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Diffusion limitation in comparative models of gas exchange
Respiration Physiology, 91 (1993) 17-29 © 1993 Elsevier Science Publishers B.V. All rights reserved. 0034-5687/93/$05.00
17
RESP 01983
Diffusion limitation in comparative models of gas exchange * Frank L. Powell and Steven C. Hempleman Physiology Division, Department of Medicine, University of California, San Diego, La Jolla, CA, USA (Accepted 12 August 1992) Abstract. Piiper and Scheid (Resp. Physiol. 23: 209-221, 1975) compared different models of external gas exchange with performance indices defined as functions of ventilatory/perfusiveand diffusive/perfusive conductance ratios (Gvent/Gperf and Gdiff/Gperf, where Gdiff is diffusingcapacity). We expanded their analysis to include: (1)Apo, the average partial pressure gradient driving diffusion across the exchange barrier, normalized to the maximum gradient available (Pi - Pv), and (2) Jdiff, the sensitivityof total conductance to changes in Gdiff, where total conductance is the ratio of gas flux to the maximum gradient [GTOT= IVI/(Pi- Pv)]. Although the counter-current model is most efficient,it is more sensitivethan crosscurrent or ventilated pool models to changes in Gdiff. For given Gvent, Gperf and Pi - Pv, maximumGTOT may not be achieved in the counter-current model until Gdiff is over ten-fold greater than that necessary for maximum GTOT in the other models. Experimental data also shows greater Jdiff and diffusion limitation in fish than in birds or mammals. We conclude that counter-current Oz exchangecannot approach ideal levels as closely as the ventilated pool or cross-current models in nature.
Alveolargas exchange,models, diffusionlimitation;Cutaneous gas exchange;Diffusion,limitation,vertebrate external gas exchange; Models, external gas exchange, vertebrates; Parabronchial gas exchange
Piiper a n d Scheid (1972, 1975) developed a theoretical framework for analyzing the efficiency of gas exchange in different types of respiratory organs. Their analysis showed that for equivalent conditions (i.e. equal ventilation, blood flow, diffusing capacity, inspired and venous tensions a n d blood-gas dissociation curves), gas exchange performance in the models follows the order: counter-current > cross-current > ventilated pool. Their analysis also quantified limitations imposed on gas exchange by the three transport processes, ventilation, perfusion a n d diffusion. However, limitations have not been systematically investigated for the different finite ventilation models. We undertook such an investigation to better u n d e r s t a n d how changes in diffusing capacity affect gas exchange performance in the different models. In this paper we expand the Piiper-Scheid analysis by further exploring diffusion limitation in models of gill, p u l m o n a r y and cutaneous gas exchange. New contributions
Correspondence to: F.L. Powell, Department of Medicine - 0623A, 9500 Gilman Drive, University of California, San Diego, La Jolla, CA 92093-0623, USA. * Dedicated to Johannes Piiper on the occasion of his 65th birthday.
18 result from deriving, for all four models: (1)the average relative partial pressure difference that drives diffusion across the exchange barrier; and (2) the sensitivity of gas flux to changes in diffusing capacity. Some of these results have already been reported (Powell and Hempleman, 1988).
Models
We analyze four models of vertebrate external gas exchange previously presented by Piiper and Scheid (1975): (i) counter-current model of fish gills; (ii) cross-current model of avian parabronchial lungs; (iii) ventilated pool model of mammalian alveolar lungs, unicameral and multicameral reptilian and amphibian lungs; (iv) infinite pool model of cutaneous exchange. Following Piiper and Scheid (1975) we assume: (i) steady state, or constancy of all partial pressures and conductances in time; (ii) linear blood capacitance coefficients, i.e. linear dissociation curves and constant /~ (see Eq. (3) below); (iii) spatial homogeneity, i.e. no mismatch of conductances. Deviations from these assumptions have been discussed previously (Hammond and Hempleman, 1987; Piiper and Scheid, 1975, 1977) and will be considered in the Discussion as necessary to understand model analyses of experimental data. The models predict gas exchange performance as functions of conductances for ventilation with the respiratory medium, i.e. air or water, (Gvent), respiratory organ blood flow (Gperf) and diffusive transport between the respiratory medium and blood (Gdif0, equal to diffusing capacity. Conductances are defined as the factor in the Fick equation necessary to produce a given gas flux (l~l) for a given partial pressure difference (cf Piiper et al., 1971): l~I = G.Ap
(1)
Gdiff is the diffusing capacity. Gvent and Gperf are the products of convective flow rates (Vm and "~o, respectively) and their respective capacity coefficients (/~; cfi Piiper et al., 1971): G = 9.~
(2)
fl is the slope of the curve relating concentration to partial pressure in air, water or blood: /~ = AC/Ap
(3)
Gas exchange performance can be quantified from the models in terms of relative partial pressure differences of the inspired and expired medium (Pi and Pc) and venous and arterial blood (Pv and Pa):
19 Apvcm = (Pi - Pe)/(Pi - Pv)
(4)
Apper f = ( P a - P v ) / ( P i
(5)
- Pv)
Apt r = (Pe - Pa)/(Pi - Pv)
(6)
We include another Ap in our analysis, the average blood-gas (or water) partial pressure difference across the diffusion barrier (Table 1): ApD = (Pro - Pb)/(Pi - Pv)
(7)
Another useful index used by Piiper and Scbeid (1975) to c o m p a r e models is the total conductance: GTOT = l(4/(Pi - PV)
(8)
GTOT is the product of a conductance and its associated Ap, i.e. GTOT = Gdiff" Apo = Gvent.Apvcn t = Gperf. Appcr f. To quantify the effects of changes in Gdiff, Piiper and Scheid (1975) defined diffusion limitation for the various models: Ldiff = (10max - lVlact)/l~Imax
(9)
where/Vlmax is maximal l~l with infinite Gdiff and lVlact is an actual value with a finite Gdiff.
TABLE 1 F o r m u l a e for new values in a model analysis of c o m p a r a t i v e gas exchange ApD
Jdiff
Counter-current
X(1-exp{-Z})" Y (X - exp { - Z})
exp{-Z}(X 2-2X+ 1~ (X - exp { - Z}) ~
Cross-current
X (1 - exp { - Z ' } ) / Y
exp { - Z ' - Y}
Ventilated pool
X (1 - exp { - Y}) Y (X + 1 - exp { - Y})
X 2 exp { - Y} (X + 1 - exp { - y})2
Infinite pool
1 - exp { - Y} y
exp { - Y)
where:
X Y Z Z'
= Gvent/Gperf = Gdiff/Gperf =Y(I-1/X) = (1 - exp { - Y})/X
APD is the n o r m a l i z e d average gradient driving diffusion across gas exchange barriers, (Pm - Pb)/(Pi - Pv), and Jdiff is the sensitivity of total c o n d u c t a n c e to changes in diffusive conductance, bGTOT/lSGdiff. a F o r X # 1; Y / ( Y + y 2 ) for X = 1. b ForX#l;Y(l+y)2forX=l.
20 To quantify the sensitivity of the different models to changes in Gdiff, we calculated the diffusion sensitivities for each (Table 1): Jdiff =
bGToT/bGdiff
(10)
This approach has been used previously by Piiper and Scheid (1981) to analyze diffusion versus perfusion limitations of alveolar gas exchange. However, they assumed fixed alveolar gas tensions in their analysis so their Jdiff was actually that of an infinite pool model. The Jdiff we derive for the other three models allows finite values of Gvent so the ratios Gvent/Gperf and Gvent/Gdiff must be specified to compare effects of changing Gdiff/Gperf.
Model results Figure 1 shows ApD for the four models as a function of Gdiff/Gperf with Gvent/ Gperf = 1 (except for the infinite pool model which has an infinite Gvent/Gperf by definition). As expected, the average partial pressure difference across the exchange barrier increases for all models as diffusive conductance decreases relative to perfusive conductance. Considering models with finite ventilations, the relative positions are counter-current > cross-current > ventilated pool. Increasing the Gvent/Gperf ratio (not shown) moves all of the curves nearer to the infinite pool curve, which defines the limit for all models as the ratio approaches infinity. The relative position of the curves remains the same for all Gvent/Gperf but differences between them become smaller as the ratio increases. The maximum differences occur at Gvent/Gperf= 1 and this will be the case considered hereafter.
