Computer simulation of ternary diffusion in distal airways of the human lung

COMPUTER SIMULATION OF THE HUMAN LUNG*

OF TERNARY

DIFFUSION

IN DISTAL

AIRWAYS

A. BEN JEBRIA and M. BRES Unitt de Recherches de Physio-PathologicRespiratoire, INSERM (L! 68), Hbpital Saint Antoine, F- 755 71 Paris Cedex I2 (France) (Received 18 April, 1982) Gaseous diffusion plays a fundamental role in the terminal generations of airways for respiratory physiology. It has been proposed as a prime mechanism underlying stratified inhomogeneity in the alveolar space. Nevertheless, the diffusion phenomenon in the lung has often been studied using Fick’s law which is only valid for binary diffusion. Under conditions of more than two gases in a mixture, the appropriate equations for diffusion are those of Stefan. In respiration, diffusion involves at least three gases (0,, CO, and N,), and in physiological experiments, complex mixtures including heavy or light gases (SF,, He) are often added to enhance the effect of diffusion. We present in this paper the features of ternary diffusion and solve the appropriate equations for the non-steady state by a finite difference method. The simulation was performed using two models derived from the anatomical data of Weibel and Hansen-Ampaya. Moreover, four initial conditions most often encountered during current respiratory physiology tests, were used for the computations. Therefore in these four situations, O,-N,-He, O,-N,-Ar, O,-N,-SF, and O,-N,-CO, combinations were used. Our results showed that for each case mentioned above the oxygen acted differently ln ternary diffusion owing to the specific nature of the components of each mixture. Moreover, the behaviour of each component in ternary diffusion was very different from that of binary diffusion. However, this difference may be negligible when the subject breathed normal air.

Introduction Uneven intrapulmonary gaseous distribution is one of the main pathophysiological characteristics of chronic obstructive lung disease. Among the mechanisms responsible for this inhomogeneous ventilation, gaseous diffusion in alveolar space was first suspected .by Haldane et al. (1905) and by Krogh et al. (1917) who questioned whether mixing by diffusion was complete. Trying to answer this question, Rauwerda (1946) developed a simple model of the lung. A number of other mathematical models have been proposed by different authors (La Force et al., 1970; Cumming et al., 1971; Scherer et al., 1972; Paiva, 1972) which, even though based on Weibel’s data (1963), lead to inconclusive results. A critical review of these models in the literature of respiratory physiology was given by Chang et al. (1973); in particular they noticed that only two gases were included in the diffusion models. However, under conditions of more than two gases in a mixture, Fick’s law (1855) cannot be appropriate. Indeed, *Supported by a grant from INSERM (No. 82.5.028). 403 Int. J. Bio-MedicalComputing (13) (1982) 403-419 Elsevier Scientific Publishers Ireland Ltd.

404

A. Ben Jebria, :... dreres

in the lung no less than three gases take part in diffusion: oxygen, carbon dioxide and nitrogen. In addition, complex mixtures including heavy or light gases such as sulfur hexafluoride (SF,) or helium (He) are often used in respiratory physiology tests to enhance the effect of diffusion. Therefore, in these cases, the appropriate equations for the diffusion of three or multicomponent gas mixtures are those of Stefan (187 1). Toor (1957) solved these equations directly in the steady state and demonstrated that the diffusional properties of ternary gas mixtures are very different from those of binary gas mixtures. In particular, he determined the conditions under which the following phenomena may occur in ternary diffusion: (1) ‘diffusion barrier’ (the rate of diffusion of a component is zero even though its concentration gradient is not zero); (2) ‘osmotic diffusion’ (the flux of one component can exist without its concentration gradient); and (3) ‘reverse diffusion’ (the flux of one component can exist in the same direction as a concentration gradient for that component). These features cannot be accounted for in binary diffusion as described by Fick’s law. Using a simple model based on a gas film of infinitesimal thickness which was assumed,to be located immediately next to the alveolar membrane, Chang et al. (1975) confirmed the idea of Toor (1957) and called attention to possible implications of ternary diffusion in respiratory physiology. Subsequently, Model1 et al. (1976) corroborated these implications by in vitro experiments. Bres et aZ. (1977) applied Stefan’s law to conditions most often encountered during ventilation, i.e. to the non-steady state. They compared theoretical curves with those obtained experimentally (in vitro) using a simple geometrical configuration. The purpose of this study is to apply the theory of ternary diffusion under conditions close to those of respiratory physiology experiments, i.e. the non-steady state; moreover, the equations of ternary diffusion are solved by a finite difference method with an explicit scheme, using two models derived respectively from Weibel’s (1963) and Hansen-Ampaya’s (1975) anatomical data of the human lung. In addition, four initial conditions of mixtures combinations were used for computations (02-N2--He, 02-N2-Ar, 02--N2--SF, and 02-N2-CO*). Development

