Gaseous diffusion in the airways of the human lung

Respiration Physiology

1 (1966) 58-14; North-Holland Publishing Company, Amsterdam

GASEOUS DIFFUSION

IN THE AJRWAYS OF THE HUMAN LUNG

G. CUMMINGS, J. CRANK*, t Department

of Medicine,

K.

Queen Elizabeth Hospital, Birmingham

* Department

HoRsFrELDt AND I. PARKER*

of Mathematics,

University of Birmingham,

Edgbaston,

15, England

Brunel College, Acton, London, England

Abstract.The mixing of inspired gas with that already present in the lung is brought about by gaseous diffusion. It has been generally accepted that this process is sufficiently rapid as to leave no appreciable concentration gradient (stratified inhomogeneity) within the lung during quiet respiration. The shape of the alveolar plateau has therefore been explained on the basis of regional inhomogeneity. A theoretical analysis of diffusion within the lung airways has been made involving the validation of a model suitable for analysis and at the same time being compatible with anatomical information A variety of models have been analysed and critic&d, and from the analysis the conclusion has been drawn that stratified inhomogeneity within the lung is likely, and plays a part in the formation of the alveolar plateau. Alveolar gas Diffusion in the airways

Lung anatomy Pulmonary regional distribution

Ventilation of the human lung requires the transport of inspired gas to the terminal airway units down a pressure gradient produced by the muscles of the chest wall. Mixing of the inspired gas with that already present then occurs by gaseous diffusion. The importance of gaseous diffusion in the process of ventilation has been recognized since the early part of the century. It was thought by KROGH and LINDHARD (1917) that the process of mixing was slow enough to leave a gradient of nitrogen within the lungs following an inspirate of oxygen. In the subsequent expirate, the concentration of nitrogen, in its latter part, showed a gradual rise, and since this came from alveoli it was designated the alveolar plateau. This view, that stratified inhomogeneity was responsible for the slope of the alveolar plateau was accepted generally until RAUWERDA(1946) published the results of his investigations into the rate of gaseous diffusion. This work was subsidiary to his main purpose, which was the measurement of cardiac output, and his approach to the problem of diffusion was the application of mathematical analysis in situations analogous to those thought to occur within the lung. Acceptedforpublication

13 August 1965.

This work was supported in part under Contract No. AF 61(052)775 through Office of Aerospace Research (OAR), United States Air Force. 58

the European

GASDIFFUSION IN THEAIRWAYS

59

He came to the conclusion that diffusion in the lungs was sufficiently rapid as to exclude a measurable concentration gradient within a terminal airways unit one second after the establishment of the gaseous interface. He defined an airways unit as that structure fed by a terminal bronchiole, and hence having a length of 7 mm. As a result of his analysis he concluded that the current explanation of the alveolar plateau as representing stratified inhomogeneity was incorrect. The view achieved wide acceptance, and an alternative explanation for the shape of the alveolar plateau was advanced by OTIS et al. (1956) based upon mechanical factors. These authors suggested that areas of the lung with different compliance and resistance respond differently during an inspiration. An area with a low resistance and a low compliance would expand earlier and more than one with the opposite characteristics. Following an inspiration of oxygen therefore, these areas would have differing concentrations of nitrogen. The characteristics of such areas have been defined on the basis of their time constant, which is a function of their resistance and compliance; areas with small time constants respond rapidly to pressure changes and areas with long time constants respond slowly. During expiration, therefore, areas with rapid time constant would empty first, whilst areas with long time constants would empty last. This sequential emptying of areas of the lung having differing nitrogen concentrations would produce the alveolar plateau. Whilst this explanation is reasonable, it appears to us to suffer from two main defects, one experimental and one conceptual. Several authors have published experimental evidence on the effect of breath holding on the alveolar plateau (KJELLMERet al., 1959; FOWLER,1949; MILLSand HARRIS, 1965). There is general agreement that the slope diminishes with time, and that the process responsible for the change is that of gaseous diffusion. A compromise has been sought between this evidence of slow mixing by diffusion and the idea that mixing in the terminal airways units is rapid. This compromise accepts that terminal airways units equilibrate rapidly, but suggests that each unit has a different equilibrium concentration as indicated by the time constant theory. Mixing between adjacent units then occurs by diffusion and explains the change in the alveolar plateau during breath holding. Thus mixing involves the transfer outwards of gas molecules by diffusion in a system of rapidly diminishing cross sectional area, followed by diffusion inwards in an adjacent unit. Unless mixing is to occur only between neighbouring pairs of units, it is necessary that the path length of the gas molecules involved should increase, and it has appeared to us that these path lengths involved would be so long as to make the explanation offered an unlikely one. The conceptual problem concerns the configuration of the interface between the gas in the terminal airways units, which is homogeneous, and that in the airways, which is also homogeneous but at a different concentration. At this point a concentration tradient must exist and its character must change with time. The conventional explanation of the slope of the single breath nitrogen curve takes no account of this concentration gradient. Since the time constant hypothesis does not offer a satisfactory explanation on

