RC.v/CUl/o,r P/IKvio/oq~~ ( I Y78) 32. 27 4Y

@ El.\e\icr North-Holland Biomedical Press

ANALYSIS OF GAS EXCHANGE BETWEEN AIR CAPILLARIES AND BLOOD CAPILLARIES IN AVIAN LUNGS

PETER SCHEID

Abstract. A number of models is analyzed to study gas exchange between blood capillaries and air

capillaries in the avian parabronchial wall when diffusion is the only transport mechanism in the air

capillaries. The existing anatomical arrangement of blood capillaries that traverse the periparabronchial

tissue from peripherally located arterioles to draining venules at the luminal surFace appears to provide

a particularly high gas exchange efficiency.

Application of the theory to measurements in the hen using histological estimates suggests that aub-

stantial concentration gradients exist inside the air capillary gas whose magnitude vary along the para-

bronchu>. Thus at the gas inflow end of the parabronchus the partial pressure drop within the air

capillaries could amount. for both 0, and CO,. to ahout 10 IS torr at rest and to 30 40 torr during

exercise. Due to the peculiar arrangement of capillary blood tlow to the air capillaries the effects of these

gradients on gas exchange are very slight during reht. Durmg exercise. however. the diffusional resistance

Inside the air capillaries may become limiting for the over-all gas exchange. and other mechanisms may

be needed to secure respiratory gas transfer.

Avian respiration

Birds

Diffusion

Models for gas exchange

Stratification

Gas exchange in the avian parabronchial lung takes place in the periparabronchial

tissue where blood capillaries meet a meshwork of similarly fine air capillaries.

Whereas gas in the parabronchial lumen is renewed mainly by bulk flow, transfer

of O2 from this site to the gas-blood interface of the air capillaries, and vice uersa

for CO,, is accomplished mainly by diffusion in the periparabronchial airways,

that comprise the atria, the infundibula and the air capillaries. Zeuthen (1942)

and Hazelhoff (1943) have assessed this diffusional resistance to constitute a minor

28 P. SC‘HEID

impairment to the overall parabronchial gas exchange. However, the model under-

lying their calculations does not take due account of the anatomical arrangement

of blood and air capillaries (Duncker, 1971, 1972, 1974a.b; Abdalla and King,

1975). Piiper and Scheid (1973) stated, on the basis of this recent anatomical evi-

dence, that the arrangement of blood flow to the air capillaries showed functional

resemblence with a countercurrent system.

It is the purpose of this paper to develop the functional properties of gas exchange

in the air capillaries taking into account recent morphological data.

Notation

S_ybols

M, gas exchange rate (mmol mini ‘)

P, partial pressure (torr)

dp, relative partial pressure difference ( - )

D, diffusing capacity (mmol min ’ . torr - ’ )

D, diffusion coefficient (cm’. min- ‘)

Q, lung perfusion (ml. min- ‘)

N, number of parabronchi (-)

L, limitation factor (-) [cfl eq. (9)]

X, relative distance in air capillary from atria1 floor (-)

p, capacitance coefficient (mm01 . L- ’ . torr~ ‘)

1, length (cm)

r, radius of parabronchus (cm)

F, total cross-sectional area of air capillaries (cm2)

Subscripts

P, parabronchus

L, parabronchial lumen

A, air capillary

T, end of air capillary

M, blood-gas separating tissue membrane

b, blood

c, capillary

c’, end-capillary

a, arterial

V, mixed venous

GAS EXCHANGE BETWEEN AIR AND BLOOD CAPILLARIES 31

Model II with opposite flow direction to that of Model I resembles the cocurrent

system in respect of partial pressure equilibration, the partial pressure profiles being

distinctly different from those in Model I. In Model III end-capillary blood, of

partial pressure PC’, results as a mixture of capillary blood leaving the contact zone

at various lengths along the air capillary. This model thus resembles the crosscurrent

arrangement used by Scheid and Piiper (I 970) for analysis of exchange between

the bulk of parabronchial gas and the capillary blood.

Blood in the first half of the blood capillary in Model IV equilibrates with air

capillary gas in a way similar to that of Model I except for the higher average gas-

to-blood gradient of Model IV which results from the smaller diffusing capacity,

DM, allotted to this part of the blood capillary. On the course back to the periphery

of the parabronchial wall, the direction of gas exchange may be reversed (as is the

case in the example of fig. 1 A), when O2 taken up by blood from the air capillary

diffuses back into it. This situation occurs when blood capillary partial pressure

surmounts that in air capillary gas. Thus this model creates a shunt between the

effluent and influent blood which results in a decreased gas exchange efficiency of

this model as compared to Models I to III (see below).

In Model V the partial pressure gradient inside the air capillary is constant.

Exchange between gas at the air capillary end and blood is functionally equal to

that in alveoli of mammalian lungs.

IZf~i’citwq* of’ gas cl.\-thanye in thr ourious tmdt~ls

It is convenient to define the relative partial pressure differences both in air capillary

and blood

PL-PPT APA = ~

PL-Ppv

PC’ - PV Apb = -~

PL-Ppv

(2)

(3)

where P-r and PC’ denote the partial pressure at the terminal of the air capillary and

in end-capillary blood. These expressions are readily obtained from the partial

pressure profiles as shown in the Appendix, where explicit expressions are given

for Apb in all models of fig. 1.

APA, the total (relative) partial pressure drop along the air capillary, is marked

by open arrow in fig. 1. Of even higher significance is Apb from which the efficacy

of the gas exchange system may be defined. The amount of gas transferred in any

of the models of fig. 1 is proportional to the end-capillary-to-mixed venous partial

pressure difference, PC’-PV, and thus to Apb. At given values of DM/DA, DM//?bi)

and (PL-PPV), the value of Apb can thus be utilized to compare the gas exchange

efficiency of the different models of fig. 1. In this figure, Apb is indicated by black

arrow besides each partial pressure profile. It is evident that for fig. 1 the sequence

of efficiency of the models is I = II > III > IV > V.