1.o
0.8'
"'-'~'~ . . .
• ", . ~
..... Infinite Pool - - Counter--current - -Cross-current - -Ventileted Pool
'...
x. ~ .. "\'N'"'" \
0.6.
...
\
13.. <3 0.4.
~ "..' \ "'\ ~ ". "\." ~\ \ "'".. \\
0.2,
0.0
0.01
'.. ' ", . \ " " .
O.10
1.00
Gdiff/Gperf
( ~ "
10.00
"~'~ ~
""~
100.00
Fig. l. Average partial pressure difference between respiratory medium and blood for the four models (Table 1) as a function of Gdiff/Gperf. Gvent/Gperf= 1 for finite ventilation models and Gperf= 1 for the infinite pool model which has infinite Gvent by definition. Note that cross-current Apt) is similar to countercurrent ApD at low Gdiff/Gperf but more like ventilated pool Apo at high Gdiff/Gperf.
21 1.0 Infinite Pool -- Counter-current - - Cross-current - . . - V e n t i l o t e d Pool
~
. . . . .
0.8.
/ ~
. . . . . . .
/ /
0.4.
..
'
/
-.~ 0 . 6 0 0
I"t-
/
,-"~'-" ....................
0.2.
0 . 0 ............. 0.01
i 10.00
t
0.10
1.00
100.00
Odiff Fig. 2. Total conductance (GTOT) as a function of Gdiff for the four models with Gvent/Gperf = 1 for finite ventilation models and Gperf= 1 for the infinite pool model. Note the positive slope of the countercurrent curve at high Gdiffvalues where the other models have plateaued; GTOT can be increased more than the other models by increasing Gdiff in this range.
This ordering of ApD could be predicted from Piiper and Scheid's previous results (1975) with the performance index GTOT, which is proportional to Apo (cf. Eq. (8)). Figure 2 shows GTOT for the four models under the same conditions as in Fig. 1 and although the relative positions have been shown previously (Fig. 4 in Piiper and Scheid, 1975), our examination of the shapes of the curves continuously over a wider range provides new insights. The counter-current model is fundamentally different from the other three models: maximum GTOT is not achieved until very high Gdiff values. Figure 3 shows the consequences of the counter-current model's more gradual approach to maximum GTOT with increasing Gdiff in terms of Ldiff. Piiper and Scheid
1.00.8-
' ~ . . ~ •.~ ' ~ . " . \ "'~x \ . . X ~x,
..... -- ....
Infinite Pool Counter-current Cross-current Ventilated Pool
X ~
_J
°" 0.0
o.ol
':\'.:i-. I
o.lo
I
~'~, ..... ~ . . . . ~
1.oo Gdiff/Gperf
lo.oo
loo.oo
Fig. 3. Diffusion limitation (Ldif0 as a function of Gdiff/Gperf for the four models with Gvent/Gperf= 1 for finite ventilation models and Gperf = 1 for the infinite pool model. Note that counter-current Ldiff exceeds infinite pool Ldiff over most of the range in contrast to other finite ventilation models. These differences are maximum at G v e n t / G p e r f = 1 and the infinite pool curve becomes the limit for all other models as Gvent/Gperf increases.
22 (1975, 1977) have calculated Ldiff for other models, but it has only been graphed over a wide range of conductances for the infinite pool model. The counter-current model has a greater diffusion limitation than all of the models over the normal to high Gdiff/ Gperf range. Also, counter-current Ldiff exceeds the infinite pool Ldiff for most Gdiff/ Gperf, in contrast to cross-current and ventilated pool Ldiff. Ldiff for the infinite pool model is the limit for all models as Gvent/Gperf increases; with increasing Gvent/ Gperf, cross-current and ventilated pool Ldiff increase while counter-current Ldiff decreases. To make quantitative comparisons in the sensitivity of gas flux to diffusive conductance in the different models, we plot Jdiff for all of the models versus Gdiff/Gperf in Fig. 4. Jdiff is greatest for all models at the lowest Gdiff/Gperf where small increases in Gdiff result in the nearly equal changes of GTOT. This was not immediately obvious from examining the GTOT curves in Fig. 2 curves because the logarithmic abscissa attenuates their slope. The counter-current model shows greater Jdiff than the other ventilated models over most of the low to normal Gdiff/Gperf (Fig. 4). This means that increases in Gdiff/Gperf have a bigger effect on counter-current models in this range, consistent with the continuing increase in GTOT on Fig. 2 vs plateaus for the other models. The underlying cause of the counter-current model's extreme sensitivity to diffusion limitation can be understood by returning to the concept of Apo. Figure 5 plots normalized GTOT versus ApD for all of the models. GTOT is normalized to the value calculated for a given model with infinite Gdiff and Gvent/Gperf= 1, or Gperf= 1 for infinite pool model (i.e. plateau values in Fig. 2). ApD values were calculated for increasing Gdiff from 0.1 to 100 with Gperf and Gvent/Gperf= 1 (i.e. same conditions as Fig. 1). The relative partial pressure gradient necessary to drive a given normalized gas flux is lowest in the counter-current model, consistent with the concept of great-
1.0.
0.8 •
v;:~'"~',:~ ..... . •~ ' . ~ ' . "'.~'X ". \., . ~
,
,.\
O.6. --~
'\. ~ " 'x ~ . "~,.\ "..
0.4,. 0.2. 0.0 0.01
..... Inf n te Pool - - Counter-current - -Cross-current .... Ventilated Pool
"x.\ ".
0.10
\
~
~ 1.00
Gdiff/Gperf
10.00
100.00
Fig. 4. Sensitivity of total conductance to diffusive conductance (Jdiff) for the four models as a function of Gdiff/Gperf with Gvent/Gperf = 1 for finite ventilation models and Gperf = 1 for the infinite pool model. Counter-current Jdiff is the greatest of the models with finite ventilation at Gdiff/Gperf> 1 and can actually exeed infinite pool Jdiff at high Gdiff/Gperf.
23
..... ~ .-..~•... ~..~.",.. 'XN. ". "'\ \",.. 0.8 "\\ "-.. "'\ \ '... 0.6 "\ \ ".. "\ \'.,. "'\ \'. 0.4 "'\.\'... 1.0'
O
c~ no (D N
(3
E k_ O c
0.2
0.0 0.0
N~\y.
--Counter-current - -Cross-current .... Ventiloted Pool
\". ~.
I
I
I
I
0.2
0.4
0.6
0.8
1.0
L~PD Fig. 5. GTOT, normalized to the maximum for a given model, as a function of ApD for each of the four models. Conditions as in Figs. 1 and 2, from which the data is derived, with Gvent/Gperf= 1 for models with finite ventilation and Gperf= 1 for the infinite pool model. The counter-current model can maintain higher relative gas fluxes (normalized GTOT) with lower diffusive driving pressures (ApD) than the other models. Maximum GTOT is achieved by ApD = 0.1 in all models except the counter-current which shows continued GTOT increases at low ApD.
est efficiency in that model. More interesting, however, is the linearity of the countercurrent plot compared to the plateaus of normalized GTOT at low ApD for the other models. Only the geometrical arrangement of exchange elements in the counter-current model is capable of utilizing low partial pressure gradients for effective gas transport. It is in this low ApD range that the counter-current model shows larger absolute GTOT (el high Gdiff range on Fig. 2). The ability of the counter-current model to continue increasing diffusive transfer at low average driving pressures explains its high efficiency.