60

G. CUMMINGel al.

these two important points, we considered that some other explanation for the slope of the alveolar plateau might be possible. The original suggestion of Krogh, that gaseous diffusion created a concentration gradient appeared to us more likely, and that a further investigation into diffusion was indicated. This view was supported by the work of Roes et al. (1955). Diffusion is influenced by the character of the initial interface, about which there are two main views. The first suggests that gas travelling at high velocity in the upper airways has a parabolic profile, and as a result a thin pencil of inspired gas is drawn into the terminal unit. Radial diffusion then results in rapid mixing and a concentration gradient is unlikely. The other view assumes that a square wave interface is established in the airways, but that diffusion down the airways is sufficiently rapid to exclude a gradient. The choice between a parabolic front and a square wave front is best made from anatomical considerations. The larger airways branch by dichotomy. The cross-sectional area of each pair of branches is greater than that of its parent, and the total cross-sectional area of all the branches increases progressively with distance from the trachea (WEIBEL, 1963). Consequently the linear velocity of gas flow diminishes progressively until in the alveolar ducts, flow velocity is less than 1 mm per sec. A velocity profile certainly exists in the trachea, and is probably transmitted through successive divisions. We thus have a situation where the diffusion path of molecules in the axial stream becomes progressively shorter, whilst the velocity of axial flow becomes slower. Radial diffusion will tend to obliterate the velocity profile and create a square wave front when the rate of diffusion laterally is greater compared with the rate of flow forwards. As an example, a bronchiole of 1 mm in diameter has a forward gas velocity of 5 mm by radial diffusion per sec. In a cylinder of this size the time taken for 90 OJ,equilibration is about 5 msec (CRANK, 1957), so that at this point radial mixing is forty times faster than forwards flow. Since a bronchiole of 1 mm in diameter would supply several terminal airways units, it seems likely that the interface within a terminal unit will have a square wave form. Since the gas front is square, mixing can only occur by axial diffusion. Although the airways are arranged in a parallel manner, gaseous mixing occurs in a serial manner, so that any inadequacy of mixing within a terminal unit will result in a concentration gradient. This does not deny the possibility that units may have different equilibrium concentrations, resulting from different degrees of ventilation. In view of the conflicting evidence about the time course of gaseous diffusion, and its possible importance in the explanation of the alveolar plateau, we have carried out an analysis of diffusion in situations similar to those expected in the lung. The problem is to create a geometrical model which is susceptible of mathematical analysis, and compatible with lung anatomy. We shall describe the anatomical information we have used, then show geometrical models from the simplest to the most sophisticated we have considered, with a criticism of the validity of each model, and the solution to the diffusion situation within each.

61

GAS DIFFUSIONIN THE AIRWAYS The mathematical numerical

data appear

treatment

of each model forms

an appendix,

so that only the

in the text.

