Particle-induced indentation of the alveolar epithelium caused by surface tension forces

Abstract

Physical contact between an inhaled particle and alveolar epithelium at the moment of particle deposition must have substantial effects on subsequent cellular functions of neighboring cells, such as alveolar type-I, type-II pneumocytes, alveolar macrophage, as well as afferent sensory nerve cells, extending their dendrites toward the alveolar septal surface. The forces driving this physical insult are born at the surface of the alveolar air-liquid layer. The role of alveolar surfactant submerging a hydrophilic particle has been suggested by Gehr and Schürch's group (e.g., Respir Physiol 80: 17–32, 1990). In this paper, we extended their studies by developing a further comprehensive and mechanistic analysis. The analysis reveals that the mechanics operating in the particle-tissue interaction phenomena can be explained on the basis of a balance between surface tension force and tissue resistance force; the former tend to move a particle toward alveolar epithelial cell surface, the latter to resist the cell deformation. As a result, the submerged particle deforms the tissue and makes a noticeable indentation, which creates unphysiological stress and strain fields in tissue around the particle. This particle-induced microdeformation could likely trigger adverse mechanotransduction and mechanosensing pathways, as well as potentially enhancing particle uptake by the cells.

interaction between inhaled particles and the lung is of great interest in lung physiology (13). It is likely that it triggers a variety of subsequent physiological and pathophysiological events, including physical and chemical insults on the epithelium, uptake by alveolar macrophage or by alveolar type-I, type-II pneumocytes, as well as transepithelial translocation, both transcellularly and pericellularly. The nature of this interaction depends on the physicochemical characteristics of particles (e.g., size, shape, surface characteristics, core properties).

As soon as inhaled particles touch the lung alveolar surface, the first event to occur is that the particles encounter surfactant at the air-liquid interface. Pulmonary surfactant, located in the liquid layer covering the alveolar walls, generally lowers surface tension; this, in turn, reduces the contact angle between the particle and the air-liquid interface. In the physiological range of surface tension (<30 dyn/cm²), most of particles are hydrophilic and submerged in the alveolar lining layer called hypophase (50, 51). If the particle diameter is greater than the thickness of the hypophase, the surface tension forces that act at the particle and air-liquid interface border push the particle toward the epithelium, as shown by Gehr and Schürch's and colleagues [e.g., Schürch et al. (51, 52), Gehr et al. (12), Geiser et al. (16, 18)]. As a result, submerged particles push against the epithelial surface creating, thereby, indentation on the soft alveolar septal tissue (Fig. 1). This indentation likely creates unphysiological stresses and strains, which may, in turn, cause important pathophysiological consequences. While numerous potential biological consequences of this phenomenon are listed and discussed in detail in the discussion section, an observation we have recently made, which motivates this study, is as follows. We recently visualized innervations of alveolar septa with sensory neurons (31) [and also by others (9, 25, 63)]; this leads to an idea that it is highly conceivable that even a microstress exerted by a particle deposited around sensory neurons could be large enough to trigger firing those afferent neurons, causing subsequent functional effects.

Fig. 1.

Transmission (A and B) and scanning (C and D) electron micrographs of puffball spores deposited on alveolar surfaces. Notice that the spore particle is totally covered by the surface lining layer and submerged. The epithelium was indented even by the particle's spiny protrusions (at these locations, the particle is separated from the capillary by 100 nm). A, alveolar lumen; C, capillary; EN, endothelial cell; EP, epithelial (type 1) cell; LC, leukocyte; LL, osmophilic lining layer material. Scale bars = 2 (A and D), 0.5 (B), or 5 μm (C). [From Geiser et al. (18)].

The objective of this study is to elucidate the mechanisms by which deposited particles exert mechanical forces on the septal tissue and to estimate the extent of physical insults exerted on the alveolar epithelium. The quantitative assessment of these forces and alveolar tissues deformation may give us an insight into new mechanosensing as well as mechanotransducting pathways.

METHODS

Surface tension is the key driving force to submerge a deposited particle into the alveolar liquid layer and to force the particle into alveolar tissue. The surface tension force depends on a number of factors, including 1) pulmonary surfactant concentration, 2) alveolar surface area, 3) the wettability of a particle, which is governed by the particle surface characteristics and liquid-air surface tension, 4) the thickness of an alveolar liquid layer called hypophase, and 5) the extent of particle indentation into the alveolar wall tissue, which depends on tissue stiffness. Because the surface tension forces, the extent of indentation, and the tissue restoring forces are all mechanically coupled, it is necessary to define the governing equations for the indentation process, which are based on the laws of interfacial phenomena and the laws of the solid mechanics.