Figure 2 serves to compare the efficiency of the models over a wide range ot

DM/pbQ (abscissa) and for some values of DM;DA (curve parameter). The following

features are apparent.

(i) For very low values of DM,‘DA (c’.cJ. no significant diffusion resistance in air

capillaries) the efficiency is identical in all systems. In fact. for z.ero diffusion resis-

tance (DA ---f -x ) all models become functionally identical with the Ventilated Pool

Model of Piiper and Scheid (1075) used for analysis of alveolar gas exchange in

homogeneous lungs. The curve for DM/DA = 0.01 virtually coincides with that for

DM~DA = 0.

(ii) For any value of DMM/DA and DMi[jbQ the efficiency in Models I and II is

identical [c;f. eqs. (Al4) and (A20) in Appendix]. This result W;IS unexpected to me

since the efficiency of the cocurrent system (with convective flow on both sides of

the membrane) is allegedly poor, whereas the countercurrent system has the highest

efficiency known (Piiper and Scheid. 197.5). However, this fundamental difference

in gas exchange efficiency obtains only for models with convective flow of both

media. To demarcate from these models those in which gas transport is by diffusion

in one medium, the terms countercurrent-lib-c and cocurrent-lilir were adopted.

Despite identical efficiency, the partial pressure profiles in both Models I and II

generally differ considerably (cj: fig. I). In particular. the total partial pressure

drop inside the air capillary (-IPA, open arrow in fig. I) is much higher in Model I

than in II. Thus this partial pressure drop, which has been used by Zeuthen (1941)

0.8

@b

0.6

0 0.1 0.2 0.5 1 2 5 10 20 50 1

DM /Pbb

Fig. 2. Relative partial pressure in blood. a measure to compare the gas transfer efficacy of the various

models. against the conductance ratio. DM,/%~. Curves calculated for three values of DM;DA. Encircled

numbers refer to the models displayed in fig. I.

GAS EXCHANGE BETWEEN AIR AND BI.oOD (‘APILLARIES 29

Theory

I. MODELS FOR GAS EXCHANGE BETWEEN AIR CAPILLARY AND BLOOD CAPILLARY

General proprties

It is assumed that respiratory gases are transported by d$firsion within the air spaces

of the periparabronchial tissue, consisting of the infundibula and air capillaries;

by d#uion in the tissue barrier separating gas from blood: by cornwtion within

the blood.

The following quantities are required for a quantitative treatment of gas exchange

(typical units in brackets; c:$ Piiper of a/., 1971):

Conduuctanws [mmol . min ’ . torr- ‘1 (c;f: Piiper and Scheid, 1975, 1977):

(I) Diffusive conductance in the gas phase of the air capillary:

DA = D. pg. F/lA (1)

where D represents the diffusion coefficient of the gas under study [cm’.min- ‘1; fig, is the capacitance coefficient of the gas phase, equalling 0.05 10 mmol . L I . torr- 1

for all ideal gases at 41 C (typical body temperature of the duck >nd hen); F and

1~ are cross-sectional area and length. respectively, of the air capillary;

(2) Conductance of diffusive transfer in the liquid phase through the gas-to-blood

tissue membrane, DM ;

(3) Convective conductance of blood, fiba, where [jb [mmol L ’ torr- ‘1 is

the capacitance coefficient in blood of the gas under study (equal to the slope of

the blood dissociation curve), and Q [ml. mini ‘1 is the blood flow to the blood

capillary.

Partial pressures (torr): (1) In the parabronchial lumen (and the atrium) at the

segment from which the air capillary departs, PL; (2) in mixed venous blood, Pv.

The following idealizing assumptions are made and will be discussed later:

(1) The parabronchial lung is functionally homogeneous, meaning that the ratio

DM//3bQ is constant along a given air capillary and that both ratios DM/DA and

Dlcl//IbQ are constant within a given parabronchus as well as in different para-

bronchi.

(2) The cross-sectional area of the air capillaries does not vary in radial direction.

(3) The air capillary may be represented by a straight, non-branching tube

running through the parabronchial wall from the atria1 floor to the interpara-

bronchial septum.

(4) There does not exist a radial partial pressure gradient within the parabronchial

lumen and the adjoining atria.

(5) The system is in steady state, implying constancy in time of PL, PV, and Q.

(6) bb is independent of partial pressure (linear blood dissociation curves).

30

M0Ll~Jl.s NIlU(I’XY/

The special arrangements

the five models that have

I’. S(‘Hl:II)

between blood capillaries and air capillaries underlying

been analyzed are shown in fig. I. In Models I and II

(fig. 1 A and B) the blood capillary contacts the air capillary along its entire course,

blood flow direction being opposite in the two. In Model IV (fig. ID) the blood

capillary follows the air capillary to its origin and then returns, diffusional exchange

taking place in both legs of this hair-pin. In Model III (fig. IC) blood contacts the

capillary only at a short segment along the air capillary length whereas in Model V

(fig. I E) gas has to overcome the entire diffusional resistance offered by air capillary

gas since the blood capillary contacts the air capillary only at the terminal end.

Gas exchange in these models is quantitatively analyzed in the Appendix. In fig. I

the partial pressure profiles in air capillary and blood capillary of each model are

depicted for a selected set of conductance ratios, DM~DA and DM/BbQ. The profiles

for Model I show much resemblance to those of a countercurrent system (Piiper

and Scheid. 1973). particularly the overlap of gas and blood partial pressure ranges.

I have, therefore, adopted the term countercurrent-like arrangement for this system.

A MODEL I B MODEL II

Countercurrent - hke Cocurrent - like IC’ MODEL Ill

Crosscurrent -like

PL

I

I

PT

4-l

PC. P, 1

I3 PC,

P - P, pn

pb d

PL

PL- Pv PT

0 PV 0 X 1

D MODEL IV

Recurrent Loop

PL

E MODEL V

Terminal Contact

PL

PL

PT

on/P& = 5

DM/DA = 5

Fig. I Models analyzed for diffusion limitation in parabronchial air capillaries.

