Probing the impact of axial diffusion on nitric oxide exchange dynamics with heliox

Probing the impact of axial diffusion on nitric oxide exchange dynamics with heliox. Appl Physiol nitric oxide (NO) is a potential noninvasive index of lung inﬂammation and is thought to arise from the alveolar and airway regions of the lungs. A two-compartment model has been used to describe NO exchange; however, the model neglects axial diffusion of NO in the gas phase, and recent theoretical studies suggest that this may introduce signiﬁcant error. We used heliox (80% helium, 20% oxygen) as the insufﬂating gas to probe the impact of axial diffusion (molecular diffusivity of NO is increased 2.3-fold relative to air) in healthy adults (21–38 yr old, n (cid:1) 9). Heliox decreased the plateau concentration of exhaled NO by 45% (exhala-tion ﬂow rate of 50 ml/s). In addition, the total mass of NO exhaled in phase I and II after a 20-s breath hold was reduced by 36%. A single-path trumpet model that considers axial diffusion predicts a 50% increase in the maximum airway ﬂux of NO and a near-zero alveolar concentration (C A NO ) and source. Furthermore, when NO elimination is plotted vs. constant exhalation ﬂow rate (range 50–500 ml/s), the slope has been previously interpreted as a nonzero C A NO (range 1–5 ppb); however, the trumpet model predicts a positive slope of 0.4–2.1 ppb despite a zero C A NO because of a diminishing impact of axial diffusion as ﬂow rate increases. We conclude that axial diffusion leads to a signiﬁcant backdiffusion of NO from the airways to the alveolar region that signiﬁcantly impacts the partitioning of airway and alveolar contributions to exhaled NO. 50 ml/s (C E NO,50 ) and thus the elimination rate. Linear regression is performed through a sequence of 3 consecutive data points, and the intercept, which has been previously inter- preted as J (cid:2) aw NO , and the slope, which has been previously interpreted as C A NO , are presented according to 4 different ﬂow rate ranges.

gas exchange; model; exhaled breath NITRIC OXIDE (NO) performs many important functions in the lungs and has been regarded as a potential noninvasive marker of lung inflammation (2). The characteristics of NO gas exchange are unique compared with other endogenous gases because exhaled NO is thought to have a significant alveolar and airway source (6,8,15,22). A two-compartment model is commonly used to characterize NO exchange dynamics for healthy and diseased lungs (9,11,13,20,21,23,24,27,29), by partitioning exhaled NO into airway and alveolar contributions using three flow-independent NO exchange parameters: maximum flux of NO from the airways (JЈaw NO ), the diffusing capacity of NO in the airways (Daw NO ), and the steady-state alveolar concentration (CA NO ). However, the two-compartment model considers only convection of NO in the airways as a transport mechanism and has neglected axial diffusion of NO in the gas phase to preserve mathematical simplicity.
Recently, our laboratory (17) and others (31) separately demonstrated theoretically that axial diffusion may play an important role in NO transport. During exhalation, the concentration of NO is higher in the airways compared with the alveoli, creating a gradient for diffusion of NO from the airways to the alveolar region (the opposite direction of convection). Thus the alveolar region may serve as a sink for NO produced in the airways, and neglecting axial diffusion may introduce significant error in the estimation of the flow-independent NO parameters such as JЈaw NO and CA NO . On the basis of our earlier theoretical result (17), we hypothesize that axial diffusion of NO in the gas phase significantly reduces exhaled NO concentration by "backdiffusion" of NO from the airways to the alveolar region. The backdiffusion of NO may falsely lower the estimate of airway flux (i.e., JЈaw NO ) and elevate the steady-state alveolar concentration (i.e., CA NO ).
