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A mechanical approach to the longitudinal dispersion of gas flowing in human airways
0021-929lM5 c
13.00 +
1985 Perpmon
00
Press Ltd.
A MECHANICAL APPROACH TO THE LONGITUDINAL DISPERSION OF GAS FLOWING IN HUMAN AIRWAYS* ABDELLAZIZ
BEN JEBRIA
Laboratoire de Physiologic. CniversitC de Bordeaux-II-146 France
rue Leo Saignat. 33076-Bordeaux Cedex.
Abstract-The atm of this work is to contribute to elucidating the mechanism underlying gas mixing in the human pulmonary airways. For this purpose, a particular attempt is made to analyse the fluid mechanical aspectsofgaseous dispersion using bolus responsemethods. The experiments were performed on five normal subjectsby injection of IO cm3 bolus of He, Arand SF6 into the latter part of the inspired airstream. in such a way that the whole bohts entered the inspiratory flow and was recovered during the following expiration. The results. presented in a logarithmic plot ofdimensionless variance (dispersion of the output bolus) against the Peclet number. show that gaseous dispersion is only slightly dependent on the nature ofthe tracer gas but is strongly related to flow velocity. This is in agreement with the theory of turbulent or disturbed dispersion; houe\.er. it:eems that Taylor laminar dispersion does not play a significant role in the airways.
NO>IENCLATURE
Subscripts i
It, H x :
C’p
tube radius concentration of tracer gas diameter of a tube or an airway coefficient of molecular diffusion laminar dispersion coefficient turbulent dispersion coefficient disturbed dispersion coefficient frequency of oscillation constant coefficient equal to 0.73 length of a corresponding generation length over which appreciable changes in concentration occur axial length of total airways in which the bolus is penetrated number of airways per generation peclet number Reynolds number radial coordinate cross-sectional area time mean residence time average velocity of a fluid penetration volume flow rate friction velocity reduced time angular frequency axial coordinate generation number
Greek lerrers I
; \’ P
oz 1 6, ; 70
Received
Womersley parameter resistance coefficient dynamic viscosity kinematic viscostty density variance of C-curve variance of the time concentration curve arbitrary variable of a function friction stress
May 1984: in recked form Norember 1984. *This work was supported by a grant from INSERM (No. 84.5001).
j 0
input number of reading concentration patterns output INTRODUCTION
the interesting biomechanical aspects of the lung, gas flow and airway resistance have received much attention. However, few attempts have been made to analyse the fluid mechanical aspects of gaseous dispersion in the airways, despite its considerable role in respiratory physiology. One of the pending problems is that of the mechanisms leading to the uneven distribution of inspired gas throughout the lung. How much of this inhomogeneity is the result of unequal flow in different regions (regional inhomogeneity)? How much is the result of a non-equilibrium of gas by diffusion in alveolar sacs (stratified inhomogeneity)? And how significant a role could gaseous dispersion play in these two mechanisms? One of the promising approaches towards understanding the distribution of inhaled gas throughout the lung, has been that of mathematical models. Since the pioneer work of Rauwerda (1946), a number of other mathematical models have been proposed by different authors (La Forceand Lewis, 1970; Cumming et al., 1971; Scherer er al., 1972; Paiva, 1973) to quantify the role of stratified inhomogeneity in the alveolar plateau slope; their results remain controversial. More recently, it has become apparent that gas transport in the lung cannot be described by the association of pure convection and pure molecular diffusion, if one considers the central airways. Some authors (Yu, 1975; Ultman er al., 1978; Ben Jebria et al., 1981) have demonstrated theoretically and experimentally that gaseous dispersion in the proximal airways plays an important role, because of the complicated nature of the motion prevailing in this zone of the lung. Indeed, during a single breath. flow conditions in the Among
399
ABDELWZIZ BEN JEBRIA
400
large airways are different from those in the central airways and even more different from those in the more distal airways. In fact, the repeated branching tube network of the bronchial tree from the trachea to the periphery, the increase in the total cross-sectional area of the many parallel pathways, and the irregular geometry of the airways are the features necessary for an understanding of gas flow in the lung. In a person breathing normally, the inspired gas enters the lung through the upper airways (oropharynx or nasopharynx, and larynx); below this level, the flow generates an intense turbulent jet in the trachea which propagates even further down. Moreover, in the successive finite bronchial tubes, each junction disturbs the flow, and these eddies do not dissipate before the next blending. Consequently, in the upper, large and medium airways, longitudinal dispersion occurs through the interaction of axial convective velocities and complex motions due to the turbulent eddies and swirling. In the most distal airways and alveoli, gas transport occurs mainly by molecular gaseous diffusion, since convection decreases in the presence of the considerable increase of the total cross-sectional area. Our main concern, in this paper, is with gas mixing in conducting airways, to the exclusion of the distal airways and alveoli. A particular attempt is made to analyse the fluid mechanical aspects of the problem using a bolus response method; this is to point out the importance of the longitudinal dispersion of inert tracer gas in the pulmonary mixing of normal subjects. This bioengineering approach per se contributes to the understanding of the mechanisms underlying the gas transport process in the lung. THEORETICAL CONSIDERATIONS
Dispersion of a tracer gas in a laminarjlow
During Poiseuille laminar flow, when a tracer gas is instantaneously introduced into a fluid moving steadily through a pipe, its velocity is greater near the center than near the walls, because of a varying parabolic profile effect over its cross-section (Fig. 1). Now, as Rohrer (1915) pointed out, if there is a variable longitudinal concentration of the components of the
fluid, this leads to variable radial concentration and to increased diffusion. In fact, in the regions where the tracer molecules are carried downstream less rapidly. the lower velocity allows these molecules to diffuse radially. In such a case, this phenomenon can be quantified and the concentration is given by the following equation, usually known as Taylor’s equation (Taylor, 1953).