GAS EXCHANGE BETWEEN AIR AND BLOOD CAPILLARIES 33

and Hazelhoff (1943) to assess the limitation to gas exchange imposed by air capillary

diffusion (see below). appears to be not well suited for estimation of the efficiency.

The slope of the curves PA(X) at the beginning of the air capillary, (dPA/dx)x = (),

which is proportional to the gas exchanged, as is the area between PA(X) and Pb(X),

is identical for both. As is evident from the difference between PA(X) and Pb(X)

along the air capillary, gas is exchanged in Model II preferentially in the initial

parts whereas it is more evenly distributed over the air capillary length in Model I

(c:/: tig. I).

(iii) In general. the efficiency of Models I (and II), III, IV and V are different.

This difference is most pronounced in an intermediate range of DM/pbi) Model III

is always less efficient than I and II, and Model V even less than III. Model IV

behaves functionally like Models I and II for small values of DM/fibQ and becomes

identical with Model V for high values of DM//IbQ. Its efficiency can thus be higher

or lower than that of Model III, depending on DM/flbQ.

In the absence of diffusion resistance in t. 4 air capillary (DA + n) all five models

of fig. I behave functionally like the alveolar model. The impairment imposed on

gas exchange by diffusion in the gas phase may thus fictitiously be added to the

membrane resistance, and an apparent membrane diffusing capacity. DM*, in an

analog alveolar model of otherwise identical parameters may be defined which

lacks air capillary diffusion resistance (DA + z) but behaves functionally identically

with the real model (DA finite).

The relative partial pressure difference in blood, dpb. in the idealized, alveolar

model is ((:/I Piiper, 1962; Piiper and Scheid, 1975)

dpb = 1 - exp ( - DM*//Ib@ (4)

and thus

DM* = -/?bG.ln {l-dpb] (5)

Since both the real model for the air capillary and the analog (= alveolar) model

are assumed to have identical gas exchange performance, dpb of both must be

identical and DM* for any model can be calculated by introducing the respective

dpb (Appendix) into eq. (5). DM* thus calculated will in general fall short of the

true DM value and the discrepancy between both reflects the diffusion resistance

inside the air capillaries.

II. GAS EXCHANGE IN THE COMPOSITE PARABRONCHUS

So far gas exchange was considered in the model of an isolated air capillary with

its blood supply. The parabronchus consists of a number of these elements, arranged

in series with respect to parabronchial gas flow. This results in parabronchial gas

34 P. SCHEID

partial pressures, PL, that vary along the parabronchial axis and are not constant

as was assumed for the model analysis of fig. 1.

With varying PL along the parabronchus, PC’ varies, too. But for the homogeneous

parabronchus, with constant DM/pbQ throughout (cf: Assumptions), Apb does not

vary. Therefore, the apparent diffusing capacity, DM*, which is apt to describe the

effects of diffusion impairment inside the isolated unit of air capillary, may be used

to assess this effect on gas exchange in the parabronchus as a whole.

Lkvitiny efjkct qf air capillary d@sion

To evaluate the impairment imposed on gas exchange by diffusion in the air capil-

laries, the rate of parabronchial gas exchange, &I. may be compared with that, h;I T,,

in a parabronchus of identical parameters except for infinite DA (no diffusion

resistance in air capillaries).

To calculate h;r, the apparent diffusing capacity, DM*, may be used. In the cross-

current system applicable to parabronchial gas exchange (Scheid and Piiper, 1970;

Piiper and Scheid, 1975)

PbQ - -( 1 - exp ( - DM*/fibQ)) 1

PgV J (6)

where PI constitutes inspired partial pressure and ‘? parabronchial ventilation..

Using eq. (4), this can be re-written as

ti = (PI-PV)./?gV. I-exp { [- $Apb]j

In the absence of diffusion resistance in air capillaries, DM* = DM, and thus (for

unchanged PV)

- $(l -exp(-DM/fibQ)) II Combination of eqs. (7) and (8) allows to calculate a limitation factor

LA = IQ-&I 7-=_l-

l-exp{- s.Apb)

M, g( I- exp ( -

(9)

DM/@bQ))

which may be used for a quantitative evaluation of the impairment on parabronchial

gas transfer imposed by diffusion resistance in the air capillaries.

Figure 3 is a plot of LA against DM//?bQ for some selected values of DM/DA and

for figi’/fibQ = 1. The sequence of LA for the different models is very similar to the

sequence of the corresponding values of Apb (d: fig. 2): the model with lower LA

at a given set of parameters has a higher value Apb as would be expected from eq.

(9). In general, LA increases with increasing DM/DA (increasing diffusion resistance

in air capillaries) and, for a given DM/DA, decreases with increasing DM/flbQ.

GAS EXCHANGE BETWEEN AIR AND BLOOD (‘APILLARIES 35

LA

80

(%I

60

0.1 0.2 0.5 1 2 5 10 20

44lPtl~

50 loo

Fig. 3. Limitation imposed on gas transfer by diffusion resistance m air capillaries against the conduc-

tance ratio DM:/ibQ. Curves plotted for three values of DM:DA. Symbols as m fig. 7.

In the definition of LA [eq. (9)] the effect on gas exchange rate by air capillary diffusion is investigated while PV is kept constant. It may, however, be more relevant to keep the total gas exchange rate constant (which at steady state equals the metabolic rate) and express the effects ofdiffusion limitation in the air capillary as changes in PS. Thus removal of this diffusion resistance would result in a mixed venous partial pressure, PV, From eqs. (6) to (9) an expression for this partial pressure can be found

(Pv, -Pv) = (PI-Pii). LA (10)

Application to experimental data

Dif;fsing capacity DA

The theory will now be applied to experimental data, both morphological and physiological, to assess the limiting role in gas exchange played by diffusion in the gas phase of the air capillaries. Table 1 contains ranges of morphometrical estimates obtained for a variety of birds by H.-R. Duncker (persona1 communication). These values are utilized to calculate air capillary diffusing capacity, DA, from eq. (1). The total cross-sectional area, F, was assumed to equal one half of the area of the luminal wall of parabronchi (cf: Zeuthen, 1942; Hazelhoff, 1913). Thus