of the ternary diffusion model

1. Bimy and ternary diffusion features Because of its complex nature, the theory of ternary diffusion is less used than that of binary diffusion in respiratory physiology, even though the basic equations have long been available. For binary gas diffusion, the corresponding basic equation is described according to the classical Fick’s law:

N, =

-D12Sz

(1)

Ternary diffusion

405

where Nr is the flux (rate of transfer) of gas 1 in moles/unit time, D12 the diffusion coefficient between gases 1 and 2 in cm a s-’ , S the diffusion area in cm2, Cr the concentration of diffusing gas 1 in moles/unit volume and x the distance in cm. Equation (1) is usually called Fick’s first law. Differentiating with respect to x and taking into account the conservation of mass yields: 1 dS

ac,

+smx In the simple case of a constant dS -= dx

(2)

diffusion area, i.e.

0,

one gets Fick’s second law:

ac, -D,2 at-

a2cl a2

-

It can easily be seen from the theory of binary gas diffusion that the movement of each gas depends only on its own concentration and on the diffusion coefficient; moreover, the two gases have flux rates which are equal and opposite in direction. Because of this, the two components are in equilibrium at the same time. Now, consider the case of three gases; the diffusion equations, worked out by Stefan (1871) can be written:

in which, C,, C,, C3 are the concentrations of the three gases in moles/ unit volume; ar, a2, a3 are the ratios Cr/(C, + C2 + Ca), C&C’, + CZ -I- Ca), C&C, + CZ + C3) 3 = 1; IV,, IV,, IV3 are the respective fluxes with rangingfromOto1witha1+a2+a N, + N2 + N3 = 0; and D12, Dz3, D31 are the binary diffusion coefficients. It can be seen from Eqns. (4) (5) and (6) that ternary diffusion is more complicated than

A. BenJebria, M.Bres

406

binary diffusion; furthermore, this difficulty is more important when the diffusion area is not constant: that is the case of the human lung. Nevertheless, merely by inspection of Eqns. (4) (5) and (6), it can be seen that the flux of one component can exist without a concentration gradient for that component or even in the same direction as the concentration gradient. Those features, called by Toor (1957) osmotic diffusion and reverse diffusion, cannot be accounted for by Fick’s law. 2.

Anatomicalshwture

The lung model used for computer simulations of ternary diffusion is based on the morphometric data given by Weibel (1963) on the one hand, and by Hansen et al. (cm 1

Distance 0.6

0.5

0.4

0.3

0.2

01

0

0.1 L

0.2 I

0.3 I

0.4

0.5 L

0.6 1c

J i0

50

. 40 z u 30

2 0 lz

20

10

.O 16

’ 17 '16 '19 20'212223242

i

Generation Number Fig. 1. Graphical representation of the distal airways models accompanied by schematic architecture of transitory and respiratory airways. From left to right Weibel (1963) geometry followed by Hansen-Ampaya (1975) geometry.

407

Ternary diffusion TABLE I MORPHOMETRIC DATA OF DISTAL AIRWAYS HANSEN-AMPAYA (RIGHT)

Length (cm)

TO QXIBEL (LEFT) AND TO

Dimensions of Hansen-Ampaya (1975)

Dimensions of Weibel(l963) -Generation No.

ACCORDING

Crosssection

Generation No.

