Modification of a sigmoidal equation for the pulmonary pressure-volume curve for asymmetric data

To the Editor: A sigmoidal equation to characterize the pulmonary pressure-volume (P-V) curve has been proposed by Venegas and coworkers (3) and is widely used for computerized analysis of P-V curves. It is based on a mathematical model introduced by Paiva et al. (1), amended to fit the entire curve even below functional residual capacity (FRC). The equation

where a represents lower asymptotic volume, b the upper asymptotic volume or vital capacity, c the pressure at the point of highest compliance (“true” inflection point of the sigmoidal curve), and d an adjustment parameter, provides a close proximity between observed and calculated P-V relations by using the Levenberg-Marquardt fitting procedure. This function is symmetrical with respect to the inflection point, but there is no physiological reason justifying the symmetry, prompting Venegas et al. to further develop the model so that it could be applied to asymmetric data as the product of a normal distribution-based recruitment function [R(P)] times a function representing the compliant properties of the already-recruited lung above FRC, such as the one described by Salazar and Knowles (2) with A corresponding to total lung capacity (TLC) and parameters B and k reflecting the elastic stiffness of the lung. Venegas et al. (3) concluded that this would result in the following

Choosing arbitrary parameters, Venegas et al. presented a simulated model of the shape of the resulting curve. However, the increase from four to five fitting parameters was judged to be disadvantageous.

The following comments have to be made about the mathematical modeling. In Eq. 4, the value of V for P = c is infinite, since the exponent -(P - c)/d will become zero and e⁰ = 1; therefore, the denominator will be zero, meaning that the function has a singularity at P = c. Using the same arbitrary values (TLC = 1, B = 1, k = 0.1, d = 2.5, and c = 15), we graphed of the function represented by Eq. 4 (see our Fig. 1).

To be mathematically correct, the denominator of Eq. 4 should read 1 + e^[-(P-c)/d] instead of 1 - e^[-(P-c)/d], which will never become zero, and thus the graphic of the function would have the same shape as that presented in Fig. 7 of the article by Venegas et al. (3).

Another concern is regarding Eq. 3; the original equation by Salazar and Knowles (2) was expressed as

with V₀ being the maximum lung volume, which is equal to a + b observed in Eq. 1. Therefore, if we substitute Eq. 5 for Eq. 3 into the corrected Eq. 4, we get with only four fitting parameters.

When we use the same arbitrary values for V₀ = 1, k = 0.1, c = 15, and d = 2.5 in Eq. 6, we obtain a function whose graphic is also plotted in Fig. 1, which can also be compared with the graphic from Eq. 1, with a = 0, b = 1, c = 15, and d = 2.5.

Another point that needs clarification is the concept of convergence used to show that, in the upper limit of pressures, where e^-(P-c)/d <<1, Eq. 1 converges to Eq. 3 with A = a + b, B = be^c/d, and k = d, since convergence is strictly related with the definition of limit.

Considering the fact that 0 < e^-(P-c)/d < 1 for P > c and that e^-(P-c)/d converges to zero when P takes higher values, Eq. 1 will converge toward the horizontal upper asymptote a + b.

Equation 3 has similar asymptotic behavior, within the same range of pressure values, and this function will also have a horizontal upper asymptote A; thus A = a + b.

In Eq. 3, parameter B must always be positive and its value will influence the shape of the curve V = -e^-kP before suffering a vertical translation associated to the , and k is a measure of adjustment. The same kind of geometrical transformations also occurs in Eq. 1.

With this kind of consideration, it is understandable that a correspondence exists between Eq. 1 and Eq. 3. The authors have shown that Eq. 1 and Eq. 3 have the same asymptotic behavior, but the way the parameters B and k were derived still remains unclear.

Equation 6 represents the correct mathematical model; however, it neither proves the underlying assumptions nor the physiologic correctness. These steps have yet to be done in future investigations.