To probe the impact of axial diffusion experimentally, the present study utilized heliox as the insufflating gas during a single-breath maneuver. Heliox is a mixture of 80% helium and 20% oxygen that increases the molecular diffusivity of NO 2.3-fold in the gas phase relative to air while introducing only a minimal impact on upper airway gas mixing because of a lower density (See METHODS and DISCUSSION for details). We found that heliox decreased the concentration of NO in the exhaled breath during all three phases of the exhalation profile. Analysis of the data by a single-path trumpet model that considers axial diffusion suggests that backdiffusion of NO from the airways to the alveolar region significantly impacts the partitioning of exhaled NO into airway and alveolar contributions. width of the phase I and II peak calculated by taking the volume at which the exhaled concentration is larger than 50% of C NOpeak z Axial position in the lungs (cm) z ϭ L Axial position at the end of the trumpet z ϭ L ϩ A single node just beyond the end of the trumpet

Experiment
Subjects. Nine healthy adults (five women) participated in this study, and their characteristics are presented in Table 1. All subjects had an forced expiratory volume in 1 s (FEV 1) to forced vital capacity (FVC) ratio Ͼ 0.80, and those with a history of smoking at any time, cardiovascular, pulmonary, or neurological diseases were excluded. The Institutional Review Board at the University of California, Irvine approved the protocol, and written, informed consent was obtained from all subjects. Protocol. Each subject performed two types of exhalation maneuvers in both air and heliox. When air is replaced with heliox as the insufflating gas, the molecular diffusivity, D NO,i, of NO and gas density change significantly. DNO,i increases by ϳ2.3-fold (i.e., DNO,heliox Ϸ 2.3 DNO,air) (14), and gas density decreases by approximately threefold (12). The former enhances gas-phase diffusion of NO (the impact is primarily in the small airways and is the desired effect for this study), whereas the latter reduces mixing and axial dispersion of NO primarily in the larger airways, which has a negligible impact on this study (see DISCUSSION for details).
The first maneuver was three repetitions of a single-breath maneuver that included a preexpiratory 20-s breath hold followed by a decreasing flow rate (from ϳ6 to ϳ1% of vital capacity per second) (29) to measure C NOpeak, W50, VI,II, and AI,II, and to estimate several flow-independent NO-exchange parameters. A positive pressure of Ͼ5 cmH2O was maintained to prevent nasal contamination during the breath hold (1). A Starling resistor (Hans Rudolph, Kansas City, MO) with a variable resistance was used to progressively decrease the flow rate during the exhalation. A schematic of the experimental apparatus has been previously presented (29). The second maneuver was a vital capacity maneuver performed in triplicate to collect plateau NO concentration on the basis of the American Thoracic Society guidelines (1) in which the exhalation flow rate was maintained at a target of 50 ml/s. Before performing the breathing maneuvers in heliox, each subject inhaled heliox during quiet tidal breathing for 2 min to washout nitrogen in the functional residual capacity. To determine the potential impact of nitrogen in the residual volume, each subject also performed the preexpiratory breath hold maneuver with a 20-s breath hold in the presence of heliox without tidal breathing heliox before the maneuver. After measurement of the indexes of NO-exchange dynamics (4,5,25), general spirometry such as FVC, FEV 1, and FEV1 by FVC were measured in all subjects (Vmax229; Sensormedics, Yorba Linda, CA) by using the best performance (see Table 1) from three consecutive maneuvers.
Airstream analysis. A chemiluminescence NO analyzer (NOA280, Sievers, Boulder, CO) was used to measure the exhaled NO concentration. The instrument was calibrated on a daily basis by using a certified NO gas (45 ppm NO in 100% N 2 for air calibration and 45 ppm NO in 100% He for heliox calibration; Sievers). The zero-point calibration was performed with a NO filter (Sievers) immediately before the collection of a profile. Calibration with 80% of carrier gas (either nitrogen or helium as in the case of air or heliox, respectively) balanced with oxygen resulted in a negligible change in the response of the instrument (Ͻ2% for helium). The flow rate and pressure signals were measured via a pneumotachometer (RSS100, Hans Rudolph). The pneumotachometer was calibrated daily, set to provide the flow in units of STPD, and accounted for changes in gas properties (e.g., viscosity) when using heliox.
