theoretical analysis of the effects of diffusivity on pulmonary gas transport and mixing

Annals o f Biomedical Engineering, Vol. 7, pp. 375-386, 1979 0191-5549/79/050375-12 $02.00/0 Printed in the USA. All rights reserved. Copyright �9 1980 Pergamon Press Ltd.

THEORETICAL ANALYSIS OF THE EFFECTS OF DIFFUSIVITY ON PULMONARY GAS

TRANSPORT AND MIXING

David A. Scrimshire and

Robert J. Loughnane

Department of Production Technology and Production Management

Biomedical Engineering Section University of Aston in Birmingham Gosta Green, Birmingham, England

The effects o f different tracer gas diffusivities upon pulmonary gas transport and mixing have been examzned by means o f a physico-mathematical lung model. Specifically, it is demonstrated how the expired alveolar plateau slope o f a tracer gas gives an indication o f the magnitude o f the end expiratory concentration differences existing in the acinus. Further, by modifying the initial analysis slightly (t~ allow for a finite f lux o f gas across the alveolar wallj it has been in- dicated how more marked stratified inhomogeneities are associated with the transport o f soluble rather than isoluble tracer gases.

INTRODUCTION Because of the practical difficulties of making direct measurements in the

more distal regions of the bronchial tree it is at present only possible to infer what gas concentration differences exist in the acinus during the breathing cycle from expired data. Whilst Engel and his co-workers (3) have succeeded in sampling gas at terminal bronchiolar level in dogs, the experimental proto- col requires open chest surgery and cannot be applied routinely in man. Moreover, as pointed out by the authors, the physical size of the catheter used may interfere significantly with local gas flows.

For a reasonably accurate interpretation of single breath washout (or washin) curves to be made, it is obviously necessary to employ a suitable quantitative model. Although considerable controversy has existed over the exact specification of internal boundary conditions to apply (12), a recent model, embodying a consensus of contemporary findings, has demonstrated

Requests for reprints may be sent to David A. Scrimshire, Department of Production Technology and Production Management, Biomedical Engineering Section, University of Aston in Birmingham, Gosta Grenn, Birmingham B4 7ET, England.

375

376 David A. Scrimshire and Robert 3". Loughnane

close correspondence between simulated and actual experimental results (1 1). In the present s tudy use is made of this model to examine the effects of both tracer gas diffusivity and blood solubility on gaseous transport and mixing within the normal lung, and specifically to relate concentration differences obtaining in the acinar region to those directly measurable at the mouth.

M e t h o d s

The physical form of the model is derived from a modified version of Weibel's "Model A," and consists o f the last 13 generations of the bronchial tree. The equations governing the transport of gases into and out of the model may be written as:

aF D {a2F 1 ~ aFJ Q aF (1) at = -7+Say ay s ay

where F =- F ( y , t) is the fractional concentration of inspired tracer gas at distance y from the beginning of the model and at time t after the start of the respiratory maneouvre; S = S (y) is the total cross-sectional area of the model at distance y from the portal end; D is the binary molecular diffusion coefficient between the inspired and residual gases; and Q is the volumetric gas flow rate.

In order to solve Eq. 1, suitable boundary conditions must be specified. At the mouth there is a constant flow of inspired tracer gas and this may be written as:

F (0, t) = 1.0 for t 1 ~ t ~ T/2 (2)

During inspiration the contr ibution from diffusive mixing at the model entrance is considered negligible in comparison to the convective mixing and this implies that

=0 = 0 for ~- < t ~ T (3)

At the distal end of our model it is required that there is no flux of input gas across the alveolar wall, y = L, when considering an insoluble tracer. Such a requirement may best be achieved by ensuring that the total contribution from both diffusive and convective gas fluxes is zero, implying that a balance exists between these two transport processes. In order to formulate this condition in mathematical terms we need to define the total flux function G(y, t), that is,

�9 aF G(y, t ) = Q F - D S - - (4) ay

Theoretical Analysis of Diffusivity 377

where G -= G(y,t) is the total flux (made up of both a diffusive and a convective contribution) at distance y from the beginning of the model and at time t after the start of the respiratory maneouvre.

