 0.10
cr -
FIG. 3.
Variation
of
(j with
cS
and
TV.
X\W' 0.30
0 24
I
0.1s
0 FIG. 1.
Mucus transport
in lung.
0.12
a
=
0.75
a,
q
0.95
AP’ 0.06
0
0
To
=
lo2
w.
=
- 5.0
I
I
I
I
I
0.02
0.04
0.06
0.06
0 .I0
6s -
FIG. 4.
Variation
of
0
with
E<
and
;.
Proc.6th
800
Int.
Confi on Mathemutical
REFERENCES Barton, C. and Raynor (1967). Analytical investigation of cilia induced mucus Bull. Math. Biophysics, 29, 419-428.
Blake, J.R. (1975). On the movement of mucus J. Biomechanics, 8. 179-190. in the lung. flow.
Barton, C. et al. (1968). Peristaltic flow in Bull. Mathematical Biophysics,s, tubes. 663-680. Blake, J.R. and H. Winet (1980). On the mechanics of muco-ciliary transport. Biorheology, 17, 125-134. Infinite models for Blake, J.R. (1971). J. Fluid Mech., 2, ciliary propulsion. 209-222. Blake, J.R. (1973). Mucus flows. Biosciences, l7_, 301-313.
Math.
Modding
Lardner, T.J. and W.J. Shack transport. Bull. Math. 325-335.
(1972). Cilia Biophys., 34,
An investigation of the Miller, C.E. (1966). movement of Newtonian liquids initiated and sustained by the oscillation of mechanical cilia. Proc 5th Cong. Appl. 715-720. Mech. Shukla, J.B., R.B. Parihar, B.R.P. Rao and S.P. Gupta (1980). Effects of peripherallayer viscosity on peristaltic transport of a bio-fluid. J. Fluid Mech., 97. 225-237.
Vol. I I,
MUCUS TRANSPORT
pp.797-800.
IN
0895.7 I77/88 $3.00 + 0.00 Pergamon Press plc
1988
THE LUNG
Manju Agarwal Department of Applied Science, Institute of Eng. and Technology, Aliganj Extension P.O., Post Bag-l, Lucknow J.B. Shukla Department of Mathematics, Canada T6G 2Gl
University
of Alberta,
Edmonton,
Alberta,
Abstract. In this Paper, the mucus transport in the lung due to metachronal wave motion of the cilia and longitudinal wave motion of the epithelium caused by coughinq is studied by taking into account the effect It is shown that as the of shear force at the mucus air interface. parameters characterisinh these factors increase, the mucus flow rate increases. By taking into account the variation of viscosity across the mucus layer (two layer model), it is also pointed out that the flow rate decreases as the viscosity of mucus layer in contact with the air mucus interface increases. Keywords.
Mucus; metachronal
wave; viscosity. Generally the viscosity of the mucus blanket chanqes appreciably with depth due to air flowing past the blanket and temperature gradient across it increasing the viscosity towards the air mucus interface. In the following analysis, since the mucus blanket layer is of an order of magnitude thicker than the length of the cilia, the fluid transport in the serous layer is assumed to be negligible.
INTRODUCTIDY Cilia are small hairlike projections on the epithelial cells lined all along the bronchial respiratory tract, the inner lininq of which is covered by mucus blanket. The mucus secreted from the underline cells, is a hiohlv viscous fluid, the main function of which.is‘to clear the inhaled air of unwanted particles and to bring it to near body temperatures. In recent decades the mechanism of cilia motion and mucus transport have been studied by many investigators [Miller, 1966, 1967, 1969; Barton and Raynor, 1967; Blake 1971, 1973, 1975; Lardner and Shack, 1972; Blake and !a!inet,19801.
The non-dimensional equations governing the slow motion of mucus layer, under long wave in the moving frame can length approximation, be written as follows;
(1)
It may be noted here that thounh the mucus transport in tubles due to metachronal wave along a ciliated surface (peristaltic wave) has been studied TLardner and Shack,. 19721. no attempt has been made to apply this con&t in the study of motion of mucus blanket due to cilia in the lung. In view of the above, we study in this paper the transport of mucus in the lung by taking into account the followino factors: 1.