The anatomical information The structure of the airways of the human lung has been described by several authors (WEIBEL, 1963; MILLER, 1937; VAN HAYEK, 1960). There are differences in detail, but in general the airways may be described as being a system of dichotomous branchings, but with irregular terminations. For instance a terminal bronchiole may be reached after 12 branchings, whilst another may be found after 25 branchings. Consequently the time taken for a molecule of gas to travel from the carina to the terminal bronchiole will differ according to the path taken. This point will be discussed further. Zcross

section (cm2)

300 I

534 $144 1600 3220 5880 11800

Length (cm)

Diam. (cm)

TB

0.165

0.060

RBl

0.141

0.054

RH2

0.117

0.050

RB3

0.099

0.047

‘4%

0.083

0.045

AD2

0.070

0.043

AD3

0.059

0.041

area

I

Fig. 1. Representation of a terminal airways unit. Dimensions to scale but branching angles formalised. The structures beyond the terminal bronchiole are more homogeneous, and a typical structure is shown diagrammatically in fig. 1. This diagram represents a section through a terminal airways unit, and the dimensions are appended. The number of alveoli on each duct and sac represents the average number present in such a unit and their dimensions (but not shape) are appropriate. The dichotomous character of branching is shown. The alveoli increase in number from the first order respiratory bronchiole RB, to the alveolar ducts AD, _ 3. The third order of alveolar ducts has also been described as the alveolar sac. The creation of a geometrical model from this structure involves some difficulties. The first of these is what dimension to take for the area of cross section. If we consider a single alveolar duct, the gas interface will be established by mass movement within the lumen of the duct. This represents the minimum cross section. Radial diffusion will however be rapid, the interface will move laterally into the alveoli, and complete mixing will occur over this short distance. The total diameter of duct plus two alveoli

62

G. CUMMING et al.

represents the maximum cross section. Axial diffusion is possible however, only down the central duct, and we have therefore adopted the area of this as representing the cross section of the geometrical model, considering the errors involved in this assumption later. Using this measurement, the representation of increasing areas within the last 2.5 cm of airways is shown in fig. 2. It will be seen from this that the smaller the distance from the end of the airways, the closer can the structure be represented by a cone. The linear distance from the terminal bronchiole to the end of the alveolar sac is about 0.7 cm, and it is convenient to find the gradient across this distance when comparing different models, so that direct comparison is possible. The total volume of Dwnensrons of the terminal

1.0 Radius

0.5 0 of summed cross

airways

a5 1.0 section (cmJ

1.5

Fig. 2. Cross sectional area of the airways over the terminal 2 cm.

airways proximal to the point 0.7 cm from the terminal sac is about 200 ml, so that most of the lung volume is contained in the terminal 0.7 cm. The inspiration of one litre of gas establishes a gas interface at the level of the first order alveolar duct, on an average, using the dimensions of Weibel. This means that the average diffusion is about 2 mm (see fig. 12). The selection of a geometrical model

The diffusion equilibria reached in the ensuing models will be shown in a standard manner, so that the linear distance is 0.7 cm, and the times of diffusion are 1 set, 5 set, 10 set and 30 sec. Where equilibrium is complete in a short time, the longer times are not shown. MODEL NUMBER

1

The simplest case is that of a cylinder of infinite length, with a square wave gas interface having a gradient of unity initially. Since the cylinder is of infinite length mixing will not penetrate to the extreme ends, and therefore no molecules will be lost, nor

63

GAS DIFFUSION IN THE AIRWAYS

will reflection of the diffusing molecules occur from the ends. The results from analysis of this system are shown graphically in fig. 3. This case is clearly inapplicable within the lungs. Infinite

cylinder

1.0. 0.9 08 1.0

0 0 a.1 0.2 0.1 Linear distance lcmsl

03

a2

0.3

Fig. 3. Results of Analysis of Model Number 1. On the ordinate is plotted the fractional concentration of nitrogen. The abscissa shows linear distance from the original interface. At zero time the vertical line indicates that on the left there is unit concentration of nitrogen and on the right zero concentration of nitrogen. After one second diffusion has occurred so that at the original interface position the concentration of nitrogen is 0.5, or 50%. On the left of the cylinder 0.35 cm from the interface, nitrogen concentration has fallen to 0.7, and at a similar point on the right concentration is 0.3. Thus a concentration gradient of 40 % of nitrogen exists over this 7 mm of cylinder. After an infinite time concentration is everywhere uniform, at 50 %. Closed cylinder

-0

0.1

0.2

0.3 0.4 Lhear distance

a5 (cm )

0.7

Fig. 4. Results of analysis of Model Number 2. MODEL NUMBER

2

In this case the cylindrical model is of a finite length of 0.7 cm with both ends closed. The initial conditions are the same as in Model Number 1, and the results of the analysis are shown in fig. 4.

