Abstract
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 onedimensional trumpet model of the lungs to assess the impact on NO exchange dynamics, particularly the impact on the estimation of flowindependent 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
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 coworkers (24, 26) first combined experimental results with a twocompartment 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 twocompartment model (17, 26) to characterize NO exchange dynamics. Use of the twocompartment 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 twocompartment 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 flowindependent parameters include the maximum flux of NO from the airways (J _{awNO}), the diffusing capacity of NO in the airways (D_{awNO}), and the steadystate alveolar concentration (Ca _{ss}). 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 gasexchange 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 SF_{6}) from the lungs, as well as the mechanisms underlying the positive slope of the alveolar plateau of CO_{2} and N_{2} (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 flowindependent 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 flowindependent parameters. We developed a onedimensional 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 J _{awNO} and D_{awNO}; however, this result is strongly dependent on a significant production of NO in the small airways.
Glossary
 A_{I,II}
 Area under the curve in phases I and II of the exhaled NO profile, parts per billion (ppb)/s
 A_{aw}
 Total surface area of airway space, cm^{2}
 A_{c,aw}(z)
 Crosssectional area of airway space, cm^{2}
 A_{c,a}
 Total crosssectional area of alveolar space, cm^{2}
 C_{NO}
 Concentration of NO in the gas phase of the lungs, ppb
 Ca_{ss}
 Steadystate alveolar concentration of NO, ppb
 C_{exh}
 Exhaled NO concentration, ppb
 C^{*}_{exh}
 Modelpredicted exhaled concentration, ppb
 C_{NOplat}
 Plateau exhaled NO concentration at a constant exhalation flow rate, ppb
 D_{awNO}
 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 · s^{−1} · ppb^{−1}
 D′_{awNO}
 Diffusing capacity of NO in the airway per unit axial distance, ml · s^{−1} · cm^{−1}
 Da_{NO}
 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 · s^{−1} · (ml NO/ml gas)^{−1}] and is equivalent to pl · s^{−1} · ppb^{−1}
 D′_{aNO}
 Diffusing capacity of NO in the alveoli per unit axial distance, ml · s^{−1} · cm^{−1}
 D_{NO,air}
 Molecular diffusivity (diffusion coefficient) of NO in air, cm^{2}/s
 J_{awNO}
 Maximum total volumetric flux (ppb · ml · s^{−1} or pl/s) of NO from the airways
 J′_{awNO}
 Maximum total volumetric flux of NO from the airways per unit axial distance, ppb · ml · s^{−1} · cm^{−1}
 Ja_{NO}
 Maximum total volumetric flux (ppb · ml · s^{−1} or pl/s) of NO from the alveoli
 J′_{aNO}
 Maximum total volumetric flux of NO from the alveoli per unit axial distance, ppb · ml · s^{−1} · cm^{−1}
 L
 Length of airway in trumpet model (27.40 cm)
 N(z)
 Number of alveoli per unit axial distance
 N_{t}
 Total number of alveoli
 N_{max}
 Maximum number of alveoli at any axial position
 n_{III}
 Number of data points in phase III of the exhalation profile
 ρ
 Molar density of the gas in the lungs (mol/cm^{3}); constant
 RMS
 Rootmeansquare error between experimental data and model prediction
 V˙e
 Volumetric flow rate of air during expiration
 z
 Axial position in the lungs, cm
METHODS
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.1
A and is based on Weibel's anatomic data (28). We will reserve the term “twocompartment” model to describe the model previously reported to describe NO exchange dynamics (24) in which the alveolar region is a single wellmixed 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
), for the concentration [in parts per billion (ppb)] of NO in the airway gas phase (C_{NO}) is obtained from a mass balance over a differential volume of length Δz:
The lefthand side of Eq. 1 represents NO accumulation in the gas phase. The first term in the righthand side ofEq. 1 represents axial convection, the second term represents axial diffusion with variable crosssectional area, the third term represents the airway production and adsorption rate of NO, and the last term represents the alveolar production and adsorption rate of NO. For z < 26.8 cm, there is no alveolar contribution to NO exchange, and N(z) = 0. This sets the fourth term on the righthand side of Eq. 1 to zero (no alveolar contribution), and the airway contribution includes the entire surface area, i.e., the term [1 −N(z)/N _{max}] = 1. Forz > 26.8 cm, N(z) progressively increases (see Fig. 1 B) with z such that the magnitude of the third and fourth term on the righthand side ofEq. 1 decreases and increases, respectively. For example, at the end of the trumpet, N(z) =N _{max}, and there is no contribution from the airway source (i.e., 1 −N _{max}/N _{max} = 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)/N _{max}, which is 1 − 143 × 10^{6}/298.1 × 10^{6} = 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. J _{awNO}, D_{awNO},J a _{NO}, and Da _{NO} 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 C_{NO}(z,t) that satisfies the governing equation, the airway is divided into K sections with K+1 node points in the axial position (zcoordinate). Each node is separated by a 0.2cm interval (a smaller interval of 0.1 cm interval did not affect the solution; see ). 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 stepsize 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 × 10^{−5}, and the requested maximal error tolerance for the dependent variable, C_{NO}, was 1.0 × 10^{−7}.
