INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

DURING THE LAST FEW YEARS, several models have been used for the interpretation of exhaled nitric oxide (NO). Tsoukias and George (13) and Pietropaoli et al. (8) proposed qualitatively similar models that had an airway and an alveolar compartment. They can account, for example, for the marked flow dependence of end-expired NO concentration (10). NO concentration in airspaces results from a balance between 1) excretion from tissues, 2) diffusion into blood, and 3) ventilation. By parametric model fitting on experimental NO curves as a function of expired flow and independent measurement of NO diffusion into blood, Pietropaoli et al. separately assessed airway and alveolar contributions to NO excretion (_NO). Nevertheless, the models of both Pietropaoli et al. and Tsoukias and George neglect transport by axial molecular diffusion between airways and alveoli. This mechanism necessarily participates on concentration homogenization, especially at low flow rate, because it transports NO molecules from the airway to the alveolar compartment, which results in less NO molecules available for expiration.

Using a model of gas transport that includes convection (bulk flow) and axial molecular diffusion (6) in a geometry derived from the symmetrical model of Weibel (17), we propose to assess the influence of axial diffusion on exhaled NO in two types of experiments: the expired flow dependence on end-expired NO (by using several tests, each with constant expired flow) and the NO expiration profile with change of expired flow, i.e., a variable flow during expiration (14, 15).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

General Equation of Transport in a Symmetrical Model of the Lung

(1)

The last term of Eq. 1 is the NO source term: the difference between _NO (in ml/s) and NO diffusion into blood (D_NO · C) per unit volume. D_NO is expressed in ml · s¹ · ppb¹. We separate airways and alveoli contributions to _NO and D_NO as

where _NOAw and _NOAl are airway and alveolar _NO, respectively, and D_NOAw and D_NOAl are airway and alveolar NO transfer factor, respectively.

Geometrical Boundaries

Numerical Computation

Source Terms

Airway and alveolar wall areas at node i are computed as follows.

Airways. At node i belonging to generation n, the airway wall area is computed as the lateral surface of a cylinder of radius r_i (17) and length equal to 0.02 cm (internodal distance), multiplied by 2ⁿ. Toward the alveolar end, the airway wall surface decreases dramatically and becomes nil at the level of the alveolar ducts. On the basis of Weibel's data, the total airway wall area is equal to 2.2 m² (17).

Alveoli. The alveolar wall surface at node i (A_alvi) may be approximated by
where 0.02(S_i  s_i) is the total alveolar volume at node i, and s_a/v_a is the ratio between the wall surface and the volume of an alveolus. In the present model, the total alveolar wall area is equal to 75 m².

View larger version (14K): [in this window] [in a new window]   Fig. 1.   Distribution of airway (A) and alveolar wall (B) area as a function of axial distance from alveolar end and of generation number. Distributions are expressed as percentage of the total area.

Numerical Values of NO Parameters

Flow dependence of end-expired NO. Table 1 shows the parameters chosen as the source terms. These are the means of values deduced from seven subjects with an airway-alveolus compartment model (8).

Dynamic change of flow during a single exhalation. NO parameter values are summarized in Table 2. _NOAw and D_NOAw are given by Tsoukias et al. (14) and correspond to the individual experimental curve shown on Fig. 5 of their article. For this subject, they also deduced a steady alveolar NO concentration (alveolar _NO/alveolar D_NO) of 1.82 ppb. With alveolar D_NO = 1.46 × 10⁶ ml · s¹ · ppb¹ (Table 1), we have alveolar _NO = 2.66 × 10⁶ ml/s.