64

G. CUMMING et

a/.

This model is similar to that selected for evaluation are in agreement. The shown in the appendix. that of the sine-cosine mer requires less work

by Rauwerda,

and the analyses

mathematical approach to the analysis has been different as The two types of solution used, that of the error function and are equally valid provided sufficient terms are taken. The forfor numerical evaluation at small times, whilst the reverse is

the case for large times. The results from this second model show that concentration gradients within such a structure would be extremely small during the time of a normal breath and it is on this basis that the conventional view of diffusion rate largely rests. The application of such a model, with two closed ends 7 mm apart to the anatomical situation within the lungs is of doubtful validity. MODEL NUMBER 3 This cylindrical model has one end closed and the other end open, and is of infinite length. The initial conditions are such that an interface of unity gradient is established initially say 2 mm from the closed end, and the gradient after 1 and 5 set are computed. The results are shown in fig. 5.

Cylinder

open at one end

L/near distance Icmsl

Fig. 5. Results of analysis of Model Number 3.

The gradient over the 7 mm from the closed end after one second is 8.5% as compared with 0.1:/i in Model Number 2. This difference illustrates the importance of selecting an appropriate model, since small changes in the initial conditions bring about large changes in concentration gradient. The foregoing models, being cylindrical, are inappropriate even as a first approximation to the conditions in the lung, as shown in figs. 1 and 2. Rauwerda considered that the lung structure more nearly resembled a conical segment of a hollow sphere. It will be seen from fig. 2 that the smaller the dimensions of the airways unit considered, the more valid is the assumption of a conical shape.

GAS DIFFUSION IN THE AIRWAYS

MODEL

NUMBER

65

4

This conical model with closed ends is shown in fig. 6, and it is shown by Rauwerda, and confirmed by us that the diffusion equilibrium in this system was similar to that in the closed cylinder shown in fig. 4. Thus this model with its approximation to anatomical configuration supports the idea that the diffusion equilibrium is rapidly reached. It fails however to satisfy the anatomical requirement that one end should be closed and the other end open, and in contact with an infinite suppIy of gas, in this case oxygen. MODEL

NUMBER

5

This model satisfies the condition that one end of the cone should be open, and this end is in contact with an infinite supply of well-stirred oxygen. The interface between the nitrogen in the cone and the oxygen above it is established initially at various Ievels as indicated in fig. 6. The analysis shows that the concentration of nitrogen within the cone tends rapidly to zero, instead of attaining a finite equilibrium concentration depending on its geometry. The model does not behave as a terminal lung unit and requires modification. The defect lies in the end condition where the open end is in contact with well stirred oxygen and will clearly result in the rapid removal of the nitrogen. The functional requirement is that the open end should be in contact with an infinite source of oxygen, and that loss from the cone should be entirely by diffusion and not by stirring. This situation could be represented by a cylinder of infinite length fixed to the open end of the cone. The mathematical problem thus presented is susceptible of solution, but is somewhat complicated. As we have said this solution is inadmissibIe because the condition that no nitrogen should reach, and hence be removed across the open end of the cone cannot be satisfied when its relation to anatomy is considered. The open end represents a terminal bronchiole, and if some nitrogen moiecules reach this point, then diffusion beyond it wiil necessarily occur. Thus rapid equilibration and the absence of nitrogen molecules at the open end are incompatible. For the analytical solution of Model Number 5 to be acceptable some means of resolving this difficulty has to be found. In anatomical terms we must define a point in the airways at which no nitrogen molecules will be found following an inspirate of oxygen and the lapse of an appropriate time interval. Such a point could be defined with some certainty in the pharynx, but this is unacceptabIe since the lungs as a whole cannot with justification be regarded as a cone. A segment of lung which contains 16 terminal units has an effective shape as shown in fig. 2, and has an entry bronchiole which is 2.1 cm from the terminal air sac. The chances of the appearance of a finite number of molecules at this point ten seconds after an inspirate of pure oxygen is small, and a model with these dimensions more nearly satisfies the open end conditions. This is our Model Number 6, The fact that

66

G. CUMMING

ef

al.

the model deviates more grossly from a cone is a disadvantage and the effects of this will be discussed later. A r=o

A

/\

Fig. 6. Geometric configuration of Model Numb& 4. Both ends of the cone are closed. rl = 0.13 cm, rz = 0.13-0.83 cm, r3 = 0.83 cm.