Model analysis of experimental data We applied the analysis to experimental oxygen exchange data collected by others from yellowfin tuna (Thunnus albacares; Bushnell and Brill, 1992), white Pekin ducks (Anas platyrhynchos; Kiley etal., 1985), and humans (Hammond and Hempleman, 1987). Men were sitting or pedaling on a bicycle ergometer. Ducks were standing or running on a treadmill. Tuna were ram ventilated in flow-through tanks at 25 °C with their spinal nerves blocked to prevent locomotion. Exercising tuna data were extrapolated from measurements on alert resting and swimming dogfish (Piiper and Scheid, 1977). These data sets were selected for completeness of the necessary variables collected under hypoxic conditions. In hypoxia, the oxyhemoglobin equilibrium curve is approximately linear (cfi assumption (ii) of constant fl), Gdiff is measured most accurately and inhomogeneity effects are minimized (cfi Piiper and Scheid, 1977). Diffusive conductances are corrected for any inhomogeneities when possible and ventilations are cor-
24 rected for d e a d s p a c e . W e are n o t a w a r e o f any experimental d a t a meeting these requirements for the infinite pool m o d e l (i.e. c u t a n e o u s gas exchange). T h e s e m e a s u r e d a n d calculated variables are s u m m a r i z e d in Table 2 a n d results are shown in Fig. 6. Resting d a t a are shown on the left panel o f Fig. 6. G p e r f is n o r m a l i z e d to a c o m m o n value o f one for all cases so they can be c o m p a r e d . G d i f f / G p e r f is quite similar in all three cases a n d GTOT varies as expected from differences in the ideal efficiencies o f the models. H o w e v e r , the degree to which the different m o d e l s actually a p p r o a c h their ideal p e r f o r m a n c e follows the inverse order. GTOT in the t u n a is only 62 ~'o o f the counter-current ideal, d u c k GTOT is 81 ~ o f the c r o s s - c u r r e n t ideal and h u m a n GTOT is 92~o o f the ventilated p o o l ideal (i.e. L d i f f = 0.38, 0.19 and 0.08, respectively). The intrinsically m o r e efficient m o d e l s o p e r a t e farther from their o p t i m a and they are on steeper parts o f the curves in Fig. 6. Consequently, Jdiff increases with the intrinsic efficiency of the three models. Exercise decreases the species differences o b s e r v e d at rest. A t exercise vs rest, b o t h d u c k s and h u m a n s show larger diffusion limitation ( L d i f f = 0.29 and 0.17, respectively), while the t u n a shows a smaller diffusion limitation ( L d i f f = 0.27). G d i f f increases with exercise in all cases but so does Gperf. Thus, Gdiff/Gperf, which is the important value, does not increase significantly. The result is that all three species are on steeper parts o f the GTOT curves a n d show similar sensitivities to changes in G d i f f (i.e., Jdiff) at these m o d e s t levels o f exercise.
TABLE 2 Experimental data for model analysis of oxygen exchange in hypoxic resting and exercising man (ventilated pool; Hammond and Hempleman, 1987), duck (cross-current; Kiley et al., 1985) and tuna in 25 °C water (counter-current, Bushnell and Brill, 1992)
Mass (kg) Pio2 (Torr) IVlo2(/~mol.min- 1.kg- 1) (L.min- 1.kg- 1), I) (L.min- 1.kg- l) Gdiff (#moi.min- l.Torr- 1.kg- l)b Gvent/Gperf Gdiff/Gperf
0.32 0.10 0.14 27 0.21 1.1
Man
Duck
Tuna
68 78
2.4 84
1.4 90
1.2 0.62 0.25 53 0.68 1.1
0.48 0.47 0.28 38 0.68 1.1
1.7 2.1 0.60 68 1.4 1.0
0.47 6.9 0.12 18 0.99 1.6
0.69~ -1.5c 1.9c
a VA calculated from iVlco2and P a c o 2 in man; parabronchial V calculated from ~rl and measured dead space in ducks; gill ~z measured with dye dilution technique. b DLo2 corrected for heterogeneity (0 weighted) in man;. Do: corrected for heterogeneity and cardiac output in 2.4 kg ducks: Do2 (/~ mol.min- t-Torr- l.kg- 1) = Q (L/min).97 + 25 (Hempleman and PoweU, 1986; and unpublished data); Do2 calculated for counter-current model, i.e. not To2. Not measured in tuna but increased from resting values by factors measured in dogfish on transition from alert rest to swimming (Piiper et al., 1977).
25 HYPOXIC REST
1.0
1.0
.... Infinite Pool
0.8-
--tuno :..:d~g~
HYPOXlC EXERCISE
~
-""'
f..'"
/:" / /
O.B
-4--, 0.60
0.6
(.9 0.4.
0.4
/ eo,0~ ...........
0.2.
//~.o8
'
.
...... d
.
.
l
f
f
_
.
0.2 .. ,if'"
0.0 0.01
0.10
1,00
Gdiff
10.00
100.00
0.0 .... 0.01
I
0.10
1
t
1.00
10.00
100.00
Gdiff
Fig. 6. GTOT for the infinite pool model, tuna (counter-current model), duck (cross-current model) and man (ventilated pool model) as a function of Gdiff for the measured conditions given in Table 2. Gperf is normalized to a common value of one for all cases. Left panel: resting tuna show the largest diffusion limitation and greater sensitivity of IVlo2 to Gdiff while duck values are intermediate to those of tuna and man. Right panel: diffusion limitation and sensitivity of/Vlo2 to Gdiff increase in man and duck with hypoxic exercise, in contrast to tuna. In both cases, Gdiff would have to increase more in tuna than in duck or man to reach the GTOT plateau, representing more ideal performance.
Discussion
This study extends to the previous contributions of Piiper and Scheid (1975, 1977, 1981) by deriving: (1)the relative average blood-gas partial pressure difference driving diffusion, ApD, and (2)the sensitivity of gas flux to changes in diffusive conductance Jdiff, both for all four comparative models of gas exchange. ApD is useful because it predicts the gradient driving diffusive gas exchange in different vertebrate respiratory organs, given conductances and inspired and venous tensions. Hence, the analysis provides a framework to consider problems like the physiological significance of differences in physiologic and morphometric diffusing capacities that have been reported for a variety of animals. Jdiff has utility in quantifying how effective a given change in diffusing capacity is at changing gas flux in different exchange models for given inputs. As shown in Fig. 6 and discussed below, the Gdiff/Gperf range over which Jdiff is significantly > 0 could have important biological implications.
Critique. The utility of these models for studying the physiology of diffusion limitation depends on assumptions going into their derivations, which have been discussed in detail elsewhere (Piiper and Scheid, 1975, 1977, 1989). For example, the diffusion barrier is assumed to be a simple membrane obeying Fick's law. Actually, Gdiff is a lumped parameter including, among other things, the kinetics of 02 and CO2 reactions in blood and diffusive resistance in the respiratory medium. These factors may be important when comparing animals having different body temperatures and breathing media of different diffusivity and capacitance (i.e. air vs water). Our criteria for selecting experimental data to analyze (cf Results) generally satisfied the model assumptions. The human data set was the most complete and, therefore, internally consistent (Hammond and Hempleman, 1987). Gdiff values from the
26 Kiley et al. (1985) duck study were significantly lower than estimates corrected for heterogeneity so we used corrected values accounting for changes in cardiac output from another duck study (Hempleman and Powell, 1986). Gdiff values corrected for heterogeneity are not available for fish but Piiper and Scheid (1989) noted that the agreement between morphometric and physiologic Do2 in fish is consistent with negligible functional inhomogeneity. Data from paralyzed, ram ventilated tuna were selected as the only complete set in hypoxia in the literature (Bushnell and Brill, 1992). The conductances are in the range expected for slowly swimming tuna, which is probably as close to rest as are ducks standing on a treadmill or humans sitting on a bicycle ergometer. Gdiff changes with exercise were predicted from a presumably similar baseline of "alert rest" (as opposed to "quiescence") in dogfish (Piiper and Scheid, 1989). This increase probably reflects gill capillary recruitment (Farrell et al., 1980) which we assume occurs in tuna also.