General Conditions for the Simulations

Flow dependence of expired NO. In these simulations, inspiratory and expiratory volumes were always 1 liter, inspiratory flow was always 500 ml/s, and expiratory flow varied. We considered the NO concentration value at the onset of the model after 1 liter expired (the end of expiration in our simulations) as the output variable. In the following, the term "end-expired NO concentration" refers to this output variable. For expired flows of 50 and 2,000 ml/s, we performed 1) simulations with a uniformly distributed NO production and 2) simulations with weighted contribution of generations 0 to 2 to _NO. Keeping the overall NO airway excretion given in Table 1, we determined, by successive iterations, the weighting of the three first generations necessary to obtain, with an expiratory flow of 50 ml/s, a midexpired (10 s after the beginning of expiration) concentration at the onset of generation 3 roughly equal to 50% of the concentration at the onset of generation 0 (9). In the following, the term "nonuniform NO production" always refers to simulations with this generation 0-2 weighting.

Dynamic change of flow during a single exhalation. With the above defined initial conditions, we considered one inspiration of 4.4 s with 500 ml/s flow, followed by a breath hold of 20 s (zero flow in Eq. 1), followed by a 17-s expiration. The expired flow pattern mimics the experiment depicted on Fig. 5A of Tsoukias et al. (14), i.e., a linear decrease from 420 to 140 ml/s for the first 3.9 s, a linear decrease from 140 to 70 ml/s for the next 5.7 s, and a linear decrease from 70 to 50 ml/s for the remaining 7.4 s. We performed simulations by considering uniform and nonuniform NO production, respectively.

Differences Between Simulation and Recommended Experimental Procedures

Influence of End-Inspiration Breath Hold on End-Expired NO Concentration

Sensitivity to NO Parameters

Flow dependence of expired NO. We performed double pairs of simulations as described in Flow dependence of expired NO, at a constant expired flow of 50 and 2,000 ml/s, respectively, in the following situations: 1) D_NOAw = 0; 2) _NOAw of Table 1 multiplied by two and five, respectively; and 3) _NOAw of Table 1 multiplied by five and alveolar _NO = 0, only for uniform NO production.

Dynamic change of flow during a single exhalation. Again, when we considered both uniform and nonuniform NO production, we performed simulations as described in Dynamic change of flow during a single exhalation in the following situations: 1) D_NOAw = 0 and D_NOAw of Table 2 multiplied by two, respectively; and 2) _NOAw of Table 2 multiplied by two and five, respectively, for uniform NO production and multiplied by two for nonuniform NO production.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Figure 2 shows the NO concentration profiles at mid-expiration, for an expiratory flow of 50 ml/s. Curve 1 is the curve obtained with a uniform distribution of NO production. Curve 2 corresponds to a weighted contribution of generations 0 to 2 to NO production. The weighting was computed to obtain a concentration value at the onset of generation 3 (arrow at 7 cm of axial distance) approximately equal to half the concentration value at the model onset (arrow at 27 cm of axial distance). This corresponds to a contribution of generations 0 to 2 of 20% of total _NOAw.

View larger version (13K): [in this window] [in a new window]   Fig. 2.   Nitric oxide (NO) concentration profiles at midexpiration as a function of axial distance from alveolar end for a constant expiratory flow of 50 ml/l, obtained by solving Eq. 1. Curve 1 corresponds to uniform distribution of NO production. Curve 2 corresponds to weighting NO production in generations 0 to 2 (see text for details). Vertical arrows indicate the onset of generation 0 (Gen 0)and onset of generation 3 (Gen 3). ppb, Parts per billion.

Figure 3 shows NO concentration profiles as a function of the axial distance from alveolar end for generations 7 to 23 (Fig. 3A) and NO expired concentration at the onset of the model as a function of time (Fig. 3B). In Fig. 3A, we show the initial conditions before the test (curve 1), end-inspiratory concentration (curve 2), and concentration after a 20-s breath hold (curve 3). In Fig. 3B, curves 1 and 2 represent concentration without and with a 20-s breath hold, respectively. These simulations were performed with 2 s of inspiration and 2 s of expiration at a constant flow of 500 ml/s.

