## Abstract

The following is the abstract of the article discussed in the following letter:

The relatively recent detection of nitric oxide (NO) in the exhaled breath has prompted a great deal of experimentation in an effort to understand the pulmonary exchange dynamics. There has been very little progress in theoretical studies to assist in the interpretation of the experimental results. We have developed a two-compartment model of the lungs in an effort to explain several fundamental experimental observations. The model consists of a nonexpansile compartment representing the conducting airways and an expansile compartment representing the alveolar region of the lungs. Each compartment is surrounded by a layer of tissue that is capable of producing and consuming NO. Beyond the tissue barrier in each compartment is a layer of blood representing the bronchial circulation or the pulmonary circulation, which are both considered an infinite sink for NO. All parameters were estimated from data in the literature, including the production rates of NO in the tissue layers, which were estimated from experimental plots of the elimination rate of NO at end exhalation (E_{NO}) vs. the exhalation flow rate (V̇_{e}). The model is able to simulate the shape of the NO exhalation profile and to successfully simulate the following experimental features of endogenous NO exchange: *1*) an inverse relationship between exhaled NO concentration and V̇_{E}, *2*) the dynamic relationship between the phase III slope and V̇_{E}, and *3*) the positive relationship between E_{NO} and V̇_{E}. The model predicts that these relationships can be explained by significant contributions of NO in the exhaled breath from the nonexpansile airways and the expansile alveoli. In addition, the model predicts that the relationship between E_{NO} and V̇_{E} can be used as an index of the relative contributions of the airways and the alveoli to exhaled NO.

*To the Editor*: Exhaled nitric oxide (NO) reflects airway inflammation. By measuring NO at different flow rates the relative contributions of the airways and alveolar compartment may be estimated, as demonstrated by Tsoukias and George (1). They proposed a two-compartment model of pulmonary NO output in an effort to explain several fundamental and experimental observations. Total output of NO (E_{NO}) is: In this equation V̇_{E,ee} is the flow at end exhalation, C_{alv,ee} is an estimate of the alveolar concentration, *J̄*_{t:g,air} is the average flux of NO from airway tissue to luminal air, and *A*_{s,air} the total surface area of airway tissue. By plotting the V̇_{E,ee} values on the *x*-axis and E_{NO} values on the *y*-axis of an *x-y* diagram, the slope of the regression line between measurements at different flow rates is supposed to be consistent with the NO concentration of the alveolar compartment, while the intercept with the *y*-axis is consistent with the NO flux from airway tissue (Fig. 6 in Ref. 1).

However, NO output is calculated as the product of the constant expiratory flow and the NO concentration. Therefore, there is a mathematical relation between the variables on the *x*-axis and the *y*-axis. A necessary condition to derive meaningful results from linear regression and its associated correlation coefficient is that the variables on the *x*-axis and *y*-axis are independent, i.e., have no mathematical relation. In the present model this can be achieved by rearranging the above equation: where C_{ee} is the NO concentration measured in exhaled air at end exhalation. By plotting C_{ee} on the *y*-axis and 1/V̇_{E,ee} on the *x*-axis, the slope of the line between measurements now is consistent with the NO flux. Meanwhile, the intercept with the *y*-axis is consistent with the alveolar NO concentration. Because an indefinitely high V̇_{E,ee} minimizes the bronchial contribution to the total NO output, the NO concentration measured in exhaled air then approximates the concentration of the alveoli.

To analyze whether such a new arrangement in this two-compartment model changes the end results significantly, we measured end-expiratory NO concentrations three times at four different flow rates (30, 50, 100, and 200 ml/s), using a NIOX (Aerocrine, Stockholm, Sweden). Study subjects were four healthy and six asthmatic individuals of various ages (5–43 yr) and sex (4 male, 6 female). Afterward, we derived the alveolar NO concentrations [parts/billion (ppb)] and bronchial NO fluxes (nl/s) following the two equations described above. The rearranged equation led to small but significant shifts between the alveolar and bronchial contribution. The mean (SD) alveolar concentration according to Tsoukias and George (1) compared with our analysis was 4.19 (1.19.) vs. 5.58 (1.82) ppb (*P* < 0.05). Also, the mean (SD) bronchial NO flux differed: 1.00 (0.60) vs 0.90 (0.51) nl/s (*P* < 0.05). The mean (SD) difference between the two methods in alveolar NO concentration was 1.39 (1.25) ppb and in bronchial NO flux was 0.10 (0.09) nl/s (see Fig. 1).