2.
3.
?JLp=0
As the movement of mucus is assumed to be qoverned by metachronal (peristaltic) wave due to cilia motion, the longitudinal wave due to muscular activitv of the epithelium bv couqhinq and expiration/inspiration of the air-in the _ lunq, the following boundary conditions in nondimensional form are prescribed,
Effect of metachronal wave alonq the ciliated surface by representing it with peristaltic wave. Effect of shear stress on the mucus air interface due to air motion (insoiration , and expiration). Effect of motion of eoithelial surface due to muscular activity caused by coughing. MATHEMATICAL
(3)
ar
w = w. sin Z-1 x - 1
MODEL
u = WI r=hS
The physical situation of the movement in the lung is idealised by a circular tube geometry, the inner wall of which is ciliated (see Fig.1) It is noted that there exists two distinct fluid layers, the scrous layer consisting of watery liquid with much lower viscosity than the mucus blanket lying on the tip of cilia.
at
r=h
at
r=h
2
- dw p dr = ~~ sin 2~ x
u=Wir=h$
797
s
(4)
798
Proc.
6th Ini.
Conf
on Mathematical
Modeilinn
hs = 1 + cs sin 2~ x h=
a +
E
1 (hz-h2)(Wo I3 = I
(5)
sin ~II x
Equations (l)-(5) have been written by using the following dimensionless parameters and variables [see Figure 1 for symbols]. r=g
%
“=!L c,
u=u’,
,
ES
0
J5
i=“,
c1
%
ij =
S
S
and
~=~
S
aS
where X,R and x,r are dimensionless stationary and moving coordinates corresponding t;,d;m;ns;onal coordinates (XI,R')_ and 1s dlmenslonless time; n(r) is viscosi;y function; ns is viscosity of the
stress and velocity.
is interface
~~ sin 2nx c
is the peristaltic
being
the flux
q = (q' 1 m:C
in the moving
system)
which Ap
=
(6)
I1
;
q
is given
'f2 f 01
to the waves
can be written
.
( hl
hlI
r 2 hs
hi
hl = a
at
r = h
and
r = hs,
as follows.
= al + el sin 2~ x
(15)
is the dimensionless where a =al/as. al 1 mean radius of the central interface and is the dimensionless amplitude of e1 = b,/as
2
(1 - $2
0 = (1 - a*)[(1 +$)
- 2(1+WoCs) 2+ 3CE
(8)
(9) sin 2n x dx
h (r
for
(16) +F T
11 - (1 - $)%
30
I2 = 1
for
the wave at this interface. After using Equation (14) in Equation (8) to Equation (12) we have from Equation (13), the expression for 0 as follows.
where
I1 =
>1
S
on integration and taking p(O) - p(1) = 0, we get I2+'3
(8).
(14)
hl
2r w dr
O I1
(13)
where the viscosity of the layer in contact Assuming the with the air is considered large. form of the waves at the interface r = hl, is
1;s
q=r
C2)
time
us
by
q =
+ ; ($
by Equation
1
uS
wave
h - =. sin 2n x / ' $ dr r n flux
is given
U1 ;=__=;
similar
The dimensionless
q
q + (1 -a*)
;=--1Us -
h r ' J211 (r-$idr
- 2,
- hr)dr
In such a case,
shear
Integrating equation (1) and using corresponding boundary conditions (4) for W, we get the dimensionless velocity
W = (W. sin 2n x-l)
($
It is noted here that the expression for flow rate in the fixed coordinate system given by Equation (13) is valid for any general radial To study the effect of variation of viscosity. viscosity variation more simply and analytically we consider the case of Step-wise Variation of Viscosity as shown in Figure 1.
