- Email: [email protected]
The importance of a source term in modeling multibreath inert gas washout
~.~0N PHp~
ELSEVIER
Respiration Physiology 103 (1996) 99-103
Short communication
The importance of a source term in modeling multibreath inert gas washout Peter W. Scherer
a,b,*, Joseph
D. Neff a,¢, James E. Baumgardner b,c, Gordon R. Neufeld b,c
a Department of Bioengineering, University of Pennsylvania, School of Engineering andAppliedScience, 3320 Smith Walk, Philadelphia PA 19104-6392, USA b Department of Anesthesia, University of Pennsylvania, School of Medicine, Philadelphia PA 19104, USA c Department of Anesthesia, Philadelphia Veteran's Affairs Medical Center, Philadelphia PA 19104, USA Accepted 11 September 1995
Abstract The single path model (SPM) of airway gas transport with a distributed blood source term was used to simulate multiple breath inert lung gas washout of N 2, He, and SF6 after total body equilibration with these gases. Normalized phase III inert gas washout slopes were computed for each breath and compared with published experimental data obtained under similar conditions on human subjects. The model predicts a normalized slope asymptote which agrees with experimental results within two standard deviations or less of the mean, depending on the lengths and diameters assumed in the acinar airways of the SPM. In the model and in the human subject data, the asymptote represents the development of a quasi-steady state in which the volume of inert gas exhaled at the mouth is equal to the volume transported into the acinar airways by the pulmonary blood during each breath. The present study indicates that at least in the steady state, airway inhomogeneity is not essential to model lung washout data, and that a distributed blood source term in the SPM yields good agreement with experiment. Keywords: Gas exchange, alveolar slope; Inert gas, expired slope; Model, expirogram, alveolar slope; Washout, lung gas
1. Introduction When inert gases that have been equilibrated with body tissues are washed out of the lung, a quasisteady state is reached after 20 or 30 breaths and beyond (Lundin, 1953). In this steady state, during each breath, the volume of inert gas which flows out
* Corresponding author. Tel: (215) 898-7214, Fax: (215): 5732071, E-mail: [email protected].
from the mouth on expiration is equal to that delivered into the acinar airways from the pulmonary capillary blood. Since the solubility of inert gases in blood is low, these volumes are quite small, remaining constant at about 3 ml (BTP) per breath for the first 20 min of washout, and gradually declining thereafter (Lundin, 1953). Although the contribution of this blood source to the shape of the inert gas washout curve is small during the first few breaths, it becomes much more important as the steady state is reached. Since previous attempts to model multi-
0034-5687/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 3 4 - 5 6 8 7 ( 9 5 ) 0 0 0 8 0 - 1
100
P.W. Scherer et al./ Respiration Physiology 103 (1996) 99-103
breath inert gas washout (Crawford et al., 1985; Paiva and Engel, 1984), have not included a blood source term, we studied this process using the single path model (SPM) with a source term.