(1) Taylor (1953) and later Aris (1956) showed that under certain conditions this mechanism (coupling between axial convection and radial diffusion) can be described as longitudinal dispersion governed by an effective coefficient of diffusivity ( 9i,,m) obeying the onedimensional axial transport equation (‘I Taylor (1953) found that the laminar dispersion coefficient depends on the mean axial velocity of the fluid. the diameter of the tube and the molecular diffusion coefficient, as follows ..z.Jz
9
rm=D++
It is apparent from this equation that glaa increases very rapidly as flow rate and tube diameter increase. Moreover, g,,,,,, is very much larger than the molecular diffusion coefficient (D) and it can be seen that the lower the molecular diffusivity of the inert tracer gas. the higher the dispersion coefficient for any flow velocity. Consequently, Qr,,,, is closely dependent on the nature of the tracer gas flowing in the tube.
Dispersion of a tracer gas in turbulentjlow
Unlike laminar flow, when a tracer is injected instantaneously into a fluid moving through a straight tube at high flow rate under turbulent flow conditions. the particles are distributed randomly in all directions. Thus, the velocity cannot be descr%ed. analytically since it fluctuates randomly, both in magnitude and in direction (Fig. 1). It has been recognised that the transition to turbulence in a circular pipe begins at the same value of the Reynolds number (about Re = 2300). In these conditions, Taylor (1954) showed that the analogous problem of dispersion in turbulent flow can be solved in the same way as that in laminar flow. Thus, the turbulent dispersion coefficient is expressed as follows S?I”? = 10.1 au* = 7.14au&
time -_o
later
Fig. 1. Velocity profiles of gaseous dispersion in steady laminar and turbulent flow after an instantaneous injection of a bolus of a tracer gas.
in which, u+=m=&? ,,/:-2logy
= -0.4+4logRe
(4)
Gas flowing in human airways
401
instantaneous flow and volume were monitored during the whole respiratory cycle. Re =f ud=--. The seated subject, provided with a nose clip, V P breathed air continuously through a mouth piece It can be seen from equation (4) that the relationship connected through a valve system to inspiratory and between the turbulent dispersion coefficient and the expiratory lines. He controlled his respiratory flows by velocity is linear rather than quadraticas in the laminar looking at an oscilloscope. placed directly in front of dispersion coefficient in equation (3). In addition .$Z,., him. After some practice, the subject was able to is independent of the molecular diffusion coefficient, breathe at a constant flow rate. The bolus was then unlike Qla,,,. injected at a predetermined level of the tidal volume during inspiration, in such a way that the whole bolus Dispersion of' a tracer gas in disturbed flow entered the inspiratory flow and was recovered during the following expiration. The dispersion of the tracer In the successive branched airways, flow conditions gas depend on the flow rate, on one hand, and on the are extremely complicated and secondary motions degree of the penetration of the bolus in the airways, persist even at low values of Reynolds number (Pedley on the other. Thus, the subjects were asked to breathe et al.. 1977). Thus, at each airway bifurcation, seconthree times independently and respectively at rates of dary swirlin motions distort the axial velocity profiles, 200,300 and 400 cm3 s- ’ (inspiratory flow rate equal and their effect&m dispersion are similar to those of to the expiratory one). For each flow rate and each turbulence. Hence, dispersion in branching networks tracer gas, a wide range of penetration volume was like airways is difficult to analyse mathematically. investigated (from 80 to 300 cm3). The latter was taken However, Scherer et al. (1975) measured benzene vapor to be the portion of the tidal volume included between dispersion in a five generation symmetric geometrical the beginning of the bolus injection and the end of model of the trachea-bronchial tree; they computed an inspiration. effective axial diffusivity coeficient (disturbed disperThe flow rate was measured with a Fleisch pneumosion coefficient) which was expressed by the following tachograph and the tracer concentration with a formula ‘Centronic’ mass-spectrometer using the ‘Spectrum’ gdis =D+Kud (5) mode; this was to obtain the best possible precision at or low concentrations. Both sets of apparatus were 5?,,JD = 1 + K Pe. connected to the analog-digital converter of a With PDP 1l/O3 digital computer. and
ud
ud
Pe=D. As can be seen from equation (5) the disturbed dispersion coeficient is proportional to the product of the mean flow velocity and the airway diameter. Moreover molecular diffusion probably plays a negligible role in the more proximal airways. This is a result qualitatively similar to turbulent dispersion in a circular pipe. according to Taylor’s theory (1954).
METHODS
Experimenral
procedure
of bolus-response
The biomechanical objective of the research was limited to the study of gaseous dispersion in the central airways of the trachea-bronchial tree. For this purpose, the tracer bolus response technique seemed a promising. useful, and helpful test for understanding the mechanisms of gas mixing occuring in the lung. Thus, the experiments were performed on five healthy subjects by injection of 10 cm3 bolus of helium, argon and sulphur hexafluoride into the latter part of the inspired airstream, using an apparatus described in the form of a technical note and published elsewhere (Ben Jebria et al.. 1981). The tracer bolus was automatically injected into the carrier gas by an electrically operated valve, the opening period of which was set at 0.04 s. The concentration of the tracer gas as well as the
Data analysis
If the tracer gas is injected instantaneously (without any delay) into a fluid moving down a pipe, at x = 0, the solution of equation (2) which gives the tracer distribution as a function oft and x is, from Levenspiel and Smith (1957), C(x, t) = *exP(
-s)+
(6)
At position x = f.p, and on rearranging equation (6) the concentration of tracer gas as a function of time is given by 1 C(t) = 2&Y tlVP)(~luLP) x exp
(
-
(1-
.
rit/vp)2
4(VtlVP)(gluLP)
>
The use of equation (6) or equation (7) for our problem assumes that the central airways are assimilated to a single tube with mean axial velocity and cross-sectional area equal to that of the zeroth generation (trachea). Assuming that the output tracer concentration as a Gaussian function and the dispersion as the variance of the equation (7), then
ALIDELLAZIZ BEN JEBRIA
402
If the mean residence time is as follows t=
Jb; CC(e)de / j;
C(C)dC.