We require that G(L, t) = 0 for all t, and from Eq. 4 the following zero flux boundary conditions may be derived:

3F = + DS(L) F for tl ~< t ~< T = L =L 2

and

=L DS(L) F =L for ~ < t ~ T (6)

In the above equations tl is the time required for the inspired gases to traverse the upper 10 generations, and T is the total duration of the respiratory cycle. The numerical technique used to solve the gas transport equation has been fully detailed elsewhere (11) along with an exhaustive examination of stability and convergence (12).

In order to specifically relate the end expiratory concentration differences, existing throughout the length of the model, to the alveolar plateau slope measured at the mouth, we employ the following equations:

FEI (t) = F(O, t-tx) (7)

FEN 2 (t) = 0.8 (1.O-F(O,t-tl)) (8)

where FEI (t) is the expired fractional concentration of input tracer gas

measured at the mouth and at time t, and FEN 2 (t) is the expired nitrogen

concentrat ion measured at the mouth and at time t.

Results Figures 1 and 2 show the input gas concentrations within the model at

end inspiration and end expiration respectively for three tracers having molecular diffusion coefficients of 0. I cm 2/sec, 0.315cm 2/sec and 0.76cm 2 /

sec corresponding to SF6/N2 Ne/N2 and He/N2 mixtures. As intuitively expected it can be seen that at end inspiration the heavier gas (SF6) has penetrated deeper into the model than that of the lighter gases (Ne or He), and hence it has a greater dead space volume. Furthermore, the heavier gas displays a more marked concentrat ion stratification in the acinar region. The actual concentration differences between the ends of the model at end expiration are 0.42% for He, 0.6% for Ne and 1.2% for SF6, with the greatest gradients occuring over the last 0.5cms of the model length.

1 . 0

SF6

Ne

z O p. <~

I - z uJ

z O O.5

,-I

z O I - r <l:

u.

378 David A. Scrimshire and Robert J. Loughnane

0 l 1.0

DISTANCE, cms

FIGURE 1.

1.9

End inspiratory input gas concentration/distance profiles corresponding to the three tracer gases SF6, Ne and He. Note that the left hand side o f the horizontal axes corresponds to the "por ta l " or "mou th " end o f the model and the right hand end to the distal alveolar wall.

Z

<~ n-

z

0.158

uJ ro z o r~ _i

z 0 I--

M=

0.15

0.14 L I 0 1.0 1.9

DISTANCE, cms

FIGURE 2, End expiratory input gas concentration/distance profiles corresponding to the three tracer gases SF6, Ne and He. Note that greater stratif ied concentration gradients are associated with the heavier gases.

Th eore tical Analysis o f Diffusivity 3 79

The resulting concentrations of the three tracer gases being expired through the "mouth" end of the model are given in Fig. 3 and show that the heavier the gas, the greater the dead space volume and the greater the alveolar plateau slope. The plateau slopes for the three tracer gases were calcu- lated on the basis of an extrapolation to 500mls expired (i.e., percent per 500mls) and were found to be 2.9% for He, 3.4% for Ne and 4.8% for S F 6 .

DISCUSSION The theoretical results presented show that the expired "alveolar plateau"

slope of a tracer gas does indeed give an indication of end expiratory concentration differences in the acinus. Moreover, the magnitude of the slope, and the degree of stratified inhomogeneity, increase as the molecular weight of the input gas increases (that is, D decreases). Such results concur with the experimental findings of previous workers (2, 4, 6, 8, 9, 14) who also concluded that gases having lower diffusivities should reach equilibrium more slowly, and further hypothesized that expired concentration differences arose as a consequence of similar differences obtaining in the more distal regions of the bronchial tree.