Empirical data analysis. Experimental single-exhalation profiles with preexpiratory breath hold were characterized empirically by C NOpeak, W50, total VI,II, defined as the minimum point (zero slope or dCENO/dV ϭ 0) in the exhalation profile (29), and AI,II ( Fig. 1). In addition, the plateau concentration at a constant flow rate of 50 ml/s (CENO,50), as previously described by the American Thoracic Society (1), was also determined.

Model Development and Simulation
Two-compartment model. A previously described two-compartment model was used to estimate four flow-independent NO exchange parameters: , and 4) mean airway tissue NO concentration, CawNO (ppb) (equal to the ratio of JЈawNO to DawNO). A detailed description of the mathematical algorithm to calculate the parameters has been previously described (29).
Trumpet model. The impact of axial diffusion was assessed utilizing a "trumpet model" of the lungs (17) by comparing simulations that included (D NO,air ϭ 0.23 cm 2 /s for air or DNO,heliox ϭ 0.52 cm 2 /s for heliox) and excluded (DNO,air ϭ 0) axial diffusion. The structure of the trumpet model (see Fig. 2) is based on Weibel's anatomic data (32) (see APPENDIX for the governing equation and boundary conditions).
Development of a parameter estimation routine is beyond the scope of this work. Hence, mean values of flow-independent parameters from the nine subjects, as determined from the two-compartment model, were utilized in the trumpet model simulations. The parameters were then adjusted to predict two indexes that characterize the single-breath maneuvers (29). These include A I,II and the dynamic shape of phase III. The dynamic shape of phase III is characterized by Fig. 1. Definitions of CNOpeak, W50, VI,II, and AI,II are presented in a schematic of a representative exhalation nitric oxide (NO) profile using the single-breath technique with a preexpiratory breath hold and a decreasing exhalation flow rate. The distinction between phases I and II and phase III is the point of zero slope (minimum point) in the exhalation profile, as previously described (28). All parameters are defined in the Glossary. the normalized root mean square error between model prediction and experimental data, R III, defined by In this fashion, R III is the mean fractional deviation of the model prediction to the experimental data (i.e., the gold standard) in phase III. In a fashion similar to R III, A*I,II values are also referenced to an optimal or gold standard determined from experimental data. The new variable is denoted Ā III and is defined as where Â I,II is the area under the curve in phase I and II of the composite experimental profile (Fig. 3). Thus Ā I,II is interpreted as the fractional deviation from the gold standard (same as R III).
Statistics. Data were analyzed by a paired t-test. All variables were assumed to be normally distributed. All statistical tests were performed on the untransformed data. A P value of Ͻ0.05 was considered statistically significant. Figure 3 presents the composite experimental profile for the single-breath maneuver with a 20-s preexpiratory breath hold with a decreasing exhalation flow rate. The composite profile was attained by taking the mean exhaled concentration at equivalent exhaled volume intervals in the nine healthy subjects. A decrease in the concentration of NO exhaled in all three phases of the single-exhalation breathing maneuver was observed when heliox was the insufflating gas, although C NOpeak was not statistically lower (see Fig. 4 below).

RESULTS
C NOpeak , W 50 , V I,II , and A I,II for all nine subjects are presented in Fig. 4 to demonstrate specific differences in the exhaled NO profile between air and heliox breathing. C NOpeak (Fig. 4A) was not impacted by the presence of heliox; however, W 50 , V I,II , and A I,II (Fig. 4, B-D) were all significantly reduced (mean values of 26, 23, and 36%, respectively).
There were no differences in C NOpeak , W 50 , and V I,II with or without 2 min of tidal breathing heliox before the singleexhalation maneuver. Mean (SD) values for C NOpeak , W 50 , and V I,II in the absence or presence of the 2-min heliox tidal breathing were 52 (20) and 53 (30) ppb, 139 (28) and 124 (27) ml, and 544 (88) and 485 (55) ml, respectively. In addition, mean (SD) values for C*E NO,50 in the absence or presence of the 2-min heliox tidal breathing were 10.8 (5.94) and 11.4 (6.53) ppb, respectively, and were not statistically different. Â I,II for air and heliox were 12,800 and 8,150 ppb⅐ml, respectively.