(9)
Then the variance is as follows
Finally, for practical reasons, the variance of the timcconcentration is estimated as described by Lcvenspiel and Smith (1957), by substituting the integrals of equation (10) by finite sums as follows zt;cj
2
ut
xt,cj
2
( >
I--
(11)
-.
ZCj
ZCj
The expression of equation (11) is useful for computing the variance of the time-concentration curve from experimental data. And in this case l? = ( if/ Vp)2a:
Fig. 2. Schematic representation of the bolus response method: input signal (bolus of 10 cm’ injected during inspiration) followed by output mass spectromcter%hannel (bolus
(12)
measured in front of the lips during expiration).
As can be seen in Fig. 2, it is never possible to obtain experimentally a unit impulse input signal; thus the experimental dispersion is considered as: At# = uk -Uf.
RRSULTS
First of all, we have to distinguish between the large and medium airways. Indeed, Fig. 3 shows that for penetration volumes lower or equal to 140cm3corresponding to the zone above the fourteenth generation according to Weibel (1963)-there are no differences in dispersion between helium, argon and sulphur hexafluoride. In contrast, it can be seen in Fig. 3 that for a penetration volume ranging from 140 to 300cm3 (corresponding to the zone between the fifteenth and eighteenth generation) the dispersion of He, Ar and SF, is different. Nevertheless, in both cases, gaseous dispersion decreases as flow rate increases; but this decrease is much less in the first case. Now, as demonstrated by Fredberg (1980), the relationship between the mass transfer coeficient (in our case u2) and the Peclet number is similar to the relationship between the normalized pressure drop and the Reynolds number for the airways (Jaffrin and Kesic, 1974); in other words, there is an analogy between diffusional resistance and, flo\)r resistance.
Now, according to the theory developped by Levenspiel and Smith (1957), cr2 can be evaluated by substituting in equation (8). the theoretical function of C by the formula of equation (7). Thus
D z
u2=8
2
(
+2;
(13)
>
or, solving for g/uL, $=b(&5i-1).
(14)
Finally, as can be seen in Table 1, we have calculated the values of the longitudinal dispersion coefficient in the zeroth generation (trachea) according to the following formula: 9 = Q(&G
-l)u,L,
in which u2 and (u, 3: i/S.)
(1%
were measured.
Table 1. Comparison between the dispersion coefficientsobtained in this work and those of Scherer and Taylor. V,p: V, < 14Ocm’ and V,p: 140cm3 c V, G 300 cm” Dispersion coefficients (cm’ s- ‘) Flow rate
Tracer He
Cl&-
V,P
VaP
Scherer’s coetlicient
200 300 400
102.47 135.37 152.22
173.31 189.36 194.19
104.26 156.00 207.73
200
102.22 133.75 151.49 102.98 133.50 153.94
156.35 168.19 178.34 128.51 145.65 156.22
103.66 155.40 207.13 103.55 155.29 207.02
Ar % 200 SF6
Present work
2
Taylor’s laminar coefficient 131.58 295.06 523.92 523.32 1177.23 2092.70 1162.59 2615.71 4650.08
403
Gas flowing in human airways
+
5. I:
4
rrocn+
i %, ‘.., \I ‘..._, ?.
-.
3. ;; i 2. E z kIa E 6
2_
%.
‘.... “...,
Thus, the experimental results of gaseous dispersion (in terms of 17’)are shown in Figs 4 and 5, which are a logarithmic plot of dimensionless variance vs the Peclet number in the trachea. For each flow rate and for each tracer gas, the data plotted are the arithmetic average of dispersion obtained in all subjects. The principal results obtained are that : (1) for penetration volumes lower or equal to 140 cm3 and for each flow rate, gaseous dispersion for the three tracer gases (He, Ar, SFJ is independent of the xeroth Peclet number (Fig. 4); (2) for penetration volumes ranging from 140 to 300 cm3 and for each flow rate, gaseous dispersion is dependent on the Peclet number in the trachea (Fig. 5). These results mean that in the tirst case (in the more proximal airways) dispersion is only related to flow velocities; however, in the second case (near the distal airways), dispersion is on the one hand related to flow velocities and to physical properties of tracer gases on the other. This finding is in agreement with the theoretical prediction of turbulent dispersion of Taylor (1954) as well as with the in-oitro experimental results of Scherer et al. (1975). In addition, although molecular diffusion affects longitudinal dispersion in the middle airways (140 cm3 < VPQ 300 cm3) it does not imply that this result is in contrast with that found
3oocm’
+
‘k.. %. . .“‘3 ....,,., b..t .. . “.... \ i He ....... . .. “.. .X_.‘. . ...... i Plr ‘+ SF6
r 300
) 200
400
300 Flow
I
Rate (cf-+’
Fig. 3. Gaseous dispersion (variance of bolus) plotted against flow rate for the three tracer gases (He, Ar, SF& Vertical bars indicate mean values f SD.
Pe
Kj2
lo2
2
2
4
I
1
9 7 9 9,02
2
3
4
5
6 799,0’
1
Fig. 4. Logarithmic ptot of gaseous dispersion vs the Pecletnumber in the trachea. Results obtained in the zone lower or equal to 140 cm3 of penetration volumes (VP). V,, Vsand Ps represent the flow rates of 200,300 and 400 cm3 s- ‘, respectively.
Fig. 5. Same as Fig. 4 except that these results are obtained for penetration volumes ranging from 140 to 300 cm’.
AEIDELLAZIZ BEN JEBRIA
404
by Scherer et al. (1975). Indeed, according to equation (5), when the penetration volume increases (then the cumulative cross-section of the airways) the velocity decreases gradually and the mixing by diffusion becomes more and more important.
underlying gas mixing?
for
If one refers to the values of Reynolds number in the geometry of the airways, one can expect that during spontaneous breathing, the flow in the human lung may be laminar, and that the motion occurs with a parabolic velocity profile. Thus, as suggested by some authors (Pack et al., 1977); Van Liew and Mazzone, 1977; Horsfield et al., 1977) the Taylor mechanism of dispersion in a fully developed laminar flow may be applicable. However, as far as we know, the two conditions set forth by Taylor (1954) to apply equation (3). have been somewhat overlooked. They are 4L a
ua % D 6.9.
