## Slope of Phase III Is Being Overinterpreted

*To the Editor:* In 1975, Paiva (2) reported the results of an analysis of a two-compartment model of ventilation. The model included different volume expansions and different time courses of emptying of the two compartments, and Paiva computed gas concentrations and slopes of phase III for a multibreath washout of a test gas. He noted that the slopes of phase III, normalized by mean concentration in the expired gas (*S*
n), increased nearly linearly with breath number for several breaths and approached an asymptotic value at large breath numbers. Later, Paiva and Engel (3) analyzed the diffusional transport at a branch with different flows in the branches. The analysis revealed a diffusional pendelluft between the branches that contributed to gas mixing. Paiva et al. (4) noted that this mechanism produced a value of *S*
n that was relatively large for the first breath and quickly reached a limiting value. They denoted this mechanism in which diffusional transport occurs between convective pathways as diffusion-convection-dependent inhomogeneity (*dcdi*) to distinguish it from convection-dependent inhomogeneity (*cdi*), in which transport occurs only along pathways. They argued that these two mechanisms were distinguishable in plots of*S*
n vs. breath number (*n*) by virtue of the following characteristics: the contribution of *dcdi* is large for *breath 1*, increases slightly during the next few breaths, and remains constant for higher breath numbers, whereas the contribution of *cdi* continues to increase with *n*. A small but persistent stream of papers has followed. Differences between plots of *S*
n vs.* n *for gases with different diffusivities and changes in these differences with changes in gravity or posture have been interpreted according to these ideas.

Recently, I was asked to review a paper from this school. In my review, I expressed my concern that data on *S*
n were being overinterpreted. Rather than lying in the bushes as a reviewer, it seems better to express these concerns publicly, and that is the purpose of this letter.

My belief that plots of *S*
n vs.* n *are being overinterpreted is based on the results of more modeling. A slightly extended, but still extremely simplified, model of nonuniform ventilation illustrates this point. The model is a compartmental model with different volume expansions and different time courses of emptying for the compartments. The model is described as follows. Total lung volume, nondimensionalized by end-inspiratory lung volume, is denoted as Vl. In the washout maneuver, each breath extends from Vl = ½ to Vl = 1. The lung is divided into *m* compartments with equal volumes at Vl = 1. The volume of the *i*th compartment, nondimensionalized by its end-inspiratory volume, is denoted V_{i}. The nondimensionalized compartmental volumes are assumed to be quadratic functions of Vl.
_{i} vs. Vl, compartmental volume at end-expiration is* a _{i}
*, the slope of the plot is

*b*, and the curvature is 2

_{i}*c*. The values of the coefficients are subject to constraints. First, the constraint that V

_{i}_{i}= 1 at Vl = 1 requires that

*a*+

_{i}*b*/2 +

_{i}*c*/4 = 1. Second, the constraint that the sum of the compartmental volumes equals lung volume requires that Σ

_{i}*a*=

_{i}*m*/2, Σ

*b*=

_{i}*m*, and Σ

*c*= 0. Because of the constraint on V

_{i}_{i}, only two of these three equations are independent.

At the beginning of the washout maneuver, the concentration of the test gas in the lung is uniform and taken to be 1. The concentration in the*i*th compartment at the end of the *n*th inspiration, denoted C_{i}(*n*), is the concentration during the previous breath C_{i}(*n* − 1) times the volume ratio *a _{i}
*; the concentration in the expired gas, denoted C(Vl,

*n*), is the sum over compartments of the product of C

_{i}(

*n*) and the contribution of the

*i*th compartment to expired volume, −(1/

*m*)dV

_{i}/dVl; and the slope of phase III, denoted

*S*(

*n*), is the derivative of C(Vl,

*n*) with respect to expired volume. The mean concentration in the expired gas, denotedC̄(

*n*), is the value of C(Vl,

*n*) midway through the expiration at Vl = ¾.

*S*n(

*n*), is the ratio

*S*(

*n*)/C̄(

*n*). This is a simplified model in which dead space is neglected and diffusion is ignored, except that the compartments are assumed to be well mixed. It is the same as Paiva's earlier model (2) but generalized to

*m*compartments.

Two three-compartment models, described by the two sets of parameter values shown in Table 1, are denoted*models A* and *B*. Plots of V_{i} vs. Vl for these models are shown in Fig. 1, and plots of*S*
n vs. *n* are shown in Fig.2. The plot of *S*
nvs.* n *for *model A* is like the plots for humans reported by Crawford et al. (1), and the plot for*model B* is like the plots for rats reported by Verbanck et al. (5). The human data have been interpreted as showing a dominant *cdi* mechanism, and the rat data have been interpreted as showing a dominant *dcdi* mechanism. Of course, the values of *S*
n shown in Fig. 2 are entirely the result of the *cdi* mechanism because the *dcdi*mechanism is not included in the model.

In a two-compartment model, ventilation and course of emptying in the two compartments are completely coupled; the values of the parameters for the second compartment are entirely determined by the values for the first, and only two of the six coefficients in *Eq. 1
* can be chosen independently. A three-compartment model allows a little more freedom. Four of the nine parameters of that model can be chosen independently. With the different distributions of ventilation and time courses of emptying that are allowed in a multicompartment model, a wide variety of plots of *S*
n vs.* n *can be obtained, and these include plots that have been interpreted as signatures of the *dcdi* mechanism.

