From Pressure-Volume Relationship to Volume-Energy Relationship: A Thermo-Statistical Model for Alveolar Micromechanics

The connective tissue fiber system and the surfactant system are essential and interdependent components of lung elasticity. Despite considerable efforts over the last decades, we are still far from understanding the quantitative roles of either the connective tissue fiber or the surfactant systems. Through thermo-statistic considerations of alveolar micromechanics, the author introduced a thermo-statistic state function “entropy” to analyze the elastic property of pulmonary parenchyma based on the origami model of alveolar polyhedron. By use of the entropy for alveolar micromechanics, from the logistic equation for the static pressure (P)-volume (V) curves including parameters a , b, c, and k ( a set of equations was obtained to define the internal energy of lungs (U L ) and its corresponding lung volume (V L ). Then, by use of parameters a , b, c, and k, an individual volume-internal energy (V L − U L ) diagram was constructed from reported data in patients on mechanical ventilation. Each V L − U L diagram constructed was discussed that its minimal value and its shape parameter b/k represent quantitatively the energy of tissue force and the energy of surface force. Furthermore, by use of the V L − U L relationship, the hysteresis of lungs estimated by entropy production was discussed as dependent on the difference

, a set of equations was obtained to define the internal energy of lungs (U L ) and its corresponding lung volume (V L ). Then, by use of parameters a , b, c, and k, an individual volume-internal energy (V L − U L ) diagram was constructed from reported data in patients on mechanical ventilation. Each V L − U L diagram constructed was discussed that its minimal value ( ) 2 o U c a b = + and its shape parameter b/k represent quantitatively the energy of tissue force and the energy of surface force. Furthermore, by use of the V L − U L relationship, the hysteresis of lungs estimated by entropy production was discussed as dependent on the difference in the number of contributing pulmonary lobules. That is, entropy production might be a novel quantitative indicator to estimate the dynamics of the bronchial tree. These values obtained by combinations of parameters of the logistic P-V curve seem useful indicators to optimize setting a mechanical ventilator. Thus, it is necessary to develop easy tools for fitting the individual sigmoid pressure-volume curve measured in the intensive care unit to the logistic equation.

Introduction
Measuring pressure-volume (PV) curves have been used in setting mechanical ventilation to quantify the elastic properties of lungs. Classical physiology has explained that the elastic property originates from the basic components of the lung skeleton: 1) a continuous network of elastic fibers in the alveolar walls that connects arteries, airways, and interstitial space (the tissue force), and 2) the force generated at the surface between the air and the alveolar surfactant lining layer (the surface force) [1]. The quasi-static PV curve of the respiratory system describes the mechanical behavior of the lung parenchyma during inflation and deflation, and measuring PV curves has been used in research to quantify the elastic properties of lung parenchyma [2]. Despite considerable efforts over the last decades, we are far from understanding the quantitative indicators of either the connective tissue fiber or the surfactant systems for optimization of mechanical ventilation.
Elasticity is measured in the excised lung as the relationship between lung volume and pressure measured at the closed airway (PV curve). Experimental study revealed, as shown in Figure 1, that the PV curve of an excised lung filled with air (dashed lines) is quite different from that of a degassed lung refilled with saline solution (solid lines) [1] [2]. It has been recognized that the curve of the saline-filled lung mainly represents the properties of the tissue components whereas that of the gas-filled lung includes both the tissue properties and the surface tension exerted along the alveolar walls. During inflation, the work done  the folding and unfolding of paper bags for the "origami model" of alveolar polyhedron [6], by use of which Min explained the elastic properties of lungs as entropy change of alveolar micromechanics [7]. The entropy change of alveolar microstructure is less in dissipating energy than the stretching change of tissue as well.
Reports have indicated that PV curves are fit to the exponential equation such as ( ) where V is volume, P is transpulmonary pressure, and A, B, and K are constants [8]. However, it has generally been recognized that the exponential equation is poorly fitted for data including the low range of lung volumes, particularly for lungs in acute respiratory distress syndrome (ARDS).
To overcome this poor fitting other equations including first or third-degree polynomials were proposed, but the increased number of parameters had little or no physiological meaning [9]. In 1998, Venegas et al. found a comprehensive sigmoid equation (the logistic equation) in dimensionless form describing PV curves in deflation and inflation with good fitting [10]. The logistic equation has been recognized to fit PV curves of a whole range sufficient even in patients with idiopathic pulmonary fibrosis (IPF) [11].
In this paper, after thermo-statistic considerations on alveolar micromechanics an entropy model of the lungs was constructed for describing the static PV and its corresponding lung volume V L revealed a logistic expression in dimensionless form as follows: where a , b, c, and k are parameters fitted for each static PV curve [10]. Parameter a has units of volume and corresponds to the lower asymptote volume, which approximates the physiological variable of residual volume ( a ≈ RV). Parameter b, also in units of volume, corresponds to the total change in volume between the lower and upper asymptotes, which approximates the physiological variable of vital capacity (b ≈ VC). Parameter k corresponds to a parameter for normalizing transpulmonary pressure P tp to dimensionless form.