MODEL NUMBER

r=O

Fig. 7. Geometric configuration of Model Number 6. The cone is open at r = rl, rl = 0.5 cm, r2 = 2.1-2.5 cm, r3 = 2.6 cm.

6

In this model the cone is larger, the diffusion distance being a maximum of 2.1 cm, is open at its narrow end and in contact with an infinite quantity of well-stirred oxygen (see fig. 7). The analysis of this model yields the results shown in fig. 8. The gradient after 1 sec. over the terminal 7 mm is a function of the depth at which the initial interface is estabEffect of interface positron on difrusion gradient arref 1saamd (Open system)

0.7 I 6

zl

s

ilo. a a 0.3 $02 2 0.1 0 05

1.0 Linear

I.5 distance (cm)

20

Fig. 8. Results of analysis of Model Number 6.

2.5

GAS DIFFUSION

IN THE AIRWAYS

67

lished, and varies between 15% when the interface is just inside the first respiratory bronchiole to 4.5% when it is in the second order alveolar duct. Fig. 9 shows the effect of time on the concentration gradients when the original interface is established at 2.4 cm, or 0.2 cm from the end of the alveolar sacs. From the results shown in figs. 8 and 9, it is clear that this model does not adequately Change of diffusion gradient with time. Interface at 2.4 cm (Open system)

Linear

distance

km)

Fig. 9. Effect of time on gradients in Model Number 6 when original interface was established 2.4 cm (re in fig. 7).

at

Effect of interface position on diffusion gradient after 1 second (Closed system)

Linear distance (cm)

Fig. 10. Results of analysis of Model Number 7.

describe the observed behaviour of thelungs. Firstly, the slopeof thegradient diminishes with time, in accord with experiment, but the absolute concentration of nitrogen also falls with time whereas in the lungs it tends towards an equilibrium value which does not then change with breath holding. This is due to the fact that in the model, nitrogen is diffusing out of the system and being lost in the well-stirred oxygen, thus denying the initial requirements.

68

G. GUMMINGet

al.

A second objection to the model is that the actual concentrations of nitrogen are too low, for instance with an interface position of 2.4 cm, a concentration would be expected in the lungs of about 600/, whereas a value of only 30% is computed. The reason for this lies in the inadequacy of a cone to represent the anatomical conformation of the size of unit considered. A model which tends towards an equilibrium concentration, thus dealing with the first of the objections, is that of a closed cone. MODEL NUMBER 7

This is a cone, closed at both ends, with an interface between oxygen and nitrogen established initially at different distances down the cone. The results of the analysis of this model are shown in fig. 10. Comparison of fig. 8 and fig. 10 indicates that the gradients over the terminal 7 mm in the closed system and in the open system are identical after 1 sec. The effect of time on the gradient is however different, the closed system tending towards equilibrium, (fig. 11) whilst the open system tends towards zero.

linear

distance

(cm)

Fig. 11. Effect of time on concentration gradients in Model Number 7.

The known behaviour of the lung appears to require a model with some characteristics of both open and closed systems. For short times, as in ventilation at normal speeds, when the time of contact of an interface would be about 2-3 sec. there is little difference in the predicted gradients in the two models, and either can be used with equal validity. For long times, when the lung approaches an equilibrium concentration its behaviour approximates to a closed system whereas on anatomical grounds this is untrue. It appears that the anatomical arrangement confers on the terminal airways unit a behaviour which lies intermediate between that of Model Number 6 and Model Number 7. The absence of reflection from a closed end implies that the actual rate of equilibration will be slower than in Model 7 and the failure to lose a significant

