Anales AFA Vol. 33 Nro. 1 (Abril 2022 - Julio 2022) 1-5

https://doi.org/10.31527/analesafa.2022.33.1.1

Física Estadística y Física Biológica

PERFILES DE DENSIDAD DE RADÓN EN POROS ALVEOLARES: UNA APLICACIÓN

DEL FORMALISMO DE ECUACIÓN INTEGRAL

RADON DENSITY PROFILES IN ALVEOLAR PORES: AN INTEGRAL EQUATION

FORMALISM APPLICATION

J. C. Corona-Oran1, D. Osorio-González1, J. Mulia-Rodríguez*1

1Laboratorio de Biofísica Molecular. Facultad de Ciencias. Universidad Autónoma del Estado de

México. Instituto Literario 100. Colonia Centro, Toluca CP 50000. México.

Autor para correspondencia: * jmr@uaemex.mx jccoronao@gmail.com

Recibido: 26/06/2019; Aceptado: 22/09/2021

ISSN 1850-1168 (online)

Resumen

En este trabajo, obtuvimos perﬁles de densidad para partículas de radón cerca de la superﬁcie de la

pared alveolar como resultado del desarrollo de un modelo teórico que describe la interacción en-

tre tales partículas y las células de la pared alveolar. El modelo captura características biológicas y

ﬁsicoquímicas relevantes, como el ancho de la pared alveolar y la energía necesaria para que las par-

tículas de radón pasen a través de ella. El sistema satisface las condiciones a considerar en equilibrio

termodinámico y en condiciones normales de presión y temperatura. Resolvimos numéricamente las

ecuaciones de Ornstein-Zernike derivadas del formalismo de ecuación integral y mostramos los efec-

tos de cambiar el tamaño del alvéolo, el ancho de la pared alveolar y la energía requerida para pasar a

través de ellos en los perﬁles de densidad de radón. Nuestro método es de suma utilidad en biofísica

y otras áreas aﬁnes.

Palabras clave: radón, pared alveolar, poro alveolar, biofísica.

Abstract

In this work, we obtained density proﬁles of Radon particles near the alveolar wall surface bythe

development of a theoretical model that describes the interaction between such particles and the al-

veolar wall cells. The model captures relevant biological and physicochemical characteristics such

as the width of the alveolar wall and the energy required for the Radon particles to pass through it.

The system satisﬁes the conditions to be considered in thermodynamic equilibrium and under normal

pressure and temperature conditions. We numerically solved the Ornstein-Zernike equations derived

from the integral-equation formalism and showed the effects of changing the alveolus size, the alveo-

lar wall width, and the energy required to pass through itsRadon density proﬁles. Our method is a

helpful tool in biophysics and other related areas.

Keywords: radon, alveolar wall, alveolar pore, biophysics.

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– CO2gas 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 Å[1]. 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 µm 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 pro-

posed to model molecular interactions between Radon and pulmonary alveolar walls; Molecular Dy-

namics (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 descrip-

tion of Radon gas behavior is a signiﬁcant challenge in science since it has a very low concentration in

the atmosphere, approximately 6×10−11 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, criti-

cal 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 co-

rrespond 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 works [4] (Table 1).

The Lennard-Jones potential has the mathematical form:

βuRR(r) = 4ε

kBTσ

r12

−σ

r6(1)

TABLE 1: Show the Lennard-Jones parameters.

ε/kBσ/ A

292.0 4.145

The parameter values of σRand ε/kBfor 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 kBT, where kB(≈1.38×1023 Kg

m2s−2K−1) is the Boltzmann constant, and Tis the temperature in Kelvin. The energy parameter

ε∗is deﬁned by the dimensionless expression:

ε∗=ε

kBT(2)

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 σR. 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].

βuRa(r∗)) = 4ε∗u0−(u0+ε)r∗6

4ε∗+ (u0+ε)r∗12 ,(3)

with

r∗=r−σRa

ω,(3a)

σRa =(σR+σa)

2.(3b)

In the above deﬁnition, ωis the half-width of the potential, the parameter u0is the height, ε

is the depth of potential, and σRa means the arithmetic average of the diameters of Radon σRand

Alveoli σarespectively. 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.

FIG. 1: Radon-alveoli interaction. U0and ωare tunable parameters for the potential model.

We used the integral-equation formalism from liquid theory to describe the local density distribu-

tions functions

g(n)(r1,r2,...,rn) = 1

ρnρ(n)(r1,r2,...,rn)

=VnN!

ZnNn(N−n)!Zdrn+1...rnexp(−βu(r1,r2,...,rn))

(4)

where Zis the partition function in the canonical ensemble, Vis the system volume, and Nis the

number of particles. The Eq. (4) is the probability to ﬁnd nparticles in a particular conﬁguration

(r1,r2,...,rn). In the case of n=2, we have the radial distribution function (RDF)

N(r1,r2) = ρ2

N(r1,r2)

ρ2

N(r1)ρ2

N(r2).(5)

The geometric interpretation of the RDF is given in Fig. 2.