(16)
We shall test the validity of these two conditions using Weibel’s anatomical data (1963). However, some rearrangements are needed to apply them in the lung. The condition us/D g 6.9 yields Va(2)
(17)
S(z)D ’ ‘*‘*
With S(r) = n(z)na’(z), equation
L
9
4ztW4
(17) becomes 6.9
W)
or,
As regards the other condition, 4L/a & us/D. if to simplify matters (the initial concentration being of c solution is the length) infinite = 1/2Co erf(x/2fi) (Crank, 1970). About 83 % of the change in concentration occurs for an abcissa x -2fi
thus
(19)
--.44
t_
U
Now taking into account equations equation (19) can be written as
and the condition
(3) and (20),
4Lja % us/D yields
16
u/(z) J 480B--F
us(z)
(23)
As a numerical example, the two conditions equations (18) and (23) give
DISCUS!SION
What is the mechanism
or
(22)
Q=400cm3s-1
of
n(z)u(zw 74 n(z)!(z) Q 96.
Taking the symbol< to mean a ratio of 1: 5, the first condition is fulfilled from 0 to generation 6, and the second from generation 10 to higher generations. Thus there is no overlapping of the two conditions; therefore, there is no place in the airways where Taylor’s laminar dispersion coefficient can be applied. However, this result does not exclude the fact that the mixing in the middle airways could occur by the coupling between radial diffusion and axial convection (as it has been noticed by Wilson and Lin ( 1970) ); and therefore gas transport can be described transitorily by equation (1) in the intermediate region of the lung. Implication of gaseous dispersion in the mechanism underlying the new mode of ventilation
Recently, there has been widespread opinion which recognizes that adequate alveolar ventilation and gas exchange can be maintained with tidal volumes which are smaller than the anatomic dead space at high oscillatory frequencies. However, it remains to elucidate the mechanism contributing to gas transport during high frequency ventilation. Among the hypothetical ideas contributing to an understanding of this phenomenon, some qualitative mechanisms, such as enhanced mixing in the central airways combined with molecular diffusion in the periphery of the lung, are advanced. Nevertheless, the explanation of the fluid mechanical phenomena which generate this augmented diffusion remains controversial. The question is what is the nature of the flow? Is it simply oscillated laminarly, or oscillated with turbulence; or is there a steady state turbulent flow configuration? Fredberg (1980) suggested that, in thecentral airways, the period of an oscillation is so much greater than the lifetime of a turbulent eddy, and that the turbulent flow is present at any instant. This turbulence is responsible for the quasi-steady state corresponding to the disturbed dispersion. In addition, even though the presence of laminar flow in the periphery of the airways is likely, w = 27rf) the unsteadiness parameter a (a = a&; is very small; thus in terms of a these laminar situations are also improbable. More recently, in reviewing the mechanisms of gas transport during high frequency ventilation, Chang (1984) suggested five modes of transport, namely: (1) longitudinal dispersion, due to turbulent eddies and secondary swirling motions occuring in the large and medium airways; (2) convective exchange existing in the large and medium airways due to asymmetrical
Gas flowing in human airways
profiles; (3) convection by high-frequency pendelluft due to the recirculation of air among units of imbalances time constants; (4) direct ventilation of some alveoli by bulk flow convection when the lung units are situated near the airway opening; (5) dominant molecular diffusion near the alveolocapillary membrane. However, these modes of gas transport are not mutually exclusive and can certainly interact. In order to provide better understanding of the mechanisms of gas transport during high frequency oscillation, future research is necessary both experimentally (in oitro and in oiuo) and theoretically. velocity
Acknowledgement--The author gratefully thanks Nadine Capdeville for her secreterial assistance and for typing the manuscript. REFERENCES .
Aris, R. (1956) On the dispersion of a solute in a fluid flowing though a tube. Proc. R. Sot. 235,67-71. Ben Jebria, A., Mallet, A., Steimer, J. L., Hatzfeld, C. and Boisvieux, J. F. (1981) Finite element simulation of gas transport in proximal respiratory air ways: Comparison with experimental data. Comput. biomed. Res. 14.493-505. Ben Jebria, A., Walton C., Mensch B. and Hatxfeld C. (1981) Injecteurs d’emboles de gaz inerte et de gaz marqu6. Bull. Eur. Physiopath. Resp. 17, 261-267. Chang, H. K. (1984) Mechanisms of gas transport during ventilation by high frequency oscillation. J. appl. Physiol. 56, 553-563. Crank, J. (1970) The Mathematics o/ Difision. Clarendon Press, Oxford. Cumming, C., Horsfield D. K. and Preston S. B. (1971) Diffusion equilibrium in lungs examined by nodal analysis. Resp. Physiol. 12, 329-345. Fredberg, J. J. (1980) Augmented diffusion in the airways can support pulmonary gas exchange. J. appl. Phpsiol. 49, 232-238. Horsfield. K., Davies, A. and Gumming, G. (1977) Role of conducting airways in partial separation of inhaled gas mixtures. J. appl. Physiol. 43, 391-396. Jaffrin, M. Y. and Kesic, P. (1974) Airways resistance: a fluid mechanical approach. J. appl. Physiol. 36, 354-361. La Force, R. C. and Lewis, B. M. (1970) Diffusional transport