CE NO,50 is presented in Table 2 for air and heliox. Subject 6 could not adequately perform the constant-exhalation flow-rate maneuver for both air and heliox, and subject 5 could not adequately perform the constant-exhalation flow-rate maneuver in the presence of heliox. Mean CE NO,50 was significantly decreased by 45% in the presence of heliox (8.0 ppb compared with 14.6 ppb).  increased (JЈaw NO ϭ 1,100 pl/s) and CA NO is set to zero (both at the boundary z ϭ L ϩ and by setting JЈA NO equal to zero because JЈA NO is the product of CA NO and DA NO ; see Glossary and APPENDIX).
In the absence of axial diffusion, the trumpet model reproduces both A I,II and the dynamic shape of phase III compared with the two-compartment model (data not shown). The pres-ence of axial diffusion significantly reduces the concentration of NO in all phases of the exhalation profile but does not impact C NOpeak (Fig. 5A). The presence of heliox (increasing molecular diffusivity of NO from 0.23 to 0.52 cm 2 /s) further reduces the concentration of NO in all phases but does not impact C NOpeak (Fig. 5B). For both air and heliox, values for Ā I,II and R III significantly deviate from zero and are equal to 0.45 and 0.10 and 0.50 and 0.21, respectively (Fig. 5, C and D).
Increasing JЈaw NO increases the concentration of NO in all phases of the exhalation profile and thus impacts A I,II and R III . Increasing JЈaw NO by 1.5-fold (JЈaw NO ϭ 1,100 pl/s) in the presence of axial diffusion (D NO,air ϭ 0.23 cm 2 /s), holding Daw NO and DA NO constant (5.03 and 1,467 pl⅐s Ϫ1 ⅐ppb Ϫ1 , respectively), and decreasing the alveolar concentration to zero (CA NO ϭ 0 and JЈA NO ϭ 0) provides the optimal simulation (Ā I,II and R III , improve to 0.25 and 0.04, respectively; see Fig.  5C). When this same set of flow-independent parameters is used to simulate heliox (D NO,heliox ϭ 0.52 cm 2 /s), the trumpet model is able to simulate phase I and II very accurately (Ā I,II improves from 0.50 to 0.03), but the impact on phase III is negligible (R III is unchanged from 0.21).

DISCUSSION
This is the first study to probe the impact of axial diffusion on NO exchange dynamics in the lungs experimentally by  manipulating the rate of axial diffusion in the airways with heliox as the insufflating gas. Our results demonstrate that the exhaled NO concentration when heliox is the insufflating gas is decreased in all three phases of the single-exhalation breathing maneuver with a preexpiratory breath hold as well as the CE NO,50 . In addition, W 50 of the exhalation profile, V I,II , and A I,II after a breath hold are substantially reduced. These experimental observations in the presence of heliox are consistent with a reduced amount of NO eliminated in the exhalation breath and result in a 34% decrease in the estimate for JЈaw NO according to a two-compartment model that neglects axial diffusion. When axial diffusion is considered in an alternate model (trumpet model), optimal parameter values predict a 50% increase in JЈaw NO and a near-zero CA NO and JЈA NO .

Impact of Heliox as the Insufflating Gas
NO in the airways is transported by both molecular diffusion (resulting from concentration gradients and random motion of individual NO molecules) and convection (resulting from bulk flow of the insufflating gas). With bulk flow only in the axial direction (i.e., in the absence of mixing), NO transport may be characterized in terms of the Peclet number, Pe, a dimensionless index. Pe corresponds to the ratio of NO mass transport rate by convection (bulk flow) to that by molecular diffusion. Furthermore, Pe is inversely proportional to the molecular diffusivity of NO in the gas phase, D NO,i (i denotes the insufflating gas, either air or heliox) and directly proportional to the product of bulk gas velocity and a characteristic length (usually selected as the airway diameter). The importance of molecular diffusion, relative to convection, increases with decreasing Pe and increasing D NO,i .