Length (cm)

(cm’)

(cm’) 17 18 19 20 21 22 23

0.141 0.117 0.099 0.083 0.070 0.059 0.050

300 534 944 1600 3220 5880 11800

Crosssection

16 17 18 19 20 21 22 23 24 25 26

0.097 0.097 0.088 0.066 0.051 0.058 0.043 0.041 0.030 0.028 0.022

128 229 484 883 1991 5225 7697 9790 6170 2050 655

(1975) on the other. Therefore, we used two lung models: the first one (Weibel, 1963) is represented as a regular dichotomy of the airways as shown in Fig. 1 (left); the second one (Hansen et al., 1975) is less regular, as shown in Fig. 1 (right). This figure was constructed on the basis of the morphometric data listed on Table 1. It can be seen from Fig. 1 and Table 1 that we have focused our interest on the study of ternary diffusion only in the distal airways of the trachea-bronchial tree. Indeed, for a person breathing normally, the inspired and expired gases are transported in the airways in two different functional and structural zones of the lung, namely: (1) the upper and central airways, through which the transfer of respiratory gases is essentially the result of a pressure gradient between upstream and downstream. This is accomplished by many complicated motions, such as convection with swirling and either turbulent or laminar flow; and (2) the terminal (or distal) airways, in which gaseous diffusion plays the main role. 3. Initial and boundary conditions The following assumptions were used for computations: (1) longitudinal gas mixing occurs by molecular ternary diffusion alone (without convection); (2) radial diffusion is instantaneous; (3) diffusion across alveolar walls is not existent; and (4) the morphometric model is not expansible, i.e. it is independent of time. As shown in Fig. 2, the terminal airways of the human lung are schematically represented by their smoothed summed cross section (trumpet). For the sake of simplicity, four diagrams resume the initial conditions used for the simulation. Each

408

A. Ben Jebria, hf. Bres

--

Fig. 2. Schematic representation of four cases of initial conditions for simulations. These situations are often used in physiology respiratory tests. The last one represents the normal breathing situation. Note that the distal airways are represented by their cumulative cross-section area which is similar to a trumpet.

one is composed of two sides containing different mixtures of gases diffusing into one another. In the first situation, a subject whose lung contains 21% O2 and 79% Nz, inhales a mixture of 21% O1 and 79% He; in the second, he inhales a mixture of 21% Oz and 79% Ar; and in the third, he inhales 21% O2 and 79% SF+ These three situations are most often used in respiratory physiology experiments to enhance the effect of diffusion. In the last situation presented in Fig. 2, the subject breathes normal air (21% O2 and 79% N,) and his alveolar gases are composed of 15% 02, 79% Nz and 6% COz. This situation is very close to the lung ventilation condition. Thus, in each of these four situations, a mixture of three gases is present, namely: 02-N2-He, 02-N2-Ar, 02-Nz-SF6, 02-N2--C02.

Ternary diffusion

409

4. Numerical procedure for ternary diffusion simulation Because of their complex nature, no analytical solution for Eqns. (4) (5) and (6) can be found even when the diffusion area is constant. Thus, these equations were solved using the finite difference method with an explicit scheme on a PDP 11/03 digital computer. We will try to develop this method of resolution. For that purpose let us consider: (7) al + a2 + a3 = 1

(8)

From Eqns. (7) and (8) substituting IV3and a3 in Eqns. (4) and (5) yields:

(9)

S

$ =+-(a2NI 12

- aJV2) + L(-N2

- a2NI + aI&)

00)

032

By rearranging these lead to the following forms:

(12) Now, for the sake of simplicity, we define the following expression as: 1

(Y=--D12

7

1 013

1

1

D21

023

=---

410

A. Ben Jebn’a, M. Bres

Then,

ac,

= N,(-cun, - P)+N2(%)

%

sz =4(7az)+Nz(-Ya,

(13)

- 6)

(14)

By eliminating N2 from equations (13) and (14), Ni can be written as:

N,,=

S a7wG

-(w

+ P)(ra1

+ 6)

(15)

Now,

N,AC$SAx

ac, =%

ax

aa,

and

N,=C

x2

-c$

aa, zSh

axand

(16)