405
in the human lung. J. appl. Physiol. 28, 291-298. Levenspiel, 0. and Smith, W. K. (1957) Notes on the diffusion type model for the longitudinal mixing of fluids in flow. Chem. Engng Sci. 3, 227-233. Pack, A., Hooper, M. B., Nixon, W. and Taylor, J. C. (1977) A computational model of pulmonary gas transport incorporating effective diffusion. Resp. Physiol. 29, 101-124. Paiva, M. (1973) Gas transport in the human lung. J. oppl. Physiol. 35, 401410. Pedley, J. J., Schroter, R. C. and Sudlow, M. F. (1977) Gas flow and mixing in the airways. Bioengineering Aspects oj the Lung, Vol. 3, pp. 163-265. Dekker, New York. Rauwerda, P. E. (1946) Unequal ventilation of different parts of the lung and the determination ofcardiac output. Thesis, Groningen, Netherlands. Rohrer, F. (1915) Der stromungswiderstand in den menschelichen atemwegen und der Einfluss der unegelmassigen nerzweigung des bronchial systems auf den atmungsverlauf in Verschifdenen lunge&e&ken. Pfluegers Arch. ges. Physiol. 162, 225-299. Scherer, P. W., Shendalman, L. H. and Greene, N. M. (1972) Simultaneous diffusion and convection in single breath lung washout. Bull. math. Biophys. 37, 393412. Scherer, P. W., Shendalman, L. H., Green, N. M. and Bouhuys, A. (1975) Measurements of axial diffusivities in a model of the bronchial airways. J. a&. 38. . . Phvsiol. _ 719-723. Taylor, G. (1953) Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Sot. 219A. 186-203. Taylor, G. (1954) The dispersion of matter in turbulent flow through a pipe. Proc. R. Sot. 223A, 446468. Taylor, G. (1954) Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. R. Sot. 22SA, 473-477. Ultman, J. S., Doll, B. E., Spiegel, R. and Thomas, M. W. (1978) Longitudinal mixing in pulmonary airways. Normal subjects respiring at constant flow. J. appl. Physiol. 44, 297-303. Van Liew, H. D. and Mauone, R. W. (1977) Mixing in flowing gas. Resp. Physiol. 30, 27-34. Weibel, E. (1963) Morphometry ofthe Human Lung. Springer, New York. Wilson, E. A. and Lin, K. H. (1970) Convection and diffusion in the airways and the design of the bronchial tree. Airway Dynamics Physiology and Pharmacology (Edited by Bouhuys, A.) pp. 5-19. C. Thomas, Springfield. Yu, C. P. (1975) On equation of gas transport in the lung. Resp. Physiol. 23, 257-266.
13.00 +
1985 Perpmon
00
Press Ltd.
A MECHANICAL APPROACH TO THE LONGITUDINAL DISPERSION OF GAS FLOWING IN HUMAN AIRWAYS* ABDELLAZIZ
BEN JEBRIA
Laboratoire de Physiologic. CniversitC de Bordeaux-II-146 France
rue Leo Saignat. 33076-Bordeaux Cedex.
Abstract-The atm of this work is to contribute to elucidating the mechanism underlying gas mixing in the human pulmonary airways. For this purpose, a particular attempt is made to analyse the fluid mechanical aspectsofgaseous dispersion using bolus responsemethods. The experiments were performed on five normal subjectsby injection of IO cm3 bolus of He, Arand SF6 into the latter part of the inspired airstream. in such a way that the whole bohts entered the inspiratory flow and was recovered during the following expiration. The results. presented in a logarithmic plot ofdimensionless variance (dispersion of the output bolus) against the Peclet number. show that gaseous dispersion is only slightly dependent on the nature ofthe tracer gas but is strongly related to flow velocity. This is in agreement with the theory of turbulent or disturbed dispersion; houe\.er. it:eems that Taylor laminar dispersion does not play a significant role in the airways.
NO>IENCLATURE
Subscripts i
It, H x :
C’p
tube radius concentration of tracer gas diameter of a tube or an airway coefficient of molecular diffusion laminar dispersion coefficient turbulent dispersion coefficient disturbed dispersion coefficient frequency of oscillation constant coefficient equal to 0.73 length of a corresponding generation length over which appreciable changes in concentration occur axial length of total airways in which the bolus is penetrated number of airways per generation peclet number Reynolds number radial coordinate cross-sectional area time mean residence time average velocity of a fluid penetration volume flow rate friction velocity reduced time angular frequency axial coordinate generation number
Greek lerrers I
; \’ P
oz 1 6, ; 70
Received
Womersley parameter resistance coefficient dynamic viscosity kinematic viscostty density variance of C-curve variance of the time concentration curve arbitrary variable of a function friction stress
May 1984: in recked form Norember 1984. *This work was supported by a grant from INSERM (No. 84.5001).