Geometric and Material Model and Associated Assumptions

To keep the problem tractable without a loss of generality, we consider the following approximations. We assume that a particle is rigid and spherical with a radius of R ranging from 0.125 to 0.5 μm. We also assume that the indentation occurs orthogonal to the solid-liquid surface. The tissue is considered as a semi-infinite medium, and its mechanical behavior is simplified by assuming that material is linear elastic with elasticity modulus E, ranging from 10 to 30 kPa (11) and Poisson's ratio within the values from 0.1 to 0.5.

Throughout the analysis, it is assumed that surface tension γ is constant in each calculation, ranging from 10 to 50 dyn/cm. This range corresponds to conditions in the lung where the alveolar walls are covered by a liquid layer with normal or with dysfunctional lung surfactant. For this range of γ, the extent of tissue indentation is primarily determined by a balance between surface tension force, F_γ, which pushes the particle toward the tissue, and resisting tissue elastic force, F_el, due to the resulting tissue indentation. For a given particle size and characteristics of lung surfactant and tissue elasticity, there is a maximum particle indentation u_max. Both forces can be expressed by complex nonlinear functions of u_max; F_γ includes geometric parameters, as described below, while F_el depends on the tissue mechanical characteristics and the contact conditions between the particle and tissue. For simplicity, the contact is assumed to be frictionless during the indentation. We consider the process to be quasi-static; hence, inertial and viscous effects are neglected in the present analyses. Other forces, such as gravity and buoyancy forces, are small¹ and they are considered negligible for the size of particles that we considered in this study (51).

Surface Tension Acting on a Particle

Because the alveolar surface is covered by a liquid layer, a particle forms a particle-air-liquid interface as soon as it lands on the alveolar surface. There are three interfaces: the particle (P)-liquid (L), the particle (P)-air (A), and air (A)-liquid (L) with three associated interfacial tensions, γ_PL, γ_PA, and γ_AL (Fig. 2). The air-liquid interfacial tension, γ_AL, is also called the surface tension and is denoted as γ for simplicity. The angle between γ and γ_PL at the contact point is called the contact angle, θ. At equilibrium (46),

$${\gamma }_{PA}={\gamma }_{PL}+\gamma \cos \theta$$ (1)

Angle θ defines “wettability” of a particle; the particle is “wetted” by the liquid when 0 < θ < π/2, and “completely wetted” when θ = 0. On the other hand, the particle is “unwetted” when π/2 < θ ≤ π.

Fig. 2.

Particle-air-liquid interface. The interfacial tensions associated with the interfacial energies are denoted as follows: particle-air as γ_PA, particle-liquid as γ_PL, and air-liquid (i.e., the surface tension) as γ_AL, or simply as γ. The balance of the interfacial tensions at a common point where three surfaces meet defines Young's equilibrium equation. The three-phase line denotes the line where particle surface interfaces with air-liquid surface.

The magnitude of θ depends on surface characteristics (surface-free energy) of the particle and γ. For a latex (polystyrene) particle interacting with pulmonary surfactant (DPPC) (51), the relationship between cosθ and γ, spanning a representative physiological range, is shown in Fig. 3. A similar relationship between θ and γ can be obtained for a polymethylmethacrylate (PMMA) particle. The graphic representation of these relationships is called a Zisman plot (10, 68). This plot shows that the “wetting coefficient”, cosθ is equal to 1 at low surface tensions (i.e., γ < γ_crit), denoting “complete wetting,” while for higher surface tension (γ > γ_crit), cosθ decreases linearly with increasing γ. The nonlinear effects due to the line tension are negligible² over the whole range of γ (37, 38). The typical range of alveolar air-liquid interfacial tension under normal breathing conditions is 0 < γ < 30 dyn/cm, where γ is modulated by the lung surfactant (64). The Zisman plot demonstrates that in this range, both latex beads and PMMA particles are completely wetted by the liquid layer.

Fig. 3.

Zisman plot. The Zisman plot defines a characteristic relationship between contact angle and surface tension. Both a latex (polystyrine) particle and a polymethylmethacrylate (PMMA) particle interact similarly with monolayer of 1,2-dipalmitoyl-sn-3-glycerophosphorylcholine pulmonary surfactant (DPPC). Thus, the Zisman plots of those particles almost coincide, and they both reach perfect wetting (θ = 0) at the critical surface tension of γ_crit = 26.27 dyn/cm. However, a PMMA particle interacting with aqueous solution reaches θ at lower γ_crit = 22.25 dyn/cm.

Knowing the relationship between θ and γ, we are now in a position to estimate the normal surface tension force F_γ acting on the particle (47, 48, 51),

$$F_{\gamma }=2\pi R\gamma \sin \phi \sin \left(\theta -\phi \right)$$ (2)

where ϕ is the polar angle, denoting the position of the particle-air-liquid contact point on the surface (Fig. 4), which is given as

$$\cos \phi =1-\frac{{\delta }_{hypo}}{R}-\frac{z_0}{R}-\frac{u_{\max }}{R}$$ (3)

where δ_hypo and z_o denote the thickness of the hypophase [∼0.1 μm on average or much less depending on the location in the alveolus, according to P. Gehr (personal communication)] and a capillary rise, respectively.