F = 0.5.2rcr.lp.N

where r constitutes the radius of the parabronchial lumen; lp, the length of an

36 P. SCHEID

TABLE I

Range ofmorphometrical data assessed from a variety of birds* to estimate air capillary diffusmg capacity _~ .- ~~~~ ~~~~-

Minimal Maximal

estimate estimate

Number of parabronchi in both lungs N 700 300

Length of parabronchus (cm) IP 2.5 3.0

Radius of parabronchial lumen (cm) r 0.0’ 0. I Length of air capillary (cm) IA 0.02 0.05

Diffusion coeflictents

(cm’ .min ‘)

n 0) 13.20**

n < 0: Y.hO**

Diffusing capactty of air capillary (DA),,, 0.42 9.52

(mmol.min I ‘torr~ r) (DA),,,. 0.31 6.92

* Personal communication by H.-R. Duncker.

** Binary diffusion coefficients in NL. at 41 C. calculated usmg eq. ( I I I I ) of Reid and Sherwood (1966).

individual parabronchus; and N, the number of parabronchi in the paleopulmo of

both lungs.

Membrane dt#Using capacity, DM, and pwfi.uiw conductance, DbQ

The values for flbQ and (DM*),> during rest listed in table 2 are taken from Scheid

and Piiper (1970; <f:f: Piiper and Scheid. 1975) and were obtained for the unanesthetized

chicken breathing spontaneously a hypoxicchypercapnic mixture. Since (DM*),~

was calculated assuming a model in which air capillary diffusing capacity, DA, was

TABLE 2

Effects of diffusion in air captllaries on parabronchial gas exchange during rest and exercise

(mmol.min~ r)

;o ( mmol min ’ torr r ) DM* (mmol mini ’ torr ’ ) DM (mmol~min~‘.torr~ r)

DA (mmol.min~‘.torr r)

/?g\i (mmol~min~‘.torr~‘)

LA (“,,) Pt - PV (torr)

Pa - PV (torr)

PV, - PV (torr)

PV, -PV

Pa-P?

Rest Exercise

20 IX

0.34 I.3

02 CO? 02 c-01

I 0 - 0.x

0.034 0.13 0.063 _

0.067 1.13

0.42 0.31

0.039

I.3 0.6

55 -26

32 - 8

0.7 - 0.2

_ _ 0.67 Il.3

0.42 0.31

0.78

13.5 12.x

90 -30

60 -15

I’ - 3.8

0.02 0.02 0.20 0.25

For details see text.

GAS EXCHANGE BETWEEN AIR AND BLOOD CAPILLARIES 37

infinite, this value may underestimate the true membrane diffusing capacity, DM, and constitutes the apparent value of eq. (5). The true value for (DM)~, was obtained from (DM*)~~ by calculating dpb from eq. (4) and DM from eqs. (A12) and (AlA). As can be seen from table 2, DM deviates from DM* by only 6”;).

To calculate (DM)~~~, a ratio (DM)~~~/(DM)~~ = 16.8 was assumed which equals the ratio of Krogh’s diffusion constants for both gases in water at 37 C (Kawashiro

et ul., 1975).

For exercise a 20-fold increase in O2 uptake, tioJ, was assumed. Ventilation was

taken to increase in proportion with h;loz and both Q and DM to increase by a factor of 10. Although no experimental values are available for this increasein the con- ductances, it appears to be necessary for them to be elevated in order that the lung

can cope with the increased demand.

The partial pressure differences, (PI - PV) and (Pa - PV), during rest are those

obtained by Scheid and Piiper (1970) whereas the values during exercise were assumed.

Partial prrssure projk The partial pressure profiles for O2 and CO, in air and blood capillaries have been

calculated for Model I using the data of table 2 and eqs. (Al 0) and (Al 1) of the

Appendix (fig. 4).

Under the most favorable conditions. when DA = (DA)_ (dashed lines in fig. 4).

the partial pressure profiles resemble those expected for absent diffusion resistance.

A. REST IO ---------__T_----

8. EXERCISE ----------

B

Fig. 4. Partial pressure profiles in blood and air capillaries for Model I, plotted against the length of

the exchange area. Continuous lines, DA = (DA),,“; dashed lines. DA = (DA),,,~~. A. rest; B, exercise.

Upper half, 0, : lower half, CO,. Arrows indicate range of partial pressures in gas (open arrow) and blood

(closed arrow). The stippled arrow reflects the difference in the range of blood partial pressure between

(DA)mn and (DA),,,~~.

38 P. SC‘HEID

Only for CO, during exercise is there a signi~cant drop in air capillary partial pressure under this condition.

If, on the other hand, DA is at its minimal value, there does exist a marked partial pressure drop inside the air capillary gas for both CO, and 0, during rest and exercise as indicated by the open arrows in fig. 4. During either rest or exercise, the drop for CO? is more pronounced than that for 0,; conversely, the drop is enlarged during exercise.

This gradient inside the air capillary is not necessarily related to the i~npairment of gas exchange due to diffusion resistance in the air capillary (see above). A better yet qualitative indication for this impairment can be obtained from the relative partial pressure difference in blood, dpb, which is indicated for the case of DA = (DA)“,~~ by the black arrow in fig. 4 in the convention adopted from fig. 1, and by the stippled arrow for DA = (DA)_. As DA is increased from (DA),,,~,, to

(DGmax 3 dpb increases as the difference between the two arrows. Since (DA),,,~~ is functionally close to infinity, this increase is an estimate of the increased efficiency of gas exchange when the high diffusion resistance in the air capillaries is removed. Thus a particularly prominent increase in efficiency may be obtained during exercise, and, conversely, a prominent impairment to gas exchange by air capillary diffusion is expected under these conditions.