2. Methods The SPM based on the convective diffusion equation OC
(v+ v )-g +
OC
0 ( D A OC) = a z --i-- T z + S(z,t)
(1)
has been described previously (Scherer et al., 1988). The blood source term S(z,t) for inert gases has the form NA(Z) S(z,t) = OBh---==---- (C B - C(z,t)) NT
(2)
where QB is cardiac output (cm3/sec), NA(z) is the number of alveoli/generation at generation z, N T is the total number of alveoli in the entire lung (Haefeli-Bleuer and Weibel, 1988), A is inert gas solubility in blood (ml gas/ml blood/atm at BTP), C B is mixed venous inert blood gas concentration (atm), and C is alveolar inert gas concentration (atm). Equation 1 was solved by an implicit finite difference method using the boundary conditions 0C/Oz = 0 at z = 24 (alveolar end of model) at all times, C = 0 at z = 0 (the mouth end) during inspiration, and 0C/0z = 0 at z --- 0 during expiration. The anatomical dimensions used in the SPM are those proposed by Haefeli-Bleuer and Weibel (1988). The experimental data we chose to simulate is that of Crawford et al. (1985) who conducted a careful study in humans of multiple breath inert gas washout. They measured normalized phase III slopes of N 2, He, and SF6 on washout after equilibration with body tissues. Unless stated otherwise, all numerical simulations were performed using a tidal volume of 1.0 liter and a breathing frequency of 12 breaths per minute at a constant and equal inspiratory and expiratory flow rate, as used by Crawford et al. (1985) The molecular diffusivities used for N 2, He and SF6 in air were
0.25, 0.725 and 0.101 cm2/sec, respectively. The solubilities used for the inert gases in blood were 0.0146, 0.01 and 0.0076 ml gas. (ml blood) - 1 . atm-1, respectively. Cardiac output was fixed at 110 cm3/sec. The initial conditions (t = 0) for each simulation were that the airways were filled with inert gas at the concentration used for body tissue equilibration and the mixed venous blood inert gas concentration, C B, was in equilibrium with the inert gas in the airways. The computation simulated the breath by breath inhalation of pure 0 2 (C inert gas = 0 at mouth on inhalation) for 50 breaths. For the He and SF6 simulations, C a was held constant at its initial value throughout the entire washout process. For N2, C B was reset equal to the end expiratory 24th generation airway value for the first 5 breaths then held constant at a value of 0.3 atm to yield a volume of 2.36 ml N 2 excreted per breath when the steady state was reached (see discussion). For each breath, the inert gas washout curve seen at the mouth was computed and the normalized slope (dC/dV/CME, where C ME is mixed expired gas concentration) based on the last 30% of phase III was determined following the method of Crawford et al. (1985). In the SPM, the effect of variations in acinar airway geometry was studied by varying airway diameter and length. Total acinar cross-sectional area in each generation was multiplied by a scaling factor /3, while the length of each generation was adjusted to maintain the original volume (Schwardt et al., 1991, Schwardt et al., 1994). /3 was varied over the range 1.0 + 0.6 for N 2 and He. This corresponds to the mean + two standard deviations in Haefeli-Bleuer and Weibel's (1988) data on acinar airway lengths and diameters. For SF6, /3 was varied over the range of 1.0 + 0.4, since larger variations resulted in normalized slope values far greater than the experimental data.
3. Results Figs. 1 and 2 show the SPM predictions of breath by breath normalized phase III washout slope compared with the experimental values. Also shown are the SPM predictions of the multibreath washouts without a source term (SPMO), and, on Fig. 1, the numerical predictions of Paiva and Engel (1984) as
P. W. Scherer et al. / Respiration Physiology 103 (1996) 99-103
reproduced by Crawford et al. (1985) for N 2 washout, using a multipath asymmetric acinar model without a blood source term. Although there is considerable variation in the experimental data, for a given subject the normalized slopes (NS), with units of L-1, start low for the first breath with values between 0.05 and 0.15 and, for N z, rise to asymptotic values around 0.15-0.35 between the 20th and 30th breaths. The mean values from Crawford et al. (1985), shown by black circles in Fig. 1, start around 0.1 for the first breath and rise to around 0.25 after 25 breaths. The transition period to the steady state found in the SPM simulations is about the same length as that seen in the human subject data. The curves for He and SF6 washout are similar to that of N 2, but at steady state the NS values for He are lower than for SF6 due to the higher diffusivity of He. Limitations in the mass spectrometer used by Crawford et al. (1985) prevented data collection for He and SF6 beyond the 15th breath.
101
For N 2 and SF6, the effect of the acinar cross-sectional area scaling factor/3 is seen to be quite large. For the first N 2 breath, the SPM (with /3 = 1) gives a NS value of 6.3 X 10 - 4 L -1 with the source term, and 5 x 10 - 6 L -1 without it, while the asymmetric acinar model of Paiva and Engel (1984) gives a value of 8.5 X 10 -3 L -1. When /3 was reduced to 0.4 for N 2, the SPM yielded a NS of 0.14 for the first breath, and produced NS values close to the mean of the experimental data over the entire breath number range. As washout continues and the steady state is reached, the asymptotic SPM predictions for the NS of N 2 lie between 0.19 (for /3--- 1) and 0.32 (for /3 = 0.4), which fall within two standard deviations or less of the mean of the Crawford et al. (1985) data. In contrast, the predictions of the SPMO and the asymmetric acinar model do not rise to a plateau, but remain constant, and are much lower than the asymptotic values of Crawford's data. It is quite likely, however, that adding a source term to the
Nitrogen Washout 0.40
0.40 0.35
0.35
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o
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. . . . . . . . . . . . . . . .