(17)

Therefore, substituting Eqns. (16) and (17) in Eqn. (15) leads to the following equation:

-aal

Aa -

(W’ At

Aa

1=

7a1+

aaz+fl-

Aa,

6

tw

ffrw2

7a1+

6

411

Ternarydiffusion

By eliminating N, from Eqns. (13) and using Eqns. (16) and (17) the following equation can be deduced: -Y~Z

(Ad2 Aa =

Aa,

-

cfaz+ P

At

7aI -I- 6 -

Aa,

(19) wla2

~.

aa2

+

P

We subdivide the distal airways in small compartments, and let n be the number of these compartments; So if we put: al = U,

anda

= V,,

then Eqns. (18) and (19) can be written as following:

[email protected] + %)(vn+l - cl)/2 FU,

7(%+,

= -

o(v,+,+

+ Q/2

%)/2+B-

+ 6 cfY(u,+

-

1 +

-(%+, U,)

(%+I

7 (%+ 1 + Q/2

v,)

7

+

Q/4

(20)

+ s

(21)

Now, by approximating dU,/dt and dV,/dt in the first class by time discretization and taking into account the variation of summed airways cross-sectional area S, then the governing equations finally used for simulation were:

Un(t+At)=

U,(t)+2

Fun_, S,,_, -Funs,, ~-

At

s n-l + 4~

(Ad2

FV,_,

S,_,

-FV,S,

G(t + At) = v,(t) + 2 Sn-1 + Sn

At -(Ax)’

(22)

(23)

412

A. Ben Jebria. M. Bres

According to Eqn. (8) and putting a3 = IV,, then:

iv, = 1 -(U,

+ &)

(24)

Equations (20) and (21) together with Eqns. (22), (23) and (24) form the solution of ternary diffusion Eqns. (4) (5) and (6). Simulation and results

The simulation was performed using the values of diffusion coefficients listed in Table 2; a time increment At = 0.4 X 10V4s and a spatial increment Ax = lo-* cm were chosen in such a way that the inequality D At/(Ax)* < l/2 is satisfied for the stable solution (Collins, 1973). In each of our four mixtures (Fig. 2) we calculated the behaviour of the three components with the computer program. For the first three mixtures (02-N,--He, 02-N&U and 02-N2-SF,J, Figs. 3-5 show the theoretical curves of the three component concentrations of each mixture for Weibel geometry (left) and for Hansen-Ampaya geometry (right). In each figure, the concentration of each gas is plotted versus distance or generation number after 2 s of computation on one hand (upper figure) and versus time at the end of the anatomical model on the other hand (lower figure). It can be noticed, as it is shown in Figs. 3--5, that although oxygen concentration was initially the same in the tank and in the lung (Fig. 2) after 2 s of calculation the predicted oxygen concentration in the terminal airways at equilibrium is modified; this modification is intimately dependent on the components of the mixture. For the helium mixture (Fig. 3), it can be seen that the oxygen fractional concentration at equilibrium, fell from 21% to 13% for Weibel geometry and from 21% to ??

TABLE 2 BINARY DIFFUSION COEFFICIENTS (0) USED IN COMPUTATION DIFFUSION, AT 37°C AND 101.3 X 10’ Pa Gas pair

D (cm’ s-l)

References

0,-N,

0.230 0.772 0.205 0.097 0.173 0.780 0.210 0.105 0.181

Ben Jebria et al. (1979) Ben Jebria et al. (1979) Ben Jebria et al. (1979) Ben Jebria eta!. (1979) Ben Jebria et al. (1979) Walker etal. (1961) Roberts (1963) Worth et al. (1978) Worth et al. (1978)

O,-He O,-AI O,-SF, o,--co, N,-He N,-Ar N,-SF, N, -CO,

OF TERNARY

413

Ternary diffusion

Generat .

17

;

18

aon

Number

19

20

Generatoon 16

21 .22.23,

17

Number

18

19 .20.21.22.23.24~

0.4.