My objective here is not to propose an alternative model for interpreting the slope of phase III or to claim that diffusion, including the *dcdi* mechanism, does not play a role in determining the slope of phase III. There is ample evidence that diffusion plays a role in gas transport, and the physical basis for the*dcdi* mechanism is clearly sound. Dr. Paiva has made a determined effort to use accurate models of the geometry of the acinus. However, assumptions about the distribution of ventilation are still required in the model. In addition, the slope of phase III not only depends on the distribution of ventilation, it also depends on the distribution of time constants and the correlation between ventilation and time constants. The distribution of ventilation at the small scale at which most of the nonuniformity occurs has not been described thoroughly. Less is known about the time course of emptying and the correlation between ventilation and time course. The mechanisms that lie behind nonuniform ventilation and time of emptying are unknown. My objective is only to state that, at this point, I do not think we have an adequate foundation for interpreting plots of*S*
n vs. *n*.

- Copyright © 2001 the American Physiological Society

## REFERENCES

## REPLY

*Only experiments and new intra-acinar morphometrical data in humans will tell whether the “slope of phase III is being overinterpreted” . . . Or not To the Editor:*The letter of Ted Wilson helps to shed light on the interpretation of the phase III slope, and I would like to make a suggestion along his lines. Nevertheless, his letter also requires a clarification and two corrections.

#### Clarification.

Traditionally, the slope of phase III or alveolar plateau refers to vital capacity (VC) single-breath washouts (SBW). Usually, the measured gas is nitrogen, after a VC inspiration of oxygen followed by a VC expiration at constant flow. We will only be concerned here with the slope of phase III during multiple-breath washouts (MBW). The traditional end-expired volume at each breath is functional residual capacity. This point is important, as it was shown (1-5) that almost all the phase III slope of a VC SBW is due to inhomogeneities occurring near residual volume and total lung capacity. MBW involves mechanisms acting only in the range of normal breathing.

#### Corrections.

The MBW in humans has not “been interpreted as showing a dominant*cdi* mechanism” (1-1-1-4, 1-10) and “the plot of *S*
n vs.* n *for *model A*” is not “like the plots for humans reported by Crawford et al. (1-1).” The basis for computation of *dcdi* and*cdi* contributions is that, contrary to *cdi*, the back-extrapolation to *breath 0* of *S*
nshould not be zero for *dcdi* models. This is clearly shown in Fig. 6 of Crawford et al. (1-1) and Fig. 3 of Verbanck et al. (1-10).

#### Suggestion.

Even if Dr. Wilson states that his objective “is not to propose an alternative model for interpreting the slope of phase III,” the interest of extending to three compartments the two-compartment model published in 1975 (1-6) can only be evaluated by comparison with experiments. Unfortunately, Dr. Wilson simulated experiments with 20 successive inspirations up to TLC that were never performed (they may not be feasible, even in trained subjects). I suggest that he use the three-compartmental model to simulate the slopes from 1-liter inspiration from FRC and to compute the model parameters from parametric fitting on the published data. Then, using the same parameters, he can evaluate the predictive power of the model by checking whether the simulated slopes with increasing tidal volume or end-expiratory lung volume are in agreement with experiments (1-3,1-4). My guess is that the number of compartments has to be increased to four, five, or more. Perhaps I have a too negative preconception of models of data because my research has been focused on models of systems (1-7).

#### Discussion.

Whatever the number of compartments characterized by the curves of Fig. 1 and obtained from experimental data, the physiological meaning of these curves is of interest. Because MBW performed in microgravity were identical to those performed on the ground (1-8), the mechanical characteristics of the compartments are gravity-independent. Furthermore, because 1 s of end-inspiratory apnea significantly changes the normalized slopes (1-2), the curves of Fig. 1 should correspond to inhomogeneous properties of lung periphery. However, I think that different time constants, as suggested by Dr. Wilson, are not relevant for the model, because, in those zones, time constants are much smaller than the breathing period. Obviously, inhomogeneous elastic properties (compliances) in the lung periphery may play a role, and we fully agree that “the distribution of ventilation at the small scale at which most of the nonuniformity occurs has not been described thoroughly.” New imaging techniques may be decisive in this field.

Finally, and whatever the number of compartments, the model suggested in the letter cannot simulate slopes depending on gas diffusivity, except by introducing dead spaces, specific for each gas. It would be even more difficult to imagine this type of model simulating slopes that are larger for the more diffusive gas, as in rats (1-9). We recall that the very same model based on rat morphometrical data simulated realistic He and SF_{6} slopes of SBW and MBW without requiring fitting of any parameter, i.e., based on the measured lung structure and on the classical laws of physics. It would have been maliciously miraculous that the computer predicted the correct values. Unfortunately, detailed morphometrical data of human acini are not yet available. This precludes a similar study in humans. I hope that the present correspondence will stimulate anatomists to perform such important work.

#### Conclusion.

I am confident that *1*) both *cdi* and*dcdi* mechanisms are important in describing ventilation inhomogeneities in the human lung, *2*) the models we have been using are realistic, and *3*) the slope of phase III is not being overinterpreted. I think that a consensus on the subject will come more from experiments and new intra-acinar morphometrical data in humans than from new modeling.

I thank Ted Wilson for his letter and for not “lying in the bushes” as a reviewer.

- Copyright © 2001 the American Physiological Society