Comparison with previous studies. Piiper and Scheid (1977) have studied the effects of diffusion on several partial pressure gradients ratios. Apt r is an extremely useful index of gas exchange efficiency, being a normalized form of the alveolar-arterial difference which is a standard for alveolar gas exchange. In an alveolar model, Apt r c a n be interpreted as the fraction of resistance to gas transfer between inspired air and venous blood attributed to alveolar gas-capillary blood exchange (Piiper and Scheid, 1972). However, in the more efficient cross-current and counter-current models, Aptr can actually assume negative values and it cannot be easily interpreted as a fractional resistance. A negative Apt r emphasizes that the geometrical arrangement of these more efficient models results in performance only possible in alveolar exchange by active secretion. Piiper and Scheid (1977) derived the contribution of a model's spatial arrangement to Apt r as Apt r = Apo when Gdiff is infinite. More generally, they define: Apt r =
Apo + Apdifr
(11)
Apdiff Can be greater or less than Aptr but it is always greater than zero. APdiffrepresents the relative increase from the ideal expired-arterial gradient caused by a finite Gdiff; Apo, which we calculate here, is the relative blood-gas (or water) gradient with a finite Gdiff. With knowledge of the normalization value Pi - Pv, ApD is useful for exercises like comparing the diffusing capacity requirements between different models. As discussed under Models, Jdiff, the sensitivity coefficient for changes in gas flux with changes in diffusing capacity, was originally derived for the infinite pool model by Piiper and Scheid (1981). However, they used it to analyze alveolar gas exchange assuming constant alveolar gas tensions. Jdiff for the ventilated pool model here incorporates the effect of changes in ventilation and perfusion on alveolar gas and requires specifying Gvent/Gperf to determine the sensitivity of gas flux to changes in Gdiff (cf. Table 1). Piiper and Scheid (1975) applied their analysis to resting experimental data and their general conclusions about oxygen exchange are consistent with ours. The hypoxic canine data they used is similar to the human data we selected for the ventilated pool
27 analysis. Their avian data was from hypoxic and hypercapnic chickens. Those chickens presumably had higher ventilatory drive than the hypoxic ducks but Gvent/Gperf was similar for both of our cross-current analyses. Their counter-current analysis used normoxic dogfish but it is interesting that their general conclusions are similar to ours for hypoxic tuna. In contrast to birds and mammals, fish show significant diffusion limitation in resting normoxia as well as hypoxia (Piiper and Scheid, 1989).
Physiological significance.
Applying the analysis to experimental data from hypoxic resting animals reveals clear differences in performance between the models as they occur in nature (Fig. 6). At rest, oxygen exchange in air breathing man is very close to its theoretical maximum and humans would benefit relatively little from further increases in Gdiff (i.e. Ldiff and Jdiff are small). In contrast, water breathing tuna could benefit greatly from increased Gdiff and the air breathing ducks represent an intermediate case. Exercise reduces these species differences but Gdiff would still have to increase about five-fold more in the tuna than in duck or man to maximize gas flux at the measured Gvent, Gperf, Pi and Pv (Fig. 6). Although evolution does not have a goal of optimal design, natural selection frequently results in the optimal function of a given structural design (Lindstedt and Jones, 1987). Our observations raise the questions " H o w did the tuna evolve with the intrinsically most efficient gas exchange model operating farthest from ideal conditions?" More specifically, how does the tuna's evolutionary solution compare with other possibilities like (1)more optimal performance in less efficient models or, (2) smaller but more ideally matched conductances in the counter-current model? The model analysis allows us to explore the physiological consequences of these alternatives.
1. Larger conductances in less efficient models To consider the first alternative, we calculated what GTOT would be for the three finite ventilation models given the conductances measured in the resting tuna (cf Table 2). GTOT was 0.55 and 0.44 in cross-current and ventilated pool models, respectively, compared to 0.62 in the counter-current model (i.e. the value shown in Fig. 6). Next, we considered how conductances would have to increase in the cross-current or ventilated pool models to yield a GTOT of 0.62. Increasing gill ventilation was not considered because the increased work of breathing water with high density and viscosity and low 02 capacitance would presumably be selected against in nature. Increasing Gvent would also require increasing Gperf to maintain Gvent/Gperf near the ideal value of 1. Assuming the tuna Gvent/Gperf value, increasing Gdiff is not a viable solution with the ventilated pool model, maximum GTOT with infinite Gdiffis only 0.49. However, a cross-current model could reach a GTo'r of 0.62 if Gdiff/Gperf increased three-fold. Reasons why this may not have occurred are considered later.
2. Better conductance matching in the most efficient model Figure 6 is useful for considering the alternative of more ideal conductance ratios in the counter-current model. Gvent/Gperf is already very near the ideal value of 1 so
28
again we focus on Gdiff/Gperf. We calculated that Gdiff/Gperf would have to increase almost ten-fold in the tuna to come as close to optimal performance as man does and Gdiff/Gperf is already greatest in the tuna. Recall that counter-current Ldiff and Jdiff were larger than for the other models over a significantly greater range of Gdiff/Gperf (cf. Figs. 3 and 4). For tuna to increase Gdiff ten-fold, respiratory organ capillary volume, surface/volume ratio and/or exchange barrier thickness would have to change significantly from the general vertebrate plan. The strength of capillaries and developmental program for vascular branching may preclude such evolutionary changes. Also, the gills are involved in many more physiological functions besides oxygen exchange (McDonald et al., 1991) so increasing their surface area may compromise other vital processes like CO2, water or ion homeostasis. On the other hand, increasing Gdiff/Gperf by ten-fold through decreasing Gperf would not only move Gvent/Gperf farther from the ideal value of 1, but it would require tissue oxygen delivery and exchange in fish to be much more efficient than in other vertebrates. This would presumably necessitate a different and more efficient design of systemic oxygen transport than that found in the rest of the vertebrates. Perfusion limitations are not larger in fish than in terrestrial vertebrates (Piiper and Scheid, 1975), so the required increase in perfusive oxygen delivery could not likely be achieved through separation of pulmonary and systemic circulations. In conclusion, the analysis provides quantitative arguments against models other than counter-current being a realistic evolutionary solution for oxygen exchange in fish. The counter-current model achieves the highest gas flux for given conductances, which is important for water breathers with significant stratification at their gas exchange surfaces and high costs of breathing. Furthermore, the analysis showing significant diffusion limitation in counter-current models over a wide range of Gdiff/Gperf suggesting reasons that gill-O 2 exchange may not be optimized in nature. More ideal conditions with smaller conductances may be precluded by (1) the evolutionary history of the vertebrate cardiovascular and respiratory systems, and (2)the sensitivity of the counter-current model of Gdiff changes over such a wide range (cf. Fig. 2 and Fig. 6). History imposes boundary conditions on the natural selection of structure-function relationships so Gdiff/Gperf may not be able to increase enough to reach ideal countercurrent conditions through random mutations (cf. Fig. 6). Acknowledgements. We thank Dr. A.T. Gray for discussions about the material presented. This work was supported by NIH Grant HL 17731.
References Bushnell, P.G. and R.W. Brill (1992). Oxygen transport and cardiovascular responses in Skipjack Tuna (Katsuwonus pelamis) and Yellowfin Tuna (Thunnus albacares) exposed to acute hypoxia. J. Comp. Physiol. 162: 131-143. Farrell, A.P., S. S. Sobin, D.J. Randall and S. Crosby (1980). Intralamellar blood flow patterns in fish gills. Am. J. Physiol. 239: R428-R436.
