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Departments of 1 Chemical Engineering and Materials Science and 2 Biomedical Engineering, University of California, Irvine, California 92697
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Nitric oxide (NO) appears in the exhaled breath and is a potentially important clinical marker. The accepted model of NO gas exchange includes two compartments, representing the airway and alveolar region of the lungs, but neglects axial diffusion. We incorporated axial diffusion into a one-dimensional trumpet model of the lungs to assess the impact on NO exchange dynamics, particularly the impact on the estimation of flow-independent NO exchange parameters such as the airway diffusing capacity and the maximum flux of NO in the airways. Axial diffusion reduces exhaled NO concentrations because of diffusion of NO from the airways to the alveolar region of the lungs. The magnitude is inversely related to exhalation flow rate. To simulate experimental data from two different breathing maneuvers, NO airway diffusing capacity and maximum flux of NO in the airways needed to be increased approximately fourfold. These results depend strongly on the assumption of a significant production of NO in the small airways. We conclude that axial diffusion may decrease exhaled NO levels; however, more advanced knowledge of the longitudinal distribution of NO production and diffusion is needed to develop a complete understanding of the impact of axial diffusion.
gas exchange; model; exhaled breath
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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NITRIC OXIDE (NO) is an important physiological mediator within the lungs and has potential clinical importance as a noninvasive marker of lung inflammation as well (2). However, NO exchange dynamics in the lungs are not yet fully developed, primarily because of the fact that exhaled NO has both airway and alveolar contributions (3, 5, 13, 21). Tsoukias and co-workers (24, 26) first combined experimental results with a two-compartment model in an attempt to describe both alveolar and airway sources. This was followed by similar models described by Pietropaoli et al. (12) and Silkoff et al. (23) as well as new breathing techniques that used the two-compartment model (17, 26) to characterize NO exchange dynamics. Use of the two-compartment model to enhance our understanding of a range of inflammatory diseases such as asthma (23), cystic fibrosis (18), and allergic alveolitis (8, 9) quickly followed.
The important feature of the two-compartment model is the partitioning of exhaled NO into airway and alveolar contributions and characterizing NO exchange with as few as three parameters that do not depend on the exhalation flow rate. These flow-independent parameters include the maximum flux of NO from the airways (JawNO), the diffusing capacity of NO in the airways (DawNO), and the steady-state alveolar concentration (CAss). To maintain conceptual and mathematical simplicity, all of the models presented thus far have neglected axial diffusion in the gas phase and considered only convection of NO in the airways as a mode of transport. However, there is ample evidence in the literature suggesting that axial diffusion in the gas phase is an important gas-exchange mechanism, particularly in the very small airways and alveoli. Previous investigators have demonstrated that axial diffusion can play an important role in describing the washout of inert gases (He and SF6) from the lungs, as well as the mechanisms underlying the positive slope of the alveolar plateau of CO2 and N2 (10, 11, 14, 15). NO contrasts with these other gases in that it has both an airway and an alveolar source. Thus the importance of axial diffusion on NO exchange dynamics has yet to be investigated. Because of the fact that NO has a source in the peripheral lung (where the relative impact of axial diffusion is greatest), we hypothesize that axial diffusion may play a role in NO gas exchange and affect the estimation of flow-independent NO exchange parameters.
The goal of this study is to assess the impact of axial diffusion on NO gas exchange and, in particular, on the estimation of the previously described flow-independent parameters. We developed a one-dimensional model ("trumpet model") of NO gas exchange based on the symmetrical bifurcating structure of Weibel (28). We evaluated the performance of the model in the presence and absence of axial diffusion by comparing it to experimental exhaled NO data in the literature. We conclude that axial diffusion may decrease the exhaled concentration of NO, and the relative importance is inversely related to exhalation flow rate. The potential loss of NO in the exhaled breath may lead to an underestimation in both JawNO and DawNO; however, this result is strongly dependent on a significant production of NO in the small airways.