FIG. 2: The radial distribution function is interpreted as the conditional probability of ﬁnding a particle at a

distance r2given that another is in the position r1. In the ﬁgure, we deﬁne r =r2−r1.

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

h(r) = c(r) + ρZh(r)c(r−s)ds.(6)

Here, h(r) = g(r)−1 and c(r)are the total and direct correlation functions, respectively. Another

essential equation establishes the relationship between the potential interaction and the pair correla-

tion function

g(r) = exp[−βu(r) + h(r)−c(r) + b(r)].(7)

We introduced another correlation function named bridge function, b(r), and assumed an appro-

ximation to it, called the closure equation. We used the Hypernetted Chain closure, which mathema-

tically is written like

b(r) = 0.(8)

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

S(q) = 1+4πρ Zg(r)sin(qr)

qr r2dr.(9)

The matrix formulation was used to generalize the formalism to mixtures. The OZ in matrix

formulation has the following form

hαβ (r) = cαβ (r) + ρZhαβ (r)cαβ (r−s)ds,(10)

gαβ (r) = exp[−βuαβ (r) + hαβ (r)−cαβ (r) + bαβ (r)].(11)

The importance of correlation functions is that thermodynamic relationships can be obtained di-

rectly [14].

The internal energy and the pressure can be calculated through the expressions

U=3

2NkT +2πρ Zu(r)g(r)r2dr,(12)

P=NkT

−2

3πρ2Zu′(r)g(r)r2dr.(13)

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.

FIG. 3: The radial distribution function of Radon particles.

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

FIG. 4: The radial distribution function of Radon particles.

FIG. 5: The radial distribution function of Radon particles.

FIG. 6: The radial distribution function of Radon particles.

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 con-

centrations 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, con-

trolled by the parameter ω, 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; howe-

ver, 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

CO2and O2take place. Our results agree with experimental reports where it has been shown that

for attached particles, the activity median aerodynamic diameter is approximately 100−200 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].

REFERENCES

[1] ICRP. Human Respiratory Tract Model for Radiological Protection. ICRP Publication 66. 24 (1994).

[2] C. Desplechain, B. Foliguet, E. Barrat, G. Grignon y F. Touati. The pores of Kohn in pulmonary alveoli.

Bulletin europeen de physiopathologie respiratoire 19, 59-68 (1983).

[3] A. D. Grifﬁths, S. D. Chambers, A. G. Williams y S. Werczynski. Increasing the accuracy and tempo-

ral resolution of two-ﬁlter radon–222 measurements by correcting for the instrument response. Atmos.

Meas. Tech. 9, 2689-2707 (2016).

[4] J. Corona, F. Zaldivar, L. Mandujano-Rosas, F. Méndez, J. Mulia y D. Osorio-Gonzalez. Theoretical

Model to Estimate the Distribution of Radon in Alveolar Membrane Neighborhood. J. Nucl. Phy. Mat.

Sci. Rad. A. 4, 59-68 (2016).

[5] V. M. Rosenoer, M. Oratz y M. A. Rothschild. Albumin: Structure, function and uses (Pergamon Press,

1977).

[6] C. Fry y M. Thoennessen. Discovery of the astatine, radon, francium, and radium isotopes. At. Data

Nucl. Data Tables 99, 497-519 (2013).

[7] G. Rudorf. Über einige numerische Konstanten der Radiumemanation und deren Beziehung zu denen

der Edelgase. Zeitschrift für Elektrochemie und angewandte physikalische Chemie 15, 748-749 (1909).

[8] R. W. Gray y W. Ramsay. CXXII.—Some physical properties of radium emanation. J. Chem. Soc.,

Trans. 95, 1073-1085 (1909).

[9] W. Herreman. Calculation of the density of liquid radon. Cryogenics 20, 133-134 (1980).

[10] J. P. Hansen e I. R. McDonald. Theory of Simple Liquid (Academic Press London, 2006).

[11] B. Hribar-Lee y M. Lukšiˇ

c. Replica Ornstein–Zernike Theory Applied for Studying the Equilibrium

Distribution of Electrolytes across Model Membranes. J. Phys. Chem. B 122, 5500-5507 (2018).

[12] B. Hribar-Lee. The application of the replica ornstein-zernike methodology for studying ionic membrane

equilibria. Acta Chimica Slovenica 59, 528-535 (2012).

[13] L. I. Bracamontes, O. Pizio, S. Sokolowski y A. Trokhymchuk. Adsorption and the structure of a

hard sphere ﬂuid in disordered quenched microporous matrices of permeable species. Mol. Phys. 96,

1341-1348 (1999).

[14] E. Rutherford. The Boiling Point of the Radium Emanation. Nature 79, 457-458 (1909).

[15] G. Butterweck, C. Schuler, G. Vessù, R. Müller, J. W. Marsh, S. Thrift y A. Birchall. Experimental

Determination of the Absorption Rate of Unattached Radon Progeny from Respiratory Tract to Blood.

Radiat. Prot. Dosim. 102, 343-348 (2002).