a revised model of gas transport in human lungs

A revised model of gas transport in human lungs David A. Scr imshire

Department of Production Technology and Production Management (Biomedical Engineering Section], University of Aston, Gosta Green, Birmingham B4 7ET, UK (Received February 1979)

A critical reappraisal is made of the boundary conditions assumed in contemporary models of gas transport in the human lungs. It is demonstrated that the previously assumed zero concentration gradient at the alveolar wall does not guarantee zero f lux for an insoluble tracer gas at this point and, more importantly, causes an unrealistically rapid equili- bration of gaseous concentrations to occur. In view of these major shortcomings, a revised set of boundary conditions are proposed which are shown to yield results in close agreement wi th experimental findings.

I n t r o d u c t i o n

The primary function of the human lungs is to remove carbon dioxide from, and add oxygen to, the bloodstream. The process is achieved by bringing ambient air and arterial blood into intimate contact in small air sacs, termed alveoli, which are clustered around the last few generations of the bronchial tree. The bronchial tree itself may be likened to a dichotomonously branching system of tubes, with the trachea ('windpipe') being denoted as generation-0, followed by a further 23 generations of similar airway tubes. Although the cross-sectional area of any individual daughter tube is always less than that of a mother, the total cross- sectional area of the airways increases at each generation, and for an average lung follows the pattern depicted in Figure 1.1,2

In order for respiration to take place, gas molecules must travel from the trachea, through the bronchial tree, to the alveolated region; that is from generation-0 to generation- 23. Both convection and diffusion contribute to this process, with convection dominating in the major airways (i.e. from generation-O to -10) and diffusion in the peripheral airways (i.e. generationA 5 to -23).

In an attempt to distinguish between the roles of these transport mechanisms, physiologists have resorted to the use of mathematical models which describe the behaviour of a tracer gas being inspired into an indigenous gaseous mixture.a-13 The physical dimensions of lung architecture invariably assumed are due to Weibel, 1 and describe a typical normal bronchial tree,in terms of total airway cross- sectional area versus generation number as shown in Table L It is interesting to note from the data that a tremendous increase in total area occurs over the last four generations of branching. For example, some 90% of the total bronchial

tree volume is contained within generations-20 to -23, which represents a longitudinal distance of only 2.2 mm.

By utilizing Weibel's data, all previous theoretical studiesa, S, 7-z° have concluded that an inspired tracer gas should reach equilibrium very rapidly during the breathing cycle, and no concentration gradients should exist at the end of expkation. Experimentalists, in contrast, have consistently demonstrated that gaseous mixing cannot be

0

E 0

-6 t - .o_

In

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6

0

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Figure 1 sectional areas plotte d against distance down bronchial tree

I

Distance down bronchial airways,(cm) Pictorial representation of model, showing airway cross-

0307-904X/79/030289-06/$02.00 © 1979 IPC Business Press Appl. Math. Modelling, 1979, Vol 3, August 289

Gas transport in human lungs: D. A. Scrimshire

Table I Physical lung model

S L Z

2.54 12.00 o 1.70 4.07 1 1.56 1.62 2 1.46 0.65 3 1.81 1.09 4 2.27 0.91 5 2.89 0.77 6 3.73 0.65 7 5.08 0.55 8 7.00 0.46 9 9.80 0.39 10

14.33 0.33 11 21.05 0.28 12 32.53 0.23 13 50.73 0.20 14 82.60 0.17 15

131.60 0.14 16 242.18 0.12 17 522.00 0.10 18

1307.00 0.08 19 2946.00 0.07 20 5510.00 0.08 21

15 328.00 0.05 22 26 216.00 0.04 23

S, total cross-sectional area of all airways in generation Z measured in cm =. L, length of a typical airway in generation Z measured in cm. Z, Weibel's notation for generation number or order of branching; generation-O being the trachea.

instantaneous, and that concentration differences must be present in the bronchial tree at the end of a normal breath. Indeed, by employing tracers of different diffusivity (for example H e and ST6) , it has been shown that heavier gases give rise to more pronounced concentration stratifications and that such phenomena become more marked in patients suffering from chronic lung disorders.

Intuitively, it might be suspected that this apparent inaccuracy of the analytical models is related to their inherent gross approximation of pulmonary geometry; which is in essence a one-dimensional analogue of the bronchial tree. Such a possibility has, however, been con- sidered by several groups of workers. Baker et al., 9 for example, found that the rate of tracer gas mixing was insensitive to any single or multiple change in airway dimensions. Moreover, even when a more complex expand. ing and contracting model is assumed similar rapid equilibrium is still evident, as is apparent from the detailed studies of Pack e t al. 11,12 Indeed, in a recent p~per Paiva 13 has stated that the only significant effect of considering rigid rather than compliant models would be that input gas concentrations will tend to be slightly underestimated.