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SPM, no source term .
.
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5
10
15
20
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30
35
40
45
-0.05 50
Breath Number
Fig. 1. SPM predictions of the normalized slopes of breath by breath N 2 washout are compared to experimental data. The mean + one SD of the normalized slopes from the experiments of Crawford et al. (1985) are shown. SPM simulations (dashed lines) for/3 = 1 asymptote to just below one SD from the data mean, while simulations at /3 = 0.4 asymptote close to Crawford's mean values. Numerical simulations without a source term are seen to remain flat and far below the experimental data, for both Paiva and Engel's model (1984) (P&E line) and the SPMO at /3 = 1. For/3 = 0.4, the SPMO prediction (not shown) is a straight horizontal line at NS - 0.14 L -1 .
P.W. Scherer et al./ Respiration Physiology 103 (1996) 99-103
102
Helium and SF6 Washout
050 0.45
1 050
0.40
0.40
0.35
0.35
~,~ 0.30
0.30
0.25
0.25
0.20
0.20
0.15
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0.10
0.10
0.05
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............................................
0.00
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10
15
20
25
30
35
40
45
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Breath Number
Fig. 2. SPM predictions of the normalized slopes of breath by breath SF6 and helium washout are compared to experimental data. Solid lines and symbols represent SF6, dashed lines and open symbols represent helium. The mean -t- one SD for the experimental data of Crawford et al. (1985) are shown for the first 15 breaths (circles), along with the mean + one SD of data from previously published steady state He and SF6 experiments (Neufeld et al., 1991, squares). The SPM predicts that the curves for He are insensitive to the /3 factor while SF6 normalized slopes increase rapidly for/3 < 1.0. For a given /3 value, the asymptote of the SPM simulation for He always lies below that of SF6 in agreement with experiment. The simulation with zero blood source and /3 = 1 lies well below the experimental data.
Paiva and Engel model would yield much better agreement with the experiment.
4. Discussion The SPM numerical results are strongly dependent on the presence of a blood source term and the value of /3, the acinar total cross-sectional area scaling factor. Values of /3 greater than unity are seen to have a negligible effect on the breath by breath value of NS for all gases, while values less than unity have an effect in inverse proportion to the molecular diffusivity of the gas. For SF6, the effect of decreasing /3 is very pronounced due to its low molecular diffusivity. Previous results in applying the SPM to healthy and diseased human lungs (Schwardt et al., 1994) suggest that values o f / 3 less than 1.0, around 0.7 or less, are necessary to obtain agreement with human gas washout data.
For a given inert gas and /3 value, the NS predicted by the SPMO remains very small and constant throughout the entire simulation, at a value approximately equal to the first breath prediction of the SPM. As noted by Crawford et al., (1985) this is due to the fact that if one normalizes the airway concentration C(z,t) by its initial value Co, then each breath is mathematically equivalent with the same initial and boundary conditions and the NS, which depends on C / C 0, becomes independent of breath number. A similar breath number insensitivity occurs in the SPM with variation of the mixed venous inert gas concentration C a in the blood source term. If C(z,t) is normalized by C B, then each breath becomes equivalent and the plot of NS vs. breath number becomes insensitive to CB; e.g. for a given gas, one obtains the same steady state asymptote independent of C B. The inspiratory and expiratory flow waveform does have an effect on the NS. The SPM predicts
P.W. Scherer et al./Respiration Physiology 103 (1996) 99-103
that steady flows as used by Crawford et al. (1985) produce higher values of NS than sinusoidal flows. This is because a steady flow produces higher convective gas velocities in the acinus during the beginning of inspiration which prevent acinar gas from diffusing back towards the mouth and equilibrating by diffusion as quickly as occurs with a sinusoidal wave form.