Fig. 4.

A schematic view of particle-hypophase-tissue interactions. At the equilibrium the surface tension force F_γ is balanced with the elastic tissue restoring force F_el, and this equilibrium force causes a deformation in tissue with the maximum indentation depth of u_max. The increase in thickness of hypophase in the proximity of the particle (comparing to far-field thickness δ_hypo) is caused by capillary rise, z_o. The angle ϕ measured with respect to the vertical axis defines the angular position of the point where three surfaces meet, where tan(β_o) is the slope (in vertical plane) of the air-liquid interface at the particle surface, and R is the particle radius. The wetting angle is denoted as θ.

The a priori unknown values of ϕ, z_o, and u_max are determined iteratively, following the computational steps as follows: 1) for a given ϕ, F_γ is calculated from Eq. 2, and z_o is calculated from the solution of the Young-Laplace equation (46) (see Eq. 4 below); 2) for known F_γ, u_max is calculated form the analytical solution (see Eq. 14, below) of the Hertz problem (21); and 3) by varying ϕ, the solution for z_o and u_max is found iteratively to fully satisfy the Eq. 3 within a prescribed tolerance.

Capillary Rise and Menisci Profile Around Small Spherical Particles

At the surface of a spherical particle, the three-phase line is a circle that is uniquely defined by the polar angle, ϕ (Fig. 5A). The equilibrium value ϕ is a priori unknown, and it can be calculated from Eq. 3 once z_o and u_max are determined. The capillary rise, z_o, is obtained from the hydrostatic Young-Laplace equation for the meniscus (46):

$$\Delta p=\rho gz-\gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right)=0$$ (4)

where Δp denotes the pressure difference across the liquid-air interface (we set it to be equal to zero because the system is assumed to be at equilibrium and there is no lateral flow), ρ is the mass density of the liquid,³ g is the gravity acceleration, and z(r) is the capillary rise of an axisymmetric surface at distance r from the (vertical) z-axis. We showed above that gravity effects are negligible for submicron particles. There are two finite radii of curvature involved at the spherical particle-fluid-air-interface in the three-dimensional problem: one denoted as R₁ in the plane shown in Fig. 5A, and the other denoted as R₂ in the orthogonal plane. Both radii R₁ and R₂ depend on the shape of the axisymmetric liquid-air surface; they can be expressed as functions of z(r), as

$$R_1=\frac{{\left(1+z{\prime }^2\right)}^{\frac{3}{2}}}{z″}and{R}_2\frac{r{\left(1+z{″}^2\right)}^{\frac{1}{2}}}{z\prime }$$ (5)

where z′ = dz/dr and z″ = d²z/dr². The boundary conditions refer to the slopes, which are defined at the three-phase line and in the far field: 1) at radius r₀ = Rsinϕ, as the slope of the air-liquid interface at the sphere surface (Fig. 4); and 2) at r >> R, the slope of the air-liquid interface in the vertical plane approaches to zero. These boundary conditions in mathematical form are

$$\begin{array}{ll}z\prime =\tan {\beta }_0\hfill & atr=r_0\hfill \\ z\prime =0\hfill & \text{as}r\to \infty \hfill \end{array}$$ (6)

where β_o = θ + ϕ − π (Fig. 4). The vertical position of the three-phase line on the surface of the spherical particle is not a priori known and is implicitly defined by Eq. 3.

Fig. 5.

The geometry of the interface between air-liquid and particle surfaces, and the geometry of the particle indentation into alveolar wall tissue. A: capillary rise. At the radius, r, the air-liquid surface is elevated by z(r) above the surface where pressure across the surface is zero. The capillary elevation z(r) is function of two radii of curvature of air-liquid surface, R₁ and R₂ (see Eqs. 5 and 6). The capillary rise has a maximum value at the particle surface, denoted in Fig. 4 as z_o, where z_o is the vertical distance from the free surface at far field to the three-phase line. The capillary rise is due to a net upward force produced by the attraction of the liquid to a solid surface. B: deformation of tissue caused by particle indentation. The spherical contact pressure distribution, q(r), reaches a maximum value q_o at the vertical axis of symmetry where the maximum indentation u_max occurs. The radius of the contact surface is denoted as a, and the indentation force is equal to surface tension force F_γ.

