quasi-static pulmonary pressure-volume curves (P-V curves) have been used to characterize the mechanical behavior of the total respiratory system in research and clinical settings. In a typical P-V curve, a region of a steep (nearly linear) slope with greater compliance (the volume change per pressure change) exists between high and low pressure ranges where the curve flattens with very low compliances. The local compliance (i.e., compliance at a specified value of pressure) corresponds to a local gradient of the pulmonary P-V curve. High compliance is associated with both distension of open alveoli of the lungs and recruitment of collapsed alveoli of the lungs (9). To understand the properties of the respiratory system as well as their changes observed in clinical studies, various P-V equations have been proposed (3, 5, 9-12, 14, 15). Parameters in the equations are determined by curve fitting the equation to a set of data of interest. The parameters in a model equation should have some physiological interpretation based on mathematical characterization; therefore, it is important that the parameters be defined with precision in order for the equation to be used for a quantitative characterization of the pathophysiology of a patient as well as a guide for therapy and improved patient care (1, 2, 8, 9).

Recently, a sigmoidal form of a P-V model equation

was proposed by Venegas, Harris, and Simon (15) (hereinafter, to be referred to as V-H-S) with the following characteristics: 1) the parameter a represents the lower bound of the volume; 2) the parameter b represents a difference between the upper and the lower bound of volume; 3) the parameter c indicates the inflection point of the curve; and 4) the parameter d is associated with the width in the pressure range within which most of the volume change occurs.

The symmetric equation with the four parameters was proven to be in excellent agreement with data from both healthy humans and dogs and those with lung injury. A more recent report, based on analyses of 24 sets of inflation and deflation P-V curves from patients with the acute respiratory distress syndrome (ARDS), also shows that the application of the sigmoidal equation reduces inter- and intraobserver variability in the clinical evaluation of P-V curves (8). Data analyses of P-V curves have been based on the magnitude of pressure at a certain point of a specified curve, such as the lower inflection point (LIP; loosely defined as pointed out in Ref. 8), the pressure at the point of maximum compliance increase (decrease), P_mci(mcd), in addition to the magnitude of a, b, c (= true inflection point), and d. The objective of this article is to examine the mathematical basis of the sigmoidal equation and apply a dimensional analysis that may help relate clinical data to the corresponding shape and range of the pulmonary P-V curves, particularly using nondimensional parameters constructed from the parameters of the sigmoidal equation and the data range in terms of inflation/deflation work. First, the paper briefly describes the mathematical development and examination of the sigmoidal equation; this is followed by a section that analyzes available P-V data based on the parameters developed in the previous sections. Relations of the sigmoidal equation with other model equations are discussed in the last section.

Glossary

a	Lower bound of V in V-H-S (ml) = VL
b	Difference between the upper and the lower bound of V in V-H-S (ml) = VU − VL
c	Pressure at the inflection point of the curve in V-H-S (cmH2O) = P0
d	Width in the pressure range within which most of the volume change occurs in V-H-S (cmH2O) = 1/α(VU − VL)
P	Pressure (cmH2O)
Pcu(cl)	Upper (or lower) corner pressure, intersection between a tangent to the P-V curve at the inflection point, P = P0, and the two horizontal volume asymptotes, VU and VL
Pmci(mcd)	Pressure at the point of maximum compliance increase (decrease)
P0	Pressure at the inflection point of the curve
Pgrad	(VU − VL)/(dV/dP)max
P̄	P/P0 − 1
V	Volume (ml)
VU	Upper asymptote (ml)
VL	Lower asymptote (ml)
V1	Lower bound of volume integral (ml)
V2	Upper bound of volume integral (ml)
ΔV	VU − VL = b (ml)
V̄	V−(VU+VL)/2ΔV/2
W1–2	Work during expansion and compression processes (cmH2O · ml)
W̄1–2	Nondimensional work during expansion and compression processes
α	Constant of proportionality in Eq. 2a(ml−1 · cmH2O−1)
ω	ΛP̄/2
Λ	αP0 (VU − VL), nondimensional

DEVELOPMENT OF NONDIMENSIONAL SIGMOIDAL EQUATION

Equation 1 implies that, at the upper and the lower asymptotes of the volume that occur as P → ∞ and P → −∞, respectively, the lungs are either fully distended or collapsed. In other words, dV/dP (local compliance) approaches zero as the volume approaches the asymptotes. Therefore, a continuous function, V(P), satisfying these conditions is a solution to the following first-order differential equation

where α is a positive constant. A solution to the first-order differential equation (Eq. 2a) is the sigmoidal equation for the pulmonary P-V curve of V-H-Swhere P₀ is a pressure at the midpoint (inflection point) of the curve (for derivation, see ).