View larger version (11K): [in this window] [in a new window]   Fig. 3.   A: NO concentration in generations 7 to 23 as a function of axial distance from alveolar end obtained by solving Eq. 1. Curve 1 is the initial NO profile before inspiration. Curve 2 is the NO profile after 2-s inspiration at a constant flow of 500 ml/s. Curve 3 is the NO profile after a 20-s breath hold. B: NO concentration at the onset of generation 0 as a function of time during expiration at a constant flow of 500 ml/s. Curves 1 and 2 are NO expiratory profiles without and with a 20-s breath hold, respectively.

Figure 4 shows the NO concentration profiles in generations 7 to 23 during inspiration (Fig. 4A) and expiration (Fig. 4B) as a function of the axial distance every 0.2 s (no breath hold). Curve at t = 0 (Fig. 4A) corresponds to curve 1 of Fig. 3A. Curves at t = 2 s (Fig. 4A) and t = 0 (Fig. 4B) correspond to curve 2 of Fig. 3A.

View larger version (19K): [in this window] [in a new window]   Fig. 4.   NO concentration profiles in generations 7 to 23 during inspiration (A) and expiration (B) as a function of axial distance from alveolar end obtained by solving Eq. 1 with constant inspiratory and expiratory flows equal to 500 ml/s. Curves are shown every 0.2 s (from 0 to 2 s). A: curve at time (t) = 0 is the steady-state profile before the test (curve 1). B: curve at t = 0 is the NO profile after 2-s inspiration at 500 ml/s (curve 2).

Differences Between Simulation and Recommended Experimental Procedures

Simulation of Experiments

Flow dependence of expired NO. Table 3 gives the experimental results derived from the data of Pietropaoli et al. (8) and Tsoukias and George (13) (lines 1 and 2, respectively) and summarizes the results of the simulations for expiratory flows of 50 and 2,000 ml/s, respectively, both for uniform and nonuniform NO production (lines 3 to 9).

View larger version (26K): [in this window] [in a new window]   Fig. 5.   End-expired NO as a function of (constant) expired flow. The gray zone and the dashed line are the range and mean of experimental values, respectively (see text for details).  and , Obtained by solving Eq. 1 with Table 1 values for NO parameters, with [NO molecular diffusion coefficient (D) = 0.22 cm2/s] and without (D = 0) axial diffusion, respectively, and uniform NO production.  and , Values corresponding to  and , respectively, with nonuniform NO production.

Dynamic change of flow during a single exhalation. Figure 6 shows the expired NO concentration profile as a function of time with the decreasing expired flow pattern described above, i.e., linear decrease from 420 to 140 ml/s for the first 3.9 s, a linear decrease from 140 to 70 ml/s for the next 5.7 s, and a linear decrease from 70 to 50 ml/s for the remaining 7.4 s. The gray outline on each panel is the experimental curve from Fig. 5 of Tsoukias et al. (14). Figure 6A corresponds to simulations with uniform NO production: curve 1 (solid line) with the NO parameters of Table 2; curve 2 (dotted line) with the assumption that D = 0; curves 3 and 4 (dashed lines) correspond to _NOAw multiplied by two and five, respectively. The arrows indicate the corresponding peak values at the onset of expiration. Simulations 1, 2, and 3 shown in Fig. 6B are the equivalent of those in Fig. 6A but for nonuniform NO production. The peaks are out of scale, and their values are indicated explicitly beside the vertical arrows.

View larger version (20K): [in this window] [in a new window]   Fig. 6.   Expired NO concentration as a function of time with a decreasing expired flow (see text for details). Solutions of Eq. 1 are considered with uniform (A) and nonuniform (B) NO production. Gray outlines show experimental curves (14). Curve 1 (solid line) on both panels was obtained by solving Eq. 1 with Table 2 NO parameters values. Curve 2 on both panels (dotted line) was obtained by assuming D = 0 in Eq. 1. A: curves 3 and 4 (dashed lines) considered NO production in the airways (NOAw) from Table 2 multiplied by 2 and 5, respectively. Horizontal arrows show peak NO concentration for each curve B: curve 3 (dashed line) considered QNOAw from Table 2 multiplied by 2. The peak of curves 1, 2, and 3 are out of scale, and their numerical values are given beside the vertical arrows.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Compartmental models, such as those previously used to describe NO transport (8, 11, 13), have been very useful to interpret several tests in respiratory physiology. However, they use differential equations that only allow computation of time-dependent concentrations in each compartment. Present simulations correspond to solutions of Eq. 1, which is a partial differential equation and enables the consideration of simultaneous convection and diffusion that operate in a structure based on the symmetrical lung model of Weibel (17).