Glossary
| AI,II | Area under the curve in phases I and II of the exhaled NO profile, parts per billion (ppb)/s |
| Aaw | Total surface area of airway space, cm2 |
| Ac,aw(z) | Cross-sectional area of airway space, cm2 |
| Ac,A | Total cross-sectional area of alveolar space, cm2 |
| CNO | Concentration of NO in the gas phase of the lungs, ppb |
| CAss | Steady-state alveolar concentration of NO, ppb |
| Cexh | Exhaled NO concentration, ppb |
| C*exh | Model-predicted exhaled concentration, ppb |
| CNOplat | Plateau exhaled NO concentration at a constant exhalation flow rate, ppb |
| DawNO | Diffusing capacity (ml/s) of NO in the entire airway tree, which is
expressed as the volume of NO per second per fractional concentration
of NO in the gas phase [ml
NO · s 1 · (ml NO/ml gas)1] and is equivalent to
pl · s1 · ppb1
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| D'awNO | Diffusing capacity of NO in the airway per unit axial distance,
ml · s1 · cm1
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| DANO | Diffusing capacity (ml/s) of NO in the alveoli, which is expressed as
the volume of NO per second per fractional concentration of NO in the
gas phase [ml
NO · s1 · (ml
NO/ml gas)1] and is equivalent to
pl · s1 · ppb1
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| D'ANO | Diffusing capacity of NO in the alveoli per unit axial distance,
ml · s1 · cm1
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| DNO,air | Molecular diffusivity (diffusion coefficient) of NO in air, cm2/s |
| JawNO | Maximum total volumetric flux
(ppb · ml · s1 or
pl/s) of NO from the airways
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| J'awNO | Maximum total volumetric flux of NO from the airways per unit axial
distance,
ppb · ml · s1 · cm1
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| JANO | Maximum total volumetric flux
(ppb · ml · s1 or
pl/s) of NO from the alveoli
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| J'ANO | Maximum total volumetric flux of NO from the alveoli per unit axial
distance,
ppb · ml · s1 · cm1
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| L | Length of airway in trumpet model (27.40 cm) |
| N(z) | Number of alveoli per unit axial distance |
| Nt | Total number of alveoli |
| Nmax | Maximum number of alveoli at any axial position |
| nIII | Number of data points in phase III of the exhalation profile |
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Molar density of the gas in the lungs (mol/cm3); constant |
| RMS | Root-mean-square error between experimental data and model prediction |
 E |
Volumetric flow rate of air during expiration |
| z | Axial position in the lungs, cm |
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Model development.
The structure of the trumpet model used to describe both the airways
and the alveolar region of the lungs is shown in Fig. 1A and is based on Weibel's
anatomic data (28). We will reserve the term
"two-compartment" model to describe the model previously reported
to describe NO exchange dynamics (24) in which the alveolar region is a single well-mixed compartment and not distributed axially, as in the trumpet model described in this study. The following
governing partial differential equation (additional details of the
derivation presented in the APPENDIX), for the
concentration [in parts per billion (ppb)] of NO in the airway gas
phase (CNO) is obtained from a mass balance over a
differential volume of length 
z:
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(1) |
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is volumetric
flow rate of air (cm3/s),
J'awNO and
J'ANO are the
maximum airway and alveolar fluxes per unit axial distance,
respectively, of NO
(ppb · ml · s1 · cm1),
and D'awNO and
D'ANO are the airway and
alveolar diffusing capacities per unit axial distance, respectively
(ml · s1 · cm1).
Units for J'awNO,
J'ANO,
D'awNO, and
D'ANO are described in
greater detail in the GLOSSARY. The trumpet is assumed to
be rigid; thus Ac,aw(z) and
Ac,A are not a function of time. Air
passes through the trumpet on inspiration at z = 0 and
is assumed to exit the trumpet at CAss. On
exhalation, air enters the trumpet at z = length of
airway in trumpet model (L) at
CAss. During both inspiration and
expiration, the volumetric flow rate of air is assumed to be constant
with respect to axial position.