At high bulk gas velocities (typical of the larger airways), fluid mixing at airway bifurcations (called axial dispersion) also impacts NO transport rates. Previous work (3,16,26) demonstrates that axial dispersion is dependent on flow rate, geometry, and the fluid's physical properties, with mathematical characteristics similar to molecular diffusion. Scherer et al. (16) and others (3,26) have described the combined effects of molecular diffusion and axial dispersion in terms of an effective diffusivity, D* NO,i . These experiments used nitrogen (with similar physical properties to air) as the insufflating gas. From these data, D* NO,i was correlated as the sum of D NO,i and an axial dispersion term, to account for molecular diffusion and gas mixing at airway bifurcations, respectively. The axial dispersion term was proportional to the product of gas velocity and airway diameter. When air is the insufflating gas, the similarity of air and nitrogen allows axial dispersion's impact to be approximated for an airway branch with known gas velocity and diameter.
When Pe is large (e.g., Ͼ10), convection dominates NO transport, relative to molecular diffusion, but axial dispersion retains some influence. For heliox, the precise impact of axial dispersion under these conditions remains unclear. Reynolds number, Re, is a dimensionless index, directly proportional to gas density, and for a specific insufflating gas, Re is also proportional to Pe. Because the data of Scherer et al. and others (3,16,26) imply that the relative importance of axial dispersion increases with Re, a gas with lower density will have lower Re and will exhibit less mixing. Therefore, the impact of axial dispersion is expected to be less important for heliox than for air at large Pe. Conversely, when Pe is small (e.g., Ͻ0.1) molecular diffusion dominates over both convection and axial dispersion.
At typical exhalation flow rates (e.g., 250 ml/s), the precise impact of replacing air with heliox as the insufflating gas in the larger airways (i.e., generation Ͻ5) requires further investigation. However, in the smaller airways, convection and gas mixing are much less important and molecular diffusion dominates. Hence, replacing air with heliox significantly enhances molecular diffusion in the smaller airways and provides a means to study the impact of axial diffusion of NO in the lungs.

Impact of Axial Diffusion on Exhaled NO Concentration
When heliox was used as the insufflating gas in the present study, exhaled NO concentration was decreased in breathing maneuvers spanning a range of static and dynamically changing flow rates. This result is consistent with previous (17) and present simulations (Fig. 5) of a trumpet model that considers axial diffusion and is due to diffusion of NO down the concentration gradient (i.e., from high to low concentration) from the airway to the alveolar region.
The presence of heliox did not impact C NOpeak but reduced W 50 , V I,II , and A I,II . These findings were confirmed by the trumpet model simulations. The reduced A I,II is due to the reduced W 50 (thinner peak causes a small area). The reduced W 50 and V I,II are due to depletion of NO in the smaller airways while a constant C NOpeak is maintained (see Fig. 1, which describes how W 50 and V I,II are determined). The depletion of NO is a direct result of enhanced axial diffusion of NO from the airways to the alveolar region. The constant C NOpeak is due to a minimal impact of axial diffusion in the larger airways during breath hold. C NOpeak occurs in the first part of the exhaled breath, which is the largest distance from the sink (i.e., the alveolar region) because of axial diffusion. If breath hold time were increased such that a new steady state was achieved in the airway compartment, a lower C NOpeak should be observed. We confirmed this by simulating a 1-h breath hold time using the trumpet model, and C NOpeak decreased by 2% (146 ppb, the ratio of JЈaw NO /Daw NO or Caw NO , to 143 ppb). Thus concentrations of NO throughout the airway tree during a breath hold are potentially impacted; however, for breath hold times less than ϳ1 min, the impact is only observed in the small airways.

Impact of Axial Diffusion on Flow-Independent NO Parameters
Previous results from our group and others (17,31), as well as simulations in the present study (Fig. 5), demonstrate that axial diffusion results in a loss of NO produced in the airways to the alveolar region. The result is a decrease in the estimated source of NO from the airways (i.e., JЈaw NO ). This hypothesis is confirmed by the 34% reduction in JЈaw NO (735 pl/s compared with 485 pl/s) predicted by the two-compartment model, which neglects axial diffusion when heliox replaces air as the insufflating gas and rate of the axial diffusion is increased by 2.3-fold. Thus, to simulate the exhaled concentration observed experimentally with a model that considers axial diffusion, the endogenous source of NO in the airways (i.e., JЈaw NO ) needs to be increased. The present study suggests that 1.5-fold increase (50%) of JЈaw NO in the presence of air (from 735 to 1,100 pl/s) and a 2.3-fold increase (130%) in the presence of heliox (485 to 1,100 pl/s) combined with a zero alveolar concentration and source (decrease in CA NO from ϳ2.5 ppb to 0, and JЈA NO ϭ 0) is necessary to compensate for the effects of axial diffusion in all three phases of the exhalation profile (Fig. 5).
