Respiratory system model for quasistatic pulmonary pressure-volume (P-V) curve: inflation-deflation loop analyses

A respiratory system model (RSM) is developed for the deflation process of a quasistatic pressure-volume (P-V) curve, following the model for the inflation process reported earlier. In the RSM of both the inflation and the deflation limb, a respiratory system consists of a large population of basic alveolar elements, each consisting of a piston-spring-cylinder subsystem. A normal distribution of the basic elements is derived from Boltzmann statistical model with the alveolar closing (opening) pressure as the distribution parameter for the deflation (inflation) process. An error minimization by the method of least squares applied to existing P-V loop data from two different data sources confirms that a simultaneous inflation-deflation analysis is required for an accurate determination of RSM parameters. Commonly used terms such as lower inflection point, upper inflection point, and compliance are examined based on the P-V equations, on the distribution function, as well as on the geometric and physical properties of the basic alveolar element.     

General characteristics of the sigmoidal model equation representing quasi-static pulmonary P-V curves.

A pulmonary pressure-volume (P-V) curve represented by a sigmoidal model equation with four parameters, V(P) = a + b[1 + exp[-(P - c)/d]](-1), has been demonstrated to fit inflation and deflation data obtained under a variety of conditions extremely well. In the present report, a differential equation on V(P) is identified, thus relating the fourth parameter, d, to the difference between the upper and the lower asymptotes of the volume, b, through a proportionality constant, alpha, with its order of magnitude of 10(-4) to 10(-5) (in ml(-1). cmH(2)O(-1)). When the model equation is normalized using a nondimensional volume, (-1 < < 1), and a nondimensional pressure, (=(p/c) - 1), the resulting - curve depends on a single nondimensional parameter, Lambda = alphabc. A nondimensional work of expansion/compression, (1-2), is also obtained along the quasi-static sigmoidal P-V curve between an initial volume (at 1) and a final volume (at 2). Six sets of P-V data available in the literature are used to show the changes that occur in these two parameters (Lambda defining the shape of the sigmoidal curve and (1-2) accounting for the range of clinical data) with different conditions of the total respiratory system. The clinical usefulness of these parameters requires further study.     

A mathematical derivation of the P-V relation at end-systole

A dynamical model of the left ventricle as a thick-walled cylinder contracting radially is used to derive the P-V (pressure-volume) relation in the left ventricular cavity during contraction. It is shown how the mathematical results derived could apply to experimental results.     

Comparative study of four sigmoid models of pressure-volume curve in acute lung injury

Background  

The pressure-volume curve of the respiratory system is a tool to monitor and set mechanical ventilation in acute lung injury. Mathematical models of the static pressure-volume curve of the respiratory system have been proposed to overcome the inter- and intra-observer variability derived from eye-fitting. However, different models have not been compared.     

Lower inflection point and recruitment with PEEP in ventilated patients with acute respiratory failure.

The lower inflection point (LIP) on the total respiratory system pressure-volume (P-V) curve is widely used to set positive end-expiratory pressure (PEEP) in patients with acute respiratory failure (ARF) on the assumption that LIP represents alveolar recruitment. The aims of this work were to study the relationship between LIP and recruited volume (RV) and to propose a simple method to quantify the RV. In 23 patients with ARF, respiratory system P-V curves were obtained by means of both constant-flow and rapid occlusion technique at four different levels of PEEP and were superimposed on the same P-V plot. The RV was measured as the volume difference at a pressure of 20 cm H(2)O. A third measurement of the RV was done by comparing the exhaled volumes after the same distending pressure of 20 cm H(2)O was applied (equal pressure method). RV increased with PEEP (P < 0.0001); the equal pressure method compares favorably with the other methods (P = 0.0001 by correlation), although individual data cannot be superimposed. No significant difference was found when RV was compared with PEEP in the group of patients with a LIP < or =5 cm H(2)O and the group with a LIP >5 cm H(2)O (76.9 +/- 94.3 vs. 61.2 +/- 51.3, 267.7 +/- 109.9 vs. 209.6 +/- 73.9, and 428.2 +/- 216.3 vs. 375.8 +/- 145.3 ml with PEEP of 5, 10, and 15 cm H(2)O, respectively). A RV was found even when a LIP was not present. We conclude that the recruitment phenomenon is not closely related to the presence of a LIP and that a simple method can be used to measure RV.     

