Abstract
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 +
 mechanical properties
 lung compliance
 lung recoil
 acute respiratory distress syndrome
 pneumoconstriction
quasistatic pressurevolume (PV) curves have been used in research and in the clinical setting to quantify the elastic properties of the lungs and respiratory system, particularly with respect to changes in surfactant composition (1, 17, 23), lung recoil (7, 19, 25), and degree of alveolar derecruitment (24). Quantification of the curve typically consists of the compliance or specific compliance from the slope of the curve at a given pressure or over some volume range, the measurement of recoil pressure at a given fractional volume, or the measurement of the fractional lung volume remaining at a given inflation pressure. However, because of the nonlinear shape of the PV curve, the values of these parameters and the changes observed in these parameters can vary substantially depending on the arbitrary pressure or volume used in the comparison. Furthermore, accurate estimation of these parameters may require the collection of data at precise points of the curve or, if such data are not available, extrapolation of the data within a section of the PV curve, while a substantial fraction of the data is ignored.
Describing the lower portion of the PV curve could also be important in the management of patients with the acute respiratory distress syndrome (ARDS). The inflation limb of the respiratory system PV curve has recently been proposed to identify a safe range of ventilatory pressures during mechanical ventilation in patients with ARDS (2, 26). It has been postulated that ventilatorinduced lung injury, occurring in ARDS, is caused by overdistension of alveoli at high transpulmonary pressures and/or by increased forces caused by the repetitive recruitment and derecruitment of alveolar units (10, 18, 31, 32). In ARDS the inflation limb of the PV curve has a sigmoidal shape, with a point of rapid change in upward curvature, which will be referred to as “lower corner pressure” (P_{cl}),1and a point of rapid change in downward curvature, which will be referred to as “upper corner pressure” (P_{cu}). Physiologically, P_{cl }is thought to correspond to the pressure at which a maximal alveolar recruitment occurs, whereas P_{cu} is thought to represent the pressure above which maximal elastic distension of the lung parenchyma is approached. Mechanical ventilation delivered with airway pressures kept within the range from P_{cl} to P_{cu} is thought to limit alveolar overdistension and maximize recruitment of alveolar units (2, 26). In practice, these points of maximum curvature are often determined by eye from a plot of the PV curve, a method that is not only imprecise, but also highly subjective.
Curve fitting of the PV relationship is one approach to solving these problems. The curve fit permits more accurate extrapolation of the curve over a desired data range. To facilitate comparisons between curves obtained from different subjects or under changing conditions, the volume data are often normalized by total lung capacity (TLC), defined as the lung volume at an arbitrary inflation pressure ranging from 25 to 40 cmH_{2}O (4). Because individual data sets rarely reach the exact maximum pressure, TLC values may be more objectively obtained by curve fitting the data. In addition, physiologically meaningful parameters obtained from the fitted model may better characterize the PV curve over its full range, rather than at an arbitrary specified point. Although only valid for volumes >50% TLC (27), the exponential equation proposed by Salazar and Knowles (28) has been most widely utilized to characterize these volumes, and the parameters thus obtained correlated with changes in pulmonary elasticity with aging and smoking (5, 6, 9) and with emphysema, asthma, and interstitial fibrosis (1315). The lower portion of the curve, which has relevance for assessing alveolar recruitment and air trapping, is not described by these models. Polynomial and other models that describe the entire PV curve have been limited by the lack of physiological significance of the parameters (8).
Fitting an equation to experimental or clinical PV data provides a systematic method to characterize PV curves and derive objective parameters from them. The purpose of this communication is to present a simple form of a sigmoidal equation that fits with remarkable accuracy the inflation and deflation limbs of PV curves obtained under a variety of experimental and pathological conditions and yields physiologically useful parameters. When the pressure and volume data are expressed in dimensionless form normalized by parameters obtained from the model, the data collapse onto a single comprehensive PV relationship.
METHODS
A sigmoidal equation of the static PV curve was formulated as
Inflation limbs of the PV curves were obtained by manual inflation with calibrated syringes in 8–10 discrete volume steps until an airway opening pressure (Pao) close to 30 cmH_{2}O was reached. Deflation limbs of the PV curves were obtained after an inflation to TLC by withdrawing volume in discrete steps back to atmospheric airway pressure. For each lung volume, Pao was recorded after ∼5 s to allow the pressure to reach a quasisteadystate value. Data from dogs in which ARDS was induced by intravenous infusion of oleic acid consisted of transpulmonary pressure, calculated as Pao minus esophageal pressure, vs. total absolute lung volume, estimated as inflation volume plus functional residual capacity (FRC) measured by helium dilution. Human data consisted of inflation volume above FRC plotted against Pao. Left lung PAoccluded openchest dog data consisted of absolute lung volume, measured with a positron camera, vs. Pao (30). PV curves were fitted by Eq. 1 in a personal computer using the LevenbergMarquardt iterative algorithm to minimize the sum of squared residuals. The algorithm was set to run until the resulting sum of squared residuals changed by <0.0001, yielding estimates of the parameters a, b, c, andd and the bestfit coefficientR ^{2}.