5

_-e--e-

0’

Nl /

__-%

u’ 0.2: . . . . . . .

o/’ o

0.2 . ...*

/ ‘.;“”

,

,

/--

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ot . . . . . . . . . . .

, I 12

3

4

Distance

Ov,,

Time

5

L 60

o/’

1’ ‘.,‘.. *............................................... I I 12

[mm)

(WC)

3

4

Distance

OoU

02

5

6

(mm)

2 Time

(set)

Fig. 3. Results of simulation using the helium mixture (first case of Fig. 2). Fractional concentration of He, N, and 0, versus distance or generation number after 2 s of diffusion (top); fractional concentration of the same three gases computed at the end of the anatomical models versus time (bottom). From left to right, behaviours of the three component concentrations according to Weibel and Hansen-Ampaya geometry, respectively.

14.5% for Hansen-Ampaya geometry; moreover, the fractional nitrogen concentration fell also from 79% to 31.5% for Weibel geometry and from 79% to 45.5% for HansenAmpaya geometry. However, it can be noticed that at equilibrium, the sum of oxygen and nitrogen concentrations was smaller than the helium concentration for Weibel geometry; the contrary was observed for Hansen-Ampaya geometry. For the argon mixture Fig. 4 shows a slight increase in oxygen concentration (21% + 21.5%), for Weibel as well as for Hansen-Ampaya geometry; at the same

A. Ben Jebria, hf. Bres

414 Generation 17

Number

18

19

2

3

20

Generat 21

OR’8 0

1

0.4

16

. 01’ 4

Distance

0

(

22.23

0.8

1.2 Time

5

60

11

10

1.6

2

0

19

21.22.23242q

20

I

12

3 Distance

(mm)

(set)

ICI” Number

0.4

0.6

1.2 Time

4

5

6

1.6 (set)

2

(mm)

Fig. 4. Results of simulation using the argon mixture (second case of Fig. 2). Same as Fig. 3 with AI replacing He.

time nitrogen concentration fell slightly: (79% + 58%) for Weibel geometry and (79% + 66%) for Hansen-Ampaya geometry. For the sulphur hexafluoride mixture, it can be seen that oxygen concentration increased notably from 21% to 23% for Weibel geometry and from 21% to 22% for Hansen-Ampaya geometry; however, nitrogen concentration underwent a very slight fall after diffusion: (79% 3 67%) and (797o -+ 7 1.5%) for Weibel and Hansen-Ampaya geometry respectively. Now, let us consider the fourth situation of ternary diffusion mentioned in Fig. 2. It can be noticed as it is shown in Fig. 6 that although nitrogen concentration was initially the same in the tank and in the terminal airways, nitrogen concentration in

415

Ternary diffusion

Generation

I

17

0

0.6,

1

Generat ion Number

Number t9

18,

20.

4 5 3 Distance (mm)

2

16

21 ,22.23,

60

--__

--__

--a_

----____

19 .20. 21 .22.23.24.25~

5 3 4 Distance (mm)

---___

--!r

NL_.

0.6 -

a6.

0.4 -

OA.

o,2 *_*_.__.__*..................................

19

12

0.6,

---__

l?

6

--a_

.

0, . . . . . ..-a’ o2 . . . . . . . . . . .._.._........................................02

SF6_ 0 0

4

0.4

0.6

1.2 Time

1.6 (set)

2

o0

0.4

Fig. 5. Results of simulation using the sulphur hexafluoride as Fig. 3 with SF, rtiplacing He.

0.6

1.2 Time

1.6 (set)

2

mixture (third case of Fig. 2). Same

the lung fell from 79% to 78.6% after 6 s of diffusion; moreover, oxygen concentration increased by 0.8% and carbon dioxide concentration decreased by 0.4% compared to binary diffusion. Thus, in this case the presence of CO2 at a low concentration does not notably affect the equilibrium of the other gases of the mixture. Discussion 1. Validity of the numerical method The complex nature of Eqns. (4)

(5) and (6) prevented us from solving them

A. Ben Jebria, hf. Bres

416

Generation 0.80 I

11

18

,

Number ,

19

,

20

,21,22,23,

1

0.79

__

1 -w

-z

_---_------_____-_

---w-w

?b

0.76J

0

0 0.21 ._ .I. : m L

..-.._ : *.Q..

r;

-...