There is assumed to be no flux of NO across the end of the alveolar sacs (axial gradient is zero), which is equivalent to assuming equilibrium between alveolar NO production and adsorption. This has been shown to be a good assumption for exhalations that last >10 s, including any breathhold time (5, 24). The airway is assumed initially to have a uniform zero NO concentration.
Because Weibel's symmetrical bifurcating model utilizes a generation number within the transition (generations 17–19) and respiratory (generations 17–23) regions, we sought values for N(z) that would provide a similar distribution of alveolar space over the same axial dimensions. This is detailed in Fig. 1 B, where the open bars represent the number of alveoli at the generation number and axial position in the Weibel model, and the shaded bars represent the number of alveoli and axial position (0.2cm intervals) in our trumpet model. Additional details regarding the boundary conditions and numerical solution are in the .
Experimental data and model simulation.
To simulate a series of different breathing maneuvers, we utilized previously estimated values for the flowindependent NO parameter values from the twocompartment model: J _{awNO}(ppb · ml · s^{−1}or pl/s) = 640; D_{awNO} (ml/s or pl · s^{−1} · ppb^{−1}) = 4.2; J a _{NO}(ppb · ml · s^{−1} or pl/s) = 3,638; Da _{NO} (ml/s or pl · s^{−1} · ppb^{−1}) = 1,467 (17, 25). Note that Ca _{ss} =J a _{NO}/Da _{NO}. We determined the impact of axial diffusion on NO gas exchange by performing simulations in the presence (D _{NO,air} = 0.23 cm^{2}/s) or in the absence (D _{NO,air} = 0) of axial diffusion. We then compared the performance of the model to two different types of experimental breathing maneuvers in the literature.
Breathing maneuver 1 is a single exhalation (vital capacity maneuver) preceded by a 20s breath hold in which the exhalation flow rate decreases approximately linearly in time. This maneuver was first described by our group (26) and can be used to estimate D_{awNO}, J _{awNO}, and Ca _{ss} from a single breathing maneuver. We combined the experimental breathing maneuvers from 10 healthy adults (each repeated five times) as previously reported (17) into a single composite profile representative of healthy adults. The exhalation flow rate decreases approximately linearly in time with an average relationship equal to 200 − 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 (A
_{I,II}) of the exhalation profile (Fig. 2
A) (17, 18, 26). The boundaries of phases I and II in the exhalation profile are defined as the start of exhalation (V˙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
Breathing maneuver 2 is a vital capacity maneuver without a breath hold in which the exhalation flow rate is held constant. We will examine data from our group at both 50 and 250 ml/s, as recommended by the American Thoracic Society (ATS) (1) and the European Respiratory Society (6), respectively, and also that reported by Silkoff et al. (22) over a much wider range of flow rates (4.2–1,550 ml/s). In this case, we are interested in predicting the plateau exhaled concentration (C_{NOplat}) in phase III (alveolar plateau) of the exhalation profile.
For both breathing maneuvers, inspired volume is assumed to be 4 liters at an inspiration flow rate of 2 l/s. In all cases, the experimentally reported value will have an asterisk (*). For example, the experimentally determined area under the curve in phases I and II will be denoted A ^{*} _{I,II}.