&jkct qf’uir capillary- d&$4sion resistutm ott hlooci gu.ws

This impairment can more quantitatively be expressed by the limitation factor, LA. LA can be calculated for Model I using the values for Bbif, DM, DA and {$g%’ listed in table 2 together with eqs. (Al4) and (9). For both rest and exercise the minimal values of DA from table 1 were used to assess maximum effects of air capil- lary diffusion.

During rest, LA is about 1 “o for both 0, and C02, meaning that gas exchange rates could be increased by about 1 “<, if DA were increased to infinity. During exercise,

LA is increased for both gases to about l3”,,. The effects of air capillary diffusion on blood gases was estimated from LA using

eq. (10). For both 0, and COz only small effects of air capillary diffusion on PV are calculated at rest which increase, however, substantially during exercise. During both rest and exercise the absolute change in PV is about 3 times larger for 0, than for COz. However, when this value is related to the total arterial-to-mixed venous partial pressure difference, very similar values result for both gases.

Calculation of LA has similarly been performed using recent experimental values offibQ, pgv, and DM* in the resting duck (R. E. Burger, M. Meyer, P. Scheid, un- published). The resulting limitation factors for 0, and CO2 are about twice the values assessed for the hen (table 2) and are thus very similar.

GAS EX<‘HANGE BETWEEN AIR AND BLOOD CAPILLARIES 39

Discussion

I. MODEL ANALYSIS

For the analysis it was assumed that diffusion is the only mechanism for gas transport

in the air capillaries. In fact, even in those avian species in which there exist

anastomoses between air capillaries of neighboring parabronchi (Duncker, 1971;

King, 1966) convective movement through them would require a pressure difference

between adjacent parabronchi (Hazelhoff, 1943) for which there is no basis.

Quantitative analysis in all models is based on a number of simplifying assump-

tions which in part were made because of lack of precise knowledge and in part to

avoid confusing mathematical complications. Of particular interest are the following

assumptions.

(1) The total cross-sectional area of the air capillaries was assumed to be constant

in radial direction (assumption 2) although an increase might be expected in the

direction towards the periphery of the periparabronchial tissue. Thus the cross-

sectional area of the air capillaries at their origin from the parabronchial lumen

constitutes a lower limit; and so does DA that is calculated on this basis. This entrance

region to the air capillaries is conceivably the most significant part since all gas has

to pass this cross-section.

(2) Branching of the air capillaries has been neglected (assumption 3). Such

branching would have no effect on diffusion unless the total cross-sectional area

of air capillaries changed in the direction from the parabronchial lumen towards

its periphery (see above); and unless the curvature of the air capillary path con-

stituted a significant increase in the air capillary length. For this latter reason the

thickness of the periparabronchial tissue may be a minimum estimate for IA, and

DA calculated therefrom may thus overestimate the true diffusing capacity of the

air capillaries.

(3) It was assumed that no concentration gradients exist in radial direction

either in the air capillaries or in the parabronchial lumen, including the atria (as-

sumption 4). For the air capillary this seems to be justified because of its small diam-

eter. This diameter is, on the other hand, large as compared with the free path of

the gas molecules at atmospheric pressure so that influences by the proximity of

solid walls on gas diffusion are not expected (Weis-Fogh, 1964). But also for the

parabronchial lumen the assumption of absent radial concentration gradients may

be met since convective movement of parabronchial ventilation aids diffusion in

homogenizing gas concentrations. More serious may be the assumption that the

gas concentration within the parabronchial lumen equals that of the radially adjoin-

ing atria. However, since the dimensions of the atria are of the same order or below

the radius of the parabronchial lumen, secondary motions induced by ventilatory

flow are expected to provide enough convection inside the atria to counteract partial

pressure gradients in them (R. C. Schroter, personal communication).

40 P. SC‘HEID

(4) The dimensions of the parabronchus and its air capillaries were assumed to

be fixed. However, smooth muscle that occurs abundantly in the parabronchial wall

could adjust the parabronchial lumen. Thus during flight, when the demands on

gas exchange are maximal. the smooth muscle could relax to increase the diameter

of the parabronchial lumen which would not only reduce the air flow resistance

through the parabronchi (Molony rt ul., 1976) but would effectively reduce the

length of the air capillaries, and thus increase DA. Furthermore, rhythmic contrac-

tions of these muscles could result in convective movement of gas in the air capillaries

and thus effectively reduce the gas exchange resistance as was proposed by Akester

(1971).

Atu~ton~ical urrat~getmwt of’uir and blood cupilluries md the ai~~quatrJuttc,tiorlul tnodel

Gas exchange between air capillaries and blood capillaries depends on the relative

anatomical arrangement of the two capillary systems and it is this arrangement

which finally decides which of the models of fig. 1 is the most appropriate for this

problem. The arrangement of blood capillaries and air capillaries has recently been

studied experimentally in much detail (Duncker, 1971~ 1974a; Abdalla and King,

1975). These authors agree that the air capillaries anastomose profusely to form a

meshwork (<t:f: King, 1966). Different views, however, are held among investigators

on the structural arrangement of blood capillaries in the parabronchial wall. Duncker

(1971, figs. 37 and 38; 1972) assumes blood capillaries to emerge from arterioles at

the interparabronchial septum and to drain into venules close to the atria1 floor.

On their course these blood capillaries anastomose freely to form a meshwork

resembling that of air capillaries. Later, Duncker (1974a, fig. 2; 1974b, fig. 6) reported

that the blood capillaries run more or less straight, without anastomosing, connecting

arterioles in the periphery and venules at the parabronchial lumen. Abdalla and

King (1975, fig. 24) agree with Duncker (1971) on rich anastomoses between blood

capillaries. In their model, however, most intraparabronchiai arterioles and venules

penetrate to some extent into the parabronchial wall. Capillary blood flow in this

model may thus be oblique to the radial direction.