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N(z)/Nmax] = 1. For
z > 26.8 cm, N(z) progressively
increases (see Fig. 1B) with z such that the
magnitude of the third and fourth term on the right-hand side of
Eq. 1 decreases and increases, respectively. For example, at
the end of the trumpet, N(z) = Nmax, and there is no contribution from the
airway source (i.e., 1 Nmax/Nmax = 0). The
alveolar source at this axial position is the fraction of the alveolar
surface area not utilized in the previous axial positions. The fraction
is equal to 1 N(z)/Nmax, which is 1 143 × 106/298.1 × 106 = 0.48. In other words, 48% of the alveolar source of NO occurs at this axial position because of the fact that 48% of the alveoli are
present or partitioned to this position. Radial velocity
gradients are neglected (plug flow), and only the exchange of NO is
considered. JawNO, DawNO,
JANO, and
DANO are assumed to be constant and uniformly distributed per unit volume.
Model solution.
The governing equation is solved numerically using the method of lines
(16, 20). This method uses a finite difference relationship for the spatial derivatives and ordinary differential equations for the time derivative. Thus, to seek
CNO(z,t) that satisfies the governing
equation, the airway is divided into K sections with K+1 node points in
the axial position (z-coordinate). Each node is separated by
a 0.2-cm interval (a smaller interval of 0.1 cm interval did not affect
the solution; see APPENDIX). The first node is just before
the mouth (z = 0), and the last node
point (z = L) represents the end of the
alveolar sacs. Then, a stiff integration algorithm including error
estimation and time step-size control is used to ensure accuracy of the
solution (16, 20). In all simulations, the accuracy of the
independent variable (t) was set to 1.0 × 105, and the requested maximal error tolerance for the
dependent variable, CNO, was 1.0 × 107.
Experimental data and model simulation.
To simulate a series of different breathing maneuvers, we utilized
previously estimated values for the flow-independent NO parameter
values from the two-compartment model: JawNO
(ppb · ml · s1
or pl/s) = 640; DawNO (ml/s or
pl · s1 · ppb1) = 4.2; JANO
(ppb · ml · s1 or
pl/s) = 3,638; DANO (ml/s or
pl · s1 · ppb1) = 1,467 (17, 25). Note that
CAss = JANO/DANO.
We determined the impact of axial diffusion on NO gas exchange by
performing simulations in the presence
(DNO,air = 0.23 cm2/s) or in
the absence (DNO,air = 0) of axial
diffusion. We then compared the performance of the model to two
different types of experimental breathing maneuvers in the literature.
10 × t (ml/s)
where t is time (s) and the total expiratory time is 15 s.
For breathing maneuver 1, we are interested in predicting
the dynamic shape of the profile. Consistent with our previous
approach, this will include the volume of NO exhaled in phases I and II of the exhalation profile and the dynamic shape of phase III (17, 18, 26). The volume of NO that accumulates in the airways during
the breath hold has previously been characterized by the area under
curve in phases I and II (AI,II) of the
exhalation profile (Fig. 2A)
(17, 18, 26). The boundaries of phases I and II in the
exhalation profile are defined as the start of exhalation
(E > 0) and when the slope of exhaled concentration with time is equal to zero (26). Consistent with
our previous report, the root mean square error (RMS) will be used as
an index of the goodness of fit for the dynamic shape of phase III
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(2) |
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Fig. 2A is the experimental composite exhalation profile for breathing maneuver 1 (17). Fig. 2B presents the corresponding simulations of the trumpet model in the presence and absence of axial diffusion. Note that, in the absence of axial diffusion, the trumpet model is able to reproduce both AI,II and the dynamic shape of phase III (RMS = 0.94 ppb). This is an important feature of the performance of the trumpet model compared with the two-compartment model that was used to estimate the values used for DawNO, JawNO, DANO, and JANO used in the simulation. In the presence of axial diffusion, the peak value of NO in phases I and II is not affected; however, AI,II is reduced by ~50%, the concentration in phase III is reduced by a similar magnitude (concentration at end exhalation is reduced from 12.4 ppb to 6.24 ppb), and RMS in phase III increases to 3.0 ppb.
CNOplat is presented in Fig.
3 for breathing maneuver 2.
Both experimental and trumpet-model predicted values are shown in the
presence and absence of axial diffusion. CNOplat, using the trumpet model in the absence of axial diffusion, matches the
experimental values over a wide range of exhalation flow rates
consistent with the performance of the two-compartment model. However,
in the presence of axial diffusion, CNOplat as predicted by
the trumpet model is significantly reduced. This effect is particularly
exaggerated at lower flow rates.