GAS DIFFUSION IN THE AIRWAYS

69

quantity of nitrogen from Model 6 will imply a more rapid equalibration than is indicated by this model. The greatest objection to the acceptance of this intermediate model as explaining the observed behaviour of the lung is that of the equilibrium concentration of nitrogen which is too low. To overcome this objection requires an analysis of a model having the geometrical configuration of fig. 2. We have not attempted this, since it appears that the diffusion mixing in such a structure would be slower than in the cone we have analysed, and it is sufficient for our present purposes to show the magnitude of the minimum concentration gradient present. We have concluded that during quiet respiration there is good evidence for the existence of a concentration gradient, or stratified inhomogeneity. This conflicts with present beliefs, but supports the hypothesis of KROGH and LINDHARD (1917). Since our results do not support accepted beliefs, we would like to discuss the objections that can be levelled against the analogy between this simplified model and the human lung. In addition to those already mentioned the following exceptions can be taken. 1. The model takes no account of diffusion laterally into the alveoli leading from the ducts. 2. It assumes that the interface occupies the same position in each terminal airways unit following an inspirate. 3. It assumes that anatomical distances remain unchanged during respiration. 4. It takes no account of the fact that diffusion occurs during inspiration as well as during expiration, so that there is a moving boundary condition. 5. It assumes that the volume of gas passing down airways of similar size is identical at all inspired volumes. The complexity of the physiological system makes it unlikely that quantitative statement of the effects of these variables can be made. However, somequalitative prediction can be attempted to assess the effects of each variable on the concentration gradient. The eflects of lateral diffusion

If oxygen molecules diffuse laterally, then the number remaining for forward diffusions will be diminished. Consequently fewer oxygen molecules arrive at the remote end of the system in a given time than if no lateral diffusion has occurred. The effect of this would be to increase the predicted concentration gradient. The eflect of interface position

The anatomical arrangement of the airways implies that terminal airways units which are near to the trachea receive a greater proportion of the inspirate than those which are more distant. The reason for this is, that if we assume effective compliance to be uniformly distributed so that all units inspire the same volume of gas, units with a short transit time inspire less dead gas than those with longer transit times (SHEPHARD, 1956). Conse-

70

G. CUMMING et al.

quently the interface between dead gas and inspired gas will be established at different positions within the terminal unit, making diffusion distances and the resulting gradients different. The larger the inspirate the shorter the diffusion distance and the less the effect of dead space on its distribution. This effect is shown graphically in fig. 12. The representation of interface distribution a single graph is fraught with difficulty and involves simplifications and assumptions of dubious validity. Nevertheless fig. 12 shows in a qualitative way the order of magnitude of diffusion distances and the effect of interface distribution.

I

Distribution

of interfaces

Tidol volume (ml)

Fig. 12. Distribution of gaseous interfaces. The average position of the interface is shown by the continuous line. The vertical line shows the nearest and further interface position at different tidal volumes.

The distribution of interfaces has two effects. Firstly the degree of equilibrium will differ as between the shorter and the longer diffusion distances, though this effect will be small. Secondly, when the units are emptied during expiration the different gradients will make their contribution to the expired concentration progressively, as determined by their transit time. The effect of this will be to make the sampled concentration gradient greater than in any single unit, and the effect will be noted in the early part of expiration only. The effect on the latter part of the expired curve will be small. Change

of anatomical

distances

during

respiration

A single alveolus has a diameter of about 0.4 mm, and changes in this diameter will affect the distance over which diffusion occurs. If an inspirate of one litre is made into a lung volume of three litres, the change in linear dimensions will be q$, or 10%. The alveolus thus increases its diameter by 0.04 mm, and the average diffusion distance in the lung changes from 2 to 2.2 mm, and will thus give rise to a small increase in the concentration gradient due to the increase in diffusion distance.

GAS DIFFUSIONIN THE AIRWAYS

71

The moving boundary condition The diffusion gradients shown in the figures have been computed as if the gas interfaces were established instantaneously and no further mass movement occurred. In fact the interface advances to this position and then retires, so that the mean position of the interface over the respiratory cycle is somewhere behind that assumed in the analysis. Consequently the mean diffusion path will be greater in the lung than that assumed in the models as will the concentration gradient. Uniformity