29 Hammond, M. D. and S.C. Hempleman (1987). Oxygen diffusing capacity estimates derived from measured "~/A(~distributions in man. Respir. Physiol. 69: 129-147. Hempleman, S.C. and F.L. Powell (1986). Influence of pulmonary blood flow and 02 flux on Do2 in avian lungs. Respir. Physiol. 63: 285-292. Kiley, J.P., F.M. Faraci and M.R. Fedde (1985). Gas exchange during exercise in hypoxic ducks. Respir. Physiol. 59: 105-115. Lindstedt, S.L. and J.H. Jones (1987). Symmorphosis: the concept of optimal design. In: New Directions in Ecological Physiology; edited by M.F. Feder, A. F. Bennett and W.W. Burggren. Cambridge: Cambridge University Press, pp. 289-309. McDonald, D.G., V. Cavdek and R. Ellis (1991). Gill design in freshwater fishes: interrelationships among gas exchange, ion regulation, and acid-base regulation. Physiol. Zool. 64 (1): 103-123. Piiper, J., P. Dejours, P. Haab and H. Rahn (1971). Concepts and basic quantities in gas exchange physiology. Respir. PhysioL 13: 292-304. Piiper, J. and P. Scheid (1972). Maximum gas transfer efficacy of models for fish gills, avian lungs and mammalian lungs. Respir. Physiol. 14:115-124. Piiper, J. and P. Scheid (1975). Gas transport efficacy of gills, lungs and skin: theory and experimental data. Respir. Physiol. 23: 209-221. Piiper, J. and P. Scheid (1977). Comparative physiology of respiration: Functional analysis of gas exchange organs in vertebrates. In: Respiration Physiology II, Volume 14; edited by J. G. Widdecombe. Baltimore, MD: University Park Press, pp. 219-253. Piiper, J., M. Meyer, H. Worth and H. Willmer (1977). Respiration and circulation during swimming activity in the dogfish Scyliorhinus stellaris. Respir. Physiol. 30: 221-239. Piiper, J. and P. Scheid (1981). Model for capillary-alveolar equilibration with special reference to 02 uptake in hypoxia. Respir. Physiol. 46: 193-208. Piiper, J. and P. Scheid (1989). Gas exchange: theory, models, and experimental data. In: Comparative Pulmonary Physiology Current Concepts; edited by S.C. Wood. New York: Marcel Dekker, pp. 369416. Powell, F.L. and S.C. Hempleman (1988). Comparative physiology of oxygen transfer in lungs. In: Oxygen Transfer from Atmosphere to Tissues; edited by N.C. Gonzalez and M.R. Fedde. New York: Plenum Press, pp. 53-65.
17
RESP 01983
Diffusion limitation in comparative models of gas exchange * Frank L. Powell and Steven C. Hempleman Physiology Division, Department of Medicine, University of California, San Diego, La Jolla, CA, USA (Accepted 12 August 1992) Abstract. Piiper and Scheid (Resp. Physiol. 23: 209-221, 1975) compared different models of external gas exchange with performance indices defined as functions of ventilatory/perfusiveand diffusive/perfusive conductance ratios (Gvent/Gperf and Gdiff/Gperf, where Gdiff is diffusingcapacity). We expanded their analysis to include: (1)Apo, the average partial pressure gradient driving diffusion across the exchange barrier, normalized to the maximum gradient available (Pi - Pv), and (2) Jdiff, the sensitivityof total conductance to changes in Gdiff, where total conductance is the ratio of gas flux to the maximum gradient [GTOT= IVI/(Pi- Pv)]. Although the counter-current model is most efficient,it is more sensitivethan crosscurrent or ventilated pool models to changes in Gdiff. For given Gvent, Gperf and Pi - Pv, maximumGTOT may not be achieved in the counter-current model until Gdiff is over ten-fold greater than that necessary for maximum GTOT in the other models. Experimental data also shows greater Jdiff and diffusion limitation in fish than in birds or mammals. We conclude that counter-current Oz exchangecannot approach ideal levels as closely as the ventilated pool or cross-current models in nature.
Alveolargas exchange,models, diffusionlimitation;Cutaneous gas exchange;Diffusion,limitation,vertebrate external gas exchange; Models, external gas exchange, vertebrates; Parabronchial gas exchange
Piiper a n d Scheid (1972, 1975) developed a theoretical framework for analyzing the efficiency of gas exchange in different types of respiratory organs. Their analysis showed that for equivalent conditions (i.e. equal ventilation, blood flow, diffusing capacity, inspired and venous tensions a n d blood-gas dissociation curves), gas exchange performance in the models follows the order: counter-current > cross-current > ventilated pool. Their analysis also quantified limitations imposed on gas exchange by the three transport processes, ventilation, perfusion a n d diffusion. However, limitations have not been systematically investigated for the different finite ventilation models. We undertook such an investigation to better u n d e r s t a n d how changes in diffusing capacity affect gas exchange performance in the different models. In this paper we expand the Piiper-Scheid analysis by further exploring diffusion limitation in models of gill, p u l m o n a r y and cutaneous gas exchange. New contributions
Correspondence to: F.L. Powell, Department of Medicine - 0623A, 9500 Gilman Drive, University of California, San Diego, La Jolla, CA 92093-0623, USA. * Dedicated to Johannes Piiper on the occasion of his 65th birthday.
18 result from deriving, for all four models: (1)the average relative partial pressure difference that drives diffusion across the exchange barrier; and (2) the sensitivity of gas flux to changes in diffusing capacity. Some of these results have already been reported (Powell and Hempleman, 1988).
Models
We analyze four models of vertebrate external gas exchange previously presented by Piiper and Scheid (1975): (i) counter-current model of fish gills; (ii) cross-current model of avian parabronchial lungs; (iii) ventilated pool model of mammalian alveolar lungs, unicameral and multicameral reptilian and amphibian lungs; (iv) infinite pool model of cutaneous exchange. Following Piiper and Scheid (1975) we assume: (i) steady state, or constancy of all partial pressures and conductances in time; (ii) linear blood capacitance coefficients, i.e. linear dissociation curves and constant /~ (see Eq. (3) below); (iii) spatial homogeneity, i.e. no mismatch of conductances. Deviations from these assumptions have been discussed previously (Hammond and Hempleman, 1987; Piiper and Scheid, 1975, 1977) and will be considered in the Discussion as necessary to understand model analyses of experimental data. The models predict gas exchange performance as functions of conductances for ventilation with the respiratory medium, i.e. air or water, (Gvent), respiratory organ blood flow (Gperf) and diffusive transport between the respiratory medium and blood (Gdif0, equal to diffusing capacity. Conductances are defined as the factor in the Fick equation necessary to produce a given gas flux (l~l) for a given partial pressure difference (cf Piiper et al., 1971): l~I = G.Ap
(1)
Gdiff is the diffusing capacity. Gvent and Gperf are the products of convective flow rates (Vm and "~o, respectively) and their respective capacity coefficients (/~; cfi Piiper et al., 1971): G = 9.~
(2)
fl is the slope of the curve relating concentration to partial pressure in air, water or blood: /~ = AC/Ap
(3)
Gas exchange performance can be quantified from the models in terms of relative partial pressure differences of the inspired and expired medium (Pi and Pc) and venous and arterial blood (Pv and Pa):
19 Apvcm = (Pi - Pe)/(Pi - Pv)
(4)
Apper f = ( P a - P v ) / ( P i
(5)
- Pv)
Apt r = (Pe - Pa)/(Pi - Pv)
(6)
We include another Ap in our analysis, the average blood-gas (or water) partial pressure difference across the diffusion barrier (Table 1): ApD = (Pro - Pb)/(Pi - Pv)
(7)
Another useful index used by Piiper and Scbeid (1975) to c o m p a r e models is the total conductance: GTOT = l(4/(Pi - PV)
(8)
GTOT is the product of a conductance and its associated Ap, i.e. GTOT = Gdiff" Apo = Gvent.Apvcn t = Gperf. Appcr f. To quantify the effects of changes in Gdiff, Piiper and Scheid (1975) defined diffusion limitation for the various models: Ldiff = (10max - lVlact)/l~Imax
(9)
where/Vlmax is maximal l~l with infinite Gdiff and lVlact is an actual value with a finite Gdiff.
TABLE 1 F o r m u l a e for new values in a model analysis of c o m p a r a t i v e gas exchange ApD
Jdiff
Counter-current
X(1-exp{-Z})" Y (X - exp { - Z})
exp{-Z}(X 2-2X+ 1~ (X - exp { - Z}) ~
Cross-current
X (1 - exp { - Z ' } ) / Y
exp { - Z ' - Y}
Ventilated pool
X (1 - exp { - Y}) Y (X + 1 - exp { - Y})
X 2 exp { - Y} (X + 1 - exp { - y})2
Infinite pool
1 - exp { - Y} y
exp { - Y)
where:
X Y Z Z'
= Gvent/Gperf = Gdiff/Gperf =Y(I-1/X) = (1 - exp { - Y})/X
APD is the n o r m a l i z e d average gradient driving diffusion across gas exchange barriers, (Pm - Pb)/(Pi - Pv), and Jdiff is the sensitivity of total c o n d u c t a n c e to changes in diffusive conductance, bGTOT/lSGdiff. a F o r X # 1; Y / ( Y + y 2 ) for X = 1. b ForX#l;Y(l+y)2forX=l.