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Figure 4 presents the dynamic features of
NO accumulation and elimination for breathing maneuver 1 (inspiration, a 20-s breath hold, and expiration) in the absence (Fig.
4A) and presence (Fig. 4B) of axial diffusion. In
the absence of axial diffusion, NO accumulates uniformly
(generations 0, 12, and 16 are
indistinguishable) within the airways in an exponential fashion
(24) during the breath hold. On exhalation, the width of
phases I and II is relatively broad, reflecting significant NO levels
throughout the airways. In the presence of axial diffusion, the
concentration of NO in generations 12 and 16 again increases in an exponential fashion but approaches a smaller
concentration compared with when axial diffusion is neglected. The
concentration at the end of the trumpet (generation 23) does
not change during the breath hold, indicating a steady-state
concentration in the alveolar region of ~2-3 ppb. On exhalation,
the peak value (generation 0) is not changed, but the width
of phases I and II is narrowed, reflecting lower concentrations of NO
in the smaller airways. This finding is consistent with a smaller
AI,II as previously described.
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Figure 5 is the axial NO concentration in
the airways just before exhalation (t = 0 s) and
at end exhalation (t = 15 s) for breathing
maneuver 1 in the presence and absence of axial diffusion. Axial
diffusion does not affect (<10% change) the NO concentration in the
first 10 generations (z < 25 cm) at the end of the
breath hold. For z > 25 cm, the NO concentration
begins to decrease in the presence of axial diffusion such that
the concentrations in generations 12 (z = 25.78 cm) and 16 (z = 26.65 cm) are reduced by 18 and 50%, respectively. At end expiration, the exhaled
concentration (z = 0) is significantly reduced in the
presence of axial diffusion, consistent with Fig. 2B, and
the concentration along the airways remains lower until approximately
z = 22.5 cm. For z > 22.5 cm, the NO
concentration in the airways is larger in the presence of axial
diffusion.
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Figure 6 presents
AI,II, RMS, and CNOplat for the two
breathing maneuvers for a series of trumpet-model-simulated cases in which airway and alveolar compartment parameters are varied. Each parameter is normalized by a "gold standard" such that a value on
the y-axis of unity is optimal. AI,II
is normalized by the experimental value shown in Fig. 2A.
RMS is normalized by the minimal (or optimal) value obtained by the
two-compartment model as previously reported (17).
CNOplat is shown for an exhalation flow rate of ~50 ml/s
(61.6 ml/s, from Ref. 19), which was the mean experimental
value previously reported in 10 healthy adults and is approximately
equal to the ATS guidelines. The goal is to estimate the impact
of axial diffusion on previous estimates of flow-independent parameters
by simultaneously simulating the experimentally observed NO exchange
dynamics from both breathing maneuvers.
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The first two bars in Fig. 6 represent the trumpet model in the absence and presence of axial diffusion, respectively, when the parameter values described above in Experimental data and model simulation are used. The next four bars represent the trumpet-model prediction in the presence of axial diffusion when JANO, DANO, JawNO, and DawNO are each increased fourfold. The last bar represents the trumpet-model prediction in the presence of axial diffusion when both JawNO and DawNO are increased fourfold. This provides a measure of the sensitivity of each of the experimental endpoints to the model parameters and represents a method by which we can describe general trends on the impact of axial diffusion on the estimated of the flow-independent NO parameters.
Relative to the case in the presence of axial diffusion (second bar in Fig. 6), if JANO is increased, AI,II is increased to a value near the experimentally observed value, RMS is decreased slightly, and CNOplat is increased to near experimentally observed values. If DANO is increased, AI,II is unaffected, RMS increases slightly, and CNOplat decreases slightly. If JawNO is increased, AI,II is increased to a value above that observed experimentally, RMS is decreased, and CNOplat is increased to near the experimentally observed value. If DawNO is increased fourfold, AI,II is decreased, and the RMS and CNOplat are essentially unaffected. The last bar represents the case where both JawNO and DawNO are increased fourfold. In this case, both AI,II and CNOplat are changed to near experimental values, and the RMS is also significantly decreased. In addition, when both JawNO and DawNO are increased fourfold, CNOplat is also accurately predicted over the entire range of constant exhalation flow rates as shown in Fig. 3.