of volumejlow

The distribution of interfaces shown in fig. 12 is only applicable if the volume of gas passing down airways of similar size is identical. Such an assumption implies that the effective compliance of all regions of the lung is the same, and that during inspiration the change of compliance with volume is the same. This assumption has been shown to be incorrect by FOWLER (1964) who demonstrated the presence in the human lung of regional variations in effective compliance probably based on the difference in hydrostatic pressure between the upper and lower parts of the lung. The term effective compliance includes variations both in the elasticity of the lung, and in the applied intra-thoracic pressure. It is likely therefore that volume flow down airways of the same size will differ on a regional basis. One effect of this will be to widen the distribution of interfaces and to increase the diffusion gradient in each case, so that the computed gradient represents a minimum value. Differences in effective compliance also implies that the flow rate from different regions during expiration varies with time, and that those variations are not synchronous, so that the volume flow from different regions will vary during an expirate, increasing the slope of the alveolar plateau. Discussion It appears that two principal mechanisms are responsible for the slope of the alveolar plateau in the normal human subject, that of diffusion in the gas phase producing a stratified inhomogeneity, and regional inhomogeneity produced by mechanical factors in lung tissue, but not involving airways resistance to any marked degree. We believe that the time course of gaseous diffusion is slower than is generally accepted, and that diffusion therefore plays a more important role in the production of the alveolar plateau than is at present believed. It seems reasonable to ask what is the relative importance of stratified and regional inhomogeneity. The answer is not clear. We have not found it possible to advance a model which permits quantitative prediction of concentration gradients within the lung to see to what extent this explains the observed slope. The only evidence at present available concerns the degree to which the slope of the alveolar plateau may be abolished by breath holding (KJELLMER et al., 1959). This suggests that a slope of 1.5% may be reduced to 0.5% in 20 sec. One possible conclusion from this data is that stratified inhomogeneity represents the greater part of the slope of the alveolar plateau.

72

G. GUMMINGet a/.

If it is accepted that diffusion plays an important part in the slope of the single breath nitrogen test for inequality of ventilation, there is an implication that the single breath test and the nitrogen clearance curve measure substantially different phenomena. Provided that the respiratory rate is uniform the nitrogen clearance curve will not depend on gaseous diffusion for its shape, but only on regional distribution of inspired gas. Thus it may be that the single breath test measures mainly stratified inhomogeneity, whilst the nitrogen clearance curve measures mainly regional inhomogeneity. Such an explanation might reconcile some of the observed incompatibilities between the two tests of inequality of ventilation. Appendix THE MATHEMATICAL SOLUTION OF THE DIFFUSION EQUATION Model Number 1. The infinite cylinder. The process of diffusion is described t3C

-_= at

D

by Fick’s equation:

_a2c ax2

where C is the concentration of gas at any point x on the axis of the cylinder, and D is the diffusion coefficient between the two gases. The solution of this equation with C = C,, x < 0; c = 0, x > 0 at t = 0, has been discussed by one of us (CRANK, 1956) and may be written C(x,t,

$

=

I

cc

eMYZdy.

xl2VCDt)

A standard mathematical function, probability integral, or error function,

of which extensive tables are available usually written as erf z, where

is the

s z

erf 2 = (2/7t+)

eeY2dy.

0

The solution

to the diffusion

C(w) =

equation

may therefore

be written

in the form

iC,[l- e+&j]] .

The arithmetical results from this model are shown in fig. 3. The numerical value of diffusion coefficient used in all the calculations is that between oxygen and nitrogen at body temperature viz. 0.25 cm’. set-‘. Model Number 2. The closed, finite cylinder 1=0.7 cm,

h=0.2

cm,

(CRANK, 1956).

GAS DIFFUSION

73

IN THE AIRWAYS

The equation used by Rauwerda is similar and is as follows: n7rx . exp( -Dn2 n’t) cos 1 I

C(x,t) = Co

The results from the analysis of this model are shown in fig. 4. Model Number 3. Cylinder with one closed end.

C(x, t) = t-G

h-x h+x erf 21/(Dt) + erf 21/(Dt) .