RESULTS
Fig. 2 A is the experimental composite exhalation profile for breathing maneuver 1 (17). Fig.2 B 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 bothA _{I,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 twocompartment model that was used to estimate the values used for D_{awNO},J _{awNO}, Da _{NO}, andJ a _{NO} used in the simulation. In the presence of axial diffusion, the peak value of NO in phases I and II is not affected; however, A _{I,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.
C_{NOplat} is presented in Fig.3 for breathing maneuver 2. Both experimental and trumpetmodel predicted values are shown in the presence and absence of axial diffusion. C_{NOplat}, 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 twocompartment model. However, in the presence of axial diffusion, C_{NOplat} as predicted by the trumpet model is significantly reduced. This effect is particularly exaggerated at lower flow rates.
Figure 4 presents the dynamic features of NO accumulation and elimination for breathing maneuver 1(inspiration, a 20s breath hold, and expiration) in the absence (Fig.4 A) and presence (Fig. 4 B) 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 16again 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 steadystate 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 smallerA _{I,II} as previously described.
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. 2 B, and the concentration along the airways remains lower until approximatelyz = 22.5 cm. For z > 22.5 cm, the NO concentration in the airways is larger in the presence of axial diffusion.
Figure 6 presentsA _{I,II}, RMS, and C_{NOplat} for the two breathing maneuvers for a series of trumpetmodelsimulated cases in which airway and alveolar compartment parameters are varied. Each parameter is normalized by a “gold standard” such that a value on the yaxis of unity is optimal. A _{I,II}is normalized by the experimental value shown in Fig. 2 A. RMS is normalized by the minimal (or optimal) value obtained by the twocompartment model as previously reported (17). C_{NOplat} 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 flowindependent parameters by simultaneously simulating the experimentally observed NO exchange dynamics from both breathing maneuvers.
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 trumpetmodel prediction in the presence of axial diffusion whenJ a _{NO}, Da _{NO},J _{awNO}, and D_{awNO} are each increased fourfold. The last bar represents the trumpetmodel prediction in the presence of axial diffusion when both J _{awNO} and D_{awNO} 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 flowindependent NO parameters.
Relative to the case in the presence of axial diffusion (second bar in Fig. 6), if J a _{NO} is increased,A _{I,II} is increased to a value near the experimentally observed value, RMS is decreased slightly, and C_{NOplat} is increased to near experimentally observed values. If Da _{NO} is increased,A _{I,II} is unaffected, RMS increases slightly, and C_{NOplat} decreases slightly. If J _{awNO}is increased, A _{I,II} is increased to a value above that observed experimentally, RMS is decreased, and C_{NOplat} is increased to near the experimentally observed value. If D_{awNO} is increased fourfold,A _{I,II} is decreased, and the RMS and C_{NOplat} are essentially unaffected. The last bar represents the case where both J _{awNO} and D_{awNO}are increased fourfold. In this case, both A _{I,II}and C_{NOplat} are changed to near experimental values, and the RMS is also significantly decreased. In addition, when bothJ _{awNO} and D_{awNO} are increased fourfold, C_{NOplat} 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 J _{awNO} and D_{awNO} (case 4) and also the case in whichJ a _{NO} is increased fourfold (case 3). The presence of axial diffusion with a fourfold increase in both J _{awNO} and D_{awNO}(case 4) forces phases I and II to be taller and narrower than in the absence of axial diffusion (case 1) while keeping A _{I,II} constant. A fourfold increase inJ a _{NO} (case 3) uniformly (in time) increases the exhaled concentration in phase III, whereas a fourfold increase in both J _{awNO} and D_{awNO} (case 4) increases primarily the latter portion of phase III and is more consistent with experimental observations.