The focus of the present work is the impact of molecular diffusion on expired NO. Molecular diffusion plays a major role in gas transport in the periphery of the lung because of it being dominant in the zone peripheral to the terminal bronchioles (6). During normal breathing, it is negligible in the conducting airways. A study of Silkoff et al. (9) suggests that, in normal subjects, a large part of NO production in the airways comes from the zone between upper trachea and main bronchi. Thus, other than simulations with uniformly distributed NO production, as in compartmental models, we considered nonhomogeneous NO production by weighting the upper generations' contribution. Figure 2 shows concentration profiles at midexpiration as a function of axial distance. Curve 1 corresponds to uniformly distributed NO production, and curve 2 corresponds to a weighting of generations 0 to 2, both with the overall _NO in the airways given in Table 1. In our model, the latter case corresponds to consider 20% of NO production on generations 0 to 2, which represent <1% of the total epithelial surface. When we used the source term with values given by Pietropaoli et al. (8) (Table 1) and Tsoukias and George (13) (Table 2), the simulations qualitatively reproduce experimental findings with a smaller end-expired NO when expired flow increases (Fig. 5) (10). The NO-expired profile shows an early peak followed by a typical slope (Fig. 6) when flow decreases during a single expiration after a 20-s breath hold (14, 15). However, there is a quantitative difference between these simulations and experiments. Figure 5 shows that end-expired NO concentration is below the experimental range at any expired flow and far from the mean experimental value, especially at low flows, regardless of whether we consider a larger NO production in the first generations. Figure 6 shows that the simulated profile with the NO parameters of Table 2 (curve 1 on each panel) is below the experimental curve (gray outline on each panel). Insofar as our model describes NO transport in a more realistic way than compartment models, this disagreement may come from inadequate estimation of NO exchange parameters used in Eq. 1.

The link between the two-compartment models and our model can be done by considering D = 0 (no diffusion) in Eq. 1 with the same NO parameters (Tables 1 and 2). In this case, end-expired NO concentrations (open circles and squares on Fig. 5) are close to experimental values. Similarly, for a decreasing expired flow, the expired-NO profile becomes close to the experiments when D = 0 (curve 2 in Fig. 6, A and B). This shows the importance of molecular diffusion on NO transport, even when a larger NO production in the first generations is considered (Table 3, lines 3 and 4).

