1- INTRODUCTION
Alveoli are tiny pores or cavities inside a continuum media which behavior is similar to that of a solid or a gel. The concept is frequently associated with the cavities localized in biological tissues; nevertheless, it has also referred to other materials. The O2 - CO2 gas exchange occurs between inspired air and the lung circulating blood in the pulmonary alveolus. It has an approximately spherical shape with a mean diameter of 0.25 mm; its walls are covered by the alveolar epithelium followed by the blood capillaries epithelium (also called endothelium). The existence of pores in the alveolar tissue has been proven by scanning electron microscopy, and it has been determined that the associated endothelium membrane size is in the range of 50-60 Å, and the alveolar epithelium size is in the range from 6 to 10 Å1. Likewise, Kohn pores are localized in the internal alveolus wall, facilitating the airflow from one alveolus to another. Most of them have a diameter from 1 to 4 m in multiple mammal species.2
Rn222 is a naturally occurring inert gas whose valence electron shell is complete;3,4 thereby, it has a relatively low chemical reactivity and does not form compounds. This gas concentrates in the environment, especially interiors, and effortlessly emanates from the soil to the air. Furthermore, it can get inside the human body by water ingestion and air inhalation. Once it has reached the organism, it is conducted by the respiratory tract to alveoli, where a significant amount is expelled.4 However, the non-expelled amount decay in short-life disintegration solid products (Po218 and Po214) have a high probability of depositing in biological tissues, causing damage to the DNA by the alfa radiation emitted.
Recently,4 a potential function of semi-empirical, smooth, and continuous pairs has been proposed to model molecular interactions between Radon and pulmonary alveolar walls; Molecular Dynamics (MD) is used to determine the distribution of gas in an adjacent alveolar wall and estimate the amount of it diffusing through the alveolar membrane as a concentration function. However, there are currently no studies of Radon density profiles using the formalism of integral equations. The description of Radon gas behavior is a significant challenge in science since it has a very low concentration in the atmosphere, approximately 610 ppb. Radon emanations are short-lived due to their isotopic instability; the most stable isotope, Rn222, has a half-life of 3.82 days.5 This gas was discovered in 1900, and over the following decades, its thermodynamic properties were determined. They include standard boiling point, vapor pressure, vapor density at standard conditions, vaporization heat, critical temperature, and pressure.6-10 The critical and liquid densities have not been experimentally measured, and there are still discussions points regarding these properties.
This article aims to describe the density profile of Radon particles near the surface of the alveolar wall. The alveolar pore is assumed a spherical cavity with semipermeable walls whose diameter is more prominent than Radon particles. The permeable wall is characterized by an energy interaction potential at the highest point. The potential interaction between the alveolus wall and Radon particles is characterized by the height of the potential and the half-width of the alveolus membrane; these correspond to the energy required for a Radon particle to pass through the wall and to the thickness of the wall, respectively. The Radon particles can go across the membrane if they possess sufficient energy. We propose a theoretical model that captures the main characteristics of the interaction between a semipermeable membrane and a Radon particle.
2- METHOD
We assumed a system composed of two species: the alveolar pore and the Radon particles. We were interested in describing the Radon concentration profile concerning the wall of a single alveolus. Since we considered a diluted alveoli system, we did not consider the alveolus- alveolus interaction. The difference in sizes between the alveolar cavity and the Radon particles is such that we considered the alveolar wall as a flat wall without loss of generality. The concentrations of Rn222, usually found in the environment, were assumed under average body temperature and atmospheric pressure conditions. We assumed that Radon particles are in a highly dilute phase and can cross through the alveolar walls in both directions. Under this set of assumptions, it is helpful to consider the Lennard-Jones potential for Radon-Radon interactions. The values of the potential parameters that we used were presented in our previous works4 (Table 1).
The Lennard-Jones potential has the mathematical form:
The parameter values of and for Rn222 that we chose were previously calculated.8 The subscripts RR correspond to Radon-Radon interaction. Density and temperature correspond to the gas phase and ambient conditions, approximately. The energy units are , where (1.38x1023 Kg m2 s-2 K-1 ) is the Boltzmann constant, and is the temperature in Kelvin. The energy parameter is defined by the dimensionless expression:
Radon particles under previously mentioned thermodynamic conditions of pressure, temperature and volume, can be considered a simple fluid, characterized only by the effective diameter of the particle . Radon particles can go throughout from the inside to the outside of the alveoli and vice versa. The mathematical expression for the interaction of the alveolar membrane with the Radon particles has been reported by several authors.11-13
with
In the above definition, is the half-width of the potential, the parameter is the height, is the depth of potential, and means the arithmetic average of the diameters of Radon and Alveoli respectively. The fluid particles can cross the alveoli membrane and occupy both sides of the membrane. A graphic description of the potential can be seen in Fig. 1.