RESULTS
Equation 1 fitted equally well inflation and deflation PV curves from normal, ARDS, and pneumoconstricted lungs (Figs.13) with mean goodnessoffit coefficient (R ^{2}) of 0.997 ± 0.02 (SE). Review of the fitted parameters revealed the following results. In the dog lung the inflation P_{cl} increased from a negative value (−21 cmH_{2}O) to positive values of 3.2 and 8.7 cmH_{2}O at 30 and 60 min after induction of ARDS. In 8 of the 10 ARDS patients, inflation limb P_{cl} was also greater than zero [9 ± 6 (SD) cmH_{2}O]. The other two patients had a negative inflation limb P_{cl} (average −23 cmH_{2}O). In the left PAoccluded dogs the inflection point, c, occurred at a significantly greater pressure (P< 0.05) in the occluded left lung (5 ± 0.46 cmH_{2}O) than in the control right lung (0.78 ± 1.55 cmH_{2}O). When the data from all analyzed PV curves were normalized by the parameters derived by the fitting and plotted as (V −a)/bvs. (P −c)/d, they collapsed into a tight relationship (Fig.4) closely following Eq.1 (R^{2} = 0.997) and with residuals evenly scattered within a 5% range (Fig.5).
DISCUSSION
The major finding of this study is that a simple sigmoidal equation (Eq. 1 ) fitted with excellent accuracy the inflation and deflation limbs of experimental and clinical PV curves obtained under a variety of experimental and pathological conditions (R ^{2} = 0.997). Furthermore, when the pressure and volume data are expressed in dimensionless form and plotted as (V −a)/bvs. (P −c)/d, they collapse onto a comprehensive PV relationship.
For obvious reasons, the sensitivity of the fitting to parameters that depend on data not available is inherently poor, and fitted values of these parameters are unreliable. For example, in Fig. 2 the upper asymptote of the inflation limb in ARDS is never reached, and thus the parameter b obtained from the fitting is not reliable, even though the parameters a, c, and d may be. This limitation should not restrict the usefulness of the equation, since1) as discussed above in most cases it is not practical, or even desirable, to obtain complete data sets from both asymptotes and 2) only parameters sensitive to data within the measured range are those generally sought. As with any equation, it is important to be aware of this limitation and conduct a parameter sensitivity analysis to assess the reliability of the parameters estimated from the curve fit (11).
An important feature of Eq. 1
is the ability to objectively characterize the PV relationship and obtain accurate estimates of physiologically relevant parameters. For example, compliance at any pressure or volume, the first derivative of the equation, can be expressed as
Recently, there has been increasing interest in the lower and upper portions of the PV relationship as regions possibly representing recruitment and overdistension of alveoli, respectively. This notion has gained acceptance clinically in ARDS patients as a way to identify safe limits of ventilatory pressures for mechanical ventilation. In the literature, actual P_{c} values have been derived by eye from plotted PV data, and the definitions for them have been inconsistent. P_{c} has been defined as the points at which the PV curve consistently separates from a straight line drawn through the most linear portion of the curve (24) or where such a line intersects with lines drawn tangent to the PV curve at the lowest and highest pressures measured (2, 12). We chose to define P_{c} as the intersections between a tangent to the PV curve at its point of maximum compliance (inflection point,c) and the two horizontal asymptotesa andb, which can be readily derived fromEq. 1 as P_{c} =c ± 2d. Although this definition resembles that from Amato et al. (2), Eq. 1 could also be used to define P_{c} in different ways: as the points of maximal upward and downward curvature of the PV curve (P_{c} =c ± 1.317d), or as the points of maximal rate of change of curvature (P_{c} =c ± 2.29d). A clinically optimal definition of P_{c} remains to be determined.