It must therefore be concluded that the disparity between experimental an[t theoretical findings must lie in the way in which the, albeit approximate, physical model is represented mathematically. It is the purpose of the present paper to re-examine the mathematical formulation, and by critically scrutinizing the conventionally assumed boundary conditions, suggest a revised analysis which more faithfully represents the environment within tile bronchial tree.

Der iva t ion o f gas t r anspor t equa t ion

Considering an elemental section of Weibel's idealized physical model of the bronchial tree (see Figure 2) , the transport due to diffusion, Tdifr, at a distance,3, cm, from

6y

End of gen-23 ~ " ~

Figure 2 Sketch of model showing elemental section, length ~Sy, at distance y from beginning of 10th generation of airway branching. (For details of analysis see text.)

the end of generation-10 is given by:

aF Tdiff = --DS -- (1)

a,v where S is the total cross-sectional area at this point in cm 2 and is a function of v; Fis the fractional concentration of the input tracer gas at distance,v cm from the end of generation-10; and D is the diffusion coefficient expressed in cm2/sec.

From equation (1), and the conservation of mass, it follows that:

S at a,v (2)

where t is time in seconds. Equation (2) describes the change in the number of

molecules, situated at distance,v from the beginning of the model, due to diffusion. This number changes due to the convective air flow (Tconv), which predominates in generations-0 to -10, and may be expressed as:

Tconv = QF (3)

where (2 is the volumetric flow rate. The increase in concentration due to convection is

given by:

OF . aF S - - = - Q - - (4)

at ay

Combining the diffusive and convective terms (i.e. (2) and (4)), yields:

- - = D + ( s ) at t a g s a,v S a y

Equation (5) is usually referred to as the 'convection- diffusion equation' or the 'ga s transport equation', and its solution enables the concentration of an inspired gas, F, to be determined at any position,,v, at specific time intervals during the respiratory cycle.

B o u n d a r y cond i t ion assumpt ions

Before any solution technique can be applied, it is first necessary to specify appropriate boundary conditions at the respective ends of the model. For the insoluble tracer gas under consideration, the following specific assumptions need to be made:

(i) a constant volumetric flow-rate of gas (~ enters the model, and tile volume of the first 10 generations (with the trachea) equals F l (depending upon the initial boundary of the model assumed) (ii) a concentration of 1.0 is attributed to the inspired gases, and concentration 0.(3 to the residual lung gases

290 A p p l . Math. Mode l l i ng , 1979, V o l 3, August

(iii) no input gas should escape across the alveolar wall during inspiration or expiration (iv) during expiration, the contribution from diffusive nfixing at the portal end of the model is assumed to be negligible (v) convective transport is dominant in the upper airway generations, whilst diffusive mixing is dominant distal to the terminal bronchioles.

In order to realize these factors mathematically, it has conventionally been assumed that:

vt 7" F(0, t) = 1.0 for _ ~< t < - - (6)

Q 2

C~yyF)y V, ~<t~<T (7) = 0.0 for --:- =L Q 2

=0.0 f o r - < t ~ T (8) =o 2

/ . aF" T ~ y ) y = L =0.0 for--<t2 ~<T (9)

where T is the time for a complete respiratory cycle, and L is the axial length of tile model.

Now, whilst conditions (6) and (8) are intuitively correct, the conditions (7) and (9) are less obvious and require further verification. It is worthwhile at this stage to consider the functional form of the total flux, G, at any point within the model; that is:

OF G ( y , t) = +OF - DS - - (10)

ay

where: +-OF is the convective flux contribution; +ve for inspiration and -ve for expiration; and ( - (DS) (aF[ay) ) is the diffusive flux contribution.

Substituting, in turn, equations (7) and (9) into equation (I0) we obtain:

T G(L, t )=+OFIy=L f o r t I < t ~ < - - (I1)

2 and:

T G(L, t) = -6_Fly =L for-- < t ~< T (12)

2 It is now obvious from a scrutiny of equations (11) and

(12) that the conventional interpretation of the boundary conditions (as specified in equations (7) and (9)) does not give a true zero flux situation at the alveolar wall, i.e. at y = L. Equation (11) implies that input gas is being con- tinually drawn out of the model during the inspiratory phase, whilst equation (12) implies that a similar quantity is drawn back during expiration.

