The pressure-volume (P_L-V_L) curve of an excised lung inflated-deflated between minimum lung volume and total lung capacity (TLC) is characterized by a nonlinear relationship and is typified by a large hysteretic area indicating an irreversible energy loss. It is interesting to note that the material properties of the lung exhibit nonlinear characteristics, which cannot be linearized using an incremental, or small signal analysis. This results because of the rather large differences between the compliance of small volume perturbations and the quasi-static compliance in some areas along the quasi-static (P_L-V_L) relationship. A possible explanation of this unusual mechanical behavior of excised lungs is associated with an irreversible behavior linked to structural alterations in the lung tissue. In the past, investigators have reported changes in the configuration of lung structures at both the micro and macro level as a lung was inflated-deflated. One way to describe the sequential nature of these processes and their significance in explaining the lung’s mechanical properties is with models [1–10]. The models are very useful when comparing the compatibility of experimental evidence with theoretical predictions.

It has been postulated that the compliance of a quasi-static P_L-V_L curve, recorded between maximum and minimum volume, results from a combination of the expansion/contraction and the recruitment-derecruitment of lung units. There is evidence that the quasi-static curve can be divided into three different regions as illustrated in Fig. 1: (1) an “open” region where minimal recruitment-derecruitment of lung units occurs, (2) an “opening” region where lung units are recruited, and (3) a “closing” region where lung units are derecruited. It is understood that open lung units are capable of changing volume in all three regions [4]. Since the open region lies between the opening and closing regions, it is also assumed that a single small volume perturbation would not be capable of creating a large enough change in transpulmonary pressure to produce both opening and closing of lung units.

Experimental evidence supporting the supposition that there is minimal opening and closing of lung units in the open region of the P_L-V_L curve includes: (1) there are minimal changes in gas trapping when end-expiratory pressures are confined to the open zone [11,12]; (2) discontinuous lung sounds are not normally detected when P_L-V_L loops are recorded in the open zone [3,13]; (3) parameters of the constant phase model including G, H, and η, which are derived from the dynamic compliance (E_dyn) and viscous stress of lung tissue (R_tis) appear to be constant for P_L-V_L loops in the open zone [14,15]; and (4) measurements of normalized hysteresis (K) of all P_L-V_L loops recorded in the open zone have consistent values [2,16–18]. On the other hand, a displacement in P_L and V_L that causes that relationship to leave the open zone and enter either the opening or closing zones where lung unit recruitment-derecruitment takes place can have a considerable effect on the lung’s mechanical behavior [2,4,19].

One method that can be used to show the sequential nature of the mechanical events that occur in an excised lung as it is inflated-deflated between −5 and +30 cm H₂O is to superimpose a more rapid small volume perturbation on a slow constant ventilation rate. It can be seen (Fig. 2) that the compliance of a small loop, represented by the straight line (DE) that is generated by a volume perturbation is very different from the compliance of the quasi-static P_L-V_L curve (AEF) as the lung is inflated. This study attributes this type of a material nonlinearity to discrete structural changes in the lung’s material properties during lung inflation, which corresponds with the reopening of lung units [3,5]. Similar differences between loop and quasi-static compliance can also be observed during lung deflation at low lung volumes.

The objective of this study was to develop a comprehensive model of the recruitment-derecruitment of lung units during a complete inflation-deflation cycle. Portions of that representative model are based on previous work, which assumed that V_L represents the total volume of gas in the lung and φ symbolizes the fraction of total gas volume occupied by open lung units [5]. It was presumed that the number of open lung units at a given lung volume is equal to N_O and all open lung units have approximately the same volume (V_U), then

The differential equations derived from the fundamental relationship in Eq. (1) were used to calculate both N_O and V_U as a function of transpulmonary pressure while the lung was inflated-deflated between −5 and +30 cm H₂O.

Deflation-inflation loops were recorded as volume perturbations were superimposed on a quasi-static P_L-V_L curve of excised rat lungs that were slowly deflated and inflated between +30 and −5 cm H₂O. The normalized compliance of the quasi-static P_L-V_L curve and the normalized chord compliance of each loop were determined. Figure 7 shows an example of calculated values of NO∗ and VU∗ superimposed on the quasi-static PL-VL∗ curve of an excised rat lung. It can be seen that on the second deflation cycle to −5 cm H₂O, when the volume perturbations were added, approximately 95% of the lung units remained open above a transpulmonary pressure of +10 cm H₂O, and nearly 60% had closed between +10 and 0.0 cm H₂O. After the lung was deflated, approximately 28% of the lung units remained open or filled with trapped gas. As the lung was re-inflated, about 12% of the lung units had re-opened at a transpulmonary pressure of 12.5 cm H₂O. The remaining 60% of the lung units opened between 12.5 and 30 cm H₂O. The most rapid change in the number of open lung units occurred between 15 and 20 cm H₂O. Figure 8 shows NO∗ and VU∗ plotted as a function of VL∗. It can be seen that a nonlinear relationship exists between both VU∗ and NO∗ with respect to VL∗, and both the NO∗-VL∗ and VU∗-VL∗ relationships show a moderate amount of hysteresis.