Substituting Eq. 5 into Eq. 4, defining the capillary constant c² = 2γ/ρg, and normalizing r and z by the capillary length, c, (i.e., x = r/c and y = z/c), Eq. 4 can be transformed into a nondimensional form (47, 48):

$$\frac{y″}{{\left(1+y{\prime }^2\right)}^{\frac{3}{2}}}+\frac{y\prime }{x{\left(1+y{\prime }^2\right)}^{\frac{1}{2}}}-2y=0$$ (7)

where y′ = dy/dx, and y″ = d²y/dx². The boundary conditions (Eq. 6) can also be transformed into a nondimensional form:

$$\begin{array}{ll}y\prime =\tan {\beta }_0\hfill & atx=x_0\hfill \\ y\prime =0\hfill & \text{as}x\to \infty \hfill \end{array}$$ (8)

where x_o = r_o/c and β_o is an angle between air-liquid surface and horizontal plane at the three-phase line (Fig. 4).

The profile of the meniscus and, thus, z_o can now be calculated from the solution of Eq. 7 with boundary conditions given in Eq. 8. Since there is no analytical solution of this problem, we further simplified Eq. 7 by assuming that the dimensionless variable x is small so that xy ≪ sinβ (47). Here, β is an angle (in radial plane) between air-liquid interface and the horizontal plane at an arbitrary radius r (see Fig. 5A). Also, tanβ = y′ is a slope of the air-liquid surface. For submicron spherical particles, this condition is always satisfied because the deformation of the liquid-air interface around submicron particles is small. This assumption implicitly implies that the gravity effect is negligible,⁴ thus the term 2y in Eq. 7 can be dropped out. Therefore, in the case of submicron particles, Eq. 7 can be transformed into a system of two simple equations (47):

$$dx/d\beta =-x/\tan \beta$$ (9a)

$$dx/d\beta =-x$$ (9b)

The solution of Eq. 9a for the known initial condition β = β_o at x = x_o, is

$$x=-x_0\sin {\beta }_0/\sin \beta$$ (10)

Substituting Eq. 10 into Eq. 9b and satisfying boundary condition: y = y_o at β = β_o, the approximate solution for the profile of air-liquid interface is obtained as

$$y=y_0+\left(x_0\sin {\beta }_0\right)\text{In}\frac{\tan {\beta }_0/2}{\tan \beta /2}$$ (11)

The only remaining unknown is the value of the dimensionless capillary rise y_o = z_o/c, which is not known a priori. To determine y_o, we use the following approach.

Whereas Eq. 11 accurately describes the shape and position of the air-liquid interface in the proximity of three-phase line (i.e., the particle surface), the accuracy of this so-called “the outer solution” decreases with increasing x (away from a particle).⁵ Thus, we need some other asymptotic solution to satisfy boundary condition in far field to precisely determine y_o and, therefore, the position of the three-phase line (via Eq. 3). One convenient method for approximately determining y_o is the shooting method⁶ (47). With this method, the value of y_o is determined by matching the above outer solution, which satisfies boundary conditions at the three-phase line, with so-called “inner solution,” which satisfies the boundary condition in the far field (29). This solution yields to an approximate analytical expression of y_o, in the limit of the small Bond number ε_B² = 2r₀² /c²→ 0⁷ as

$$y_0=x_0\sin \left({\beta }_0\right)\left[\text{In}\frac{2\sqrt{2}}{x_0\left(1+\cos {\beta }_0\right)}-\lambda \right]$$ (12)

where λ is the Euler constant equal to 0.57721.

Deformation in Alveolar Tissue Caused by Particle Indentation

Under equilibrium conditions, u_max (schematically shown in Fig. 4) is calculated from a force balance between F_γ and F_el under quasi-static conditions:

$$F_{\gamma }\left(u_{\max }\right)=F_{el}\left(u_{\max }\right)$$ (13)

The force F_el is evaluated as a function of u_max using the classical Hertz solution (34, 56), which assumes no friction between the indenter and the elastic medium:

$$F_{el}=\frac{4\cdot E\cdot R^{0.5}\cdot u_{\max }^{1.5}}{3\left(1-v^2\right)}$$ (14)

The tissue deformation and stress distribution can be calculated according to the Hertzian solution for a spherical particle, in which the contact pressure q(r) is obtained in a form of a spherical distribution:

$$q\left(r\right)=q_o\sqrt{1-{\left(\frac{r}{a}\right)}^2}$$ (15)

where a is the radius of contact,

$$a=\sqrt[3]{\frac{3}{4}\frac{{\left(1-v\right)}^2}{E}{F}_{el}R}$$ (16)

and q_o is the maximum contact pressure:

$$q_o=\frac{3}{2\pi }\sqrt[3]{\frac{16}{9}\frac{F_r{E}^2}{{\left(1-v\right)}^2}{F}_{el}R.}$$ (17)

This solution is accurate only for the small deformations, i.e., when the indentation is within 10% of the particle diameter. For larger indentations, the pressure distribution significantly deviates from the spherical, leading to the solution error. This will be analyzed in detail in the discussion section.