In order to ensure a no-time condition at the alveolar wall, it would seem more obvious to equate the flux functions, given in equation (10), to zero at y = L, namely:

G(L, t) = 0.0 (13)

Rearranging equation (10) for this condition, we obtain:

= + Q- - -~F] fo r t l~< t~<- - (14) y=L DS(L) ly=L 2

for inspiration, and:

~y y=L DS(L) F f o r - < t ~ T (15)

y=L 2

for expiration.


Equations (14) and (15) represent mathematically correct 'no-flux' boundary conditions for the lung model throughout the respiratory cycle. They simply state that there must always be a balance between the convective and diffusive flux contributions at the alveolar wall for any insoluble tracer gas.

Numer ica l so lu t ion t echn ique

Having redefined the boundary conditions, it is possible to proceed to solve equation (5) by means of an explicit finite difference scheme, similar to that proposed by Bush et al. 14 Central difference approximations of derivatives are used throughout the scheme; that is:

aF (Fi,i + 1 -- Fl , i )

at At

aF (F i+l , i - -F i_ l , i ) ay 2(ZXy)

a2F _ (Fi+l,i -- 2Fi,i + F i - l , i ) ay 2 (Ay):

On substituting the above finite difference approximations into equation (5) we have, on simplifying:

Fi,i+~ =Fi,i + Dr(Fi+~,i - 2Fi.i + Fi-~.i)

+ • r(Ay)(Fi+l, i - F i - l , i ) 2 ay

(16)

Equation (16) is tile finite difference approximation to the gas transport equation. In order for the solution of equation (16) to converge to the true solution of the transport equation a stability criterion is required. By applying tile Fourier series method of stability it was found that convergence depended upon the value of r, where:

D r < (17)

and:

[~as Q] (18) xt ='(Ay) ae

This expression for r assumes a minimum value for two particular cases, that is:

Case 1: K < D When K < D, r is minimum when cos(/Sh/2) = 0 and thus:

I r < - -

2D

Case2: K > D When K > D, r is minimum when sin(/Sh/2) = 0 therefore:

D r < - -

2K 2

Thus, in o l:der to obtain a stable and convergent solution to the ntimerical approximation of the gas transport equation, it is required that:

1 i fK < D, then r < - -

2D

Appl. Math. Modelling, 1979, Vol 3, August 291


and

D i l K > D, then r < - -

2K 2

In the present contex t , a stable and convergent solut ion was obtained l o r D = 0.25, Ay = 0.02, r = 1 and hence At = 0.000 4. The value o f 0.25 cm2/sec for the di f fus ion coeff ic ient is applicable for a b inary gas m ix tu re o f oxygen and nitrogen.

Resul ts

The effect of inspiring a 500 ml breath of 100% oxygen into an indigenous 20% oxygen, 80% nitrogen mixture in the revised model is shown in Figure 3. Concentration/ distance curves are given in 0.4 see intervals, and represent the level of oxygen within the bronchial tree throughout the breathing cycle. It will be noted that concentration differences exist during inspiration, and that demonstrable gradients persist throughout expiration (see insert in Figure 3). For the breathing cycle illustrated, a 0.7% difference in tracer gas concentration exists between the 'ends" of the model at end expiration, clearly indicating that diffusive mixing is incomplete, as suggested by experi. mental findings. 16-is

The corresponding flux curves are shown in Figure 4, and illustrate that the prescribed boundary conditions do ensure that none of the tracer gas enters or leaves the model during breathing.

Discuss ion

Having proposed new boundary conditions for the alveolar end of the model that depart significantly from those previously assumed, it is cogent to enquire whether the resulting theoretical predictions stand comparison to experimental findings. Obviously, it is impossible to monitor actual gas concentrations within the lung itself, hence any comparisons must be made on the basis of gas

1"O

O 8 Detail of curve 10 E O-156[.,

_or- O . 1 5 4 ~ ,,o-1521 \ ! o15o I \

= o 0148~ ; \ 1 u 0.4 O 10 20

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P 0 2 I I

2

O 1 2 Distance down bronchial airways,(cm)

Figure 3 Fractional input gas.concentrations down bronchial tree during respiratory cycle. Curves 1 to 10 correspond to solutions F(y, t) for values of t ime t = 0 . 4 n sec (n = 1 , 2 , 3 . . . . . 9 , 10 ) . Insert shows more c lear ly the f in i te concent ra t ion di f ferences obtaining in model at end expiration - - implying that gaseous mixing is incomplete during breathing cycle even in normal subjects