Only two previous analytical attempts have been made to simulate similar results. Sasaki and Farhi (10), employing an algebraic lung model, concluded that the only effect of considering a heavy rather than a light tracer gas was that the dead space volume should be greater. Paiva et al. (7) using a more

1.0

SF 6 4.8% SLOPE

8 /

0 250 500 EXPIRED VOLUME, mls

FIGURE 3, Single-breath input gas washout curves corresponding to the three tracer gases SF6, Ne and He. Once again it is most important to note that greater alveolar plateau slopes are associated with the heavier tracer gases.


detailed lung model analysis, came to a similar conclusion, but could not demonstrate gradients within the lung at the end of expiration. It appears likely, however, that such a finding is a logical consequence of the particular internal boundary conditions employed, which have been shown to be in- appropriate for tracer gases having a f i n i t e blood solubility (11).

The particular boundary conditions applied in the present model ensure that none of the input tracer gas escapes, or indeed is reabsorbed, through the alveolar wall. As the three gases studied have very low blood solubilities ( S F 6 = 0.0067, Ne = 0.011, He = 0.0098) such an assumption would appear to be a reasonably accurate representation. Nevertheless, it is interesting to speculate how a finite gas flux across the alveolar wall would affect concentration gradients. Chang and Farhi (1) have already considered such a case in qualitative terms, and have suggested that the gas exchange process is likely to increase any stratified inhomogeneities existing in the acinus. The present model may readily be modified to accommodate a finite gas flux across the alveolar wall by slightly altering the boundary conditions, Eq. 5 and Eq. 6. The necessary modification involves changing "G = 0" to "G = k " ;

where k is the amount of input tracer gas (mls/sec) being taken up by the blood flowing in the alveolar capillaries. In fact, Eq. 5 and Eq. 6 became:

3F = + DS(L ) F =L =L DS(L) ' t2 ~ t ~ T/2 (9)

and

= L DS(L) = L DS(L) '

The effect of applying three values of G (10, 25 and 50 ml/sec) to a hypothetical gas having a diffusion coefficient of 0.25cm 2/sec (equivalent to O2/N2) is shown in Fig. 4 for end inspiration. It can be seen that the higher the G value, the greater the concentrat ion gradients for the input gas in the acinar region and the lower the alveolar gas concentration. The latter effect is to be expected since higher values o f G are associated with tracers having a higher blood solubility, which implies that they are being removed from the alveolar spaces at a faster rate by capillary blood.

Figure 5 gives the ensuing end expiratory concentrat ion within the model. It will be noted that more marked stratified inhomogeneities are associated with tracers having higher blood solubilities, and that these concentration gradients are also reflected at the " m o u t h " end of the model as is evident from Fig. 6.

A more realistic illustration of the i n d e p e n d e n t effect of input gas solubility on stratified inhomogeneities may be given by considering the specific gases argon and nitrous oxide because they have very similar diffusion coefficients (D = 0.192 for Ar, and D = 0.189 for N2 O). Because the solubility


1 .0

e I O

. 5 m m

�9 � 9

G = 1 0 . 0 G = 2 5 . 0 G = 5 0 . 0

z O u

I - ,<

I - Z IJJ r z O

. . I ,< z O D

I - r <C

I.I.

I I 0 1.0 1.9

D ISTANCE, cms

FIGURE 4. End inspiratory gas concentrat ion/d istance prof i les corresponding to three hypothet ica l tracer gases wi th d i f fe ren t so lub i l i ty factors (i.e., G = 10.0, 25.0 and 50.0). Note the lower alveolar concentrat ion levels fo r the more soluble gases.

0 . 1 4 z 0 k.- < n- I - z IJJ r z O 0.10 r j .,.i ,< z O I - (.1

1,4.

G = 25.0

I I 1.0 1.9

D ISTANCE, cms

input gas concentrat ion/distanca prof i les F IGURE 5. End exp i ra to ry corresponding to three hypothet ica l tracer gases wi th d i f fe ren t so lub i l i ty factors (i.e., G = 10.0, 25.0 and 50.0). Note that greater strat i f ied concen- t ra t ion gradients are associated wi th the more soluble gases.

0.8

0 .6

Z O m p- < ~c k- z UJ (J Z O (J z uJ

O

k- Z

0 .4

0.2

1.0

IN ALL CASES (D = 0.25) - . . . , . . . . . . . . . . . ,..~..--_

- G 25.0

G 10.0

J CONSTANT "G' FACTOR

i n = I n 1 0 0 200 300 400 500

EXPIRED VOLUME (MLS)

FIGURE 6. Single-breath nitrogen washout curves for three hypothetical tracer gases with di f ferent solubil i ty factors (i.e., G = 10.0, 25.0 and 50.0). Once again it is important to note that steeper alveolar plateau slopes are associated with the more soluble gases.