Harris et al. (8) plotted 547 P-V data points from both inflation and deflation limbs in a nondimensional P-V plane with αΔV(P − P₀) and (V − V_L)/ΔV as nondimensional pressure and volume, respectively, showing excellent agreement of the data points with the sigmoidal equation because all points clustered around the nondimensional sigmoidal equation. In our attempt to make Eqs. 2a and 3a nondimensional, we shift the origin of the coordinates (P, V) to the midpoint of the curve, located at P₀, (V_U + V_L)/2. Also, the volume is normalized so that its range falls between −1 and 1; that is, we introduce a nondimensional pressure, P̄, and a nondimensional volume, V̄, which are defined as

Equations 2a and 3a may be expressed in terms of the nondimensional variables aswhere Λ (nondimensional) = αp₀ΔV and ω = Λp̄/2.

It should be noted that the symmetric property of the sigmoidal equation becomes clear in the form of Eq. 3b; i.e.V̄(ω) = −V̄(−ω).

In characterizing P-V curves, magnitudes of various pressures at specific locations on a P-V curve are often used for clinical examinations (8, 9, 15). They are related to the nondimensional parameter, Λ, as shown below. We define the (volume) gradient pressure range, P_grad, as a pressure difference between the two intersections of a tangent to the sigmoidal curve at its point of maximum compliance (i.e., at P = P₀) and the two volume asymptotes, V_U and V_L, yielding

The two intersections are referred to as the upper (and lower) corner pressure, P_cu(cl). Also, the pressure at the point of maximum compliance increase (decrease) of the P-V curve, P_mci(P_mcd), may be specified as the points where the third derivative of V̄ with respect to P̄ is zero; henceFor derivations of Eqs. 4-6, see.

DEVELOPMENT OF NONDIMENSIONAL WORK

A quasi-static (quasi-equilibrium) process is defined as a process in which all states through which the system passes may be considered equilibrium states. Therefore, the work associated with the inflation and deflation of the lungs may also be evaluated readily along a specified quasi-static pulmonary P-V curve. Defining W_1–2 as work done during a process from an initial state 1(P₁, V₁) to a final state 2(P₂, V₂), i.e.

an integration of the right hand side using the sigmoidal equation yields the following nondimensional work,W̄_1–2, during inflation and deflation processeswhere ln indicates the natural logarithm.

The corresponding dimensional work W_1–2(cmH₂O · ml) is related toW̄_1–2 as

ANALYSIS

In our nondimensional representation of the sigmoidal equation, the parameter Λ, as the only parameter appearing in Eq. 2b, characterizes the shape of the P-V curve; whereas the work,W̄_1–2, takes into account the actual clinical data range relative to the whole range of the sigmoidal equation. Six sets of P-V data available in the literature are used to show the changes that occur in these two nondimensional parameters with different conditions of the total respiratory system. They are as follows: 1) inflation curves of a healthy anesthetized human immediately before and after an alveolar recruitment maneuver [from Fig. 4 in Jonson and Svantesson (9)], 2) inflation curve of a patient with ARDS (ARDS case 1I) [from Fig. 1 of V-H-S (15)], 3–5) inflation/deflation curves of three ARDS patients (ARDS cases 2, 3, 4) [from Harris et al. (8)], and 6) inflation/deflation curves of a supine dog before and 60 min after oleic acid-induced lung injury [from Fig. 2 of V-H-S (15)].

The data points of a healthy human and a dog were read from enlarged figures of the original papers (9, 15), which may have caused certain copying and reading errors. Therefore, instead of employing such methods as the least-square method and the least-square collocation method (4) for curve fitting, we fit the sigmoidal equation with its four unknowns (V_U, P₀, ΔV, α) by systematically changing values of V_U and P₀ (knowing that V_U ≫ V_L) and then forcing the equation to satisfy two data points (close to the two ends of a data range) until the resulting curve is visually judged to be a best fit. The two nonlinear equations from the two data points are solved by the Newton-Raphson method. The parameters of the sigmoidal equation for ARDS case 1I are listed in the original paper. For ARDS cases 2, 3,and 4, either P₀ or P_mci is listed in Fig. 2 of Ref. 8; therefore, the fittings of the sigmoidal equation are achieved by forcing the equation to satisfy two data points (close to the two ends of the data range) and V(P = 0).