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3

0 4

O 0-2

O o

O O

.5

I

q 8 7 Io ~ i f

-0"2 ~ 6 - -

o I Distance down bronchial airways, (cm)

Figure 4 Normalized gas flux curves clown bronchial tree during respiratory cycle. Curves 1--10 correspond to solutions G(y, t) for values of time t = 0 . 4 n sec (n = 1 - - 1 0 ) . N o t e that by employing particular boundary conditions cited in text a zero flux condition is assured at end of model throughout breathing cycle, closely mimicking behaviour of an insoluble tracer gas

0 8 8 -~ ~ 0 7

.uo

E o 0 5

a u 0 4

E .~_ 03 " r - ~

,,,-~_ g o.1

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Plateau store 35 % _L

. . ~ 1 I I I I

1OO 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 Expired volume, (rnl)

Figure 5 Experimentally measured expired N 2 concentration from a normal subject following a single breath of 100% oxygen. Slope of so-called 'alveolar plateau' (see text) is approximately 3.5% over terminal 500 ml expired

being expired through the mouth. Probably the most relevant, and experimentally simple, protocol to employ is the 'nitrogen washout test'. Here, a single breath of 100% oxygen is inspired, and nitrogen is continuously monitored during the subsequent expiration. Plotting concentration against expired volume yields the typical result for a normal subject shown in Figure 5. Initially, the expired gas contains no nitrogen and reflects the composition of the inspirate contained in the upper airways (so called 'dead space'). After approximately 115 ml have been exhaled a rapid increase in nitrogen concentration is observed, followed by a characteristic 'plateau', which in normals has a finite slope amounting to some 2-4% over 500 m13 s Since this portion of the washout curve corresponds to gas emanating from the distal regions of the bronchial tree, the magnitude of the slope is usually taken to be indicative of incomplete mixing.

292 AppI . Math. Model l ing, 1979, Vol 3, August

Tile model may easily be modified to facilitate a simulation of the above test by taking:

FENZ(t ) = 0.8 {1.0 --/7(0, t -- tl)} (19)

where FEN2(t) is the instantaneous nitrogen concentration at the entry of generation-0 (i.e. the trachea) during expiration. The simulated washout curve given in Figure 6 shows the model prediction of expked nitrogen concentrations following a 500 ml inspiration of oxygen, and closely resembles the actual experimental curve given in Figure 5. Specifically, the plateau of the simulated curve has a slope of 1.72% over the terminal 250 ml, which is equivalent to 3.42% if extrapolated to 500 ml expired - a typical value for normals. When the conventional boundary conditions were applied, no such realistic shape was apparent, indeed all concentration differences dis- appeared early in expiration and a perfectly flat plateau was produced as indicated by the dashed curve in Figure 6.

Whilst the previously assumed boundary condition, (OF]ay) ly =L = 0, has been shown mathematically to be unsatisfactory in defining a zero flux at the distal end of the model, this may only in part explain why tile resulting analyses could not demonstrate the anticipated small concentration gradients. On reflection, it must be ques- tioned whether any such stratifications would really be expected with (aF/ay) ]y =L set to zero, since some 90% of tile model (and lungl) volume is contained within the terminal 0.2 mm of pathway length. Such a constraint must clearly predominate all other factors in the acina region thereby forcing the concentration profiles to be flat.

The above tentative hypothesis may best be quantita. tively examined by modifying the revised boundary conditions such that varying amounts of the input tracer gas are allowed to flow either out of, or into, the distal end of the model during the breathing cycle. This may be readily achieved by specifying finite values for 'G(L, t)' in equations (1 i) and (12), thus:

G a___F[ - O Fly=L + (19) ay y=L DS(L) DS(L)

for inspiration, and:

OF - Q ~ " F I y = L - + - - (20) y =L DS(L) DS(L)

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8

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Gas transport in human lungs: D. A. Scrimsh[re

Revised (374% slope)

1 o 0 t

/ Conventionol (zero slope}

O.4

0 2 / (Sirauloted)

J I I I ! I

0 I00 200 300 400 500

Expired volume, (rat) Figure 6 Model simulation of a single breath 'nitrogen washout' test. Slope of alveolar plateau amounts to 3.42% over terminal 500 ml expired which closely approximates to that experimentally observed for normals (c.f. Figure 5). When conventional boundary conditions are applied in model, washout curve (shown dashed) is produced which has no slope reflecting unrealistically rapid diffusive mixing in alveolated region

Positive values of G represent a constant loss of tracer gas across the alveolo-capillary membrane, and negative values an influx.