j 0
input number of reading concentration patterns output INTRODUCTION
the interesting biomechanical aspects of the lung, gas flow and airway resistance have received much attention. However, few attempts have been made to analyse the fluid mechanical aspects of gaseous dispersion in the airways, despite its considerable role in respiratory physiology. One of the pending problems is that of the mechanisms leading to the uneven distribution of inspired gas throughout the lung. How much of this inhomogeneity is the result of unequal flow in different regions (regional inhomogeneity)? How much is the result of a non-equilibrium of gas by diffusion in alveolar sacs (stratified inhomogeneity)? And how significant a role could gaseous dispersion play in these two mechanisms? One of the promising approaches towards understanding the distribution of inhaled gas throughout the lung, has been that of mathematical models. Since the pioneer work of Rauwerda (1946), a number of other mathematical models have been proposed by different authors (La Forceand Lewis, 1970; Cumming et al., 1971; Scherer er al., 1972; Paiva, 1973) to quantify the role of stratified inhomogeneity in the alveolar plateau slope; their results remain controversial. More recently, it has become apparent that gas transport in the lung cannot be described by the association of pure convection and pure molecular diffusion, if one considers the central airways. Some authors (Yu, 1975; Ultman er al., 1978; Ben Jebria et al., 1981) have demonstrated theoretically and experimentally that gaseous dispersion in the proximal airways plays an important role, because of the complicated nature of the motion prevailing in this zone of the lung. Indeed, during a single breath. flow conditions in the Among
399
ABDELWZIZ BEN JEBRIA
400
large airways are different from those in the central airways and even more different from those in the more distal airways. In fact, the repeated branching tube network of the bronchial tree from the trachea to the periphery, the increase in the total cross-sectional area of the many parallel pathways, and the irregular geometry of the airways are the features necessary for an understanding of gas flow in the lung. In a person breathing normally, the inspired gas enters the lung through the upper airways (oropharynx or nasopharynx, and larynx); below this level, the flow generates an intense turbulent jet in the trachea which propagates even further down. Moreover, in the successive finite bronchial tubes, each junction disturbs the flow, and these eddies do not dissipate before the next blending. Consequently, in the upper, large and medium airways, longitudinal dispersion occurs through the interaction of axial convective velocities and complex motions due to the turbulent eddies and swirling. In the most distal airways and alveoli, gas transport occurs mainly by molecular gaseous diffusion, since convection decreases in the presence of the considerable increase of the total cross-sectional area. Our main concern, in this paper, is with gas mixing in conducting airways, to the exclusion of the distal airways and alveoli. A particular attempt is made to analyse the fluid mechanical aspects of the problem using a bolus response method; this is to point out the importance of the longitudinal dispersion of inert tracer gas in the pulmonary mixing of normal subjects. This bioengineering approach per se contributes to the understanding of the mechanisms underlying the gas transport process in the lung. THEORETICAL CONSIDERATIONS
Dispersion of a tracer gas in a laminarjlow
During Poiseuille laminar flow, when a tracer gas is instantaneously introduced into a fluid moving steadily through a pipe, its velocity is greater near the center than near the walls, because of a varying parabolic profile effect over its cross-section (Fig. 1). Now, as Rohrer (1915) pointed out, if there is a variable longitudinal concentration of the components of the
fluid, this leads to variable radial concentration and to increased diffusion. In fact, in the regions where the tracer molecules are carried downstream less rapidly. the lower velocity allows these molecules to diffuse radially. In such a case, this phenomenon can be quantified and the concentration is given by the following equation, usually known as Taylor’s equation (Taylor, 1953).
(1) Taylor (1953) and later Aris (1956) showed that under certain conditions this mechanism (coupling between axial convection and radial diffusion) can be described as longitudinal dispersion governed by an effective coefficient of diffusivity ( 9i,,m) obeying the onedimensional axial transport equation (‘I Taylor (1953) found that the laminar dispersion coefficient depends on the mean axial velocity of the fluid. the diameter of the tube and the molecular diffusion coefficient, as follows ..z.Jz
9
rm=D++
It is apparent from this equation that glaa increases very rapidly as flow rate and tube diameter increase. Moreover, g,,,,,, is very much larger than the molecular diffusion coefficient (D) and it can be seen that the lower the molecular diffusivity of the inert tracer gas. the higher the dispersion coefficient for any flow velocity. Consequently, Qr,,,, is closely dependent on the nature of the tracer gas flowing in the tube.
Dispersion of a tracer gas in turbulentjlow
Unlike laminar flow, when a tracer is injected instantaneously into a fluid moving through a straight tube at high flow rate under turbulent flow conditions. the particles are distributed randomly in all directions. Thus, the velocity cannot be descr%ed. analytically since it fluctuates randomly, both in magnitude and in direction (Fig. 1). It has been recognised that the transition to turbulence in a circular pipe begins at the same value of the Reynolds number (about Re = 2300). In these conditions, Taylor (1954) showed that the analogous problem of dispersion in turbulent flow can be solved in the same way as that in laminar flow. Thus, the turbulent dispersion coefficient is expressed as follows S?I”? = 10.1 au* = 7.14au&
time -_o
later
Fig. 1. Velocity profiles of gaseous dispersion in steady laminar and turbulent flow after an instantaneous injection of a bolus of a tracer gas.
in which, u+=m=&? ,,/:-2logy
= -0.4+4logRe
(4)
Gas flowing in human airways
401
instantaneous flow and volume were monitored during the whole respiratory cycle. Re =f ud=--. The seated subject, provided with a nose clip, V P breathed air continuously through a mouth piece It can be seen from equation (4) that the relationship connected through a valve system to inspiratory and between the turbulent dispersion coefficient and the expiratory lines. He controlled his respiratory flows by velocity is linear rather than quadraticas in the laminar looking at an oscilloscope. placed directly in front of dispersion coefficient in equation (3). In addition .$Z,., him. After some practice, the subject was able to is independent of the molecular diffusion coefficient, breathe at a constant flow rate. The bolus was then unlike Qla,,,. injected at a predetermined level of the tidal volume during inspiration, in such a way that the whole bolus Dispersion of' a tracer gas in disturbed flow entered the inspiratory flow and was recovered during the following expiration. The dispersion of the tracer In the successive branched airways, flow conditions gas depend on the flow rate, on one hand, and on the are extremely complicated and secondary motions degree of the penetration of the bolus in the airways, persist even at low values of Reynolds number (Pedley on the other. Thus, the subjects were asked to breathe et al.. 1977). Thus, at each airway bifurcation, seconthree times independently and respectively at rates of dary swirlin motions distort the axial velocity profiles, 200,300 and 400 cm3 s- ’ (inspiratory flow rate equal and their effect&m dispersion are similar to those of to the expiratory one). For each flow rate and each turbulence. Hence, dispersion in branching networks tracer gas, a wide range of penetration volume was like airways is difficult to analyse mathematically. investigated (from 80 to 300 cm3). The latter was taken However, Scherer et al. (1975) measured benzene vapor to be the portion of the tidal volume included between dispersion in a five generation symmetric geometrical the beginning of the bolus injection and the end of model of the trachea-bronchial tree; they computed an inspiration. effective axial diffusivity coeficient (disturbed disperThe flow rate was measured with a Fleisch pneumosion coefficient) which was expressed by the following tachograph and the tracer concentration with a formula ‘Centronic’ mass-spectrometer using the ‘Spectrum’ gdis =D+Kud (5) mode; this was to obtain the best possible precision at or low concentrations. Both sets of apparatus were 5?,,JD = 1 + K Pe. connected to the analog-digital converter of a With PDP 1l/O3 digital computer. and
ud
ud
Pe=D. As can be seen from equation (5) the disturbed dispersion coeficient is proportional to the product of the mean flow velocity and the airway diameter. Moreover molecular diffusion probably plays a negligible role in the more proximal airways. This is a result qualitatively similar to turbulent dispersion in a circular pipe. according to Taylor’s theory (1954).