Method: Thermo-Statistical Model of Pulmonary Parenchyma
where j T is a thermo-statistical parameter describing fluctuations of angles in the pulmonary lobule j [7]. Based on the origami model, which describes changes in volume through changes in the interalveolar angles, internal energy j u and entropy j s for the secondary pulmonary lobule of j containing j n angles are introduced as follows: where j ε is the expectation value of surface energy in the corresponding pulmonary lobule of j as The zero law of thermodynamics (if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other) can define a single parameter of temperature as T in the pulmonary parenchyma consisting of pulmonary lobules [15]. Thus, the internal energy U L and the entropy S L of the lungs were defined for the lung system as follows: 2) Helmholtz minimum free energy for the quasi-static PV relationship The equilibrium of a thermodynamic system in a controlled temperature and volume, which is applicable to the quasi-static PV relationship, is described as is the minimum as follows:

1)
Physiologically, in the static PV relationship of the respiratory system, it was described by the mechanical equilibrium among pressures (the airway pressure o P and the abdominal pressure ab P measured at closed airway) along with the components of the respiratory system, as shown in Figure 3, as follows: As a result, the transpulmonary pressure tp P is balanced with the transdiaphragmatic pressure di P as follows: Thus, transpulmonary pressure tp P changes itself along with a change in transdiaphragmatic pressure di P , which has zero condition as the standard condition at ( ) , o c V as follows: It is important to note that the elastic recoil pressure of lungs tp P behaves as a deflating force at more than o V , and an inflating force at less than o V , the same as transdiaphragmatic pressure di P does. This is quite different from the classical concept for the elastic recoil pressure of the lungs, which always works as a deflating force. Then, the elastic recoil pressure of lungs ( el P ) should be described as a difference from c as follows: Figure 3. Mechanical balance between ribcage and lungs describing the logistic PV curve. In the static equilibrium condition, the airway opening pressure is balanced with the abdominal pressure. There is an equilibrium among components of the ribcage including the transdiaphragmatic (P tp ) and the transpulmonary (P di ) pressures. It is important to note that P tp always balances to P di . (see text in detail).
The first derivative of the Equation (13) was obtained as follows: 2) The static PV curve of the lungs requires a thermodynamic equilibrium of the Equation (8) in controlled temperature and volume as follows: In the quasi-static state of the lungs, a change of internal energy d L U was obtained as follows: Thus, the change of pulmonary entropy d L T S was obtained as follows: From Equations (14) and (17), d L T S was described as follows: where U o is the internal energy at the standard condition of tp P c = , at which the lung volume was obtained as 2 o V a b = + . Then, the lung volume V L was expressed as the logistic equation as follows: ( ) 3) Two basic functions g and v for every secondary pulmonary lobule were taken out from Equations (13) and (19) as follows: Then, parameter p was obtained from the equation as follows: These functions were taken out as thermodynamic state functions in every secondary pulmonary lobule from self-similarity in the architecture of pulmonary segments and lobules (as described in the section of assumptions). Then, the internal energy and corresponding volume for the secondary pulmonary lobule j were described as follows, where oj u and oj v are the internal energy and the volume of secondary pulmonary lobule j at the standard condition, respectively. Thus, the internal energy of lung U L , the corresponding lung volume V L were defined as the summation of every contributing secondary pulmonary lobules as shown in Figure 4 as follows: As the result, the PV relationship of lungs composed of pulmonary secondary lobules was transformed to the volume-energy (V L − U L ) relationship by a set of equations as well as follows:  Based on the self-similarity in the segmental structure of lungs, each secondary pulmonary lobule was assumed to have its own logistic volume-pressure relationship. By use of the common volume-energy (g-v) relationship, the volume-energy relationship of the lungs was reconstructed as a set of equations.  Table 1 by Ferreria et al. [11] for each patient on a mechanical ventilator, which is indicated by numbers 1 to 11. Each diagram has its own minimum value of U o , and its own specific shape dependent on b/k.