We used the integral-equation formalism from liquid theory to describe the local density distributions functions
where Z is the partition function in the canonical ensemble, V is the system volume, and N is the number of particles. The Eq. (4) is the probability to find n particles in a particular configuration (r 1 , r 2 , … r n ). In the case of n = 2, we have the radial distribution function (RDF)
The geometric interpretation of the RDF is given in Fig. 2.
The relationship between the radial distribution function and the integral equations formalism is given by the Ornstein-Zernike equation (OZ). The OZ equation for a simple fluid can be written as
Here h(r) = g(r) - 1, and are the total and direct correlation functions, respectively. Another essential equation establishes the relationship between the potential interaction and the pair correlation function
We introduced another correlation function named bridge function, b(r), and assumed an approximation to it, called the closure equation. We used the Hypernetted Chain closure, which mathematically is written like
The relationship between correlation functions and thermodynamics is given by the static structure factor, S(q), which can be obtained by light scattering experiments. The mathematical expression is
The matrix formulation was used to generalize the formalism to mixtures. The OZ in matrix formulation has the following form
The importance of correlation functions is that thermodynamic relationships can be obtained directly.14
The internal energy and the pressure can be calculated through the expressions
This analysis shows the Radon concentration profile in the vicinity of the alveolar membranes and allows us to estimate the number of molecules superficially adsorbed. The thermodynamic properties of the adsorption of Radon over the alveoli membrane surface comprise work for future publications.
In the following section, we show the results obtained from the numerical solution of the Ornstein-Zernike equations systems.
3- RESULTS
In Figs. 3-6, we can observe a structural profile over the surface of the alveolar membrane. We show the results for different sizes of alveoli.As we can see, the highest point reached is symmetrically on both sides of the membrane.
The profile density over the surface is three times the bulk density. There is no relationship between the width membrane and the highest point. The increase of local density in the region corresponding to the surface of the membrane allows us to affirm that the Radon molecules are being grouped preferentially in that region. The thickness of the Radon layer adsorbed to the surface of the membrane is approximately ten Radon atoms; beyond this distance, the Radon density tends to its density in gas. The thickness of this layer depends on the width of the membrane (w), as we can see in the following figures. However, Radon layer thickness does not depend on the intensity of the energy barrier; therefore, the width barrier role is as important as potential width. We can observe that Radon particles exist inside the membrane; such particles are trapped within the membrane even though the potential barrier expels them. The Radon atoms must cross an additional energy barrier created by those Radon atoms adsorbed on the membrane to emerge from it.
4- DISCUSSION AND CONCLUSIONS
We have developed a mathematical model that qualitatively describes the Radon concentration profile near the surface of the alveolar cavities. The proposed model captures globally the observable parameters that characterize biological membranes, such as the thickness and the necessary energy that a particle requires to cross it. With these two parameters, we made a systematic study within the thermodynamic parameters of physicochemical interest that we proposed, specifically, the local density profile. The description of the density profiles on the surfaces gives us a measure of the local concentration of particles adsorbed on the alveolar wall. The results show us that the surface concentrations increase independently of the diameter of the alveolus, while the surface concentrations of Radon on the alveolar membrane are approximately three times the bulk density (away from the walls) in all cases. When analyzing the results for different values of the alveolar wall thickness, controlled by the parameter w, we observed a fraction of Radon particles contained within the alveolar membrane. Our results show that the thickness of the membrane dominates the surface profile; however, it is remarkable that there are Radon particles trapped inside the membrane. These results give us an estimate of the quantitative behavior of the concentration profiles on the alveolar walls. Let us remember that it is precisely through the alveoli that the processes of physicochemical exchanges of CO2 and O2 take place. Our results agree with experimental reports where it has been shown that for attached particles, the activity median aerodynamic diameter is approximately nm, and deposition fractions of inhaled Radon progeny can be estimated from models of particle deposition in the human respiratory tract. It has been reported that such deposition fraction is approximately 20 - 40%, with most of the deposition occurring in the alveolar region.15