20 To quantify the sensitivity of the different models to changes in Gdiff, we calculated the diffusion sensitivities for each (Table 1): Jdiff =
bGToT/bGdiff
(10)
This approach has been used previously by Piiper and Scheid (1981) to analyze diffusion versus perfusion limitations of alveolar gas exchange. However, they assumed fixed alveolar gas tensions in their analysis so their Jdiff was actually that of an infinite pool model. The Jdiff we derive for the other three models allows finite values of Gvent so the ratios Gvent/Gperf and Gvent/Gdiff must be specified to compare effects of changing Gdiff/Gperf.
Model results Figure 1 shows ApD for the four models as a function of Gdiff/Gperf with Gvent/ Gperf = 1 (except for the infinite pool model which has an infinite Gvent/Gperf by definition). As expected, the average partial pressure difference across the exchange barrier increases for all models as diffusive conductance decreases relative to perfusive conductance. Considering models with finite ventilations, the relative positions are counter-current > cross-current > ventilated pool. Increasing the Gvent/Gperf ratio (not shown) moves all of the curves nearer to the infinite pool curve, which defines the limit for all models as the ratio approaches infinity. The relative position of the curves remains the same for all Gvent/Gperf but differences between them become smaller as the ratio increases. The maximum differences occur at Gvent/Gperf= 1 and this will be the case considered hereafter.
1.o
0.8'
"'-'~'~ . . .
• ", . ~
..... Infinite Pool - - Counter--current - -Cross-current - -Ventileted Pool
'...
x. ~ .. "\'N'"'" \
0.6.
...
\
13.. <3 0.4.
~ "..' \ "'\ ~ ". "\." ~\ \ "'".. \\
0.2,
0.0
0.01
'.. ' ", . \ " " .
O.10
1.00
Gdiff/Gperf
( ~ "
10.00
"~'~ ~
""~
100.00
Fig. l. Average partial pressure difference between respiratory medium and blood for the four models (Table 1) as a function of Gdiff/Gperf. Gvent/Gperf= 1 for finite ventilation models and Gperf= 1 for the infinite pool model which has infinite Gvent by definition. Note that cross-current Apt) is similar to countercurrent ApD at low Gdiff/Gperf but more like ventilated pool Apo at high Gdiff/Gperf.
21 1.0 Infinite Pool -- Counter-current - - Cross-current - . . - V e n t i l o t e d Pool
~
. . . . .
0.8.
/ ~
. . . . . . .
/ /
0.4.
..
'
/
-.~ 0 . 6 0 0
I"t-
/
,-"~'-" ....................
0.2.
0 . 0 ............. 0.01
i 10.00
t
0.10
1.00
100.00
Odiff Fig. 2. Total conductance (GTOT) as a function of Gdiff for the four models with Gvent/Gperf = 1 for finite ventilation models and Gperf= 1 for the infinite pool model. Note the positive slope of the countercurrent curve at high Gdiffvalues where the other models have plateaued; GTOT can be increased more than the other models by increasing Gdiff in this range.
This ordering of ApD could be predicted from Piiper and Scheid's previous results (1975) with the performance index GTOT, which is proportional to Apo (cf. Eq. (8)). Figure 2 shows GTOT for the four models under the same conditions as in Fig. 1 and although the relative positions have been shown previously (Fig. 4 in Piiper and Scheid, 1975), our examination of the shapes of the curves continuously over a wider range provides new insights. The counter-current model is fundamentally different from the other three models: maximum GTOT is not achieved until very high Gdiff values. Figure 3 shows the consequences of the counter-current model's more gradual approach to maximum GTOT with increasing Gdiff in terms of Ldiff. Piiper and Scheid
1.00.8-
' ~ . . ~ •.~ ' ~ . " . \ "'~x \ . . X ~x,
..... -- ....
Infinite Pool Counter-current Cross-current Ventilated Pool
X ~
_J
°" 0.0
o.ol
':\'.:i-. I
o.lo
I
~'~, ..... ~ . . . . ~
1.oo Gdiff/Gperf
lo.oo
loo.oo
Fig. 3. Diffusion limitation (Ldif0 as a function of Gdiff/Gperf for the four models with Gvent/Gperf= 1 for finite ventilation models and Gperf = 1 for the infinite pool model. Note that counter-current Ldiff exceeds infinite pool Ldiff over most of the range in contrast to other finite ventilation models. These differences are maximum at G v e n t / G p e r f = 1 and the infinite pool curve becomes the limit for all other models as Gvent/Gperf increases.
22 (1975, 1977) have calculated Ldiff for other models, but it has only been graphed over a wide range of conductances for the infinite pool model. The counter-current model has a greater diffusion limitation than all of the models over the normal to high Gdiff/ Gperf range. Also, counter-current Ldiff exceeds the infinite pool Ldiff for most Gdiff/ Gperf, in contrast to cross-current and ventilated pool Ldiff. Ldiff for the infinite pool model is the limit for all models as Gvent/Gperf increases; with increasing Gvent/ Gperf, cross-current and ventilated pool Ldiff increase while counter-current Ldiff decreases. To make quantitative comparisons in the sensitivity of gas flux to diffusive conductance in the different models, we plot Jdiff for all of the models versus Gdiff/Gperf in Fig. 4. Jdiff is greatest for all models at the lowest Gdiff/Gperf where small increases in Gdiff result in the nearly equal changes of GTOT. This was not immediately obvious from examining the GTOT curves in Fig. 2 curves because the logarithmic abscissa attenuates their slope. The counter-current model shows greater Jdiff than the other ventilated models over most of the low to normal Gdiff/Gperf (Fig. 4). This means that increases in Gdiff/Gperf have a bigger effect on counter-current models in this range, consistent with the continuing increase in GTOT on Fig. 2 vs plateaus for the other models. The underlying cause of the counter-current model's extreme sensitivity to diffusion limitation can be understood by returning to the concept of Apo. Figure 5 plots normalized GTOT versus ApD for all of the models. GTOT is normalized to the value calculated for a given model with infinite Gdiff and Gvent/Gperf= 1, or Gperf= 1 for infinite pool model (i.e. plateau values in Fig. 2). ApD values were calculated for increasing Gdiff from 0.1 to 100 with Gperf and Gvent/Gperf= 1 (i.e. same conditions as Fig. 1). The relative partial pressure gradient necessary to drive a given normalized gas flux is lowest in the counter-current model, consistent with the concept of great-
1.0.
0.8 •
v;:~'"~',:~ ..... . •~ ' . ~ ' . "'.~'X ". \., . ~
,
,.\
O.6. --~
'\. ~ " 'x ~ . "~,.\ "..
0.4,. 0.2. 0.0 0.01
..... Inf n te Pool - - Counter-current - -Cross-current .... Ventilated Pool
"x.\ ".
0.10
\
~
~ 1.00
Gdiff/Gperf
10.00
100.00
Fig. 4. Sensitivity of total conductance to diffusive conductance (Jdiff) for the four models as a function of Gdiff/Gperf with Gvent/Gperf = 1 for finite ventilation models and Gperf = 1 for the infinite pool model. Counter-current Jdiff is the greatest of the models with finite ventilation at Gdiff/Gperf> 1 and can actually exeed infinite pool Jdiff at high Gdiff/Gperf.
23
..... ~ .-..~•... ~..~.",.. 'XN. ". "'\ \",.. 0.8 "\\ "-.. "'\ \ '... 0.6 "\ \ ".. "\ \'.,. "'\ \'. 0.4 "'\.\'... 1.0'
O
c~ no (D N
(3
E k_ O c
0.2
0.0 0.0
N~\y.
--Counter-current - -Cross-current .... Ventiloted Pool
\". ~.