Effect of intrathoracic pressure on pressure-volume characteristics of the lung in man.

Quasi-static pressure-volume (P-V) curves in normal seated human subjects were determined with pressure at the airway opening (Pa0) set below (negative pressure), above (positive pressure), or equal to ambient pressure. Dynamic compliance (Cdyn) during controlled continuous negative pressure breathing (CNPB) was also studied. Quasi-static P-V curves at negative pressure were decreased in slope, reflected a decrease in total lung capacity, and intersected the P-V curve obtained at ambient Pa0. At positive pressure the P-V curves showed an increase in slope and an increase in total lung capacity. During CNPB a fall in Cdyn was found. The fall in Cdyn was rapid and persisted for the duration of CNPB. Cdyn promptly returned to control levels when Pa0 was adjusted to ambient pressure.     

Dynamic stiffness profiles in the left ventricle.

Diastolic pressure-volume (P-V) curves were calculated on a beat-to-beat basis in the open-chest, pentobarbital-anesthetized dog, using the technique of direct transmitral flow measurement previously described. P-V curves were constructed and the slope (dP/dV) was plotted vs. pressure and time. dP/dV was used as an index of stiffness in each heart and its instantaneous changes with time were followed throughout the diastolic period. The end-diastolic P-V relation based on points from successive cycles during volume loading was found to be exponential. In contrast, the instantaneous P-V relation during any one diastolic period was not exponential. That is, the dynamic dP/dV vs. pressure plot was nonlinear. In the normal heart, stiffness was characterized in early diastole by a negative dP/dV as the ventricle continued to relax, and then frequently decreased prior to a second stiffness rise with atrial augmentation. These findings can be explained by a model containing an element whose deformation is rate dependent, i.e., a parallel viscous element. Stiffness profiles in mitral stenosis where dynamic effects are minimized substantiate this conclusion.     

A method for determining the apoplastic water volume of conifer needles

A method for direct estimation of percentage apoplastic water volume (% APO) in conifer needles is described. The method presented here, and designated the pressure-needle (P-N) method, measures the relative water content of the needles to develop a curve similar to the pressure-volume (P-V) curve. P-V and P-N curves were developed for Picea pungens Engelm. cv. Hoopsi, Pinus sylvestris L., Abies gradis (Dougl.) L., and Pseudotsuga menziesii (Mirb) Franco. The % APO estimated by the two procedures varied as much as 2-fold, while other parameters were similar. The P-V method generated consistently higher and more variable % APO than the P-N method, due to the inclusion of the apoplastic water of the stem in the P-V method. For conifers, the P-N method offers a more accurate and precise method for determining % APO.     

Influence of cycloheximide on the lung.

We examined the time course of the influence of cycloheximide on descending pressure-volume curves of excised lungs and on protein and lecithin synthesis and oxygen consumption by lung slices. We also looked at the influence of cycloheximide on granular pneumocyte ultrastructure. Excised lungs from cycloheximide-treated animals are more compliant than controls. After ventilation with air, lungs from control and cycloheximide animals show increased retractive forces and a shift to the right of the deflation P-V curve. Incubation at 38 degrees C for 30 min reverses these changes in control lungs, but not in lungs from cycloheximide-treated rabbits. There is no change in liquid delfation P-V curves after cycloheximide. Cycloheximide causes an immediate decrease of 50% in incorporation of radioactive leucine into protein by lung slices. Incorporation of radioactive palmitate into lecithin and oxygen consumption are also decreased by 50% 6 h after cycloheximide. Lamellar bodies in granular pneumocytes are smaller after cycloheximide. Cycloheximide causes a significant increase in the surface density of the lamellar body envelope. Cytoplasmic area of granular pneumocytes is increased after cycloheximide.     