The following results further illustrate the usefulness ofEq. 1 . In the normal lung, alveolar derecruitment at FRC should be minimal, whereas in ARDS, derecruitment should be substantially increased. If the location of P_{cl} on thexaxis reflects the pressure at which rapid alveolar recruitment begins, then in the normal lung P_{cl} should be negative, whereas in ARDS P_{cl} should be shifted to the right and positive. In the dog lung P_{cl} shifted from a negative value in control conditions to a positive value 60 min after induction of ARDS. Similarly, in eight of the ARDS patients P_{c} was greater than zero. These results are therefore consistent with the substantial alveolar derecruitment expected at low levels of lung inflation in ARDS. In the unilaterally left PAoccluded dogs the inflection point,c, was substantially greater in the PAoccluded lung than in the control right lung. This is consistent with the increase in lung recoil, measured at 50% TLC (30), of the PAoccluded lung caused by hypocapnic pneumoconstriction. However, the difference in lung recoil between the left and right lungs, measured as the horizontal distance between the two curves, is highly dependent on the standard volume selected in the comparison (Fig. 3).
Other PV equations.
A detailed discussion of equations used to fit PV data was presented by Murphy and Engel (20). Of these, the exponential function originally proposed by Salazar and Knowles (28) has the form
A sigmoidal model of the static PV curve of the form
A hyperbolicsigmoidal equation of the form
It is noteworthy that the single equation proposed here (Eq. 1 ) could fit this diverse collection of PV curves, particularly since Eq.1 is symmetrical2with respect to its inflection point. One would not expect a priori actual PV data to have comparable symmetry, because the mechanical phenomena occurring at each end of the curve are very different. For example, during inflation the lower part of the curve probably reflects a combination of progressive recruitment of closed alveoli and elastic inflation of open lung regions. In contrast, the upper portion of the curve may correspond to approaching the maximal elastic distension of the fully recruited alveoli. Surface phenomena also have characteristics in inflation that are different from those in deflation. It is important to note that although Eq.1 may be symmetrical, the experimental data sets used were incomplete, in that they did not include data distributed throughout the entire range of the model. Most data sets are biased toward one asymptote. Deflation limbs typically include more data from the upper asymptote; inflation limbs tend to include more data from the lower asymptote. This limitation, however, does not seem to restrict the usefulness of Eq. 1 , since in most cases it is not practical, or even desirable, to obtain complete data sets from both asymptotes. Deflation PV curves generally stop at atmospheric pressure and, in the absence of lung disease and in the intact subject, may not approach the lower asymptote at this pressure. Similarly, it may not be possible to extend an inflation PV curve to airway pressures high enough to reach the upper asymptote. Although all the data analyzed here fitted the model well, it is possible that Eq. 1 might have to be modified to fit data sets that extend significantly onto both asymptotes.
One possible modification may come from the observation that the sigmoidal Eq. 1
presented in dimensionless form as
Using these concepts, one can also formulate a PV equation capable of fitting asymmetric data as the product of a normaldistributionbased recruitment function 𝘙(P)
In summary, we have formulated a mathematical expression that fits with excellent accuracy (R ^{2} = 0.997) the inflation and deflation limbs of experimental and clinical PV curves obtained under a variety of experimental and pathological conditions. The pressure and volume data are plotted normalized by parameters obtained from the model; the data tightly follow a single comprehensive relationship. This equation provides a method to systematically characterize PV curves and objectively derive physiologically and clinically useful parameters such as vital capacity, maximal inspiratory volume, compliance at different inflation pressures, inflection pressure, and upper and lower corner pressures.
Acknowledgments
This work was supported by National Heart, Lung, and Blood Institute Grant HL38267.
Footnotes

Address for reprint requests: J. G. Venegas, Dept. of Anesthesia (CLV255), Massachusetts General Hospital, Boston, MA 02114.

Present address of B. A. Simon: Dept. of Anesthesia and Critical Care Medicine, Johns Hopkins School of Medicine, Baltimore, MD 22222.

↵1 This pressure is often referred to as “lower inflection point” in the literature (2, 20, 26). Strictly speaking, the inflection point of a curve is the point at which the curvature changes direction or sign and not the point of maximum curvature.

↵2 Strictly speaking, a functionF(x) is symmetrical with respect to the origin whenF(x) = −F(−x). In the case of Eq. 1 ,F(x) with x = (P −c)/dand transforming the function toG(x) =F(x) −F(0), the function is symmetrical with respect to the inflection point (P =c orx = 0), sinceG(x) = −G(−x). This can be readily seen from the plot in Fig. 4 and by noting that the curvatures of the function at the lower and upper corner pressures are equal but have opposite signs.
 Copyright © 1998 the American Physiological Society