A r

- - - - N20

Z O i p- <

z UJ O z 0 0.5

< z O p- O < ri- M.

382 David A. Serimshire and Robert J. Loughnane

I 0 1.0 1.9

DISTANCE, cms

FIGURE 7. End inspiratory input gas concentration/distance profi les for the transport o f argon and nitrous oxide (Ar and N20). Note the lower alveolar concentration level for N20.


of argon can be considered negligible, a value of zero is assumed for G. Nitrous oxide, in contrast, has a much greater affinity for blood, having a solubility coefficient of 0.465. Using an earlier algebraic gas exchange model (13) the value of G during a normal initial breath of 100% N20 is estimated to be approximately 5.5 ml/sec. The results for the two tracer gases are given in Figs. 7 and 8 and, as anticipated, greater end inspiratory and end expiratory input gas concentration gradients exist in the acinus for nitrous oxide. Moreover, the actual alveolar concentration level for nitrous oxide is significantly lower than that of argon for the reason previously discussed. The simulated single breath input gas washout tests for these two gases are given in Fig. 9 and demonstrate a significantly greater alveolar plateau slope for N 2 O, again reflecting conditions within the lungs.

While the boundary conditions assumed in the present analysis overcome the major problems previously experienced these are, nevertheless, somewhat difficult to interpret physically. One problem arises from the approximation of the multipath assymmetric bronchial tree geometry by means of the single path cumulative cross-sectional area representation (the so called "trumpet" model), where the alveolar wall at y = L is defined clearly and uniquely. In the actual human lung, alveoli are clustered around three generations of respiratory bronchioles, alveolar ducts and sacs at a variety of distances, L, from the carina. Consequently, at any time during the respiratory cycle it would appear likely that differences in gaseous composition, F, must

Z O k- n-

Z LU

z O

=J

z O

c.}

g.

0.15

0 .13 -

0.11 0

I I 1.0 1.9 D I S T A N C E , cms

FIGURE 8. End expiratory input gas concentration/distance profiles for the transport of argon and nitrous oxide (Ar and N20). It is readily apparent from these curves that greater end expiratory concentration differences exist for the more soluble tracer gas (i.e., N20).


1.0

Z _o I -

I.- z IJJ

z O r,3 E3 UJ n- X" X UJ

0.8

0 .6

0 .4

0 .2

ARGON - G = O (NO FLUX)

N ITROUS O X I D E - G = 5 . 5 m l s / B R E A T H

Ar

m N 2 0

(N ITROUS OXIDE)

(ARGON)

[ I i I I i 0 100 2 0 0 3 0 0 4 0 0 5 0 0

E X P I R E D V O L U M E (mls)

FIGURE 9. Single-breath input gas washout curves corresponding to the transport of argon and nitrous oxide, It isclear from thesecurves that there is both a lower alveolar concentration level and a greater alveolar plateau slope for the more soluble tracer gas (i.e., N20).

exist in the acinus, implying that the proper boundary condition to apply in trumpet type models should be some average value for G over these terminal regions. The value would be close to zero, but impossible to establish ex- actly. Alternatively, the use of models having several parallel elements, as suggested by Jones (5) may offer a partial solution to this dilemma.