5. Conclusion The agreement of the SPM steady state predictions of the normalized slope with experiment for inert gas washout in humans together with similar agreement obtained for CO 2 (Schreiner et al., 1993, Ream et al., 1995) suggests that the model is a valid representation of these processes. Spatial heterogeneities of gas concentration and flow throughout the lung during washout undoubtedly exist. The present study indicates that at least in the steady state, airway inhomogeneity is not essential to model lung washout data, and that a distributed blood source term in the SPM yields good agreement with experiment.
Acknowledgements These studies were supported in part by V A Merit Review research grant 103.8 and by NIH grant H L 33891.
103
References Crawford, A.B.H., M. Makowska, M. Paiva and L.A. Engel (1985). Convection- and diffusion-dependent ventilation maldistribution in normal subjects. J. Appl. Physiol. 59: 838-846. Haefeli-Bieuer, B. and E.R. Weibel (1988). Morphometry of the human pulmonary acinus. Anat. Rec. 220: 401-414. Lundin, G. (1953). Nitrogen elimination during oxygen breathing. Acta Physiol. Scand. 30 (suppl 111):130-143. Neufeld G.R., S.R. Gobran, J.E. Baumgardner, S.J. Aukburg, M.S. Schreiner and P.W. Scherer (1991). Diffusivity, respiratory rate and tidal volume influence inert gas expirograms. Respir. Physiol. 84: 31-47. Paiva M. and L.A. Engel (1984). Model analysis of gas distribution within the human lung acinus. J. Appl. Physiol, 56: 418-425. Ream R.S., M.S. Schreiner, J.D. Neff, K.M. McRae, A.F. Jawad, P.W. Scherer and G.R. Neufeld (1995). Volumetric Capmography in children: Influence of growth on the alvelor plateau slope. Anesthesiol. 82: 64-73. Schreiner M.S., G.L. Leksell, S.R. Gobran, E.A. Hoffman, P.W. Scherer and G.R. Neufeld (1993). Microemboli reduce phase III slopes of CO2 and invert phase III slopes of infused SF6. Respir. Physiol. 91: 137-154. Schwardt, J.D., S.R. Gobran, G.R. Neufeld, S.J. Aukberg and P.W. Scherer (1991). Sensitivity of CO2 washout to changes in acinar structure in a single-path model of lung airways, Respir. Physiol. 19: 679-697. Schwardt, J.D., G.R. Neufeld, J.E. Baumgardner and P.W. Scherer (1994). Noninvasive recovery of acinar atomic information from CO2 expirograms. Ann. Biomed. Eng. 22: 293-306.
ELSEVIER
Respiration Physiology 103 (1996) 99-103
Short communication
The importance of a source term in modeling multibreath inert gas washout Peter W. Scherer
a,b,*, Joseph
D. Neff a,¢, James E. Baumgardner b,c, Gordon R. Neufeld b,c
a Department of Bioengineering, University of Pennsylvania, School of Engineering andAppliedScience, 3320 Smith Walk, Philadelphia PA 19104-6392, USA b Department of Anesthesia, University of Pennsylvania, School of Medicine, Philadelphia PA 19104, USA c Department of Anesthesia, Philadelphia Veteran's Affairs Medical Center, Philadelphia PA 19104, USA Accepted 11 September 1995
Abstract The single path model (SPM) of airway gas transport with a distributed blood source term was used to simulate multiple breath inert lung gas washout of N 2, He, and SF6 after total body equilibration with these gases. Normalized phase III inert gas washout slopes were computed for each breath and compared with published experimental data obtained under similar conditions on human subjects. The model predicts a normalized slope asymptote which agrees with experimental results within two standard deviations or less of the mean, depending on the lengths and diameters assumed in the acinar airways of the SPM. In the model and in the human subject data, the asymptote represents the development of a quasi-steady state in which the volume of inert gas exhaled at the mouth is equal to the volume transported into the acinar airways by the pulmonary blood during each breath. The present study indicates that at least in the steady state, airway inhomogeneity is not essential to model lung washout data, and that a distributed blood source term in the SPM yields good agreement with experiment. Keywords: Gas exchange, alveolar slope; Inert gas, expired slope; Model, expirogram, alveolar slope; Washout, lung gas
1. Introduction When inert gases that have been equilibrated with body tissues are washed out of the lung, a quasisteady state is reached after 20 or 30 breaths and beyond (Lundin, 1953). In this steady state, during each breath, the volume of inert gas which flows out
* Corresponding author. Tel: (215) 898-7214, Fax: (215): 5732071, E-mail: [email protected].