I
I
I
I
0.2
0.4
0.6
0.8
1.0
L~PD Fig. 5. GTOT, normalized to the maximum for a given model, as a function of ApD for each of the four models. Conditions as in Figs. 1 and 2, from which the data is derived, with Gvent/Gperf= 1 for models with finite ventilation and Gperf= 1 for the infinite pool model. The counter-current model can maintain higher relative gas fluxes (normalized GTOT) with lower diffusive driving pressures (ApD) than the other models. Maximum GTOT is achieved by ApD = 0.1 in all models except the counter-current which shows continued GTOT increases at low ApD.
est efficiency in that model. More interesting, however, is the linearity of the countercurrent plot compared to the plateaus of normalized GTOT at low ApD for the other models. Only the geometrical arrangement of exchange elements in the counter-current model is capable of utilizing low partial pressure gradients for effective gas transport. It is in this low ApD range that the counter-current model shows larger absolute GTOT (el high Gdiff range on Fig. 2). The ability of the counter-current model to continue increasing diffusive transfer at low average driving pressures explains its high efficiency.
Model analysis of experimental data We applied the analysis to experimental oxygen exchange data collected by others from yellowfin tuna (Thunnus albacares; Bushnell and Brill, 1992), white Pekin ducks (Anas platyrhynchos; Kiley etal., 1985), and humans (Hammond and Hempleman, 1987). Men were sitting or pedaling on a bicycle ergometer. Ducks were standing or running on a treadmill. Tuna were ram ventilated in flow-through tanks at 25 °C with their spinal nerves blocked to prevent locomotion. Exercising tuna data were extrapolated from measurements on alert resting and swimming dogfish (Piiper and Scheid, 1977). These data sets were selected for completeness of the necessary variables collected under hypoxic conditions. In hypoxia, the oxyhemoglobin equilibrium curve is approximately linear (cfi assumption (ii) of constant fl), Gdiff is measured most accurately and inhomogeneity effects are minimized (cfi Piiper and Scheid, 1977). Diffusive conductances are corrected for any inhomogeneities when possible and ventilations are cor-
24 rected for d e a d s p a c e . W e are n o t a w a r e o f any experimental d a t a meeting these requirements for the infinite pool m o d e l (i.e. c u t a n e o u s gas exchange). T h e s e m e a s u r e d a n d calculated variables are s u m m a r i z e d in Table 2 a n d results are shown in Fig. 6. Resting d a t a are shown on the left panel o f Fig. 6. G p e r f is n o r m a l i z e d to a c o m m o n value o f one for all cases so they can be c o m p a r e d . G d i f f / G p e r f is quite similar in all three cases a n d GTOT varies as expected from differences in the ideal efficiencies o f the models. H o w e v e r , the degree to which the different m o d e l s actually a p p r o a c h their ideal p e r f o r m a n c e follows the inverse order. GTOT in the t u n a is only 62 ~'o o f the counter-current ideal, d u c k GTOT is 81 ~ o f the c r o s s - c u r r e n t ideal and h u m a n GTOT is 92~o o f the ventilated p o o l ideal (i.e. L d i f f = 0.38, 0.19 and 0.08, respectively). The intrinsically m o r e efficient m o d e l s o p e r a t e farther from their o p t i m a and they are on steeper parts o f the curves in Fig. 6. Consequently, Jdiff increases with the intrinsic efficiency of the three models. Exercise decreases the species differences o b s e r v e d at rest. A t exercise vs rest, b o t h d u c k s and h u m a n s show larger diffusion limitation ( L d i f f = 0.29 and 0.17, respectively), while the t u n a shows a smaller diffusion limitation ( L d i f f = 0.27). G d i f f increases with exercise in all cases but so does Gperf. Thus, Gdiff/Gperf, which is the important value, does not increase significantly. The result is that all three species are on steeper parts o f the GTOT curves a n d show similar sensitivities to changes in G d i f f (i.e., Jdiff) at these m o d e s t levels o f exercise.
TABLE 2 Experimental data for model analysis of oxygen exchange in hypoxic resting and exercising man (ventilated pool; Hammond and Hempleman, 1987), duck (cross-current; Kiley et al., 1985) and tuna in 25 °C water (counter-current, Bushnell and Brill, 1992)
Mass (kg) Pio2 (Torr) IVlo2(/~mol.min- 1.kg- 1) (L.min- 1.kg- 1), I) (L.min- 1.kg- l) Gdiff (#moi.min- l.Torr- 1.kg- l)b Gvent/Gperf Gdiff/Gperf
0.32 0.10 0.14 27 0.21 1.1
Man
Duck
Tuna
68 78
2.4 84
1.4 90
1.2 0.62 0.25 53 0.68 1.1
0.48 0.47 0.28 38 0.68 1.1
1.7 2.1 0.60 68 1.4 1.0
0.47 6.9 0.12 18 0.99 1.6
0.69~ -1.5c 1.9c
a VA calculated from iVlco2and P a c o 2 in man; parabronchial V calculated from ~rl and measured dead space in ducks; gill ~z measured with dye dilution technique. b DLo2 corrected for heterogeneity (0 weighted) in man;. Do: corrected for heterogeneity and cardiac output in 2.4 kg ducks: Do2 (/~ mol.min- t-Torr- l.kg- 1) = Q (L/min).97 + 25 (Hempleman and PoweU, 1986; and unpublished data); Do2 calculated for counter-current model, i.e. not To2. Not measured in tuna but increased from resting values by factors measured in dogfish on transition from alert rest to swimming (Piiper et al., 1977).
25 HYPOXIC REST
1.0
1.0
.... Infinite Pool
0.8-
--tuno :..:d~g~
HYPOXlC EXERCISE
~
-""'
f..'"
/:" / /
O.B
-4--, 0.60
0.6
(.9 0.4.
0.4
/ eo,0~ ...........
0.2.
//~.o8
'
.
...... d
.
.
l
f
f
_
.
0.2 .. ,if'"
0.0 0.01
0.10
1,00
Gdiff
10.00
100.00
0.0 .... 0.01
I
0.10
1
t
1.00
10.00
100.00
Gdiff
Fig. 6. GTOT for the infinite pool model, tuna (counter-current model), duck (cross-current model) and man (ventilated pool model) as a function of Gdiff for the measured conditions given in Table 2. Gperf is normalized to a common value of one for all cases. Left panel: resting tuna show the largest diffusion limitation and greater sensitivity of IVlo2 to Gdiff while duck values are intermediate to those of tuna and man. Right panel: diffusion limitation and sensitivity of/Vlo2 to Gdiff increase in man and duck with hypoxic exercise, in contrast to tuna. In both cases, Gdiff would have to increase more in tuna than in duck or man to reach the GTOT plateau, representing more ideal performance.
Discussion
This study extends to the previous contributions of Piiper and Scheid (1975, 1977, 1981) by deriving: (1)the relative average blood-gas partial pressure difference driving diffusion, ApD, and (2)the sensitivity of gas flux to changes in diffusive conductance Jdiff, both for all four comparative models of gas exchange. ApD is useful because it predicts the gradient driving diffusive gas exchange in different vertebrate respiratory organs, given conductances and inspired and venous tensions. Hence, the analysis provides a framework to consider problems like the physiological significance of differences in physiologic and morphometric diffusing capacities that have been reported for a variety of animals. Jdiff has utility in quantifying how effective a given change in diffusing capacity is at changing gas flux in different exchange models for given inputs. As shown in Fig. 6 and discussed below, the Gdiff/Gperf range over which Jdiff is significantly > 0 could have important biological implications.