Effect of simulated weightlessness on pressure-volume relationships of femoral veins in vivo of New-Zealand rabbits

Objectives: To observe the change of pressure-volume relationships of femoral veins of rabbits after simulated weightlessness. Methods: Head-Down Tilt(HDT) -20 degrees rabbit model was used to simulate weightlessness .24 healthy male New-Zealand Rabbits were randomly divided into 21d HDT group, 10d HDT group and control group, with 8 in each. The pressure-volume relationships of rabbits femoral veins were measured. Result: The femoral vein P-V relationship curves of HDT groups were shifted to larger volume change ratio than that of control group. The P-V relationship curve of the 21d HDT group was shifted more obviously than that of HDT-10d. B1 and B2 in quadratic equations of 21d HDT group were significantly higher than these value of 10d HDT group and control group during expansion (inflow) and collapse (outflow) (P<0.01). Conclusions: The femoral venous compliance increased after weightlessness simulation and the femoral venous compliance of 21d-HDT increased more obviously than that of 10d-HDT.     

Leaf hygrometer v. pressure chamber: a comparison of pressure–volume curve data obtained on single leaves by alternating measurements

Abstract. Data for the construction of pressure-volume curves were obtained by measuring water potentials of detached leaves repeatedly and alternately, with a pressure chamber and a leaf hygrometer. Good agreement between the parameters of the two resulting curves was observed. Regression lines on values after the loss of turgor were always more negative for the thermocouple data, with a maximum difference for the osmotic potential at full saturation of 0.25 MPa in Triticum and a minimum of 0.01 MPa in Populus. Neither the slopes of the regression lines nor the intercepts with the axes were statistically different. We see no reason for using one of these two unrelated methods as a standard against which the other is calibrated. Implications for the theory of pressure-volume curves are discussed.     

Quasi-static pressure-volume hysteresis in the canine respiratory system in vivo

We investigated the quasi-static pressure-volume (P-V) hysteresis of the normal canine lung in vivo by performing 15-s flow interruptions at various points throughout the breathing cycle in mechanically ventilated anesthetized paralyzed dogs. By measuring the transpulmonary pressure (Ptp) at 5 s after each interruption, we built up a quasi-static P-V loop of the lungs. We found, however, that the area of the loop was significantly smaller (by a factor of 4-6) than has been reported by others for the isolated canine lung. We also found the hysteresis loop area of the chest wall to be of similar magnitude. If we measured Ptp 10-15 s after interruption, we found it always decreased at a rate expected to result from continuing gas exchange in the lungs. We conclude that 1) the areas of the quasi-static P-V loop in vivo for the total respiratory system, as well as the lungs and chest wall separately, are significantly smaller than has been reported previously for isolated lungs and 2) continuing gas exchange in the lungs places a lower limit on the frequencies (equivalent to flow interruptions of greater than 5- to 7-s duration) at which the P-flow-V behavior of the lungs in vivo can be considered in purely mechanical terms.     

Effects of external pressures on the pressure-volume relation of the left ventricle

The effects of ventricular geometry, muscle mass, muscle elasticity and external pressures on the pressure-volume and muscle stiffness-stress relations have been quantitated on the basis of a theoretical model. Data taken from patients before and after interventions with nitroprusside and angiotensin were applied to the model in order to explain the possible causes for the marked shifts in the pressure-volume relations. The results indicate that (a) ventricular geometry does not markedly alter the pressure-volume and stiffness-stress relations unless there is a drastic change from a spherical shape to an ellipsoidal shape orvice versa, (b) increases in muscle mass and muscle elasticity of the order of 30% result in significant alterations in the P-V relations but are not the cause for the parallel shifts unless accompanied by substantial external pressures, (c) the parallel shifts in the pressure-volume relations can be accounted for entirely by the presence of external pressures without changes in muscle mass or muscle elasticity. Thus manipulation of right ventricular pressures or pericardial pressures by drug interventions may be useful in the treatment of left heart disease and the presence of such pressures must be considered in the analysis of ventricular function curves.     

胡杨、灰叶胡杨P-V曲线水分参数的初步研究

运用P-V技术初步研究了人工林胡杨、灰叶胡杨两个树种的水分参数差异,探讨了树种的抗旱性。结果表明灰叶胡杨的主要水分参数(ψ100、ψ0、ROWC、AWC、RWD、ε）明显低于胡杨,渗透调节能力和束缚水含量高于胡杨,表现对干旱环境具有较强的耐旱能力。综合评判表明灰叶胡杨对干旱的适应性较强。     

Elastic behavior of postmortem human lungs: effects of aging and mild emphysema.