mucus layer in contact with the serous layer; W. sin 2~ x is longitudinal velocity of the epithelium;
+-
The time averaged flux ij for a complete period T = x/c is given by Q
where
cus
b =$
(12)
c ’
'a Tos =-,h=~,a,-$
TO
+ r3 - 2h2r)dr
(11)
w=!c
c ’
w
C
($
fp.lis
w=!c
’
fl
IhS h 1 2C
=
x
, t=Ct’ x
x=x-t
dx 0
and
,R=R',x'=x,~=X, S
fl
sin 2~ x-l)
(l- Ef)" 2+ 3$
ES
F3 = -2a3[l Log Ul
;
-
kO!J
alI
(17)
(10)
+ &$l
-k)
+ 1 - $1
Proc. 6th Int. Conf. on Muthemutical
It is noted
i, = 1, in Equation
that for
we get the constant viscosity when ~~ = 0, W. = 0 Equation
799
Modelling
(16)
case. Further (16) reduces to
0.30 r
the flow flux for peristaltic transport in a circular tube as obtained in Shukla et al. (1980). DISCUSSION
AND CONCLUSIONS
To see the effect of various parameters on the mucus flow rate Equation (16) is calculated and plotted in Figures (Z)-(4).
0
= 5.0
/L ,
0.12
It is noted from these figures that for fixed the time average flow rate 0 inTo, a, al, creases
as
effects
of variation
which
E
S
increases. of
it is noted that
In Figure
Q
(2) the
0
is shown from
w.
increases
as
increases. Similarly we see from Figure that 0 increases as ~~ increases.
0.02
0.06
0.04
0.10
0.06
-
6s
(-w,) (3) FIG. 2.
The effect of the viscosity of the mucus layer in contact with the air on the flow rates is shown in Figures (4). It is observed from this figure that 0 decreases as ; increases. From the Equation (16) it is further noted for > 0, the increase or decrease of (I with
Variation
of
0
with
~~
and
Wo.
0.30
ES
wO
depends
upon whether
it is negative
or
positive. t
0.24
1 -
0.16
-
0.12
-
8
?‘, U’ 0.06
a
= 0.75
cl,
I
0.95
AP=
-
0
wo = -5.0 jL,=50 I
0
1
0.02
004
1
1
0.06
0.06
1 0.10
cr -
FIG. 3.
Variation
of
(j with
cS
and
TV.
X\W' 0.30
0 24
I
0.1s
0 FIG. 1.
Mucus transport
in lung.
0.12
a
=
0.75
a,
q
0.95
AP’ 0.06
0
0
To
=
lo2
w.
=
- 5.0
I
I
I
I
I
0.02
0.04
0.06
0.06
0 .I0
6s -
FIG. 4.
Variation
of
0
with
E<
and
;.
Proc.6th
800
Int.
Confi on Mathemutical
REFERENCES Barton, C. and Raynor (1967). Analytical investigation of cilia induced mucus Bull. Math. Biophysics, 29, 419-428.
Blake, J.R. (1975). On the movement of mucus J. Biomechanics, 8. 179-190. in the lung. flow.
Barton, C. et al. (1968). Peristaltic flow in Bull. Mathematical Biophysics,s, tubes. 663-680. Blake, J.R. and H. Winet (1980). On the mechanics of muco-ciliary transport. Biorheology, 17, 125-134. Infinite models for Blake, J.R. (1971). J. Fluid Mech., 2, ciliary propulsion. 209-222. Blake, J.R. (1973). Mucus flows. Biosciences, l7_, 301-313.
Math.
Modding
Lardner, T.J. and W.J. Shack transport. Bull. Math. 325-335.
(1972). Cilia Biophys., 34,
An investigation of the Miller, C.E. (1966). movement of Newtonian liquids initiated and sustained by the oscillation of mechanical cilia. Proc 5th Cong. Appl. 715-720. Mech. Shukla, J.B., R.B. Parihar, B.R.P. Rao and S.P. Gupta (1980). Effects of peripherallayer viscosity on peristaltic transport of a bio-fluid. J. Fluid Mech., 97. 225-237.