Of note is the observation that the simulation indexes (i.e., R III ) for heliox are poorer than air. This is likely due to remaining simplifications of the trumpet model that are exaggerated by the differences between heliox and air. For example, the trumpet model assumes that JЈaw NO is constant per unit airway volume (resulting in a decrease in JЈaw NO per unit surface area). This model structure is consistent with reports that the lower airways do not produce as much NO as the larger airways (22) but may not represent the precise distribution. Using the optimal parameter values (JЈaw NO ϭ 1,100 pl/s and CA NO ϭ 0 ppb) and a constant exhalation flow rate of 50 ml/s, the trumpet model predicts the concentration in generation 2 (z ϭ 18.7 cm) to be 10.9 ppb, which is 71% of the exhaled concentration at the mouth (z ϭ 0, 15.4 ppb). Although somewhat higher, this value is comparable to Silkoff et al. (22), who reported that ϳ50% of exhaled NO arises from the upper airways (generations 0 -2). If a greater or smaller flux of NO is present in the airways than our model structure, then a greater or smaller, respectively, contribution of axial diffusion on exhaled NO is simulated, which introduces error in the relative impact of axial diffusion. In addition, as mentioned by Van Muylem et al. (31), other factors such as asymmetry of the airway tree and inhomogeneous ventilation may affect the exhalation profile of NO and would differentially impact the simulation of heliox compared with air because of differences in density between the two gases.
Finally, it is important to emphasize that our trumpet model structure limits the simulations of alveolar concentration to a fixed steady-state value. Hence, we cannot rule out the possibility that, even in the presence of a zero alveolar source, the alveolar concentration may be nonzero because of backdiffusion of NO from the airways. This simulation requires a more advanced model in which a separate mass balance on an alveolar compartment is coupled to the mass balance in the airway compartment.
The prediction of a zero or near-zero alveolar concentration in the presence of a zero alveolar source suggests the possibility that previous reports of CA NO values that range from 1 to 5 ppb (7,10,18,19,20) may represent NO in or near the alveolar and respiratory bronchiole region, but the source may be the airways because of backdiffusion. The most commonly used technique to estimate CA NO is a plot of V NO vs. V E (30). For exhalation flow rates Ͼ50 ml/s in adults, the slope of this plot is an estimate of CA NO and the intercept an estimate of JЈaw NO . We hypothesized that even if CA NO were zero, this plot would predict a positive slope (i.e., a nonzero CA NO ) because the impact of axial diffusion (and thus loss of NO to the alveolar region) would be reduced as the exhalation flow rate increased. To test this hypothesis and to determine the magnitude of this phenomenon, the trumpet model was used to predict CE NO,i (and thus V NO ) at five different V E values (50, 100, 150, 200, 300, and 500 ml/s) by using the optimal set of flow-independent NO parameters (JЈaw NO ϭ 1,100 pl/s, Daw NO ϭ 5.03 pl⅐s Ϫ1 ⅐ppb Ϫ1 , and CA NO ϭ 0 ppb) in the presence and absence of axial diffusion. Figure 6 demonstrates the trend in the slope and intercept of V NO vs. V E in the absence (Fig. 6A) and presence (Fig. 6B) of axial diffusion for different subgroups of the five exhalation flow rates (three of the five flow rates, i.e., 50, 100, and 150 ml/s). In the absence of axial diffusion (equivalent to the two-compartment model) with CA NO ϭ 0, the predicted slope is slightly positive (0.36 ppb) when the three lowest flow rates are used and then approaches zero as higher flow rates are used. The slightly positive value at lower flow rates as well as the slight