The elastic behavior of postmortem human lungs has been studied in an effort to differentiate the effects of normal aging from those of mild emphysema. Static pressure-volume (P-V) curves were measured in 50 lungs obtained from men 15-85 yr of age, including 12 lungs with mild-to-moderate emphysema. The emphysema was quantitatively assessed by gross and microscopic methods. The P-V relationship in all lungs is accurately described by the empirically fitted equation, P = alpha1ea2v. This expression is useful because the two parameters separate the effects of elastic behavior (alpha1) from size (alpha2) on the P-V curve. There is a close negative correlation (R = -0.94) Between age and alpha1 in normal lungs but no significant age dependence of alpha2. Further decreases in alpha1 are found in most emphysematous lungs. Alpha1 is more than 2 SEE below the age-predicted mean in five of nine lungs with minimal emphysema (1-10% by point count) and more than 5 SEE below the mean in the three more severely affected lungs. There is a close correlation (R = +0.90) between alpha1 and the alveolar surface-to-volume ratio in both normal and emphysematous lungs.     

Letters to the Editor

The following is the abstract of the article discussed in thesubsequent letter:

Venegas, José G., R. Scott Harris, and BrettA. Simon. A comprehensive equation for the pulmonarypressure-volume curve. J. Appl. Physiol. 84(1): 389-395, 1998.Quantification of pulmonary pressure-volume (P-V) curves isoften limited to calculation of specific compliance at a given pressureor the recoil pressure (P) at a given volume (V). These parameters can be substantially different depending on the arbitrary pressure orvolume used in the comparison and may lead to erroneous conclusions. Weevaluated a sigmoidal equation of the form, V = a + b[1 + e^(Pc)/d]¹, for its ability to characterize lung and respiratory system P-V curves obtained under a variety of conditions including normal andhypocapnic pneumoconstricted dog lungs (n = 9), oleicacid-induced acute respiratory distress syndrome (n = 2), andmechanically ventilated patients with acute respiratory distresssyndrome (n = 10). In this equation, a corresponds tothe V of a lower asymptote, b to the V difference between upperand lower asymptotes, c to the P at the true inflection pointof the curve, and d to a width parameter proportional to the Prange within which most of the V change occurs. The equation fittedequally well inflation and deflation limbs of P-V curves with a meangoodness-of-fit coefficient (R²) of 0.997 ± 0.02 (SD). When the data from all analyzed P-V curves were normalized by thebest-fit parameters and plotted as (V  a)/b vs.(P  c)/d, they collapsed into a single and tightrelationship (R² = 0.997). These resultsdemonstrate that this sigmoidal equation can fit with excellentprecision inflation and deflation P-V curves of normal lungs and oflungs with alveolar derecruitment and/or a region of gastrapping while yielding robust and physiologically useful parameters.

     

Quantifying succulence: a rapid, physiologically meaningful metric of plant water storage

Quantification of succulence should ideally convey information about physiological function and yet also be straightforward to measure. While important aspects of succulence and its physiological consequences may be quantified using parameters derived from pressure-volume (P-V) curves, this technique applied to succulent tissues is difficult, time consuming and generally not suitable for large comparative datasets. We performed P-V curves on leaves of 25 taxa from across Caryophyllales and compared the results with direct measures of saturated water content (SWC(meas) ), the ratio of water mass at full saturation to tissue dry mass, for the same taxa. SWC(meas) was significantly related to relative capacitance, the most physiologically relevant parameter describing tissue succulence. We developed a linear model describing SWC(meas) as a function of relative capacitance and leaf volume, which is also supported when accounting for the phylogenetic relationships among taxa. These results indicate that SWC(meas) is a suitable proxy for tissue succulence, and that both cellular properties and variation in gross morphology contribute towards a plant's relative water storage capacity. Quantifying SWC(meas) across many taxa showing variation in tissue succulence will provide a new avenue for exploring the evolutionary dynamics of this important ecological adaptation.     

On defining total lung capacity in the mouse.