Discussion
Classically static PV relationships of  Figure 1) is interpreted as being the onset of recruitment of previously unventilated pulmonary lobules. The difference between curves during inflation and deflation is called "hysteresis of lungs", which is measured by the area of PV loop and described as the entropy production (D) of Oliveria et al. [16] as follows: ( ) Based on the thermodynamic model in this paper, since p and v are thermodynamic state functions, the area of the PV loop is zero as follows: In comparing the volume-energy diagrams of deflation and inflation ( Figure   5), the minimum point (black dots in Figure 5) of the inflation curve is shifted upper and its shape transformed to a slightly wide curve. The entropy production (D) is represented as the area enclosed by the inflation and deflation V L − U L diagrams (the dark area in Figure 6). The difference between the V L − U L diagrams is caused by the difference in b, which is described as follows:   (1) or closed (0). That is, D would relate to the number of contributing lobular bronchioles reflecting dynamic conditions of the bronchial tree modified by physiological actions of the pulmonary surfactant, which is a surface-active lipoprotein complex (phospho-lipoprotein) secreted locally from cells including type II alveolar cells and Clara cells, which are non-ciliated, non-mucous, and secretory cells, in the lobular bronchioles [17].
Fitting an exponential equation to the deflation PV curves of lungs has been applied at excluding points below 50% of the total lung capacity, and has been shown useful clinically when applied to the deflation limb of the PV curve of spontaneously breathing patients. Instead, a sigmoidal model is superior when an inflation PV curve is captured in anesthetized patients because that it is necessary to fit the lower limb of the PV curve to titrate mechanical ventilation for each patient [11]. The entropy model of each V L − U L diagram may help to titrate mechanical ventilation accordingly in individual patient.

Conclusion
First, the author revised the mechanical balance among classical components of the chest wall and lungs during the static PV changes, resulting in finding that the transdiaphragmatic pressure always balances with the transpulmonary pressure. Through thermo-statistic considerations on alveolar micromechanics, a thermo-statistical model (entropy model) that would give a central role in the static PV relationship of air-filled lungs was obtained. The entropy model led to getting a set of thermodynamic state functions as common as the internal energy g and volume v in every secondary pulmonary lobule from the logistic equation for static sigmoid PV curves of the lungs. Based on reconstruction of the static PV relationship by use of g and v, the logistic PV relationship was transformed into the volume-energy relationship defined by a set of equations. An individual volume-energy diagram was drawn by use of clinically estimated parameters of each patient on the mechanical ventilator. Thus, the tissue components and the surface component may be estimated as the minimum energy and the shape of energy function respectively in an individual patient. Furthermore, the "entropy production" of lung hysteresis may be estimated by the difference between the U L − V L diagrams, and would describe the dynamics of the bronchial tree modified by the surfactant system. The V L − U L diagram is easy to construct by use of estimated parameters from the logistic equation of each static sigmoid PV curve. Therefore, the V L − U L analysis on the logistic PV curve including the entropy production might be useful to optimize setting the mechanical ventilation of individual patients in the intensive care unit (ICU). To conclude, it is necessary to develop easy tools usable in ICU for fitting the sigmoid PV curve by the logistic equation.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.