Tsoukias and George (13) pointed out the potential role of diffusion on alveolar concentration during breath hold. Indeed, Fig. 3A shows that alveolar concentration is greater after a 20-s breath hold (curve 3). However, as can be seen on Table 3 (lines 3 and 5), a 20-s breath hold has a limited impact on end-expired NO as it was shown experimentally (4). Our simulations stress the fact that diffusion plays an active role during the complete respiratory cycle, mainly during expiration. This is illustrated in Fig. 4, which shows typical concentration profiles during inspiration (Fig. 4A) and expiration (Fig. 4B) for equal time intervals as a function of distance from alveolar end. From the onset of the acinus (~1 cm from alveolar end) down to the alveolar end, there is a concentration gradient established almost immediately after the beginning of expiration that results in a backdiffusion flow toward the alveolar zone and consequently a continuous increase of NO concentration. During most of the expiration, the shape of the NO concentration profile changes very little and represents a quasi-stationary diffusion front of NO equivalent to the diffusion front of inert gases (6). This rapid quasi-stationarity during both inspiration and expiration explains that, in our simulations, NO concentration after 1 liter is expired differs little (<1 ppb at 50 ml/s) from the NO plateau measured according to recommendations for standardized procedures (12). The diffusion and convection flows are explicitly shown in Fig. 7, A and B, as functions of distance from the alveolar end at midexpiratory time at 2,000 and 50 ml/s, respectively. It appears that in both cases diffusion flow is toward alveoli (negative value) in the acinar zone. This flow transports NO molecules into the alveolar zone, with some of them diffusing into blood instead of being expired. By comparing Fig. 7, A and B, one may see that the backdiffusion flow increases when expiratory flow decreases and that it overcomes convection at low flow (Fig. 7B). This explains why the difference of end-expired NO between simulations incorporating diffusion and simulations assuming D = 0 (lines 3 and 4 of Table 3, respectively) is much greater at low flow. Increasing NO production in the first three airway generations, where diffusion is negligible, diminishes not only the relative amount of NO molecules "lost" by backdiffusion but also the NO peripheral production. The former phenomenon dominates at 50 ml/s by increasing end-expired NO by 55%; the latter phenomenon dominates at 2,000 ml/s by decreasing end-expired NO by 18% (Table 3, line 3).

View larger version (10K): [in this window] [in a new window]   Fig. 7.   Simulated convection (dashed lines) and diffusion (solid lines) NO flows as a function of axial distance from the alveolar end of the model. Negative values are flow toward alveoli, and positive values are flow toward mouth. A and B correspond to a constant expiratory flow of 2,000 and 50 ml/s, respectively. Simulated flows are shown at midexpiration, i.e., at 0.25 s (A) and at 10 s (B). Values of NO parameters are given in Table 1.

As the two-compartment models do not incorporate backdiffusion flow, they implicitly assume that all NO molecules excreted and not recaptured by blood circulation are transported by convection and expired. Use of these models to fit experimental data underestimates NO sources, especially in the airways and particularly at low flows.

The purpose of this work is to show the importance of diffusion and not to fit experimental data. However, we have estimated the impact of NO parameter variation on expired NO. One may achieve a greater concentration in the airways either by increasing _NOAw or decreasing NO transfer (D_NOAw). The latter has little effect: For uniform NO production and in the extreme situation where D_NOAw is equal to zero (line 6 of Table 3), end-expired NO concentration increases by 13% only at an expired flow of 50 ml/s. On the contrary, for uniform NO production, two- and fivefold increases in _NOAw increase end-expired NO concentration by 74 and 292%, respectively, at 50 ml/s and by 36 and 142%, respectively, at 2,000 ml/s (lines 7 and 8 of Table 3). When nonuniform NO production is considered, two- and fivefold increases in the _NOAw increase end-expired NO concentration by 83 and 332%, respectively, at 50 ml/s and by 26 and 139%, respectively, at 2,000 ml/s (lines 7 and 8 of Table 3). An increase factor between two and five would allow simulations to mimic experimental values (Table 3).