Maximal lung volume or total lung capacity in experimental animals is dependent on the pressure to which the lungs are inflated. Although 25-30 cm H2O are nominally used for such inflations, mouse pressure-volume (P-V) curves show little flattening on inflation to those pressures. In the present study, we examined P-V relations and mean alveolar chord length in three strains (C3H/HeJ, A/J, and C57BL/6J) at multiple inflation pressures. Mice were anesthetized, and their lungs were degassed in vivo by absorption of 100% O2. P-V curves were then recorded in situ with increasing peak inflation pressure in 10-cm H2O increments up to 90 cm H2O. Lungs were quickly frozen at specific pressures for morphometric analysis. The inflation limbs never showed the appearance of a plateau, with lung volume increasing 40-60% as inflation pressure was increased from 30 to 60 cm H2O. In contrast, parallel flat deflation limbs were always observed, regardless of the inflation pressure, indicating that the presence of a flat deflation curve cannot be used to justify measurement of total lung capacity in mice. Alveolar size increased monotonically with increasing pressure in all strains, and there was no evidence of irreversible lung damage from these inflations to high pressures. These results suggest that the mouse lung never reaches a maximal volume, even up to nonphysiological pressures >80 cm H2O.     

Mathematical modelling to centre low tidal volumes following acute lung injury: A study with biologically variable ventilation

Background

With biologically variable ventilation [BVV – using a computer-controller to add breath-to-breath variability to respiratory frequency (f) and tidal volume (V_T)] gas exchange and respiratory mechanics were compared using the ARDSNet low V_T algorithm (Control) versus an approach using mathematical modelling to individually optimise V_T at the point of maximal compliance change on the convex portion of the inspiratory pressure-volume (P-V) curve (Experimental).

Methods

Pigs (n = 22) received pentothal/midazolam anaesthesia, oleic acid lung injury, then inspiratory P-V curve fitting to the four-parameter logistic Venegas equation F(P) = a + b[1 + e^-(P-c)/d]^-1 where: a = volume at lower asymptote, b = the vital capacity or the total change in volume between the lower and upper asymptotes, c = pressure at the inflection point and d = index related to linear compliance. Both groups received BVV with gas exchange and respiratory mechanics measured hourly for 5 hrs. Postmortem bronchoalveolar fluid was analysed for interleukin-8 (IL-8).

Results

All P-V curves fit the Venegas equation (R² > 0.995). Control V_T averaged 7.4 ± 0.4 mL/kg as compared to Experimental 9.5 ± 1.6 mL/kg (range 6.6 – 10.8 mL/kg; p < 0.05). Variable V_Ts were within the convex portion of the P-V curve. In such circumstances, Jensen''s inequality states "if F(P) is a convex function defined on an interval (r, s), and if P is a random variable taking values in (r, s), then the average or expected value (E) of F(P); E(F(P)) > F(E(P))." In both groups the inequality applied, since F(P) defines volume in the Venegas equation and (P) pressure and the range of V_Ts varied within the convex interval for individual P-V curves. Over 5 hrs, there were no significant differences between groups in minute ventilation, airway pressure, blood gases, haemodynamics, respiratory compliance or IL-8 concentrations.

Conclusion

No difference between groups is a consequence of BVV occurring on the convex interval for individualised Venegas P-V curves in all experiments irrespective of group. Jensen''s inequality provides theoretical proof of why a variable ventilatory approach is advantageous under these circumstances. When using BVV, with V_T centred by Venegas P-V curve analysis at the point of maximal compliance change, some leeway in low V_T settings beyond ARDSNet protocols may be possible in acute lung injury. This study also shows that in this model, the standard ARDSNet algorithm assures ventilation occurs on the convex portion of the P-V curve.     

Pressure-volume relationships in non-rehydrated tissue at various water deficits

Abstract. The purpose of this paper is to examine a technique for estimating the weight at full saturation (W_s) from pressure-volume (P-V) analysis of non-rehydrated plant tissue at various water deficits. Tissue samples are typically rehydrated prior to P-V analysis to determine W_s, necessary to calculate many tissue water parameters. However, several studies have indicated that artificial rehydration may significantly alter P-V relationships, such as the plateau effect, resulting in erroneous measurements of tissue elasticity and osmotic potentials. The results of this study suggest that linear regression of P-V data at and above the turgor loss point may be used to extrapolate W_s from non-rehydrated samples at various moisture deficits, thus eliminating the plateau effect and other potential rehydration problems. Determination coefficients and standard errors of the Y-intercept indicated a strong linear relationship between tissue fresh weight and water potential (Ψ), and a high degree of predictability of W_s in all but one of the species-treatment combinations evaluated in this study, despite predawn Ψ as low as - 1.0 MPa.