The alveolar tissue, and especially epithelial cell layer, is very soft; thus, the particle indentation could be larger than the particle radius. When the radius of contact, a, reaches the sphere radius, R, further indentation does not increase the contact area and, therefore, the indentation depth increases linearly with the additional force increase. The excess force, F_plunge, with respect to the maximum Hertzian force F_Hertz,max = F_el (a = R), is

$$F_{plunge}=F_{\gamma }-F_{Hertz,\max }=F_{\gamma }-\frac{4}{3}\frac{E{R}^2}{\left(1-v^2\right)}$$ (18)

The linear increase in F_plunge for the force exceeding F_Hertz,max, i.e., for u_max, is given as

$$F_{plunge}=\frac{2ER}{\left(1-v^2\right)}\left(u_{\max }-u_{Hertz,\max }\right)$$ (19)

and therefore u_max is also linearly related to the total force F_γ > F_Hertz,max. Finally, the pressure distribution can be approximated by using the solution for the flat indenter:

$$q_{plunge}\left(r\right)=\frac{q_{o,plunge}}{\sqrt{1-{\left(\frac{r}{a}\right)}^2}}$$ (20)

and the reference pressure (at the axis of symmetry) is

$$q_{o,plunge}=\frac{F_{plunge}}{2\pi R^2}$$ (21)

The total pressure distribution for F_γ > F_Hertz,max is equal to the sum of q(r) and q_plunge(r). For known normal traction (pressure) distribution imposed by the indented particle (Eqs. 15 and 20), the stress, strain, and displacement fields can be calculated analytically by employing the Boussinesq solution (56) for the force acting on a semi-infinite body. This includes evaluation of the convolution integral over the spherical contact pressure distribution, which is described in the appendix. Because of the tensorial nature of stresses and strains, we also calculated the equivalent stress and the effective strain, which are convenient in displaying the tensorial stress and strain fields as scalar fields. The equivalent (or effective) stress used here is the von Mises stress, which in the cylindrical coordinates is

$${\sigma }_v=\sqrt{\frac{1}{2}\left[{\left({\sigma }_{zz}-{\sigma }_{rr}\right)}^2+{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2+{\left({\sigma }_{\theta \theta }-{\sigma }_{zz}\right)}^2+6\left({\sigma }_{zr}^2+{\sigma }_{r\theta }^2+{\sigma }_{\theta z}^2\right)\right]}$$ (22)

Similarly, the effective strain is defined as

$$e_{eff}=\sqrt{\frac{2}{3}\left[{\epsilon }_{zz}^2+{\epsilon }_{rr}^2+{\epsilon }_{\theta \theta }^2+\frac{1}{2}\left({\epsilon }_{zr}^2+{\epsilon }_{r\theta }^2+{\epsilon }_{\theta z}^2\right)\right].}$$ (23)

RESULTS

Degree of Particle Indentation into Alveolar Tissue by Surface Tension Forces

The degree of the particle indentation depends primarily on surface tension, alveolar tissue elasticity, particle size, and thickness of hypophase. These factors significantly change shape of air-liquid and liquid-solid interfaces and consequently, the degree of indentation. The effects of some of the factors are straightforward to predict. For instance, a decrease in the tissue elastic modulus, increases the degree of particle indentation (see Figs. 6 and 7), or when a particle is completely wetted (θ = 0, i.e., F_γ < F_γcrit), an increase in surface tension increases the particle indentation too (Figs. 6A and 7A). On the other hand, the effects of some other factors are counterintuitive. For example, the particle indentation decreases with an increase of γ when surface tension is above γ_crit (i.e., γ > γ_crit). This occurs because of a strong effect of increase of θ with increasing γ [see Zisman's plot (Fig. 3), where cosθ decreases with increasing γ]. In Fig. 6A, it is demonstrated how an increase in θ causes a decrease in ϕ and most importantly decrease in β_o. Consequently, the vertical component of surface tension force (∝ γsinβ_o) sharply decreases. Thus, according to Eq. 2, the decrease in F_γ is caused by a larger decrease in β_o compared to smaller increase in the product of γ and the length of the three-phase line, 2πRsinϕ. In other words, the effects of an increase in θ on F_γ is strong so that even a large decrease in ϕ (i.e., a large increase in the length of the three-line phase line), cannot compensate for a large decrease of β_o and z_o (Fig. 6A). Since F_γ is a strong function of these geometric factors (see Eq. 2), an increase of γ above γ_crit would result in decreasing F_γ and, consequently, a decrease in the degree of particle indentation (Fig. 6A). In this analysis, the thickness of hypophase δ_hypo is taken to be constant.

Fig. 6.