Results are shown in fig. 5. Model Number 4. Conical segment of a hollow sphere, closed at both ends.

r,=0.13

rr =0.13, C(r, t) = j

r:-rz

to

0.83,

r3 =0.83,

R, [sin A,,(r - rl ) +r,l,

- 3 Ig

cos &(r-r,)]

exp(-DLzt)

r3 -rl where

Rndl+ r3;L,2)[(1+r,r,1,2)

sin I,@, -r,)-1,(r2

-rl)

cos I,@, -rJ]

1,2(r3-r,)[(r~~,Z+l)(r~12,2+1)+(r1r3~~-1)] and 1, are the positive roots of (1 +

r,r312) sin I(r, - rr) = I(r, - rr) cos L(r, - rr) .

The same solution was used by Rauwerda but presented in a different algebraic form. Results are similar to those of fig. 4. Model Number 5. Conical segment of a hollow sphere, open at one end, r=rr,

rr =0.13, C(r, t) = (2jr) f

r,=0.13

A,, exp( -D&)

to

0.83,

r3 = 0.83,

sin I,(r-r,)

n=l

where A = (1+r,Z1,2)[r2~ncos~,(r2-rI)-sinI,(r2-rI)] ” ~,2[(r3-r1)(1+r:L~)-r3] and 1, are the positive roots of sin A(r, - rl) = r3 1 cos L(r, - rr) . The nitrogen concentration

in this model tends rapidly to zero.

Model Number 6. Conical segment of a hollow cone, open at one end.

r-r =0.5 cm,

r,=2.1

to

2.5cm,

Equation as in Model 5. Results in figs. 8 and 9.

r3 = 2.6 cm.

14

G. GUMMING et al.

Model Number 7. Conical segment of a hollow cone, closed both ends.

rl =0.5 cm,

r2 =2.1

to

2.5 cm,

r3 = 2.6 cm.

Equation as in Model Number 4. Results in figs. 10 and 12. References CRANK, J. (1957). The mathematics of diffusion. Clarendon Press, Oxford, p. 9, p. 62. FOWLER,K. T. (1964). Relative compliances of well and poorly ventilated spaces in the normal human lung. J. Appl. Physiol. 19: 937-945. FOWLER,W. S. (1949). Lung Function studies. III. Uneven pulmonary ventilation in normal subjects and in patients with pulmonary disease. J. Appl. Physiol. 2: 283-299. HAYEK, H. van (1960). The Human Lung. Hafner, New York, p. 82. KJELLMER,I,, L. SANDQVIST,and E. BER~LUND(1959). “Alveolar plateau” of the single breath nitrogen elimination curve in normal subjects. J. Appl. Physiol. 14: 105-108. KROGH, A. and J. LINDHARD(1917). The volume of the dead space in breathing and the mixing of gases in the lungs of man. J. Physiol. 51: 59-90. MILLER, W. S. (1937). The Lung. Thomas, Springfield, Illinois, p. 39. MILLS, R. J. and P. C. HARRIS(1965). Factors influencing the concentration of expired nitrogen after a breath of oxygen. J. Appl. Physiol. 20: 103-109. Errs, A. B., C. B. MCKERROW, R. A. BARTLETT,J. MEAD, M. B. MCILROY, N. J. SELVERSTONE and E. P. RADFORD(1956). Mechanical factors in distribution of pulmonary ventilation. J. Appl. Physiol. 8 : 427-443.

RAUWERDA,P. E. (1946). Unequal ventilation of different parts of the lung and determination of cardiac output. Groningen University, Groningen. REID, L. (1958). The secondary lobule in the adult lung with special reference to its appearance in bronchograms. Thorux 13: 110-115. ROHRER,F. (1915). Der Stromungswiderstand in den menschlichen Atemwegen und der Einfluss der unregelmlssigen Verzweigung des Bronchialsystems auf den Atmungsverlauf in verschiedenen Lunge&e&ken. Pfliigers Arch. ges. Physiol. 162: 225-259. Roos, A., H. DAHL~TROMand J. P. MURPHY(1954). Distribution of inspired air in the lungs. J. Appf. Physiol. I: 645-659.

Ross, B. B. (1957). Influence of bronchial tree structure on ventilation in the dogs lung as inferred from measurements of a plastic cast. J. Appl. Physiol. 10: 1-14. SHEPHERD,R. J. (1956). Assessment of ventilatory efficiency by the single breath technique. J. Physiol. 134: 630-649. WEIBEL,E. R. (1963). Morphometry of the human lung. Springer-Verlag, Berlin, p. 136.