Figure 7 is the simulated exhalation NO
profile for breathing maneuver 1 with a fourfold increase in
both JawNO and DawNO (case
4) and also the case in which
JANO is increased fourfold (case 3). The presence of axial diffusion with a fourfold
increase in both JawNO and DawNO
(case 4) forces phases I and II to be taller and narrower
than in the absence of axial diffusion (case 1) while
keeping AI,II constant. A fourfold increase in
JANO (case 3) uniformly
(in time) increases the exhaled concentration in phase III,
whereas a fourfold increase in both JawNO and
DawNO (case 4) increases primarily the latter
portion of phase III and is more consistent with experimental
observations.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The currently accepted model of NO gas exchange divides the lungs into two compartments: the airways, in which convection is the dominant transport mechanism, and the alveolar region, which is assumed well mixed. Axial diffusion as a mechanism of transport in the gas phase has been neglected for simplicity. This study has incorporated axial diffusion into a one-dimensional trumpet model of lungs to describe gas-exchange dynamics of NO in the lungs. The explicit purpose was to determine the potential impact of axial diffusion on NO exchange dynamics, particularly on the estimation of flow-independent parameters that have been used to characterize the airway and alveolar compartments. We determined that axial diffusion results in the transport of NO toward the alveoli and thus decreases the elimination of NO in the exhaled breath. The loss of NO leads to potentially underestimating both the maximum airway flux and the airway diffusing capacity for NO in models that neglect axial diffusion; however, this conclusion is strongly dependent on a significant source of NO production in the small airways.
Impact of axial diffusion on phases I and II of exhaled NO profile. Our previously described two-compartment model neglected axial diffusion and could not accurately predict the shape of the exhalation NO profile in phases I and II after a breath hold (17, 18, 26). In these reports, we suggested that axial diffusion was a potential cause, and we used the model to predict the area under the curve in phases I and II rather than the precise shape. When axial diffusion was included in the trumpet model in the present study, the shape of phases I and II was substantially altered (narrower width with a sharper peak and smoother transition to phase III); however, the shape still does not match that of the experimental data (Fig. 2). The experimentally observed shape of phases I and II remains broader (nearly 4 s compared with the model-predicted value of 2 s for the same exhalation flow rate). Thus it is apparent that additional mechanisms of gas mixing are still neglected in this simple one-dimensional model, which may be important to fully describe NO exchange mechanisms.
One important possibility is ventilation to volume heterogeneity, which results in different regions of lungs filling and emptying at different rates. During exhalation, regions of the lungs with lower concentrations (high ventilation-to-volume ratio) tend to empty first. This contributes to the positive phase III slope of inert gases, as well as phase II (transition from conducting airway space to the alveolar plateau) of the inert-gas exhalation profile (11). The impact of ventilation to volume heterogeneity on the estimation of the flow-independent parameters is not known and is a potential topic of future studies. A second possibility is the impact of a changing airway cross-sectional area during inhalation and exhalation. Our simple trumpet model assumes a rigid geometry with an effective mean value for Ac,aw(z) and Ac,A equal to that of the lungs utilized by Weibel (28), which were fixed at ~75% total lung capacity. In fact, during the breath hold at total lung capacity, Ac,aw(z) and Ac,A will be slightly larger, thus enhancing axial diffusion. Therefore, our predictions are likely to be a conservative estimate of the impact of axial diffusion. The observation that axial diffusion narrows the width of phases I and II without affecting the peak value (Fig. 2B) results in less NO being eliminated in this portion of the exhaled profile. Estimation of JawNO and DawNO is sensitive to the volume of NO eliminated in the phases I and II peak after a breath hold (26). This is evident by analyzing Eq. 1, which demonstrates that the flux of NO into the airway space from the airway wall is the difference between JawNO and DawNO · C (third term on right-hand side). At very small flow rates (<50 ml/s), the concentration in the gas phase increases (Fig. 3); thus the product DawNO · C becomes important in determining the concentration in the gas phase. At higher flow rates, the exhaled concentration depends mainly on JawNO; thus estimation of DawNO depends solely on phases I and II, whereas estimation of JawNO depends on all three phases.Impact of axial diffusion on phase III of exhaled NO profile. The concentration of NO in phase