Table 1 summarizes the relationship between the parameters appearing in Eq. 1 (a, b, c, d) and the parameters in Eqs. 2a and 2b and 3a and 3b (V_U, V_L, α, P₀). It should be mentioned here that the parameter d in Eq. 1 is related to a difference between the upper and lower asymptotes, b, through the relation 1/d = αb. To elucidate the meaning of the parameter α, we differentiate Eq. 2a with respect to volume to obtain

The sigmoidal equation, therefore, may be alternatively defined as the P-V curve for which the gradient of local compliance change with volume is constant and is equal to −2α.

Table 2 lists the numerical values of the data sets. It may be seen in the table that the magnitude of the parameter α is small with an order of magnitude of 10⁻⁴to 10⁻⁵ cmH₂O · ml.

Figure 1 depicts characteristics of the P-V curves from a healthy human before and after an alveolar recruitment maneuver, and, referring to this, we look at the general characteristics of both the dimensional (P-V) and nondimensional (P̄-V̄) curves and how they relate to one another.

Figure 1A shows good agreement between the P-V data and the sigmoidal equation. The recruitment maneuver lowers the magnitudes of P₀, P_mci, and P_cl (lower corner pressure). Figure 1B, a plot of the local compliance profile, indicates that the maneuver results in a sharper profile around the peak at P₀, as well as a higher maximum compliance. A continuous change in the local compliance in Fig.1B should also be noted even in the region in Fig.1A where the volume, as first approximation, may be seen to change with pressure linearly. Figure 1C represents the corresponding nondimensional P-V curves, Eq. 3b, over their measured data range. General characteristics of P̄-V̄ curves are listed below.

1) The nondimensional pressure, P̄, is the pressure difference, P − P₀, as a fraction of P₀. The normalization of volume shifts the upper and the lower volume asymptotes, V_U and V_L, to +1 and −1, respectively. With both the location of P₀ and the volume asymptotes made common to all P-V curves, the resulting nondimensional representation characterizes P-V curves in general in terms of a single nondimensional parameter, Λ.

2) The origin, from the definition of P̄, is at the midpoint of the curve where the dimensional values of P, V are P₀, (V_U + V_L)/2. Therefore, the first quadrant (V̄, P̄ > 0) is a region of decreasing local compliance with pressure, whereas the third quadrant (V̄, P̄ < 0) is a region of increasing local compliance with pressure.

3) The origin of dimensional P-V curves where pressure is zero is transformed into P̄ = −1 on a P̄-V̄ curve; hence, the physiological lower limit of P̄ is −1. Various pressure locations in our nondimensional representation are proportional to 1/Λ as shown in Eqs. 5 and 6. Equations 5 and 6 also imply, over the pressure range of P > 0, that there is no lower corner pressure (i.e., 1 +P̄_cl < 0) if Λ < 2 and that there is no pressure for maximum compliance increase (i.e., 1 +P̄_mci < 0) if Λ < 1.317.

4) The gradient at the origin, dV̄/dP̄(P̄ = 0), is Λ/2 (Eq. 2b). On the other hand, the location of the lower corner pressure,P̄_cl, is −2/Λ; i.e.

On the P̄-V̄ plane, therefore, the locations of both the upper and the lower corner pressure become closer to the origin as the nondimensional local compliance at the origin increases. A comparison between two curves in Fig. 1C confirms the general relation above.

5) From Eq. 4, along with the fact that one-half of the volume-gradient pressure range is equal to the difference between the pressure at the inflection point and the lower (or upper) corner pressure (i.e., P_grad/2 = P₀− P_cl), we find

The parameter Λ is twice the ratio of the pressure at the maximum compliance, P₀, to the pressure difference between P₀ and the lower corner pressure, P_cl. Referring to Table 2, we may observe that the alveolar recruitment maneuver increases Λ from 3.23 (before maneuver) to 3.49 (after maneuver). Figure 1A shows that the maneuver lowers the magnitude of P₀ (resulting in a higher recruitment rate in the lower pressure range where the compliance is increasing) and also decreases the pressure difference, P₀ − P_cl (decreasing the region of increasing compliance). The net increase in Λ is a result of these two opposing effects induced by the recruitment maneuver. The corresponding changes in Fig.1C due to the increase in Λ are described above instatement 4.

6) The equation for the expansion/compression work is presented in the previous section. For inflation processes, an evaluation of work along the P̄-V̄ curve in the third quadrant (P̄, V̄ < 0) results in a negative value. As the process enters the first quadrant (P̄, V̄ > 0) where an integration yields a positive value, the magnitude of work also starts to increase toward a positive value.

