Abstract
The following is the abstract of the article discussed in the subsequent letter:
Venegas, José G., R. Scott Harris, and Brett A. Simon. A comprehensive equation for the pulmonary pressurevolume curve. J. Appl. Physiol. 84(1): 389–395, 1998.—Quantification of pulmonary pressurevolume (PV) curves is often limited to calculation of specific compliance at a given pressure or the recoil pressure (P) at a given volume (V). These parameters can be substantially different depending on the arbitrary pressure or volume used in the comparison and may lead to erroneous conclusions. We evaluated a sigmoidal equation of the form, V = a +b[1 + e ^{−(P−c)/d}]^{−1}, for its ability to characterize lung and respiratory system PV curves obtained under a variety of conditions including normal and hypocapnic pneumoconstricted dog lungs (n = 9), oleic acidinduced acute respiratory distress syndrome (n = 2), and mechanically ventilated patients with acute respiratory distress syndrome (n = 10). In this equation, a corresponds to the V of a lower asymptote, b to the V difference between upper and lower asymptotes, c to the P at the true inflection point of the curve, and d to a width parameter proportional to the P range within which most of the V change occurs. The equation fitted equally well inflation and deflation limbs of PV curves with a mean goodnessoffit coefficient (R ^{2}) of 0.997 ± 0.02 (SD). When the data from all analyzed PV curves were normalized by the bestfit parameters and plotted as (V − a)/b vs. (P − c)/d, they collapsed into a single and tight relationship (R ^{2} = 0.997). These results demonstrate that this sigmoidal equation can fit with excellent precision inflation and deflation PV curves of normal lungs and of lungs with alveolar derecruitment and/or a region of gas trapping while yielding robust and physiologically useful parameters.
Is the Pulmonary PressureVolume Curve Symmetrical With Respect to the Inflection Point?
To the Editor: In the recent report by Venegas et al. (7), the authors propose an equation, which is, as they remark, equivalent to the exponentialsigmoid equation proposed by Paiva and coworkers (3). This equation can be shown to be a special case of a generalized growth model initially proposed by von Bertalanffy (1) and later elaborated by Richards (4) and Sager (5). The generalized function is defined by a differential equation, which transferred to the pressurevolume (PV) relationship of the lung is dV/dP = kV (
The differing shapes of these curves, the monoexponential, the exponentialsigmoid, and the asymmetricsigmoid, are due solely to the difference in n. This parameter determines the proportion of the final value at which the inflection point occurs. The inflection point, obtained by setting the second derivative of the mother function to zero, of all these curves is at V = V_{max}(1 + n)^{−1/n}. The parameter kexpresses the rate at which the value of some function of V changes, e.g., ln[(V_{max} − V)/V] for the exponentialsigmoid function. The parameter b has usually no physiological meaning and can be eliminated from the equation by adjustment of the pressure scale.
Instead of combining the two special cases, the monoexponential and the exponentialsigmoid equations, as suggested by the authors, it is possible to obtain curves of different shapes by changing the parameters in the generalized function, as illustrated in Fig.1, with two curves with differentb, k, and n. Changing d in the equation of Venegas et al. (7) results in first derivatives of the dimensionless forms resembling normal distributions with different SD. In the generalized function, as can also be seen from Fig. 1, the graphs of the first derivatives are asymmetric, with the location of the tail depending on n. This opposite asymmetry gives a basis for the possibility that the distributions of pressures differ between inspiration and expiration.
 Copyright © 1998 the American Physiological Society
REFERENCES
REPLY
To the Editor: In his Letter to the Editor, Sonander proposed an alternative function to describe the pulmonary PV relationship. He suggests that to fit asymmetric PV curves one could use a generalized sigmoidal function instead of combining a monoexponential together with an exponentialsigmoid functions, as proposed by us (12).
The proposed generalized sigmoidal function was expressed in the form used by Paiva and coworkers (11), which did not include a nonzero lower asymptote V present in most PV curves. If that parameter is included, the proposed equation can be expressed in dimentionless form
We appreciate that Sonander’s proposed equation, like our Eq. 7, could give superior fitting to PV data than our symmetric sigmoidal Eq. 5 in specific cases where the data encompass both low and highpressure asymptotes. It remains to be seen. Whether the additional parameter involved in the generalized function, and, for that matter, in our Eq. 7, can be statistically justified given the almost perfect fit obtained with the symmetric sigmoidal alone for most clinical PV data.
We disagree with the suggested advantage of using the asymmetric sigmoidal equation instead of our Eq. 7. Both equations involve the same number of parameters, and both equations result in asymmetric sigmoidal shapes. The main difference between these equations is that the asymmetric sigmoidal is empirical in nature, whereas Eq. 7 was derived from a mechanistic hypothesis: namely, that the global PV curve results from the product of the micromechanical properties of a recruited alveolar unit times the number of recruited alveolar units at each PV condition. If the measured data allowed it, using Eq. 7 could, for example, be used to extract physiologically related parameters such as the SD and mean values of a distribution of opening or closing airway pressures. The parameters derived from the asymmetric sigmoidal, although descriptive, would have no physiologically meaningful correlates.
A potential application of the asymmetric sigmoidal equation proposed by Sonander could be to describe an asymmetric recruitment function and replace the symmetric sigmodal recruitment function 𝘙(P), used to derive Eq. 7. In such application, the derivative of the recruitment function would describe a distribution of opening pressures that could be skewed instead of normal. Unfortunately, the resulting PV equation would contain yet one more parameter (n), increasing the total number to six. Given the already excellent curve fitting obtained using the fourparameter sigmoidal equation, it is difficult to expect that this application would become practical.
Footnotes

↵* There is a typo in Eq. 7 of our paper (12), the sign in front of the exponential in the denominator should have been a plus instead of a minus.

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 Copyright © 1998 the American Physiological Society