It is important to realize which of the anatomical parameters described are likely

to affect the gas exchange function of the system investigated in this paper and thus

impact on the choice of the model. In Model I the blood capillary has gas exchange

contact with the air capillary along its entire length. This would strictly apply to the

situation of straight, non-anastomosing blood capillaries, meeting straight, non-

anastomosing air capillaries, both arranged radially. At least the latter form a

meshwork; however, it is conceivable that at a given distance from the entrance

into the air capillary, the partial pressure between adjacent segments of the air

capillaries does not differ markedly. Thus for functional considerations the boundary

between adjacent air capillaries is of little significance and so is the meshwork

arrangement. Similariy. the arrangements of Duncker (1971) of a network of blood

capillaries and his later view (Duncker, 1974a, b) of straight, non-anastomosing

blood capillaries does not imply functional differences, since in both cases blood

GAS EXCHANGE BETWEEN AIR AND BLOOD CAPILLARIES 41

contacts air capillaries over their entire course. This and the blood flow direction

from peripheral to central would make Model I the primary choice for use with

Duncker’s morphological picture.

The arrangement proposed by Abdalla and King (1975) is slightly different. Here

blood emerges at various depths in the parabronchial wall and is also drained within

this mantle. Thus a blood capillary has gas exchange contact with only a fraction of

the air capillary length. If, in fact, both the arterioles and venules penetrated the

entire parabronchial wall giving off air capillaries like the steps of a ladder, Model 111

(fig. 1C) would be most appropriate. It is evident that the arrangement proposed

by Abdalla and King (1975) is intermediate between Models I and III. However,

for most values of DM/DA and DM/fibQ that might occur in avian lungs (cj: table 2)

the functional differences between these models are not very prominent (c:f: figs. 2

and 3).

A different view on the arrangement of blood and air capillaries was expressed

by Akester (1971 ; c/l his fig. 26). According to this author arterioles and venules ac-

company each other and two such systems occur, one inside the interparabronchial

septa, the other close to the atria1 floor. Blood capillaries, emerging from arterioles

and draining into venules at both sites, form a meshwork to contact the network

of air capillaries. It is conceivable that blood of a given arteriole drains into the

concomitant venule. Thus blood in the contact zone would be expected to flow in

loops, one loop from the atria1 floor, the other from the interparabronchial septum,

and both digging about half way into the parabronchial wall. It is evident that

Model IV (fig. 1) is but a poor analog of this arrangement; however, it may be

used to assess the effect of this recurrent loop arrangement on gas exchange. In

fact, this arrangement appears to be inferior to any other in which blood is provided

on one side and drained on the other of the parabronchial wall.

Zeuthen (1942) and Hazelhoff (1943) have used the Terminal Contact Model V

to evaluate the effects of air capillary diffusion on gas exchange. Their model has

not been based on microscopic observations. Their analysis will be discussed below.

Sign[/icance of anatomical arrangement

The theoretical analysis suggests that any arrangement in which blood capillaries

traverse the entire parabronchial mantle to contact air capillaries along their whole

course (Models I and II) would be most beneficial to overcome possible impairment

by air capillary diffusion. It appears to be essential to feed the capillaries on one

side and drain them at the opposite side; it is, however, not essential for gas exchange

whether arterioles run in the periphery and venules close to the lumen, or vice versa

(equal efficiency for Models I and II). Opposite blood flow with arterioles at the

parabronchial lumen would require small arteries to traverse the gas exchange tissue

whose thicker walls would occupy more space than the thin-walled venules. It may.

therefore, be concluded that the arrangement met in the parabronchial wall appears

to be most efficient for gas exchange in the presence of a diffusional resistance in the

air capillaries.

42 P. SC‘HEID

Applicahilit?! to alwolar gas r.\-change duriny strut~fkution

It is tempting to utilize the Terminal Contact Model V for analysis of alveolar gas

exchange in the presence of stratified inhomogeneities, that is, when diffusion

equilibration within the terminal lung units is incomplete. In fact, a similar model

has recently been used by Adaro and Piiper (1976) and by Sikand rt al. (1976) to

assess effects of stratification on alveolar 0, uptake in dog and man. Model V

predicts significant limitations by gas diffusion only if the air capillary diffusing

capacity, DA, is of the same order of magnitude as or below that of the tissue mem-

brane, DM (c$ lig. 2). Since Krogh’s diffusion constant for 0, in the gas phase is

about 10” to lo6 times higher than in tissue, DA would be expected to surpass DM

unless the gas has to travel in the gas phase over distances comprising alveolar

ducts and sacs. Since blood contacts alveoli all along this path, the apprapriate

model to treat stratification in an otherwise homogeneous alveolar lung would be

the crosscurrent-like Model III rather than the Terminal Contact Model V.

West rt a/. (1969) in their fig. 12 have suggested that countercurrent flow past

the alveolar ducts, from the terminal end of the bronchial tree towards the conduct-

ing airways, minimizes the impairment of gas exchange in the presence of strati-

fication. This proposed flow pattern would correspond to our countercurrent-like

arrangement of Model I, and, in fact, its efficiency exceeds that of Model III (cf:

fig. 2) which would be appropriate for the homogeneous alveolar lung. However,

it should be noted that a cocurrent-like arrangement would have the same effect

on gas exchange in the presence of stratification as the countercurrent-like flow

pattern. Furthermore, as was expressed above, analysis of stratified inhomogeneities

in terms of concentration gradients does not necessarily permit conclusions about

their impact on gas exchange.

West rt al. (1969) have also proposed that a pattern of blood flow in the terminal

lung units that corresponds to the crosscurrent-like Model III, but with more flow

past proximal alveolar units, would oppose stratification. I have calculated the

effect of blood flow distribution in Model III with the conductance values under-

lying fig. 1. When blood flow decreased linearily from twice average at the beginning

to zero at the end of the air capillary, DM being evenly distributed along the air

capillary, the efficiency increased as compared with the homogeneous Model III,

dpb increasing from 0.76 (fig. IC) to 0.81. The total relative drop in air capillary

partial pressure, APA, decreased from 0.34 (fig. 1C) to 0.26 in this example. I have

not systematically investigated the effect of blood flow distribution in Model III

on gas exchange. It is, however, worth noting from this example that an unequal

distribution of blood flow to membrane diffusing capacity may offer an advantage

over the homogeneous case, whereas in most other systems inhomogeneity is detri-

mental (Piiper, 1969; Scheid et al., 1973).