In Fig. 1C, the recruitment maneuver shifts the curve closer to the ordinate with an effect of reduced work; however, more importantly, the maneuver extends the curve further into the first quadrant, thus increasing the net amount of work over the inflation process from −0.24 (before maneuver) to −0.08 (after maneuver), as shown in Table 2.

For the deflation process, on the other hand, a negative value results from an integration from the initial high-pressure state to the final low-pressure state (that is, the upper and lower bounds of integration are opposite to the case of the inflation process). Therefore, a negative value for the nondimensional work in the deflation process indicates that the process is mostly in the first quadrant. A value ofW̄_1–2 increases toward a positive value as the process extends into the third quadrant.

In Fig. 2 are plotted both P-V curves and local compliance profiles, dV/dP − P, for the four ARDS inflation cases (case 1I to case 4I). Figure3 shows the corresponding deflation cases (case 2D to case 4D). Their P̄-V̄curves are summarized in Fig. 4. We first discuss the inflation curves and then discuss the deflation curves.

The data range for cases 2I, 3I, and 4I in Fig. 2is limited, relative to the whole sigmoidal curve. Also, compared with the data from a healthy human (discussed above for Fig. 1), both ΔV and dV/dP are substantially lower in magnitude for all four cases. The inflection pressure, P₀, as well as the data range in the region of decreasing compliance (P > P₀) are quite different among the four inflation cases. Neither P_cl nor P_mci exists for case 2I.(This observation may also be made from the value of Λ listed in Table 2; i.e., Λ = 0.69 for case 2I.) In Fig. 4, the differences are more clearly presented in terms of the data ranges with respect to the location of the inflection point (= the origin). As explained earlier, the nondimensional pressures P̄_cl,P̄_mci, and P̄_mcd shift toward the origin as the nondimensional maximum compliance dV̄/dP̄(P̄ = 0) (= slope at the origin = Λ/2) increases. Two extreme cases are case 1I [large dV̄/dP̄(P̄ = 0), a small difference between P₀ and P_cl] and case 2I [small dV̄/dP̄(P̄ = 0), P_cl < 0].

The magnitude of nondimensional workW̄_1–2 as a measure of a data range relative to the whole sigmoidal curve is listed in Table 2. A large negative value of W̄_1–2 = −0.32 for case 4Ireflects the fact that the data range is mostly limited to the third quadrant (where the compliance is increasing) in Fig. 4, whereas a small positive value of W̄_1–2 = 0.07 forcase 2I shows that its data range covers the first (where the compliance is decreasing) as well as the third quadrant equally. Comparisons among various cases may be made either by direct visual observations of the nondimensional curves or by quantitative examinations of both the magnitude of Λ, which determines the shape of the normalized sigmoidal equation, and W̄_1–2, which defines the data range. Referring to Table 2, “after maneuver” and case 4I yield similar values for Λ. A difference between the two cases appears in the magnitude ofW̄_1–2. Compared with the data of after maneuver in Table 2, the total respiratory system of case 4I operates primarily in the region of the increasing compliance. Case 3I, when compared with case 4I, has a reduced value of Λ as a result of a reduction in P₀, which in turn extends the data range into the region of decreasing compliance as evidenced by an increase in W̄_1–2. Case 2I andcase 1I are the two cases in which the magnitude of Λ is either very low or very high compared with the data of a healthy human (see before maneuver and after maneuver in Table 2).

The deflation data sets (cases 2D–4D) in Fig. 3 have a lower pressure at the inflection point (broken vertical line) compared with the corresponding inflation case. (The magnitudes of Λ listed in Table 2 indicate that P_cl for case 3D and both P_cl and P_mci for case 2D are out of the pressure range of p > 0.) According to Eq. 10,therefore, the magnitude of Λ increases for case 2D as a large drop in P_cl compensating for the decrease in P₀, whereas, for cases 3D and 4D, Λ decreases from the corresponding value of the inflation limb. These differences of the deflation curves from the inflation curves are reflected in the P̄-V̄ curves in Fig. 3 as prominent extensions into the first quadrant and also in the magnitude ofW̄_1–2 in Table 2.

The P-V curves are summarized in Fig. 5for the data of a dog before and after lung injury, with the corresponding numerical values listed in Table 2. The magnitude of Λ before injury is small for both inflation and deflation limbs with a result that P_cl for the inflation limb and P_clas well as P_mci for the deflation limb are out of the pressure range. The magnitude of Λ increases after injury for both inflation and deflation limbs, thus increasing the nondimensional local compliance at the origin on the P̄-V̄ curve. An extension of the inflation P̄-V̄ curve into the first quadrant before injury registers a positive value of 0.41 forW̄_1–2 in Table 2. Similarly, the data range of the inflation limb after injury, being limited mostly in the third quadrant, results in a negative value of −0.33 forW̄_1–2. For the deflation limb, the after-injury data range is narrower than the before-injury data range; however, with both ranges being dominantly in the first quadrant on theP̄-V̄ plane, the magnitude of W̄_1–2 is negative for the both before- and after-injury data.