nonlinear relationship (i.e., r 2 Ͻ 1.0) is due to the fact that, as the flow rate decreases, the airway compartment concentration increases (because of a longer residence time of air in the compartment) and the airway flux decreases slightly (Jaw NO ϭ JЈaw NO Ϫ Daw NO ϫ C NO ) as previously described (29). In the presence of axial diffusion with CA NO ϭ 0, the slope is 2.1 ppb for a flow rate range of 50 -150 ml/s and decreases to 0.4 ppb over the flow rate range of 200 -500 ml/s. This value for the slope (0.4 -2.1 ppb) is consistent with Fig. 6. Elimination rate of NO is plotted vs. the exhalation flow rate as predicted in the absence (A) and presence (B) of axial diffusion. In all simulations, the optimal values for the flow-independent NO exchange parameters (JЈawNO ϭ 1,100 pl/s and CANO ϭ 0 ppb) were used to simulate plateau concentration at a constant flow rate of 50 ml/s (CENO,50) and thus the elimination rate. Linear regression is performed through a sequence of 3 consecutive data points, and the intercept, which has been previously interpreted as JЈawNO, and the slope, which has been previously interpreted as CANO, are presented according to 4 different flow rate ranges. reported CA NO determined by using the slope of the relationship V NO vs. V E using the two-compartment model, which neglects axial diffusion (7, 10, 18 -20). Thus CA NO may be near zero for healthy adults, and the reported 1-to 3-ppb range determined from the slope of V NO vs. V E (7, 10, 18 -20) may be an artifact from neglecting axial diffusion.
In conclusion, in the presence of axial diffusion, the trumpet model predicts a significant backdiffusion of NO from the airways into the alveolar region, which is demonstrated experimentally by using heliox as the insufflating gas. Backdiffusion of NO from the airways to the alveolar region results in a significant loss of NO, which does not appear in the exhaled breath. Thus models that neglect axial diffusion underestimate the maximum airway flux by as much as 50%, and the alveolar concentration reported in healthy adults may reflect, in part, NO produced in the small airways. We conclude that accurate estimation of flow-independent NO exchange parameters, and thus partitioning of exhaled NO into airway and alveolar contributions, must include mathematical models that consider axial diffusion of NO in the gas phase. In addition, future studies must quantify the impact of axial diffusion for subjects with key inflammatory diseases of the airways such as asthma or cystic fibrosis, as well as diseases that involve inflammation of the alveolar region.

Governing Equation
The governing partial differential equation for CNO (ppb) is obtained from a mass balance over a differential volume of length ⌬z: where N(z) ϭ 0 for z Ͻ 26.8 cm, Nmax ϭ 143 ϫ 10 6 , and Nt is total number of alveoli in Weibel lung (298.1 ϫ 10 6 ).

Initial and Boundary Conditions
The initial condition for Eq. A1 for inspiration is CNO(z,t ϭ 0) ϭ 0, and for expiration it is equal to the concentration profile just before exhalation (either after the 20-s breath hold for breathing maneuver 1 or end-inspiration for breathing maneuver 2). The boundary conditions for Eq. 1 are as follows: Inspiration: C NO ͑z ϭ 0 Ϫ ,t͒ ϭ 0 ͑ambient air zero concentration͒ (A2) C NO ͑z ϭ L ϩ ,t͒ ϭ CA NO (A3) Expiration: C NO ͑z ϭ 0 Ϫ ,t͒ ϭ CA NO (A4) dC NO ͑z ϭ L,t͒ dz ϭ 0 ͑no flux at end of airway͒ (A5) where z ϭ 0 Ϫ refers to a single numerical node (i.e., z position) just before the start of the trumpet and t is time. The boundaries are reversed for expiration (i.e., z ϭ 0 Ϫ is always the position where air enters the trumpet). The node just beyond the end of the trumpet is expressed as CANO, which represents a fixed steady-state alveolar concentration, specifically, 2.55 ppb or 0 ppb in the present simulations.