Air-liquid and liquid-solid interfaces after particle indentation by surface tension forces. A: particle is indented more at critical surface tension, γ_crit = 26.27 dyn/cm and wetting angle θ = 0° (gray lines), than at much higher surface tension of 50 dyn/cm because of large wetting angle θ of 60°. Increase in θ excessively reduces the effect of large increase in surface tension (black lines). The thickness of hypophase, δ_hypo is 0.1 μm. B: decrease of the thickness of hypophase from 0.2 μm (gray lines) to 0.05 μm (black lines) significantly increases indentation because the thinner hypohase increases capillary rise due to decrease in angle ϕ, and increase in angle β_o. The surface tension is taken to be equal to γ_crit and θ = 0°. In all calculations, particle diameter is 0.5 μm, and alveolar wall Young's modulus and Poisson's ratio are taken to be 20 kPa and 0.5, respectively.

Fig. 7.

Depth of particle indentation as a function of surface tension, and the Young elastic modulus of the alveolar wall tissue. The equilibrium force, F_γ = F_el, and the corresponding maximum indentation, u_max, are obtained at the intersection of the lines for F_γ(u_max) the fixed surface tension, γ, and those for F_el(u_max) fixed elastic moduli, E and v = 0.5. The particle diameter is 0.5 μm. The diamond symbols denote points of static equilibrium and hence values of u_max,eq for that particular pair of forces. The increase of Young's elastic modulus, E, always increases the equilibrium force and decreases u_max,eq. A: for γ < γ_crit, both the equilibrium force and u_max,eq increase with increase of γ, while in B: for γ < γ_crit, both the equilibrium force and u_max,eq decrease.

When δ_hypo decreases (at constant γ and θ), F_γ increases, and consequently the degree of particle indentation increases (Fig. 6B). In this case, an increase in F_γ is caused by a decrease in ϕ as a consequence of larger capillary rise, z_o, for thinner hypophase. Since smaller ϕ increases the length of the three-phase line and, in addition, increases magnitude of β_o (i.e., the vertical component of the surface tension force), the increase in F_γ is amplified by these two synergetic effects. The cumulative effect, therefore, increases F_γ and the depth of particle indentation, u_max.

Because of the fact that both F_el and F_γ are nonlinear functions of the maximum particle indentation u_max, the equilibrium indentation, u_max,eq, is determined iteratively, and is represented graphically as a crossover point of the F_el–u_max and F_γ–u_max curves. Here, the F_γ–u_max relationship is obtained from Eqs. 2 and 3, while the F_el–u_max relationship is derived from the Hertzian contact (Eqs. 14 and 19). In Figs. 7 and 8, A and B, the equilibrium indentation, u_max,eq, is denoted as diamond symbols at the crossover points.

Fig. 8.

Depth of particle indentation as a function of the elastic moduli of the alveolar wall tissue and thickness of the hypophase, δ_hypo, at γ_crit (A and B), and the quantitative effects of δ_hypo on the equilibrium force, F_γ = F_el, and indentation u_max,eq (C and D). The diamond symbols denote the equilibrium force, F_γ = F_el, and indentation u_max,eq. A: force and indentation cross-over points for three thicknesses of hypophase (0.05, 0.1, and 0.2 μm), three Young's elastic moduli (E = 30, 20, and 10 kPa), and v = 0.5. The particle diameter is 0.5 μm. The decrease of the thickness of the hypophase increases both F_γ and u_max,eq. B: force and indentation cross-over points, denoted as circles and a diamond, for three Poisson's ratios (0.1, 0.3, and 0.5), E =20 kPa and δ_hypo = 0.2 μm. Increase in Poisson's ratio increases F_γ and decreases u_max, having similar quantitative effect as modest increase in Young's elastic modulus. Note that for simplicity, u_max here and in all following figures denotes equilibrium indentation u_max,eq. C: effect of the thickness of hypophase on F_γ and u_max for three Young's elastic moduli E, and v = 0.5. D: effect of Poisson's ratio for the range of v = 0.1-0.5 and for E = 20 kPa on F_γ and u_max. In C and D, F_γ = F_el vs. δ_hypo is denoted by solid lines, and the symbols indicate which Young's modulus was used; the indentation u_max vs. v is denoted by dashed lines.

In Fig. 7, u_max is plotted vs. F_γ and F_el for a 0.5-μm diameter particle (R = 0.25 μm), in the physiologically relevant range of surface tension (0 < γ < 50 dyn/cm), and the mechanical properties of the alveolar tissue (Young's modulus 10 < E > 30 kPa and Poisson's ratio of 0.5). When γ ≤ γ_crit (Fig. 7A), the extent of particle indentation, u_max,eq, increases with increasing γ for a fixed E or with decreasing E for a fixed γ. When γ > γ_crit (Fig. 7B), however, the situation is different: u_max,eq decreases with increasing γ for a fixed E and with increasing E for a fixed γ. As we discussed above, the difference between these cases is due to the difference in the γ vs. cosθ relationship depending on whether γ is below or above its critical value γ_crit (see Fig. 3, Zisman plot).