from the mouth on expiration is equal to that delivered into the acinar airways from the pulmonary capillary blood. Since the solubility of inert gases in blood is low, these volumes are quite small, remaining constant at about 3 ml (BTP) per breath for the first 20 min of washout, and gradually declining thereafter (Lundin, 1953). Although the contribution of this blood source to the shape of the inert gas washout curve is small during the first few breaths, it becomes much more important as the steady state is reached. Since previous attempts to model multi-
0034-5687/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 3 4 - 5 6 8 7 ( 9 5 ) 0 0 0 8 0 - 1
100
P.W. Scherer et al./ Respiration Physiology 103 (1996) 99-103
breath inert gas washout (Crawford et al., 1985; Paiva and Engel, 1984), have not included a blood source term, we studied this process using the single path model (SPM) with a source term.
2. Methods The SPM based on the convective diffusion equation OC
(v+ v )-g +
OC
0 ( D A OC) = a z --i-- T z + S(z,t)
(1)
has been described previously (Scherer et al., 1988). The blood source term S(z,t) for inert gases has the form NA(Z) S(z,t) = OBh---==---- (C B - C(z,t)) NT
(2)
where QB is cardiac output (cm3/sec), NA(z) is the number of alveoli/generation at generation z, N T is the total number of alveoli in the entire lung (Haefeli-Bleuer and Weibel, 1988), A is inert gas solubility in blood (ml gas/ml blood/atm at BTP), C B is mixed venous inert blood gas concentration (atm), and C is alveolar inert gas concentration (atm). Equation 1 was solved by an implicit finite difference method using the boundary conditions 0C/Oz = 0 at z = 24 (alveolar end of model) at all times, C = 0 at z = 0 (the mouth end) during inspiration, and 0C/0z = 0 at z --- 0 during expiration. The anatomical dimensions used in the SPM are those proposed by Haefeli-Bleuer and Weibel (1988). The experimental data we chose to simulate is that of Crawford et al. (1985) who conducted a careful study in humans of multiple breath inert gas washout. They measured normalized phase III slopes of N 2, He, and SF6 on washout after equilibration with body tissues. Unless stated otherwise, all numerical simulations were performed using a tidal volume of 1.0 liter and a breathing frequency of 12 breaths per minute at a constant and equal inspiratory and expiratory flow rate, as used by Crawford et al. (1985) The molecular diffusivities used for N 2, He and SF6 in air were
0.25, 0.725 and 0.101 cm2/sec, respectively. The solubilities used for the inert gases in blood were 0.0146, 0.01 and 0.0076 ml gas. (ml blood) - 1 . atm-1, respectively. Cardiac output was fixed at 110 cm3/sec. The initial conditions (t = 0) for each simulation were that the airways were filled with inert gas at the concentration used for body tissue equilibration and the mixed venous blood inert gas concentration, C B, was in equilibrium with the inert gas in the airways. The computation simulated the breath by breath inhalation of pure 0 2 (C inert gas = 0 at mouth on inhalation) for 50 breaths. For the He and SF6 simulations, C a was held constant at its initial value throughout the entire washout process. For N2, C B was reset equal to the end expiratory 24th generation airway value for the first 5 breaths then held constant at a value of 0.3 atm to yield a volume of 2.36 ml N 2 excreted per breath when the steady state was reached (see discussion). For each breath, the inert gas washout curve seen at the mouth was computed and the normalized slope (dC/dV/CME, where C ME is mixed expired gas concentration) based on the last 30% of phase III was determined following the method of Crawford et al. (1985). In the SPM, the effect of variations in acinar airway geometry was studied by varying airway diameter and length. Total acinar cross-sectional area in each generation was multiplied by a scaling factor /3, while the length of each generation was adjusted to maintain the original volume (Schwardt et al., 1991, Schwardt et al., 1994). /3 was varied over the range 1.0 + 0.6 for N 2 and He. This corresponds to the mean + two standard deviations in Haefeli-Bleuer and Weibel's (1988) data on acinar airway lengths and diameters. For SF6, /3 was varied over the range of 1.0 + 0.4, since larger variations resulted in normalized slope values far greater than the experimental data.