We conclude that a different way of applying the mathematical two-compartment model on NO output of Tsoukias and George (1) leads to small but significant shifts in the alveolar and bronchial contribution. The two methods differ especially for higher NO values (see Fig. 1), so when more inflammation is present.

Although the differences are small, for the sake of methodological correctness we would prefer the modified equation:

- Copyright © 2005 the American Physiological Society

## REFERENCE

# REPLY

*To the Editor*: The opportunity to respond to the Letter to the Editor by Rottier and colleagues is greatly appreciated. We would like to thank them for a thoughtful letter that contrasts two techniques for determining two NO exchange parameters: alveolar NO concentration (Ca_{NO}) and the maximum airway flux of NO (*J*′aw_{NO}). The solution of the two-compartment model for the exhaled NO concentration at constant air flow (Ce_{NO,V̇}) is nonlinear (2, 5), (1) where *D*aw_{NO} is airway diffusing capacity and V̇ is the exhalation flow. For mathematical simplicity, a linear approximation of *Eq. 1* can be made (2, 4): (2) To fit measurements of Ce_{NO,V̇} at constant V̇ to determine Ca_{NO} and *J*′aw_{NO}, Rottier and colleagues recommend plotting Ce_{NO,V̇} vs. 1/V̇, a technique described previously by Pietropaoli et al. (4) and referred to as the “Pietropaoli technique” (2). Our group plotted Ce_{NO,V̇}· vs. V̇, which has been described as the “Tsoukias technique” (2) and was originally introduced in a companion paper (6) to that cited by Rottier and colleagues. Although mathematical coupling occurs in the Tsoukias technique, it may lead to errors only in statistical inferences (1, 3). However, the goal of this analysis is determining unknown physiological parameters, Ca_{NO} and *J*′aw_{NO}, based on a mathematical model, by minimizing the sum of errors squared (SSE) between the experimental observations and that predicted by the model: (3) where superscripts “exp” and “mod” refer to experimental and model values of Ce_{NO,V̇}, respectively; *n* is the number of measurements; and W is the weight function, which may be dependent on V̇. Of note is the observation that if W = V̇^{2} for the Pietropaoli technique, then the objective function (right-hand side of *Eq. 2*) to determine *J*′aw_{NO} and Ca_{NO} is equivalent to the Tsoukias technique with W = 1. Hence, the Tsoukias method effectively places more weight on the exhaled concentrations collected at higher flows.

How does the choice of W(V̇) impact the error of the estimated parameters? To answer this question, a simple simulation with the following assumptions was performed: *1*) the nonlinear solution of the two-compartment (*Eq. 1*) represents the true lungs or the gold standard; *2*) two ranges of exhalation flows were examined that included those of Rottier and colleagues (30, 50, 100, and 200 ml/s) and a higher range that is more commonly found in the literature (100, 200, 250, and 300 ml/s); *3*) the true values for *J*′aw_{NO}, Ca_{NO}, and Daw_{NO} were 1,000 pl/s, 5.0 ppb, and 10 ml/s, respectively, and represent typical values inbetween a healthy subject and an asthmatic subject; *4*) Ce_{NO,V̇} was calculated using *Eq. 1*, and a normally distributed error with a standard deviation of 1 ppb was superimposed on the data to represent random error from the experimental sampling system; and *5*) *J*′aw_{NO} and Ca_{NO} were determined by the Pietropaoli and Tsoukias techniques for 500 consecutive simulations in which a new random error was superimposed on Ce_{NO,V̇} for each simulation.

The mean values of *J*′aw_{NO} and Ca_{NO} determined from the simulations were dependent on the technique employed (see Fig. 1) in a pattern consistent with that observed by Rottier and colleagues. However, the average error relative to the true values for a single simulation was actually lower for the Tsoukias technique. The differences between the techniques were greater for the low flow range 30–200 ml/s.

In summary, *J*′aw_{NO} and Ca_{NO} may be determined from exhaled NO concentrations at different exhalation flows by plotting either Ce_{NO,V̇} vs. 1/V̇ (Pietropaoli technique) or Ce_{NO,V̇}·V̇ vs. V̇ (Tsoukias technique). The Tsoukias technique places greater emphasis on high-flow data, which more accurately reflect the linear approximation of the two-compartment model. As a result, the Tsoukias technique actually provides more accurate estimates for the model-based parameters, *J*′aw_{NO} and Ca_{NO}, relative to the Pietropaoli technique.