When the particle is completely wetted (θ = 0), for γ ≤ γ_crit (Fig. 7A), the force F_γ nearly linearly decreases with the degree that the particle submerges. It is interesting to notice that for any γ ≤ γ_crit, the indentation always increases up to u_max = 0.4 μm (i.e., up to the diameter of the particle of 0.5 μm minus δ_hypo = 0.1 μm) while F_γ decreases to 0. Thus, when F_γ = 0 at u_max = 0.4 μm, the particle is completely submerged and ϕ = 180°. On the other hand, when γ > γ_crit [i.e., a particle is partially wetted (θ > 0)] (Fig. 7B), the slope of F_γ vs. u_max relationship increases (in absolute value) much faster with increasing γ, leading to a progressively lower value of u_max when F_γ = 0. In this last case, increasing θ (with increasing γ) causes β_o to decrease to 0 (see Fig. 6A), and consequently, F_γ = 0 occurs at ϕ < 180°. This, in turn, indicates that the particle is only partially submerged, even with no resistance from tissue. For simplicity, in the text that follows (and figures, other than Figs. 7 and 8, A and B), the equilibrium indentation, u_max,eq, is denoted as u_max.

Effects of Hypophase Thickness, Poisson Ratio, and Particle Size on the Degree of Particle Indentation

The effect of the thickness of hypophase δ_hypo on particle indentation is not negligible, which is demonstrated in Fig. 8A for θ = 0 and γ_crit. Decrease in δ_hypo increases both F_γ and the degree of particle indentation. This is because the decrease of δ_hypo (i.e., thinner hypophase) leads to a combined effect of an increase of the length of the three-phase line (via decreasing ϕ) and an increase of β_o, which is caused by larger capillary rise (see Fig. 6B). Altogether, these two synergetic effects result in higher equilibrium forces and thus, deeper indentation. The quantitative effects of the thickness of hypophase are summarized in Fig. 8C. Both, the equilibrium force and equilibrium indentation (denoted here as u_max instead of u_max,eq for simplicity) decrease approximately linearly with increasing δ_hypo in the range of δ_hypo from 0.05 to 0.2 μm. The force decreases faster than u_max due to a nonlinear effect of force-indentation relationship of the Hertzian contact of a rigid sphere with an elastic medium (Eq. 14).

Finally, the effect of Poisson's ratio⁸ on indentation is shown to be relatively small as demonstrated in Fig. 8B for θ = 0 and γ_crit. A decrease of the equilibrium force, F_γ = F_el, for a decrease of v from 0.5 to 0.1 is roughly equivalent to 5 kPa decrease in E. This change in F_el–u_max relationship only slightly increases the particle indentation, showing that the overall effect of v is modest. In fact, the quantitative effect of Poisson's ratio on the equilibrium force is about 16%, for a four-fold increase in v, and this increase is more pronounced at higher values of v, while the decrease in displacement is only 8% (Fig. 8D).

The effects of particle size on the equilibrium indentation u_max are examined for various tissue elastic moduli (E = 10, 20, and 30 kPa) with constant δ_hypo = 0.1 μm (Fig. 9A, Table 1) and for various hypophase thicknesses (= 0.1, 0.15, and 0.2 μm) with constant E = 20 kPa (Fig. 9B, Table 1). One of the remarkable features common in the results is that u_max behaves differently for γ < γ_crit than for γ > γ_crit, suggesting a strong influence of geometric factors, such as contact angle θ, as discussed above. It is noteworthy that the maximum indentation occurs at γ = γ_crit. As the size of the particle decreases, the magnitudes of F_γ = F_el and u_max all dramatically decrease. Also for smaller particles, the effect of E variation on u_max is dramatically reduced (Fig. 9A, top), while the effect of δ_hypo variation is amplified (Fig. 9B, top). Normalizing particle indentation u_max with particle radius R shows that the largest relative indentation u_max/R occurs for an intermediate particle size (e.g., R = 0.25 μm) at E = 10 kPa and δ_hypo = 0.1 μm (Fig. 9A, bottom). Interestingly, a much larger effect of decreased particle size on change in u_max/R is caused by decrease of hypophase thickness. Although the smallest particle causes much smaller indentation, the decrease of δ_hypo causes the largest relative imbedding of smaller compared with larger particles (Fig. 9B, bottom). In contrast, the smallest particles tested (R = 0.125 μm) with thicker δ_hypo (= 0.20 μm) does not show any indentation when γ > 45 dyn/cm.

Fig. 9.