Weibel and colleagues [21,22] were among the first investigators to use tools of morphometry to study the lung’s anatomical structure. One of their assumptions was that the number of alveoli multiplied by the average alveolar volume was equal to the fraction of total lung volume occupied by the alveoli. They also assumed that total lung area was equal to the average area of an alveolus times the number of alveoli. Weibel [21] showed that this assumption was true for most distributions of alveolar size. In 1985, Frazer et al. [4] examined nonlinearities associated with differences in the compliance of small volume perturbations superimposed on the quasi-static P_L-V_L curve. These nonlinearities were only associated with regions of the curves where there was evidence of the recruitment-derecruitment of lung units. An interesting characteristic of this type of nonlinearity is that it cannot be linearized using techniques such as small signal or incremental analysis. Later, Brancazio et al. [6] extended that model to consider how recruitment-derecruitment would be expected to influence the geometric behavior of the lungs during inflation-deflation. Their analysis was used to demonstrate the effects of lung unit recruitment-derecruitment on NO∗,VU∗ and normalized lung unit area ( AU∗) as a function of VL∗ during lung inflation-deflation. More recently, Frazer et al. [5] calculated how NO∗ varied with VL∗ as the lung was inflated by assuming that VU∗ could be estimated from PL-VL∗ curves recorded between TLC and an end-expiratory pressure of +6 cm H₂O. In that study, NO∗ was determined during lung inflation by assuming the PL-VU∗ relationship was proportional to the quasi-static PL-VL∗ loop formed by inflating-deflating the lung between +6 and 30 cm H₂O in the open zone. Results indicated that lung tissue exhibited irreversible changes in structure upon expansion, which were responsible for alterations in their lung’s mechanical properties during inflation. Much of the irreversibility could be avoided if the lung was not deflated to low end-expiratory pressures (closing zone) during subsequent PL-VL∗ cycles [3–5]. In this study, VU∗ and NO∗ were found as a function of VL∗ while the lung was deflated then inflated during a complete P_L-V_L maneuver between −5 and 30 cm of H₂O.

Figure 7 shows an example of calculated values of NO∗ and VU∗ along with the quasi-static PL-VL∗ curve of an excised rat lung. Between transpulmonary pressures of 5 and 10 cm H₂O, the difference between VU∗ on lung deflation and inflation was 27%, and the difference in NO∗ was 55%. In the volume range bounded by 35% and 55% TLC, values of VU∗ were approximately 15% greater on lung inflation than on lung deflation (Fig. 8). This corresponds to nearly a 5% increase in the linear dimension of an open lung unit during lung deflation compared to lung inflation at the same volume. The maximum difference in NO∗ between lung inflation and deflation with respect to lung volume was approximately 10% with NO∗ being greater on lung deflation. There have been both direct and indirect measurements of alveolar size made during lung expansion and contraction, which are in general agreement with the findings in this study [10,23,24].

It should be pointed out that a lung unit has not been identified anatomically and may not be represented by a specific structure such as an alveolus but may be composed by a combination of structures such as portions of alveoli, entire alveoli, alveolar ducts, bronchioles, etc., which is represented by an “average” lung unit. During lung deflation, for example, more compliant units having mechanical properties similar to those found in the lung parenchyma appeared to be derecruited first, and the units that remain open near the end of deflation appear to represent the large, less compliant, airways. Figure 7 shows that near zero transpulmonary pressure the PL-VU∗ relationship looks very much like the P_L-V_L curves of larger airways [25,26].

This study indicates that when recruitment-derecruitment occurs, the volume of an open lung unit is not directly proportional to lung volume in the opening and closing zones. As a result, there is a nonlinear relationship between VU∗ and VL∗. Where there is little evidence of recruitment-derecruitment, in the open region, VU∗ appears to be proportional to VL∗, and NO∗ remains nearly constant.

As with most models, the one described in this study has certain limitations. For instance, an implied assumption is that each open lung unit has approximately the same volume. Even though the units may be represented by different anatomical structures, it has been shown that for reasonable lung unit size distributions the model is fairly robust and assuming lung units of equal volume does not appear to contribute large errors [6,21].

Another important limitation of the model is that it does not address the time dependent behavior of lung’s tissue properties or the opening and closing of lung units [1,7,14,17,18,27,28]. It is known that the compliance of lung tissue in the open region of a P_L-V_L curve is a function of the ventilation rate. According to the constant phase model, the dynamic compliance can be expressed as C_ti = 1/Hω^(1−α), where α ≈ 0.91 and H (tissue elastance) is assumed to be independent of ω [15]. In addition, Hildebrandt [18] has shown experimentally that the compliance C_ti of P_L-V_L loops, in the open region, changes less than 10% per decade with respect to frequency. When there is recruitment-derecruitment of lung units, however, the compliance of the open units is usually much less than the quasi-static lung compliance. As a result, the modest effect of frequency on loop compliance has an even smaller influence on the percent difference between the quasi-static and loop compliance, which is used to calculate V_U and N_O.

Due to the time dependence of the lung’s tissue properties, the results presented in this study only describe changes in VU∗ and NO∗ as a function of VL∗ for a specific ventilation pattern. It would be possible, however, to use Eq. (3) to determine how the lung and lung units change volume with respect to transpulmonary pressure under a variety of conditions, which meet the theoretical prerequisites of the model.

The first requirement is that the lung units have sufficient time to open or close while the P_L-V_L operating point is in the opening or closing zones, and that units do not continue to open or close when the P_L-V_L curve enters the open zone. It appears, in Fig. 6, that when the lung was inflated, the maximum P_L values during each perturbation cycle were only slightly greater than when the lung was inflated during the second cycle at the same average quasi-static rate. If lung units did not have time to open in the opening zone during a perturbation, it would be expected that there would be a significant increase in the maximum loop P_Lmax, which would reflect the inability of lung units to open during a perturbation cycle at that volume. As the path of P_L-V_L perturbation leaves the opening zone, and enters the open zone, there is a rapid decrease in both P_L and V_L, reducing the probability of lung units continuing to open. Once a lung unit has opened, the model assumes it would remain open, and N_O would assume a constant value in the open zone. A similar argument can be made in describing the derecruitment of lung units during lung deflation. Lung units close when volume perturbations enter the closing zones at low end-expiratory pressures. An additional requirement is that the ventilation pattern must be composed of frequencies low enough that the effects of airway resistance can be neglected.