GAS EXCHANGE BETWEEN AIR AND BLOOD C‘APILLARIES 43

II. LIMITING ROLE OF AIRWAY DIFFUSION IN AVIAN LUNGS

The morphological data used to calculate the possible range of DA are very rough

estimates. Moreover, the parameters that are most unfavorable for airway diffusion

have been combined in the estimate of (DA),,,~“, and riw w-m, and thus the range

of possible DA values may have been overestimated. The physiological parameters,

particularly those assessed for exercise conditions, have been estimated with some

extrapolation from equivalent data in mammals. In particular, the increase in DM

on exercise. which is observed in man and can be attributed in part to effects of

functional inhomogeneities (Piiper, 1969) had to be adopted for calculation of

values of table 2 in order that the expected mixed venous Par values be reasonable.

As a result, this study gives only a range for the possible effects of airway diffusion

on parabronchial gas exchange.

Effects on air capillary diffusion might be expected from the reduced gas density

during high altitude flight. Since the binary diffusion coefficients are inversely

proportional to the total gas pressure, PB, DA for any gas will increase approximately

inversely with PB. Thus when all other conductances are unchanged, LA is expected

to decrease and the total gas exchange conductance to increase. This increase will

in general be not sufficient to compensate the fall in inspiratory Paz which is about

proportional to PB.

Zeuthen (1942) was the first to assess limitations on CO? exchange offered by

the air capillaries. His analysis differs from that of this study in respect of two

main points. Firstly, the model used by him is the Terminal Contact Model V

(fig. I E) and thus his results are expected to overestimate the impairment. Secondly.

he has assumed equal distribution of CO, exchange over the parabronchial length.

Pco, profiles along the parabronchus (c$ Meyer et al., 1976) show that under normal

resting conditions most CO, exchange occurs in the initial portions of the para-

bronchus. Thus less than average air capillaries have to cope with CO, exchange

and the limitation is expected to be higher than assessed by Zeuthen (1942).

Zeuthen (1942) has predicted for the hen at rest a partial pressure drop from

parabronchial air to that in the air capillaries of 0.5 torr, and has interpreted this

finding as a negligible effect of diffusion in air capillaries. From the histological

values used by him, a value of DA = 2.12 mmol . min- ’ . torr~ ’ for both lungs can

be calculated which is only one third of our maximum estimate and is thus close to

the high end of the DA range assessed here. In fact, our estimates of table 1 would

predict, on the basis of his model, a Pco2 drop of 3.25 torr for (DA)~~” and 0.15 torr

for (DA),,,~~. For a flying bird with a 25-fold increase in gas exchange, Zeuthen

(1942) has predicted a partial pressure drop for CO, of about 12 torr, whereas our

values of DA in his model would give 3.8 to 81 torr, the latter figure exceeding by

more than a factor of 2 the expected total difference between PI and PV.

I, therefore, tend to agree, from the results of his study, with Zeuthen (1942)

that “ventilation of the air capillaries will possibly be beneficial in the flying bird,

although it is hardly a necessity” although I would like to criticize the deduction

44 I’. S(‘HEID

of this conclusion for three reasons. First, his model is not appropriate. Second,

the value of DA chosen by him may well be an underestimation of the reality. In

fact, values close to (DA),~,, of table I would have led him to a different conclusion.

Third, for theoretical reasons (see above). the partial pressure drop in the air

capillaries is a very poor estimate of the impairment of gas exchange derived from

airway diffusion.

Hazelhoff (1943) has used a similar model as Zeuthen ( 1942). His estimate of DA

for O2 is about 2.5 times that used by Zeuthen for CO,. Correspondingly, his con-

clusion is that diffusion in the air capillaries is sufficient “even during very swift

flight”. The discussion to the data of Zeuthen (1942) applies to his values and

arguments as well.

Appendix

I. C’ountrrcunent-lilac arrunyrnwnt (Model I; fiy. I A)

The amount of gas crossing the membrane element, DM . dx, of the air capillary at

the distance x from the parabronchial lumen leads to a change in the diffusional

partial pressure gradient of the air capillary, dPA/dx. and at the same time to an

increase in blood partial pressure. dPb/dx, of the gas under study. Thus

-D.pg. F. {(ddjx- (ddp;\)x+dj = DM.;; [PA(x)--b(x)) (Al)

pb@Pb(x)-Pb(x+dx)) = D,.d,” IPA(x)-P~(~)~ (A2)

Using eq. (1) and introducing

a = DM/DA ; b = DM/,8bo (A3)

and the relative distance in the air capillary from its origin at the atrial floor,

X = X/IA (A4)

these equations can be re-written

d2PA ~ = a(PA-Phi dX2

dPb

dX - -b{PA-Pb;

(A%

(A61

The following boundary conditions apply

x = 0: PA = PL

x= 1: dPA/dX = 0

(A7)

(W

Equations (A5) gas and blood

GAS EXCHANGE BETWEEN AIR AND BLOOD CAPILLARIES 45

Pb = PV (A91

and (A6) can be integrated to yield the partial pressure profiles in

PA(X)-Pq k, exp (k,(l -X)) -k, exp ik,(l -X)j ___-_ ~ PL-Ppv k, exp(k+kz exp(k,)

Pb(X) - PV _~ ~ = b exp :k,(l -X)1-exp lk,tl -Xt’,

PL-Ppii ‘ ______ ~- .__~_.

k, exptk,)-k,exp(k,)

with

(AI2)

The relative partial pressure drop in air capillary, APA, and blood capillary, dpb, can be obtained from eqs. (AlO) and (Al 1)

PL-PT k~~exp(k~)-I~-k*~exp(k,)-1~ dpA =z pL_-pi; =: _~__~--- _ ._.__.__ k,exp(k,)-k,exp(k,)

(A131

exp(k,)-exp(k,) dpb =; ‘&I:.= b--p _ _-__ __ k, exp(k,)-kzexp(k,)