-...

-... *-......._

0.19

--........

0, u c ;

02 mm.................................

0.17

m c

0.15

0 ._ ” m L u.

0.06

0.03

0 1

0

2

Distance

3

4

5

6

(mm)

Fig. 6. Results of simulation using the carbon dioxide mixture (fourth case of Fig. 2). From top to bottom nitrogen, oxygen and carbon dioxide concentrations versus distance after 2 s of diffusion. Note that in this case, the simulation was performed using only Weibel geometry.,

analytically. We therefore solved the ternary diffusion equations, for the non-steady state, by a finite difference method with an explicit scheme. Stability and convergence criteria were close to those involved in parabolic equations. In that way, a spatial increment Ax and a time increment At were chosen in order to satisfy the inequality D A t/(Ax)2 < l/2. On the other hand, it is desirable for the sake of convergence that this ratio be as close to l/6 as possible (Collins, 1973). However, ??

417

Ternary diffusion

Bres et al. (1977) used this method to solve the ternary diffusion equations using a simple geometry by satisfying the preceding conditions and showed a good similarity between the predicted and the in vitro experimental curves. In addition, we confirmed the validity of the method employed in this work, by using the implicit finite element method previously performed (Ben Jebria et al., 1981) and applied it to binary diffusion by assuming that the diffusion coefficients of oxygen and nitrogen with a third gas are very similar. Note that the theoretical basis of the finite element method has been provided by Zienkewicz (1971) and by Lions (1974). Applications

t-T-r-7

Distance

(mm)

Oh-T-7-3

Morphometric

Distance

(mm)

Models

Fig. 7. Concentration ratios of He/O,, ArIO, and SF,/O, (top) and of He/N,, h/I’& and SF,IN, (bottom) plotted against distance according to Weibel geometry (left) and to Hansen-Ampaya geometry (right).

418

A. Ben Jebria, M. Bres

of this method to the linear and non-linear diffusion equations has been discussed by Fremond (1977). 2. Specific behaviours of oxygen in different mixtures The choice of the helium, sulphur hexafluoride and argon mixtures was to enhance the features of ternary diffusion with light, heavy and moderately heavy gas mixtures. As it was shown by Figs. 3-5, although oxygen concentration at the beginning of diffusion was equal in the tank and in the distal airways, at equilibrium not only was oxygen no longer constant, but its behaviour was different in the different mixtures. Figure 7 resumes the behaviour of oxygen as well as of nitrogen in each mixture and in each lung model. These behaviours in ternary diffusion are very different from those in binary diffusion. As far as the three mixtures go, it can be seen (Fig. 7) that the He/O? ratio curve is above the Ar/Oz ratio curve which is in turn above the SF,/02 ratio curve; it is the same for the He/N*, Ar/N2 and SFdNz ratio curves, respectively. Moreover, it is interesting to notice that although oxygen concentration was initially constant, it fell remarkably after diffusion, in the helium mixture. This could mean that the use of a light gas in breathing - ternary mixture - favours oxygen elimination. Nevertheless, the presence in a mixture of a heavy gas, such as sulphur hexafluoride, prevents the impoverishment of the distal airways in oxygen. These phenomena can be explained by the ternary diffusion features, namely: osmotic and reverse diffusion. ‘Ihis feature can also be observed for the (O,-Ns-COs) mixture, in which oxygen concentration increased lightly and, by symmetry, nitrogen concentration fell lightly compared to the results of binary diffusion. In this procedure, the effect of reverse diffusion is negligible, because the diffusion coefficients of these components are not very different on one hand and carbon dioxide concentration in the alveolar space is low, on the other hand. Therefore for air breathing under normal conditions the binary diffusion law can be used. 3. Influence of lunggeometry on diffusion The two anatomical lung models established by Weibel(l963) and by Hansen et al. (1975) are the most often used for gas transport mathematical models in respiratory physiology. However, the second geometry is not regular and it is may be closer to the real human distal airways. Nevertheless, Hansen-Ampaya geometry did not modify the effect of ternary diffusion features. The only difference of diffusion in the two geometries dwells in the concentration value at equilibrium because of the difference in the variation of the cumulative cross-section appropriate to each geometry. References Ben Jebria, A., Bres, M. and Hatzfeld, C., Mesure de coefficients de diffusion binaire entre l’oxyghe et un autre gaz, Bull. Eur. Physiopathd. Respir., 15 (1979) 639-647. Ben Jebria, A., Mallet, A., Steimer, J.L., Hatzfeld, C. and Boisvieux, J.F., Finite element simula-