METHODS
Experimenral
procedure
of bolus-response
The biomechanical objective of the research was limited to the study of gaseous dispersion in the central airways of the trachea-bronchial tree. For this purpose, the tracer bolus response technique seemed a promising. useful, and helpful test for understanding the mechanisms of gas mixing occuring in the lung. Thus, the experiments were performed on five healthy subjects by injection of 10 cm3 bolus of helium, argon and sulphur hexafluoride into the latter part of the inspired airstream, using an apparatus described in the form of a technical note and published elsewhere (Ben Jebria et al.. 1981). The tracer bolus was automatically injected into the carrier gas by an electrically operated valve, the opening period of which was set at 0.04 s. The concentration of the tracer gas as well as the
Data analysis
If the tracer gas is injected instantaneously (without any delay) into a fluid moving down a pipe, at x = 0, the solution of equation (2) which gives the tracer distribution as a function oft and x is, from Levenspiel and Smith (1957), C(x, t) = *exP(
-s)+
(6)
At position x = f.p, and on rearranging equation (6) the concentration of tracer gas as a function of time is given by 1 C(t) = 2&Y tlVP)(~luLP) x exp
(
-
(1-
.
rit/vp)2
4(VtlVP)(gluLP)
>
The use of equation (6) or equation (7) for our problem assumes that the central airways are assimilated to a single tube with mean axial velocity and cross-sectional area equal to that of the zeroth generation (trachea). Assuming that the output tracer concentration as a Gaussian function and the dispersion as the variance of the equation (7), then
ALIDELLAZIZ BEN JEBRIA
402
If the mean residence time is as follows t=
Jb; CC(e)de / j;
C(C)dC.
(9)
Then the variance is as follows
Finally, for practical reasons, the variance of the timcconcentration is estimated as described by Lcvenspiel and Smith (1957), by substituting the integrals of equation (10) by finite sums as follows zt;cj
2
ut
xt,cj
2
( >
I--
(11)
-.
ZCj
ZCj
The expression of equation (11) is useful for computing the variance of the time-concentration curve from experimental data. And in this case l? = ( if/ Vp)2a:
Fig. 2. Schematic representation of the bolus response method: input signal (bolus of 10 cm’ injected during inspiration) followed by output mass spectromcter%hannel (bolus
(12)
measured in front of the lips during expiration).
As can be seen in Fig. 2, it is never possible to obtain experimentally a unit impulse input signal; thus the experimental dispersion is considered as: At# = uk -Uf.
RRSULTS
First of all, we have to distinguish between the large and medium airways. Indeed, Fig. 3 shows that for penetration volumes lower or equal to 140cm3corresponding to the zone above the fourteenth generation according to Weibel (1963)-there are no differences in dispersion between helium, argon and sulphur hexafluoride. In contrast, it can be seen in Fig. 3 that for a penetration volume ranging from 140 to 300cm3 (corresponding to the zone between the fifteenth and eighteenth generation) the dispersion of He, Ar and SF, is different. Nevertheless, in both cases, gaseous dispersion decreases as flow rate increases; but this decrease is much less in the first case. Now, as demonstrated by Fredberg (1980), the relationship between the mass transfer coeficient (in our case u2) and the Peclet number is similar to the relationship between the normalized pressure drop and the Reynolds number for the airways (Jaffrin and Kesic, 1974); in other words, there is an analogy between diffusional resistance and, flo\)r resistance.
Now, according to the theory developped by Levenspiel and Smith (1957), cr2 can be evaluated by substituting in equation (8). the theoretical function of C by the formula of equation (7). Thus
D z
u2=8
2
(
+2;
(13)
>
or, solving for g/uL, $=b(&5i-1).
(14)
Finally, as can be seen in Table 1, we have calculated the values of the longitudinal dispersion coefficient in the zeroth generation (trachea) according to the following formula: 9 = Q(&G
-l)u,L,
in which u2 and (u, 3: i/S.)
(1%
were measured.
Table 1. Comparison between the dispersion coefficientsobtained in this work and those of Scherer and Taylor. V,p: V, < 14Ocm’ and V,p: 140cm3 c V, G 300 cm” Dispersion coefficients (cm’ s- ‘) Flow rate
Tracer He
Cl&-
V,P
VaP
Scherer’s coetlicient
200 300 400
102.47 135.37 152.22
173.31 189.36 194.19
104.26 156.00 207.73
200
102.22 133.75 151.49 102.98 133.50 153.94
156.35 168.19 178.34 128.51 145.65 156.22
103.66 155.40 207.13 103.55 155.29 207.02
Ar % 200 SF6
Present work
2
Taylor’s laminar coefficient 131.58 295.06 523.92 523.32 1177.23 2092.70 1162.59 2615.71 4650.08
403
Gas flowing in human airways
+
5. I:
4
rrocn+
i %, ‘.., \I ‘..._, ?.
-.
3. ;; i 2. E z kIa E 6
2_
%.