(A141

2. Cocurrent-like arrungrnwnt (Model II: fig. I B)

This arrangement resembles that of the countercurrent-like arrangement (tig. 5A), the difference being the reversed blood flow past the air capillary. Equations (A5) and (A6) apply to this system if b is replaced by -b. The boundary conditions in this model are

x = 0: PA = PL (A151

Pb = PC (A16)

x= 1: dPA/dX = 0 (A171

Integration of eqs. (A5) and (A6), with b -+ - b, and with eqs. (A 15) to (Al 7) yields

the partial pressure profiles both in air capillary gas, PA(X), and in blood, Pb(X):

PA(X)-PY exp(k,){b+k, exp(k,X)) -exp(k,)fb+k,exp(k,X)f = _______- ______ k, exp(kJ-k?exp(k,)

(A181 PL-Pf

Pqx) - PV b expel-exp(k~X)~ -expel-exp(k,X)~

PL-Ppv = * ___~__..~.._ -. .___

k, vfk,)-k, ev(k,) (A191

with k, and k, according to eq. (Al2). End-capiIlary partial pressure, PC’, can be calculated from eq. (A19) at X = 1:

PC’ - Pii exptk,)-exp(k,) Apb-:p=: .p PL-PQ b k, exp(k,)_k,exp(k,)

6420)

46 I’. X‘HtIl~

3. (‘r’o,r.Sc’lrl’l’c,tlt-lib-(’ LItWtl~J~‘IIlCtl t (Model I I1 ; fig. I C)

It is assumed that each element of air capillary length receives an equal fraction of

blood and that blood from all these elements is mixed to yield the equivalent of

end-capillary blood partial pressure, PC’.

Gas exchange between any element along the air capillary length with the respec-

tive blood flow may be treated as gas exchange in a mammalian alveolus (Piiper,

1962). The partial pressures in the blood leaving this element at the distance X,

Pb( X). and in air capillary gas. PA(X), are thus given by

Pb(X)- PV

PA(x)- PC = I -exp(- DMjpbi))

Furthermore, the amount of gas taken up by

partial pressure gradient in gas which can be

to eq. (A5)

d’PA a dX’ =b (Pb(X)- Pv]

(A?l)

this element leads to a change in the

expressed by an equation equivalent

(AZ)

This equation is to be integrated with the following boundary condition

x = 0: PA = PI. (~23)

x = I: dPA,,dX = 0 (~24)

The following solutions give the partial pressure profiles in gas and in end-capillary

blood along the air capillary

PA(x)- Pij exp ic(l -X)j +exp (-c(1 -X)]

pL_p; = exp (c) + exp ( -c) (A25)

Pb(X)- PV b.c’ exp Ic(l-X):+exp (-ccl-X))

PI. - PV a exp (c) + exp ( -cl

where

c= + J a (I-exp(-b)) b

(A26)

(A27)

The partial pressure in mixed blood leaving the air capillary, PC’, may be obtained

by integrating eq. (A26) over the air capillary length

PC’ - PV dpb = ~~

b.c exp(c)-exp(-c) pL_ppv = -y’ exp (c) + exp ( -c)

(A28)

4. Rrctirrcwt 100~ ~rrcm+‘n~mt (Model IV; fig. I D)

Blood has contact with the air capillary both in the course from the periphery to

the origin of the air capillary and back to the periphery. Membrane diffusing capa-

city, DM, is assumed to be evenly allotted to the air capillary length and, at a given

GAS EXCHANGE BETWEEN AIR AND BLOOD CAPILLARIES 47

length, to both portions of the blood capillary. The differential equations are thus

d2PA a

dX2 - = 2 ((PA-Pb)+(PA-Pb))

dPb b -=_ dX 2

(PA- Pb;

dPb = b ‘PA--b) dX 2’

(A291

(A301

(A31)

a and b are given by eq. (A3). Pb refers to partial pressure of blood in the first half, Pb in the second half of the blood capillary.

The following boundary conditions apply

x = 0: Pb = Pb (A321

PA = PL (A331

x= 1: Pb = PV (A34)

dPA/dX = 0 (A351

The general solutions of eqs. (A29) to (A35) are

Pb(X) - PV ~~_ = l+A+B.X+C.emkX+DekX PL-Ppv

Pb(X)- PV

PL-Ppv ={l+A-tBj+BX-C{g+F+I)e-*‘-D{z-

(A37j

where

k = +m+a and

(A391

A = [%rG +k)e*+ F(22 -k)epk+2k]/E NW

B = k2(&-epk)/E (A41 1

(A421 C= -(i+:--k)(ek+z)/E

D=(i+F+k)(e-“+z)/E (A431

48 P. X‘HEID

e-k- {(k’+b)+ (; +k)($ + 3ek-4k (A44)

The partial pressure in end-capillary blood, PC’. can be derived from eq. (A38)

at x = 1.

Apb = “‘-” ~ = l+[{[;+k)($+~) PL-Ppv

+(b-k’i}e-*- j@ -k)(4b: + g) +(b-kz)jek+4k]/L (A45)

5. Tcmlinal contact arrangenwnt (Model V; fig. IE)

It is assumed in this model, as in the other models. that no radial diffusion gradient

exists inside the air capillary. Gas exchange between capillary blood and gas at

the terminal end of the capillary, of partial pressure PT, can thus be calculated by

the alveolar mode1 (Piiper, 1962):

(PC’- PV) = (PT- PV)( 1 -exp (- DM//jbo)I

Furthermore mass conservation leads to

fib@ Pc’ - PV) = DA( PL - PT)

Combination of both equations yields

(A46)

(A47)

PL-PPT APA =

1 -exp(-b)

PL-Ppv = a b+a~l~-;xp ( _ b);

PC’ - PV Apb = b

1 -exp(-b)

PL-Ppv = b+ajl-exp(-b))

(A481

(A49)

The partial pressure profile in the gas phase in this mode1 is obviously linear.

Acknowledgement

I wish to express thanks to Dr. Johannes Piiper who has contributed invaluable

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