DISCUSSION
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 onedimensional trumpet model of lungs to describe gasexchange 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 flowindependent 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 twocompartment 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 modelpredicted 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 onedimensional 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 ventilationtovolume 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 inertgas exhalation profile (11). The impact of ventilation to volume heterogeneity on the estimation of the flowindependent parameters is not known and is a potential topic of future studies. A second possibility is the impact of a changing airway crosssectional area during inhalation and exhalation. Our simple trumpet model assumes a rigid geometry with an effective mean value for A _{c,aw}(z) andA _{c,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,A _{c,aw}(z) andA _{c,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. 2 B) results in less NO being eliminated in this portion of the exhaled profile. Estimation of J _{awNO} and D_{awNO} is sensitive to the volume of NO eliminated in the phases I and II peak after a breath hold (26). This is evident by analyzingEq. 1 , which demonstrates that the flux of NO into the airway space from the airway wall is the difference betweenJ _{awNO} and D_{awNO} · C (third term on righthand side). At very small flow rates (<50 ml/s), the concentration in the gas phase increases (Fig. 3); thus the product D_{awNO} · C becomes important in determining the concentration in the gas phase. At higher flow rates, the exhaled concentration depends mainly onJ _{awNO}; thus estimation of D_{awNO}depends solely on phases I and II, whereas estimation ofJ _{awNO} 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.2 A). 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. 2 B 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. 2 B and Fig. 3 that axial diffusion has a greater effect at smaller flow rates. In addition, although increasing J a _{NO}by fourfold increases the endexhaled 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.6 C).
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:
Impact of axial diffusion on flowindependent 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.,J a _{NO} andJ _{awNO}) need to be increased. We used three experimental endpoints (A _{I,II}, RMS, and C_{NOplat}) from two different breathing maneuvers that included all three phases of the exhalation to estimate the potential impact of axial diffusion on flowindependent parameters (Fig. 6). It was evident that both J a _{NO} andJ _{awNO} could increase exhaled NO concentration; however, only increasing J _{awNO} 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 inJ _{awNO} 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 D_{awNO} 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 D_{awNO}). In summary, a fourfold increase in bothJ _{awNO} and D_{awNO} 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) thatJ _{awNO} 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 inJ _{awNO}.
The potential impact of axial diffusion on D_{awNO} 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:
Representative values for A _{aw},L _{ti}, λ_{ti,air},D _{NO,ti}, and k are 9,100 cm^{2}, 0.002 cm, 0.0412, 0.000033 cm^{2}/s, and 0.69 s^{−1} (ξ = 0.29) based on Weibel's symmetrical structure (28), reported values in the literature, and a halflife of NO in vivo of ∼1 s (18, 24, 25), respectively. Using Eq. 4 , a value of 6.35 ml/s (pl · s^{−1} · ppb^{−1}) is produced for D_{awNO}, which is very close to the values used in the simulations as well as others using a twocompartment model without axial diffusion. To increase D_{awNO} 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 L _{ti} or increase in A _{aw} seems unreasonable because they are already at their realistic limits. D_{NO,ti} and λ_{ti,air} are wellcharacterized physical parameters and not likely to vary much from the values reported above; k is not well characterized, and the dependence of D_{awNO} onk is highly nonlinear. For k < 1, D_{awNO} is essentially independent of k (Eq.4 ); however, for k > 1, D_{awNO} becomes a strong positive function. Nonetheless, in order for D_{awNO}to attain values on the order of 20 ml/s, k would need to be ∼70 s^{−1} or a halflife 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 D_{awNO} 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 J _{awNO}and D_{awNO} 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 singlebreath maneuver ingeneration 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 increasingJ _{awNO} and D_{awNO} 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 D_{awNO} andJ _{awNO} 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 flowindependent 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 onedimensional 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 twocompartment 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.
Acknowledgments
This work was supported by National Heart, Lung, and Blood Institute Grant HL60636.
Footnotes

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 926972575 (Email: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
 Copyright © 2002 the American Physiological Society
Appendix
Governing equation.
The unsteadystate simultaneous convectiondiffusion equation from the trumpetshaped Weibel lung (Fig. 1) is derived by the following NO mass balance and a total mass balance, respectively, over a differential length Δz:
NO mass balance
The initial condition for Eq. 1 for inspiration is C_{NO}(z,t = 0), and for expiration it is equal to the concentration profile just before exhalation (either after the 20s breath hold for breathing maneuver 1 or end inspiration for breathing maneuver 2). The boundary conditions for Eq. 1 are as follows
Inspiration
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 modelpredicted exhaled NO profile, using the same parameters as that in Fig. 2 B, is essentially identical at grid sizes of both 0.2 and 0.1 cm.