DISCUSSION

There are other P-V equations that have been developed and utilized for analyses of clinical quasi-static P-V curves. Their (qualitative) relations with the sigmoidal equation, discussed below, serve to provide a certain degree of physiological interpretation of these empirical equations and also to show the comprehensive nature of the sigmoidal equation proposed by V-H-S.

A hyperbolic-sigmoidal equation of the following form has been proposed (10)

where k_i (i = 1, 2, 3), V_M, and V_m are parameters to be determined from P-V curves. On the other hand, an integrated form of Eq. 2amay be approximated aswhere P_R is an integration constant, thus yielding the form of Eq. 11. (For derivation, see the.) The equation may be limited in its valid range because of the approximation applied to the integral of Eq.2a.

An exponential form, proposed by Salazar and Knowles (12)

with A, B, and k as adjusting parameters, has been used extensively (5, 6, 13), particularly in high-volume ranges. Its relation to the sigmoidal equation, as well as a similar equation proposed by Paiva et al. (11), have been discussed in V-H-S. Furthermore, Eq. 3b yields the following equation in nondimensional form corresponding to Eq.12Similarly, Eq. 3b, to a first approximation, may be reduced to(For derivation of Eqs. 12a and 12b, see the.) The exponential model equations are good approximations in high- and low-pressure regions; however, their relations to the sigmoidal equation indicate that the parameters in these equations are affected by the characteristics of the P-V curve as a whole.

Various polynomial expansions of P-V curves have also been employed (3, 7, 9). A series expansion formula, if applied for tanh (ω) (ω = Λp̄/2) in Eq. 3b,results in the following equation applicable for ‖ω‖ < π/2

where B_r is the Bernoulli number with values tabulated in mathematical handbooks. It should be noted that, for a small magnitude of ‖ω‖ (i.e., ‖P/P₀‖ ∼ 1, in the region near the midpoint of the curve) the first term in Eq. 13is dominant, making a linear approximation of V̄ ∝ ω (that is, V ∝ P) valid. Hence, the region in which a linear approximation is valid decreases as the magnitude of Λ increases.

The analysis above indicates that the sigmoidal equation represents a polynomial expansion, Eq. 13, near ω = 0 (that is, near P = P₀) combined with the exponential forms,Eqs. 12a and 12b, valid in the regions near the asymptotes. There is no reason for the validity of the symmetric shape of the sigmoidal equation with respect to the inflection point (8). In terms of the differential equation, Eq.2a, the nonsymmetry implies that the magnitude of α may vary with pressure, or, in terms of Eq. 2c, the gradient of local compliance change with volume varies with pressure. However, the agreements with various clinical data suggest that quasi-static, pulmonary P-V curves are best represented by the sigmoidal equation as a first-order approximation.

In summary, general characteristics of the sigmoidal P-V equation are presented based on its nondimensional form in which the shape of the sigmoidal equation depends on the magnitude of a single parameter, Λ. On the nondimensional plane, the regions of increasing compliance and of decreasing compliance appear in the third and the first quadrant, respectively, with the inflection point located at the origin. Also, with volume normalization, the local compliance at the origin is equal to Λ/2 and is inversely proportional to the location of the lower (or upper) corner pressure (Eq. 9). A clinical data range of a specific P-V curve relative to the entire range of the sigmoidal equation is expressed quantitatively in terms of the nondimensional work, W̄_1–2, between the initial state 1 and the final state 2. The magnitude of Λ is combined with the magnitude of W̄_1–2 to distinguish P-V data of a healthy human before and after an alveolar recruitment maneuver, four cases of ARDS, and a dog before and after lung injury. The analyses presented in this report help generalize our understanding of the sigmoidal P-V equation as well as its relationship to and advantages over other empirical P-V equations. It may also provide testable hypotheses in clinical and experimental examinations of pulmonary P-V curves that will improve our understanding of respiratory system mechanics.

FOOTNOTES

• Address for reprint requests and other correspondence: U. Narusawa, Dept. of Mechanical, Industrial, and Manufacturing Engineering, Northeastern Univ., Boston, MA 02215 (E-mail:[email protected]ned.edu).

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