3. Results Figs. 1 and 2 show the SPM predictions of breath by breath normalized phase III washout slope compared with the experimental values. Also shown are the SPM predictions of the multibreath washouts without a source term (SPMO), and, on Fig. 1, the numerical predictions of Paiva and Engel (1984) as
P. W. Scherer et al. / Respiration Physiology 103 (1996) 99-103
reproduced by Crawford et al. (1985) for N 2 washout, using a multipath asymmetric acinar model without a blood source term. Although there is considerable variation in the experimental data, for a given subject the normalized slopes (NS), with units of L-1, start low for the first breath with values between 0.05 and 0.15 and, for N z, rise to asymptotic values around 0.15-0.35 between the 20th and 30th breaths. The mean values from Crawford et al. (1985), shown by black circles in Fig. 1, start around 0.1 for the first breath and rise to around 0.25 after 25 breaths. The transition period to the steady state found in the SPM simulations is about the same length as that seen in the human subject data. The curves for He and SF6 washout are similar to that of N 2, but at steady state the NS values for He are lower than for SF6 due to the higher diffusivity of He. Limitations in the mass spectrometer used by Crawford et al. (1985) prevented data collection for He and SF6 beyond the 15th breath.
101
For N 2 and SF6, the effect of the acinar cross-sectional area scaling factor/3 is seen to be quite large. For the first N 2 breath, the SPM (with /3 = 1) gives a NS value of 6.3 X 10 - 4 L -1 with the source term, and 5 x 10 - 6 L -1 without it, while the asymmetric acinar model of Paiva and Engel (1984) gives a value of 8.5 X 10 -3 L -1. When /3 was reduced to 0.4 for N 2, the SPM yielded a NS of 0.14 for the first breath, and produced NS values close to the mean of the experimental data over the entire breath number range. As washout continues and the steady state is reached, the asymptotic SPM predictions for the NS of N 2 lie between 0.19 (for /3--- 1) and 0.32 (for /3 = 0.4), which fall within two standard deviations or less of the mean of the Crawford et al. (1985) data. In contrast, the predictions of the SPMO and the asymmetric acinar model do not rise to a plateau, but remain constant, and are much lower than the asymptotic values of Crawford's data. It is quite likely, however, that adding a source term to the
Nitrogen Washout 0.40
0.40 0.35
0.35
113= 0.4 o
0.30
o
o
o
o
. . . . . . . . . . . . . . . .
0.30
-
0.25
0.25
g o
~]= 1.0
0.20
0.20
tO
0.15
0,15
.E 0
z
0.10
0,10
i
0.05
0.05 P&E
0.00 -0.05
.
m
=
" " ."
.
.
.
.
.
.
.
.
.
SPM, no source term .
.