The effect of applying six values of G; namely +50, +25, +10, -10 , -25 and - 5 0 ml]sec, is summarized in Table 2 in terms of end expiratory fractional gas concentrations in the last 14 generations of the model. It can be seen that whilst greater stratifications are produced when the tracer gas is being withdrawn through the distal end of the model, finite concentration differences still persist even when this flux is reversed and significant amounts of gas are flowing into the model. Thus, although the rate of gas equilibrium is certainly affected by flux levels at the alveolar wall, it may be concluded that the conventional practice of forcing the concentration gradient to be zero at this point must be predominantly responsible for the rapid gas mixing subsequently predicted. In contrast, the revised condition proposed in the present paper allows the concentration gradient to be a variable, whilst simul- taneously ensuring zero flux conditions at the alveolar wall throughout the breathing cycle.

Table 2 Tracer gas concentration differences within last 14 generations of bronchial tree at end expiration, for different levels of flux (G) at alveolar wall. Positive values of G represent situations in vv.hich gas flows into the model, and negative values gas flowing out of model

Gen. Distance Gas flux G (ml/sec) Z r (cm)

+50.0 +25.0 +10.0 --10.0 --25.0 --50.0

10 0.00 0.076 375 0.115 740 0.139 360 0.170 832 0.194 450 0.233 813 11 0.33 0.075 820 0.115 307 0.138 999 0.170 569 0.194 260 0.233 745 12 0.61 0.075 133 0.114 771 0.138 554 0.170 244 0.194 026 0.233 661 13 0.84 0.074 281 0.114 106 0.138 001 0.169 841 0.193 734 0.233 557 14 1.04 0.073 249 0.113 300 0.137 331 0.169 352 0.193 382 0.233 431 15 1.22 0.072 104 0.112 406 0.136 588 0.168 810 0.192 990 0.233 291 16 1.36 0.070 528 0.111 177 0.135 566 0.168 064 0.192 452 0.233 098 17 1 A8 0.069 065 0.110 035 0.134 616 0.167 372 0.191 952 0.232 919 18 1.58 0.068 014 0.109 214 0.133 934 0.166 874 0.191 593 0.232 855 19 1.66 0.066 978 0.108 405 0.133 262 0.166 384 0.191 239 0.232 664 20 1.73 0.065 994 0.107 637 0.132 623 0.165 918 0.190 902 0.232 543 21 1.79 0.065 005 0.106 865 0.131 981 0.165 450 0.190 564 0.232 422 22 1.84 0.064 621 0.106 565 0.131 732 0.165 268 0.190 433 0.232 375 23 1.88 0.064 241 0.106 268 0.131 485 0.165 088 0.190 303 0.232 329

% conc. difference 1.21 0.95 0.8 0.58 0.41 0.15

App l . Math. Mode l l ing , 1979, Vo l 3, August 293


References

1 Weibel, E. R. 'Morphometry of the human lung', Academic Press, New York, 1963

2 Hansen, J. E. and Ampaya, E. P. J. Appk PhysioL 1975, 38, 990

3 Rauwerda, P. E. 'Unequal ventilation of different parts of the lung and the determination of cardiac output', PhD thesis, University of Groningen, Netherlands, 1946

4 Cumming, G.etaL Respir. PhysioL 1966, 1,58 5 LaForce, R.C. andLewis, B.M.J. AppkPhysioL 1970,28,

291 6 Cumming, G. etal. Respir. Physiol. 1971, 12,329 7 Paiva, M. Compttt. Biomed. Res. 1972, 5, 585

8 Paiva, M. J. Appl. PhysioL 1973, 35,401 9 Baker, L.G. etaL Resp. Physiol. 1974,21,119

10 Paiva, M. Biophysile. 1972,8,280 11 Pack, A. et al. Bull. bzst. Math. Applications. Special issue:

Math. Clin. Med. 1974, 10, 20 12 Pack, A. et aL Respir. Physiol. 1977,29, 101 13 Paiva, M. Bull btst. Math. Applications, 1978, 14, 17 14 Bush, A. W. et al. In: 'Applied Numerical Modelling' (Ed. C. A.

Brebbia). Pentech Press, London and Plymouth, 1978 15 /t, lills, R. J. and Harris, P. J. Appl. PhysioL 1965, 20, 103 16 Power, G. J. AppL PhysioL 1969, 27,701 17 Kawashiro, T. etal. Respir. Physiol. 1976,28,261 18 Sikand, R. etal. J. AppLPhysiol. 1976,40,362

a revised model of gas transport in human lungs

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