In regard to the test proposed by Tsoukias et al. (14), Fig. 6A (uniform NO production) shows that the experimental NO profile after the early peak (gray outline on each panel) lies between two curves (curves 3 and 4) simulated with an airway excretion two and five times the value of Table 2, respectively. In Fig. 6B (nonuniform NO production), curve 3 (simulated with an airway excretion two times the value of Table 2) is close to the experimental curve after the peak. However, these simulations show the limits of the model in the present form: The shape of the early peak of concentration resulting from NO accumulation in the airways during a 20-s breath hold is clearly not reproduced. The simulated peak is always sharper than in experiments (Fig. 6) and larger when nonuniform NO production is considered (Fig. 6B and Table 4, lines 8 to 12). Molecular diffusion tends to decrease the peak value (Table 4, lines 2, 3, 8, and 9) and to increase its sharpness (Fig. 6). Without any mixing process, the amount of NO produced during breath hold parallels the relative epithelial surface (Fig. 1A), which gives a high NO concentration between generations 13 and 17. During expiration, this peak is spread over a relatively large volume. The diffusion process prevents NO concentration from rising during breath hold in this peripheral zone (Fig. 3A, curve 3) so that the maximum NO value is smaller and located in the first generations. The peak at expiration is thus smaller and sharper since it is spread over a smaller volume. Given that the NO peak comes from the first generations, its relative insensitivity to D_NOAw is not surprising (Table 4 lines 4, 5, 10, and 11). The insensitivity is because of the very low proportion of epithelial surface of the first generations relative to the total surface (Fig. 1A), given a very low transfer factor per unit volume in this zone. From this, it can be seen that the NO concentration in the first generations before expiration is mainly determined by diffusion, _NOAw, and breath-hold time. If we consider the amount of NO that is accumulated during breath hold, i.e., the area under the peak, instead of the peak value, the result of line 12 of Table 4, (_NOAw × 2 and a nonuniform NO production) approaches the experimental value. This is consistent with results presented on Table 3 (line 7) and Fig. 6B (curve 3) and suggests that this would be the most suitable combination of parameters to mimic experiments in normal subjects. Nevertheless, the great difference in shape between simulated and experimental NO peak suggests that some other phenomena may play a role in the first generations, such as convective mixing (including in the mouth cavity), asymmetry of the main bronchi (3), and inhomogeneous ventilation combined or not with sequential emptying. Further simulations should address these issues specifically.

The major impact of backdiffusion flow on expired NO is a consequence, in our model, of the dramatic increase of epithelial surface peripheral to generation 13 (Fig. 1A), giving a large NO production in this zone where molecular diffusion becomes the dominant mode of transport (6). This may be questioned. Findings of Silkoff et al. (9) and in vitro measurements (5, 16) suggest that, in normal subjects, NO production may be lesser in peripheral airways and larger in central bronchi. On the other hand, Dubois et al. (1) estimated that NO production occurs throughout the first 450 ml of the airways, which in the Weibel model includes, at least, respiratory bronchioles. We showed (Table 3, line 3) that a weighting of NO production in the first generations compatible with the findings of Silkoff et al. (10) does not abolish the impact of molecular diffusion.

It is worth noting that the results presented here are slightly influenced by NO production in the alveoli. If we assume NO production to be nil for a given airway excretion (lines 8 and 9 of Table 3), end-expired NO decreases by only 3.5% (1.1 ppb) and 25% (1.8 ppb) at 50 and at 2,000 ml/s, respectively. Even in the absence of any NO source in the alveoli, NO molecules coming from the airways during inspiration (by convection and diffusion) and during expiration (by backdiffusion) ensures a nonzero NO value in the alveoli. This result questions the computation of an alveolar excretion value from the estimated equilibrium alveolar concentration and an independent measurement of alveolar D_NO, as proposed by some authors (7, 8).

The present work is only theoretical, and the predicted role of diffusion on NO transport has to be established experimentally. This could be done by measuring early peak and end-expired NO at low flow and after lung equilibration with gas mixtures, which leads to very different diffusion coefficients for NO. This can be achieved by comparing NO expiratory profile when 20% of oxygen is mixed with 80% helium and 80% sulfurhexafluoride, respectively.

In summary, we simulated end-expired NO as a function of flow and NO profile as a function of time with a decreasing expiratory flow by using a model of gas transport that includes source terms, convection, and molecular diffusion. Our simulations predict that, insofar as NO production occurs in peripheral airways, axial diffusion has a major impact on expired NO and that neglecting this mechanism may lead to underestimating _NO parameters in the airways. A smaller production in the peripheral airways would temper these conclusions, but, even if further studies show a lower effect of molecular diffusion in the normal subject, the backdiffusion mechanism would remain crucial for the interpretation of increased exhaled NO due to peripheral inflammation (4).