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0.00
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I
[
[
(
t
I
I
I
[
5
10
15
20
25
30
35
40
45
-0.05 50
Breath Number
Fig. 1. SPM predictions of the normalized slopes of breath by breath N 2 washout are compared to experimental data. The mean + one SD of the normalized slopes from the experiments of Crawford et al. (1985) are shown. SPM simulations (dashed lines) for/3 = 1 asymptote to just below one SD from the data mean, while simulations at /3 = 0.4 asymptote close to Crawford's mean values. Numerical simulations without a source term are seen to remain flat and far below the experimental data, for both Paiva and Engel's model (1984) (P&E line) and the SPMO at /3 = 1. For/3 = 0.4, the SPMO prediction (not shown) is a straight horizontal line at NS - 0.14 L -1 .
P.W. Scherer et al./ Respiration Physiology 103 (1996) 99-103
102
Helium and SF6 Washout
050 0.45
1 050
0.40
0.40
0.35
0.35
~,~ 0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
E z
0.00 -0.05
0.45
............................................
0.00
b
I
)
L
i
i
~
~
)
5
10
15
20
25
30
35
40
45
-0.05 50
Breath Number
Fig. 2. SPM predictions of the normalized slopes of breath by breath SF6 and helium washout are compared to experimental data. Solid lines and symbols represent SF6, dashed lines and open symbols represent helium. The mean -t- one SD for the experimental data of Crawford et al. (1985) are shown for the first 15 breaths (circles), along with the mean + one SD of data from previously published steady state He and SF6 experiments (Neufeld et al., 1991, squares). The SPM predicts that the curves for He are insensitive to the /3 factor while SF6 normalized slopes increase rapidly for/3 < 1.0. For a given /3 value, the asymptote of the SPM simulation for He always lies below that of SF6 in agreement with experiment. The simulation with zero blood source and /3 = 1 lies well below the experimental data.
Paiva and Engel model would yield much better agreement with the experiment.
4. Discussion The SPM numerical results are strongly dependent on the presence of a blood source term and the value of /3, the acinar total cross-sectional area scaling factor. Values of /3 greater than unity are seen to have a negligible effect on the breath by breath value of NS for all gases, while values less than unity have an effect in inverse proportion to the molecular diffusivity of the gas. For SF6, the effect of decreasing /3 is very pronounced due to its low molecular diffusivity. Previous results in applying the SPM to healthy and diseased human lungs (Schwardt et al., 1994) suggest that values o f / 3 less than 1.0, around 0.7 or less, are necessary to obtain agreement with human gas washout data.
For a given inert gas and /3 value, the NS predicted by the SPMO remains very small and constant throughout the entire simulation, at a value approximately equal to the first breath prediction of the SPM. As noted by Crawford et al., (1985) this is due to the fact that if one normalizes the airway concentration C(z,t) by its initial value Co, then each breath is mathematically equivalent with the same initial and boundary conditions and the NS, which depends on C / C 0, becomes independent of breath number. A similar breath number insensitivity occurs in the SPM with variation of the mixed venous inert gas concentration C a in the blood source term. If C(z,t) is normalized by C B, then each breath becomes equivalent and the plot of NS vs. breath number becomes insensitive to CB; e.g. for a given gas, one obtains the same steady state asymptote independent of C B. The inspiratory and expiratory flow waveform does have an effect on the NS. The SPM predicts
P.W. Scherer et al./Respiration Physiology 103 (1996) 99-103
that steady flows as used by Crawford et al. (1985) produce higher values of NS than sinusoidal flows. This is because a steady flow produces higher convective gas velocities in the acinus during the beginning of inspiration which prevent acinar gas from diffusing back towards the mouth and equilibrating by diffusion as quickly as occurs with a sinusoidal wave form.
5. Conclusion The agreement of the SPM steady state predictions of the normalized slope with experiment for inert gas washout in humans together with similar agreement obtained for CO 2 (Schreiner et al., 1993, Ream et al., 1995) suggests that the model is a valid representation of these processes. Spatial heterogeneities of gas concentration and flow throughout the lung during washout undoubtedly exist. The present study indicates that at least in the steady state, airway inhomogeneity is not essential to model lung washout data, and that a distributed blood source term in the SPM yields good agreement with experiment.
Acknowledgements These studies were supported in part by V A Merit Review research grant 103.8 and by NIH grant H L 33891.
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