Ternary diffusion

419

tion of gas transport in proximai respiratory airways: Comparison with experimental data Comput. Biomed. Res., 14 (1981) 493-505. Bres, M. and Hatzfeld, C., Three-gas diffusion - Experimental and theoretical study, Pfliigers Arch., 371 (1977) 227-233. Chang, H.K. and Farmi, L.E., On mathematical analysis of gas transport in the lung, Respir. Physid., 18 (1973) 370-385. Chang, H.K., Tai, R.C. and Farhi, L.E., Some implications of ternary diffusion in the lung, Respir. Physiol., 23 (1957) 109-120. Collins, R., Analyse numerique: Application aux milieux continus, Universite Paris VII, 1973. Cumming, G., Horsfield, D.K. and Preston, S.B., Diffusion equilibrium in lungs examined by nodal analysis, Respir. Physiol., 12 (1971) 329-345. Fick, A., Uber Diffusion,Ann. Physiol. Chem., 94 (1855) 59-86. Fremond, M., La methode des elements finis - Application aux equations de Ia diffusion Ii&tires et non iineaires, Math. Comput. Simul., 19 (1977) 29-37. Haldane, J.S. and Pristley, J.C., The regulation of the lung ventilation, Z. Physiol. (Land.), 32 (1905) 225-266. Hansen, H.E. and Ampaya, E.P., Human air space shapes, sizes, areas, and volumes, J. Appl. Physiol., 38 (1975) 990-995. Krogh, A. and Lindhard, J., The volume of the dead space in breathing and the mixing of gases in the lungs of man,J. Physid. (Land.), 51 (1917) 59-90. Laforce, R.C. and Lewis, B.M., Diffusional transport in the human lung, .Z. Appl. Physid., 28 (1970) 291-298. Lions, J.L., Cows d’analyse numerique, Ecole Polytechnique, Herman, 1974. Mode& HI. and Farhi, L.E., Ternary gas diffusion - in vitro studies, Respir., Physiol., 27 (1976) 65-71. Paiva M., Gas transport in the human lung, .Z.Appt. Physiol., 35 (1973) 401-410. Rauwerda, P.E., Unequal ventilation of different parts of the lung and the determination of cardiac output, Thesis, Groningen, Netherlands, 1946. Roberts, R.C., Molecular diffusion of gases, Am. Inst. Phys., Handbook, McGraw HiB, New York, 1963. Scherer, P.W., Shendalman, L.H. and Greene, N.M., Simultaneous diffusion and convection in single breath lung washout, Bull. Math. Biophys., 37 (1972) 393-412. Stefan, J., fiber das Gleichgerwicht und die Bewegung insbesondere die Diffusion von gasgemengen,S.B. Akad. Wiss.Wien, 63 (1871) 63-124. Toor, H.L., Diffusion in three-component gas mixtures,AZChEJ., 3 (1957) 198-207. Walker, R.E., Monchick, L., Westenberg, A.A. and Favin, S., High temperature gaseous diffusion experiments and intermolecular potential energy functions, P&net. Space Sci, 3 (1961) 221-227. Weibel, E., Morphometry of the human lung, Springer-Verlag, New York, 1963. Worth, H., Niisse, W. and Piiper, J., Determination of binary diffusion coefficients of various gas species used in respiratory physiology, Respir. Physiol., 32 (1978) 15-26. Zienkiewicz, O.C., The finite element method in engineering science, McGraw-Hill, London, 1971.