A further problem associated with the zero flux boundary conditions is that the slope of the concentration gradient, OF/Oy, at the alveolar end of the model changes sign discontinuously at the end of inspiration as the flow, Q, switches over for expiration. Three alternative forms immediately suggest themselves:

1. If we allow for a transition period (or breath-holding period) between the end of inspiration and the beginning of expiration then we may circumvent the discontinuity, and replace it with a discrete changeover in concentration gradient, that is,

e

(13)


During transition or breath holding period (i.e., T/2 < t <. T/2 + TB)

aF I ~yy = 0.0, y = L

(14)

OF[ _ Q T/2 + <~r+ (15) -~Y y = L DS(L) F y=L[ for T B < t T B

2. Alternatively, we may accept the conventional zero gradient boundary

condition for inspiration and then apply the no-flux boundary condition for expiration, i.e.,

~y =O.O, tl <~ t <<. T/2 (16) y = L

and

by - DS(L) F <. T (17) y = L ' y = L

3. A further possibility worth consideration is that of a constant, but non- zero, flux at the alveoIar wail throughout expiration, that is

and

~ ! OF =+ DS(L) F , tx <~ t <~ T/2 (18) =L =L

F[ =F 1 [ , T/2 < t <<- T (19) y =L y =L

where F 1 [ is the end inspiratory alveolar wall concentration. The major y = L

argument against employing this latter set of boundary conditions must be the way in which the concentrat ion gradients in the acinus are influenced by artificially constraining the gaseous concentrat ion at the alveolar wall at the end-inspiratory level.

It must be concluded, therefore, that while many alternative forms of the boundary condition could be suggested, they all require that some specific assumption be made regarding either a concentrat ion or a concentra-


tion gradient. Moreover, this inevitably means that such values must be set at a constant level for some part of the respiratory circle, thus heavily influencing the subsequent solutions obtained to the gas transport equation. While no vigorous experimental verification has been carried out for the zero flux boundary conditions, they do produce results of the same order as those actually observed, and may therefore be said to afford a reasonable representation of the conditions existing within the actual lung.

REFERENCES

1. Chang, H.K. and L.E. Farhi. A mathematical analysis of gas transport in the lung. Respir. PhysioL 18:370-385, 1973.

2. Cumming, G., K. Horsfield, J.G. Jones, and D. Muir. The influence of gaseous diffusion on the alveolar plateau at different lung volumes. Respir. PhysioL 2: 386-398, 1967.

3. Engel, L.A., L. Wood, G. Utz, and P.T. Macklem. Gas mixing during inspiration. J. AppL PhysioL 35:18-24, 1973.

4. Georg, J., N.A. Lassen, K. Mellemgaard, and A. Vinther. Diffusion in the gas phase of the lungs in normal and emphysematous subjects. Clin. Sci. 29:525-532, 1965.

5. Jones, T.J. Theoretical Analysis o f Gas Transport and Transfer in Human Lungs. Ph.D. Thesis, University of Aston in Birmingham, Birmingham, England, 1977.

6. Kawashiro, T., R.S. Sikand, F. Adaro, H. Takahashi, and J. Piper. Study of intrapulmonary gas mixing in man by simultaneous washout of helium and sulphur hexafluoride. Respir. Physiol. 28:261-275, 1976.

7. Paiva, M. Gas transport in the human lung. J. Appl. PhysioL 35:401-410, 1973. 8. Power, G. Gaseous diffusion between airways and alveoli in the human lung. J. Appl. PhysioL

27:701-709, 1969. 9. Read, J. Stratification of ventilation and blood flow in the normal lung. J. Appl. Physiol. 21:

1521-153t, 1966. 10. Sasaki, T. and L.E. Farhi. In: Symposium on O'rculatory and Respiratory Mass Transport, a

CIBA Foundation Symposium, edited by G.E.W. Wolstenholme and Julie Knight. London: Churchill, 1969.

11. Scrimshire, D.A. A revised model of gas transport in human lungs. Appl. Math. Modelling 3: 289-294, 1979.

12. Scrimshire D.A., R.J. Loughnane, and T.J. Jones. A reappraisal of boundary conditions assumed in pulmonary gas transport models. Respir. Physiol. 35: 317-334, 1978.

13. Scrimshire, D.A. and P.J. Tomlin. Gas exchange during initial stages of N 2 O uptake and elimina- tion in a lung model. J. Appl. Physiol. 34:775-789, 1973.

14. Sikand, R., H. Magnussen, P. Scheid, and J. Piper. Convective and diffusive gas mixing in human lungs: experiments and model analysis. J. Appl. Physiol. 40:362-371, 1976.

theoretical analysis of the effects of diffusivity on pulmonary gas transport and mixing

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