‘.... “...,
Thus, the experimental results of gaseous dispersion (in terms of 17’)are shown in Figs 4 and 5, which are a logarithmic plot of dimensionless variance vs the Peclet number in the trachea. For each flow rate and for each tracer gas, the data plotted are the arithmetic average of dispersion obtained in all subjects. The principal results obtained are that : (1) for penetration volumes lower or equal to 140 cm3 and for each flow rate, gaseous dispersion for the three tracer gases (He, Ar, SFJ is independent of the xeroth Peclet number (Fig. 4); (2) for penetration volumes ranging from 140 to 300 cm3 and for each flow rate, gaseous dispersion is dependent on the Peclet number in the trachea (Fig. 5). These results mean that in the tirst case (in the more proximal airways) dispersion is only related to flow velocities; however, in the second case (near the distal airways), dispersion is on the one hand related to flow velocities and to physical properties of tracer gases on the other. This finding is in agreement with the theoretical prediction of turbulent dispersion of Taylor (1954) as well as with the in-oitro experimental results of Scherer et al. (1975). In addition, although molecular diffusion affects longitudinal dispersion in the middle airways (140 cm3 < VPQ 300 cm3) it does not imply that this result is in contrast with that found
3oocm’
+
‘k.. %. . .“‘3 ....,,., b..t .. . “.... \ i He ....... . .. “.. .X_.‘. . ...... i Plr ‘+ SF6
r 300
) 200
400
300 Flow
I
Rate (cf-+’
Fig. 3. Gaseous dispersion (variance of bolus) plotted against flow rate for the three tracer gases (He, Ar, SF& Vertical bars indicate mean values f SD.
Pe
Kj2
lo2
2
2
4
I
1
9 7 9 9,02
2
3
4
5
6 799,0’
1
Fig. 4. Logarithmic ptot of gaseous dispersion vs the Pecletnumber in the trachea. Results obtained in the zone lower or equal to 140 cm3 of penetration volumes (VP). V,, Vsand Ps represent the flow rates of 200,300 and 400 cm3 s- ‘, respectively.
Fig. 5. Same as Fig. 4 except that these results are obtained for penetration volumes ranging from 140 to 300 cm’.
AEIDELLAZIZ BEN JEBRIA
404
by Scherer et al. (1975). Indeed, according to equation (5), when the penetration volume increases (then the cumulative cross-section of the airways) the velocity decreases gradually and the mixing by diffusion becomes more and more important.
underlying gas mixing?
for
If one refers to the values of Reynolds number in the geometry of the airways, one can expect that during spontaneous breathing, the flow in the human lung may be laminar, and that the motion occurs with a parabolic velocity profile. Thus, as suggested by some authors (Pack et al., 1977); Van Liew and Mazzone, 1977; Horsfield et al., 1977) the Taylor mechanism of dispersion in a fully developed laminar flow may be applicable. However, as far as we know, the two conditions set forth by Taylor (1954) to apply equation (3). have been somewhat overlooked. They are 4L a
ua % D 6.9.
(16)
We shall test the validity of these two conditions using Weibel’s anatomical data (1963). However, some rearrangements are needed to apply them in the lung. The condition us/D g 6.9 yields Va(2)
(17)
S(z)D ’ ‘*‘*
With S(r) = n(z)na’(z), equation
L
9
4ztW4
(17) becomes 6.9
W)
or,
As regards the other condition, 4L/a & us/D. if to simplify matters (the initial concentration being of c solution is the length) infinite = 1/2Co erf(x/2fi) (Crank, 1970). About 83 % of the change in concentration occurs for an abcissa x -2fi
thus
(19)
--.44
t_
U
Now taking into account equations equation (19) can be written as
and the condition
(3) and (20),
4Lja % us/D yields
16
u/(z) J 480B--F
us(z)
(23)
As a numerical example, the two conditions equations (18) and (23) give
DISCUS!SION
What is the mechanism
or
(22)
Q=400cm3s-1
of
n(z)u(zw 74 n(z)!(z) Q 96.
Taking the symbol< to mean a ratio of 1: 5, the first condition is fulfilled from 0 to generation 6, and the second from generation 10 to higher generations. Thus there is no overlapping of the two conditions; therefore, there is no place in the airways where Taylor’s laminar dispersion coefficient can be applied. However, this result does not exclude the fact that the mixing in the middle airways could occur by the coupling between radial diffusion and axial convection (as it has been noticed by Wilson and Lin ( 1970) ); and therefore gas transport can be described transitorily by equation (1) in the intermediate region of the lung. Implication of gaseous dispersion in the mechanism underlying the new mode of ventilation
Recently, there has been widespread opinion which recognizes that adequate alveolar ventilation and gas exchange can be maintained with tidal volumes which are smaller than the anatomic dead space at high oscillatory frequencies. However, it remains to elucidate the mechanism contributing to gas transport during high frequency ventilation. Among the hypothetical ideas contributing to an understanding of this phenomenon, some qualitative mechanisms, such as enhanced mixing in the central airways combined with molecular diffusion in the periphery of the lung, are advanced. Nevertheless, the explanation of the fluid mechanical phenomena which generate this augmented diffusion remains controversial. The question is what is the nature of the flow? Is it simply oscillated laminarly, or oscillated with turbulence; or is there a steady state turbulent flow configuration? Fredberg (1980) suggested that, in thecentral airways, the period of an oscillation is so much greater than the lifetime of a turbulent eddy, and that the turbulent flow is present at any instant. This turbulence is responsible for the quasi-steady state corresponding to the disturbed dispersion. In addition, even though the presence of laminar flow in the periphery of the airways is likely, w = 27rf) the unsteadiness parameter a (a = a&; is very small; thus in terms of a these laminar situations are also improbable. More recently, in reviewing the mechanisms of gas transport during high frequency ventilation, Chang (1984) suggested five modes of transport, namely: (1) longitudinal dispersion, due to turbulent eddies and secondary swirling motions occuring in the large and medium airways; (2) convective exchange existing in the large and medium airways due to asymmetrical
Gas flowing in human airways
profiles; (3) convection by high-frequency pendelluft due to the recirculation of air among units of imbalances time constants; (4) direct ventilation of some alveoli by bulk flow convection when the lung units are situated near the airway opening; (5) dominant molecular diffusion near the alveolocapillary membrane. However, these modes of gas transport are not mutually exclusive and can certainly interact. In order to provide better understanding of the mechanisms of gas transport during high frequency oscillation, future research is necessary both experimentally (in oitro and in oiuo) and theoretically. velocity
Acknowledgement--The author gratefully thanks Nadine Capdeville for her secreterial assistance and for typing the manuscript. REFERENCES .
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