III of the exhalation profile depends on the relative contributions from both the alveolar and airway compartments (26). At very high flow rates (>500 ml/s), the residence time of a volumetric element of gas (i.e., gas bolus) is small, and exhaled NO is predominantly from the alveolar region. The progressively increasing concentration in phase III of the NO exhalation profile for breathing maneuver 1 is due primarily to the fact that the flow is decreasing linearly in time (Fig. 2A). Thus the alveolar contribution remains approximately constant (constant concentration in a collapsing balloon), whereas the contribution from the airways increases (larger residence time at slow flow rates). The presence of axial diffusion decreases the concentration of NO in phase III, but the effect is more exaggerated at lower flow rates (late portion of phase III, Fig. 2B and Fig. 3). If axial diffusion were primarily affecting the alveolar concentration, the impact on phase III would be uniform. For example, if the alveolar concentration increased by 5 ppb, then the entire phase III concentration would uniformly increase by 5 ppb. This is not the observed effect. It is clear from Fig. 2B and Fig. 3 that axial diffusion has a greater effect at smaller flow rates. In addition, although increasing JANO by fourfold increases the end-exhaled NO concentration and thus compensates for the loss of NO due to axial diffusion, this change does not significantly improve the RMS in phase III (Fig. 6C).
The relationship between the impact of axial diffusion and exhalation flow rate is best understood by comparing the velocity of axial convection to the velocity of diffusion. This ratio is captured in the dimensionless Peclet number (Pe) defined by:
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(3) |
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Impact of axial diffusion on flow-independent NO exchange parameters. It is evident that the presence of axial diffusion results in loss of NO to the alveolar region (Figs. 4 and 5). Thus, in order for the trumpet model to simulate the experimentally observed exhaled NO concentrations, the endogenous sources of NO (i.e., JANO and JawNO) need to be increased. We used three experimental endpoints (AI,II, RMS, and CNOplat) from two different breathing maneuvers that included all three phases of the exhalation to estimate the potential impact of axial diffusion on flow-independent parameters (Fig. 6). It was evident that both JANO and JawNO could increase exhaled NO concentration; however, only increasing JawNO compensated for the impact on all three phases (phases I and II are insensitive to alveolar NO production).
We then observed that a fourfold increase in JawNO was needed to simulate phase III, but this caused too large an increase in the area of phases I and II. Thus one could compensate for this by increasing consumption in the airways by increasing DawNO by fourfold without affecting phase III (flow rates are large enough during phase III in breathing maneuver 1 that NO concentration in phase III is insensitive to DawNO). In summary, a fourfold increase in both JawNO and DawNO compensates for the effects of axial diffusion in all three phases of the exhalation profile over a wide range of exhalation flow rates (Figs. 3 and 6). The obvious question becomes: are these predicted increases possible or reasonable? Our laboratory has previously shown (24, 25) that JawNO depends on the physical dimensions of the airway such as surface area and tissue thickness but is also a positive function of the production rate of NO per unit volume. Thus one possibility is simply a larger production rate of NO per unit volume of tissue to account for the possible increase in JawNO. The potential impact of axial diffusion on DawNO is more difficult to understand or justify. Our laboratory has demonstrated previously (24) that the diffusing capacity of a gas produced within the tissue (either airway or alveolar) can be estimated by the relatively simply expression:
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(4) |
ti,air is the tissue-air partition
coefficient of NO, Aaw (cm2) is the
surface area available for diffusion, DNO,ti
(cm2/s) is the molecular diffusivity of NO in the tissue,
Lti is the thickness of the tissue layer,
= 
1) is the first-order rate constant
that characterizes the rate of chemical consumption by substrates such
as superoxide. The represents a dimensionless ratio of the rate of
chemical consumption to the rate of molecular diffusion. When
consumption of the gas is negligible relative to diffusion
(k approaches zero), /tanh() approaches unity, and
DawNO approaches the more familiar definition of the
diffusing capacity of an inert gas. When diffusion is negligible (i.e.,
k becomes large), DawNO = Aaw
ti,air
1 and 1 and is a
monotonically increasing function of its argument. From Eq. 4, DawNO is a positive function of
Aaw, ti,air,
DNO,ti, and k, and it is an inverse
function of Lti. Eq. 4 provides units
of ml/s for DawNO, which are equivalent to
pl · s1 · ppb1.