Critique. The utility of these models for studying the physiology of diffusion limitation depends on assumptions going into their derivations, which have been discussed in detail elsewhere (Piiper and Scheid, 1975, 1977, 1989). For example, the diffusion barrier is assumed to be a simple membrane obeying Fick's law. Actually, Gdiff is a lumped parameter including, among other things, the kinetics of 02 and CO2 reactions in blood and diffusive resistance in the respiratory medium. These factors may be important when comparing animals having different body temperatures and breathing media of different diffusivity and capacitance (i.e. air vs water). Our criteria for selecting experimental data to analyze (cf Results) generally satisfied the model assumptions. The human data set was the most complete and, therefore, internally consistent (Hammond and Hempleman, 1987). Gdiff values from the
26 Kiley et al. (1985) duck study were significantly lower than estimates corrected for heterogeneity so we used corrected values accounting for changes in cardiac output from another duck study (Hempleman and Powell, 1986). Gdiff values corrected for heterogeneity are not available for fish but Piiper and Scheid (1989) noted that the agreement between morphometric and physiologic Do2 in fish is consistent with negligible functional inhomogeneity. Data from paralyzed, ram ventilated tuna were selected as the only complete set in hypoxia in the literature (Bushnell and Brill, 1992). The conductances are in the range expected for slowly swimming tuna, which is probably as close to rest as are ducks standing on a treadmill or humans sitting on a bicycle ergometer. Gdiff changes with exercise were predicted from a presumably similar baseline of "alert rest" (as opposed to "quiescence") in dogfish (Piiper and Scheid, 1989). This increase probably reflects gill capillary recruitment (Farrell et al., 1980) which we assume occurs in tuna also.
Comparison with previous studies. Piiper and Scheid (1977) have studied the effects of diffusion on several partial pressure gradients ratios. Apt r is an extremely useful index of gas exchange efficiency, being a normalized form of the alveolar-arterial difference which is a standard for alveolar gas exchange. In an alveolar model, Apt r c a n be interpreted as the fraction of resistance to gas transfer between inspired air and venous blood attributed to alveolar gas-capillary blood exchange (Piiper and Scheid, 1972). However, in the more efficient cross-current and counter-current models, Aptr can actually assume negative values and it cannot be easily interpreted as a fractional resistance. A negative Apt r emphasizes that the geometrical arrangement of these more efficient models results in performance only possible in alveolar exchange by active secretion. Piiper and Scheid (1977) derived the contribution of a model's spatial arrangement to Apt r as Apt r = Apo when Gdiff is infinite. More generally, they define: Apt r =
Apo + Apdifr
(11)
Apdiff Can be greater or less than Aptr but it is always greater than zero. APdiffrepresents the relative increase from the ideal expired-arterial gradient caused by a finite Gdiff; Apo, which we calculate here, is the relative blood-gas (or water) gradient with a finite Gdiff. With knowledge of the normalization value Pi - Pv, ApD is useful for exercises like comparing the diffusing capacity requirements between different models. As discussed under Models, Jdiff, the sensitivity coefficient for changes in gas flux with changes in diffusing capacity, was originally derived for the infinite pool model by Piiper and Scheid (1981). However, they used it to analyze alveolar gas exchange assuming constant alveolar gas tensions. Jdiff for the ventilated pool model here incorporates the effect of changes in ventilation and perfusion on alveolar gas and requires specifying Gvent/Gperf to determine the sensitivity of gas flux to changes in Gdiff (cf. Table 1). Piiper and Scheid (1975) applied their analysis to resting experimental data and their general conclusions about oxygen exchange are consistent with ours. The hypoxic canine data they used is similar to the human data we selected for the ventilated pool
27 analysis. Their avian data was from hypoxic and hypercapnic chickens. Those chickens presumably had higher ventilatory drive than the hypoxic ducks but Gvent/Gperf was similar for both of our cross-current analyses. Their counter-current analysis used normoxic dogfish but it is interesting that their general conclusions are similar to ours for hypoxic tuna. In contrast to birds and mammals, fish show significant diffusion limitation in resting normoxia as well as hypoxia (Piiper and Scheid, 1989).
Physiological significance.
Applying the analysis to experimental data from hypoxic resting animals reveals clear differences in performance between the models as they occur in nature (Fig. 6). At rest, oxygen exchange in air breathing man is very close to its theoretical maximum and humans would benefit relatively little from further increases in Gdiff (i.e. Ldiff and Jdiff are small). In contrast, water breathing tuna could benefit greatly from increased Gdiff and the air breathing ducks represent an intermediate case. Exercise reduces these species differences but Gdiff would still have to increase about five-fold more in the tuna than in duck or man to maximize gas flux at the measured Gvent, Gperf, Pi and Pv (Fig. 6). Although evolution does not have a goal of optimal design, natural selection frequently results in the optimal function of a given structural design (Lindstedt and Jones, 1987). Our observations raise the questions " H o w did the tuna evolve with the intrinsically most efficient gas exchange model operating farthest from ideal conditions?" More specifically, how does the tuna's evolutionary solution compare with other possibilities like (1)more optimal performance in less efficient models or, (2) smaller but more ideally matched conductances in the counter-current model? The model analysis allows us to explore the physiological consequences of these alternatives.
1. Larger conductances in less efficient models To consider the first alternative, we calculated what GTOT would be for the three finite ventilation models given the conductances measured in the resting tuna (cf Table 2). GTOT was 0.55 and 0.44 in cross-current and ventilated pool models, respectively, compared to 0.62 in the counter-current model (i.e. the value shown in Fig. 6). Next, we considered how conductances would have to increase in the cross-current or ventilated pool models to yield a GTOT of 0.62. Increasing gill ventilation was not considered because the increased work of breathing water with high density and viscosity and low 02 capacitance would presumably be selected against in nature. Increasing Gvent would also require increasing Gperf to maintain Gvent/Gperf near the ideal value of 1. Assuming the tuna Gvent/Gperf value, increasing Gdiff is not a viable solution with the ventilated pool model, maximum GTOT with infinite Gdiffis only 0.49. However, a cross-current model could reach a GTo'r of 0.62 if Gdiff/Gperf increased three-fold. Reasons why this may not have occurred are considered later.
2. Better conductance matching in the most efficient model Figure 6 is useful for considering the alternative of more ideal conductance ratios in the counter-current model. Gvent/Gperf is already very near the ideal value of 1 so
28
again we focus on Gdiff/Gperf. We calculated that Gdiff/Gperf would have to increase almost ten-fold in the tuna to come as close to optimal performance as man does and Gdiff/Gperf is already greatest in the tuna. Recall that counter-current Ldiff and Jdiff were larger than for the other models over a significantly greater range of Gdiff/Gperf (cf. Figs. 3 and 4). For tuna to increase Gdiff ten-fold, respiratory organ capillary volume, surface/volume ratio and/or exchange barrier thickness would have to change significantly from the general vertebrate plan. The strength of capillaries and developmental program for vascular branching may preclude such evolutionary changes. Also, the gills are involved in many more physiological functions besides oxygen exchange (McDonald et al., 1991) so increasing their surface area may compromise other vital processes like CO2, water or ion homeostasis. On the other hand, increasing Gdiff/Gperf by ten-fold through decreasing Gperf would not only move Gvent/Gperf farther from the ideal value of 1, but it would require tissue oxygen delivery and exchange in fish to be much more efficient than in other vertebrates. This would presumably necessitate a different and more efficient design of systemic oxygen transport than that found in the rest of the vertebrates. Perfusion limitations are not larger in fish than in terrestrial vertebrates (Piiper and Scheid, 1975), so the required increase in perfusive oxygen delivery could not likely be achieved through separation of pulmonary and systemic circulations. In conclusion, the analysis provides quantitative arguments against models other than counter-current being a realistic evolutionary solution for oxygen exchange in fish. The counter-current model achieves the highest gas flux for given conductances, which is important for water breathers with significant stratification at their gas exchange surfaces and high costs of breathing. Furthermore, the analysis showing significant diffusion limitation in counter-current models over a wide range of Gdiff/Gperf suggesting reasons that gill-O 2 exchange may not be optimized in nature. More ideal conditions with smaller conductances may be precluded by (1) the evolutionary history of the vertebrate cardiovascular and respiratory systems, and (2)the sensitivity of the counter-current model of Gdiff changes over such a wide range (cf. Fig. 2 and Fig. 6). History imposes boundary conditions on the natural selection of structure-function relationships so Gdiff/Gperf may not be able to increase enough to reach ideal countercurrent conditions through random mutations (cf. Fig. 6). Acknowledgements. We thank Dr. A.T. Gray for discussions about the material presented. This work was supported by NIH Grant HL 17731.
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