Representative values for Aaw,
Lti, ti,air,
DNO,ti, and k are 9,100 cm2, 0.002 cm, 0.0412, 0.000033 cm2/s, and 0.69 s1 ( = 0.29) based on Weibel's symmetrical
structure (28), reported values in the literature, and a
half-life of NO in vivo of ~1 s (18, 24, 25),
respectively. Using Eq. 4, a value of 6.35 ml/s
(pl · s1 · ppb1)
is produced for DawNO, which is very close to the values
used in the simulations as well as others using a two-compartment model without axial diffusion. To increase DawNO fourfold
(~20-25 ml/s), one would need to justify appropriate changes
(from values presented above) in one or more of these physical
parameters. A further decrease in Lti or
increase in Aaw seems unreasonable because they
are already at their realistic limits. DNO,ti and
ti,air are well-characterized physical parameters and
not likely to vary much from the values reported above; k is
not well characterized, and the dependence of DawNO on
k is highly nonlinear. For k < 1, DawNO is essentially independent of k (Eq. 4); however, for k > 1, DawNO becomes a
strong positive function. Nonetheless, in order for DawNO
to attain values on the order of 20 ml/s, k would need to be
~70 s1 or a half-life of <0.01 s. This does not seem
plausible on the basis of reported reaction rates of NO with substrates
present in lung tissue (19). Hence, on the basis
of anatomical and physical constraints in the lungs, it is difficult to
justify values for DawNO greater than 5-10 ml/s (as
opposed to the prediction of 20-25 ml/s predicted by the trumpet model).
One possible solution to this dilemma is the fact that the contribution
of the small airways to exhaled NO when JawNO
and DawNO have been increased fourfold in the presence of
axial diffusion remains too large compared with experimental
observations. Silkoff et al. (21) have reported that
~50% of NO arises from the upper airways (generations
0-2). In the present simulation using the trumpet model, the
NO concentration at end expiration of the single-breath maneuver in
generation 2 (z = 18.7 cm) is 52% (see Fig.
5) of the exhaled NO concentration (z = 0 cm) in the
absence of axial diffusion (i.e., consistent with Silkoff et al.).
However, in the presence of axial diffusion, independent of increasing
JawNO and DawNO by fourfold, the
concentration at generation 2 increases to >85% of exhaled
NO concentration (z = 0 cm). Thus the trumpet model in
the presence of axial diffusion predicts too much NO production in the
smaller airways. It is possible that distributing DawNO and
JawNO uniformly per unit airway volume may not
truly represent the axial distribution of NO production and diffusion in the airways. Future theoretical studies must address these important
issues to formulate a complete understanding of the impact of axial
diffusion on the estimation of flow-independent NO exchange parameters.
In conclusion, previous models aimed at characterizing NO pulmonary
exchange dynamics have neglected axial diffusion as a transport
mechanism. This study has incorporated axial diffusion into a
one-dimensional trumpet model of NO gas exchange in the lungs. We
demonstrated that, in the absence of the axial diffusion, the trumpet
model behaves very similarly to the two-compartment model. In the
presence of axial diffusion, the trumpet model predicts a significant
backdiffusion of NO from the airways into the alveolar region. This
results in a significant loss of NO that would, therefore, not
appear in the exhaled breath. The result is a potential underestimation of both the maximum airway flux of NO and the airway diffusing capacity
for NO. This result hinges on a significant production of NO in the
very small airways, for which there is evidence to the contrary. Future
theoretical work must focus on incorporating more advanced features of
NO gas exchange consistent with experimental observations, such as
spatial heterogeneity in alveolar concentration and airway NO
production, before the true impact of axial diffusion can be determined.
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Governing equation.
The unsteady-state simultaneous convection-diffusion equation from the
trumpet-shaped Weibel lung (Fig. 1) is derived by the following NO mass balance and a total mass balance, respectively, over a differential length z:
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![<FR><NU>d</NU><DE>d<IT>t</IT></DE></FR> <FENCE>C<SUB>NO</SUB><FENCE><IT>A</IT><SUB>c,aw</SUB>(<IT>z</IT>)<IT>+</IT><FENCE><FR><NU><IT>N</IT>(<IT>z</IT>)</NU><DE><IT>N</IT><SUB>t</SUB></DE></FR></FENCE></FENCE></FENCE><IT>A</IT><SUB>c,<SC>a</SC></SUB>}&Dgr;<IT>z</IT>) = [<A><AC>V</AC><AC>˙</AC></A>C<SUB>NO</SUB>‖<SUB><IT>z</IT>+&Dgr;<IT>z</IT></SUB><SUP><IT>z</IT></SUP>]](/content/vol93/issue6/fulltext/2070/img011.gif)
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(A1) |
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![× <FENCE>1 − <FR><NU><IT>N</IT>(<IT>z</IT>)</NU><DE><IT>N</IT><SUB>max</SUB></DE></FR></FENCE>&Dgr;<IT>z</IT> + (<IT>J</IT>′<SUB><SC>a</SC><SUB>NO</SUB></SUB> − D′<SUB><SC>a</SC><SUB>NO</SUB></SUB>C<SUB>NO</SUB>) <FENCE><FR><NU><IT>N</IT>(<IT>z</IT>)</NU><DE><IT>N</IT><SUB>t</SUB></DE></FR></FENCE>]<IT> &Dgr;z</IT>](/content/vol93/issue6/fulltext/2070/img013.gif)
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![<FR><NU>d</NU><DE>d<IT>t</IT></DE></FR> <FENCE>&rgr;<FENCE><IT>A</IT><SUB>c,aw</SUB>(<IT>z</IT>) + <FENCE><FR><NU><IT>N</IT>(<IT>z</IT>)</NU><DE><IT>N</IT><SUB>t</SUB></DE></FR></FENCE> <IT>A</IT><SUB>c,<SC>a</SC></SUB></FENCE> &Dgr;<IT>z</IT></FENCE> = [&rgr;<A><AC>V</AC><AC>˙</AC></A>‖<SUB><IT>z</IT>+&Dgr;<IT>z</IT></SUB><SUP><IT>z</IT></SUP>]](/content/vol93/issue6/fulltext/2070/img014.gif)
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(A2) |
z, then
letting z 
0 produces
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(A3) |
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|
(A4) |
|
(A5) |
|
(A6) |
|
(A7) |
|
(A8) |
refers to a single
node just outside of the airway. This strategy allows a numerical
solution. For example, Eq. A6 arises as diffusion is
negligible near the mouth (z = 0), and we assume a
single node (z = 0) in which there is no
diffusion of NO from the airway wall; thus the axial concentration
gradient is zero.
Grid size in model solution. We determined that 0.2 cm was the minimum grid size necessary to understand the impact of axial diffusion on the exhaled NO profile by halving the interval size to 0.1 cm and demonstrating no significant impact on the exhaled NO profile. This is demonstrated in Fig. A1 in which the model-predicted exhaled NO profile, using the same parameters as that in Fig. 2B, is essentially identical at grid sizes of both 0.2 and 0.1 cm.
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ACKNOWLEDGEMENTS |
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This work was supported by National Heart, Lung, and Blood Institute Grant HL-60636.
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FOOTNOTES |
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Address for reprint requests and other correspondence: S. C. George, Dept. of Chemical Engineering and Materials Science, 916 Engineering Tower, Univ. of California, Irvine, Irvine, California 92697-2575 (E-mail: scgeorge{at}uci.edu).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
August 23, 2002;10.1152/japplphysiol.00129.2002
Received 20 February 2002; accepted in final form 22 August 2002.
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APPENDIX
REFERENCES
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