The quantitative affects of the elastic modulus of alveolar tissue, thickness of hypophase, and the particle size on maximum indentation, u_max, as a function of surface tension γ. A: effect of Young's elastic modulus and particle size. B: effect of the thickness of the hypophase and particle size. Top: absolute value of displacements. Bottom: displacements normalized to the particle size. All calculations are performed for F_γ at γ_crit.

Table 1.

The effect of particle size, Young's elastic modulus, and the thickness of the hypophase on the equilibrium indentation force and displacement at γ_crit = 26.27 dyn/cm²

R, μm	E, kPa	δhypo, μm	umax, μm	F, nN	umax/R	F/E, μm2	F/(EumaxR)	γcrit, dyn/cm
0.125	10.0	0.10	0.1291	0.2915	1.0330	0.0292	1.8061	27.260
0.125	20.0	0.10	0.1154	0.4930	0.9235	0.0247	1.7084	27.260
0.125	30.0	0.10	0.1053	0.6444	0.8425	0.0215	1.6318	27.260
0.250	10.0	0.10	0.3044	1.4734	1.2174	0.1473	1.9365	27.260
0.250	20.0	0.10	0.2546	2.2834	1.0184	0.1142	1.7938	27.260
0.250	30.0	0.10	0.2229	2.8063	0.8916	0.0935	1.6787	27.260
0.500	10.0	0.10	0.5767	5.4672	1.1534	0.5467	1.8960	27.260
0.500	20.0	0.10	0.4533	7.6742	0.9067	0.3837	1.6928	27.260
0.500	30.0	0.10	0.3831	8.9437	0.7663	0.2981	1.5562	27.260
0.125	20.0	0.05	0.1501	0.7229	1.2008	0.0361	1.9264	27.260
0.125	20.0	0.10	0.1154	0.4930	0.9235	0.0247	1.7084	27.260
0.125	20.0	0.20	0.0420	0.1080	0.3356	0.0054	1.0286	27.260
0.250	20.0	0.05	0.2822	2.6519	1.1289	0.1326	1.8794	27.260
0.250	20.0	0.10	0.2546	2.2834	1.0184	0.1142	1.7938	27.260
0.250	20.0	0.20	0.1983	1.5704	0.7934	0.0785	1.5839	27.260
0.500	20.0	0.05	0.4740	8.2041	0.9480	0.4102	1.7308	27.260
0.500	20.0	0.10	0.4533	7.6742	0.9067	0.3837	1.6928	27.260
0.500	20.0	0.20	0.4113	6.6326	0.8227	0.3316	1.6126	27.260

Stress-Strain Fields

When a particle is indented into alveolar tissue, the tissue deforms, and consequently, a unique stress-strain field is created in the vicinity of the particle (Fig. 10). Generally, both the stress and strain diminish sharply with the distance from the particle surface. The axisymmetric formulation of the problem in cylindrical coordinates provides that σ_r ≈ σ_θ and ε_r ≈ ε_θ, where σ_r and σ_θ are radial and hoop stresses, and ε_r and ε_θ are radial and hoop strains, respectively; subscript θ denotes the hoop direction, which is different from contact angle θ. The maximum compressive stress can reach up to 25 kPa for R = 0.25 μm, and the maximum strain ∼0.4 for v = 0.5 or 0.3. The axial stress in the z-direction, σ_z, and stresses σ_r (≈σ_θ) rapidly decay from the maximum values at the particle-tissue interface, while the effective (von Mises) stress, σ_v, peaks at about 0.1 μm from the particle-tissue interface and then also diminishes sharply with the distance from the particle surface. It is interesting to notice how change in v from 0.5 to 0.3 affects the stress and strain distributions in the vicinity of the indentation surface of the particle. For v = 0.5, both σ_z and σ_r (≈σ_θ) start from the same value (∼25 kPa), whereas for v = 0.3, σ_z and σ_r (≈σ_θ) start from the different values. Similarly, for v = 0.5, both strains ε_z and ε_r (≈ε_θ) start from zero at the particle-tissue interface, whereas for v = 0.3, ε_z and ε_r (≈ε_θ) start from negative values denoting contraction of the particle-tissue interface. Especially, ε_r (≈ε_θ) switches the sign (from negative to positive, i.e., from compression to extension) with the distance from the particle surface in case of v = 0.3. These large changes in the distributions of stresses and strains result in minimal change in F_el = F_γ and u_max. The stress field for v = 0.5 shows spherical iso-σ_z distribution, which decay fast with distance from the particle-tissue interface (Fig. 10A, left). The strain field for v = 0.3 shows a complex pattern of strain sign change; for example, ε_r (≈ ε_θ) starts from a minimum negative value of approximately −0.2 at the particle-tissue interface, but soon becomes positive value of ∼0.1 at the distance of about 0.1 μm (∼0.4 R) from the surface (Fig. 10B, left).

Fig. 10.

Analytically calculated tissue deformation, stress, and strain fields under an indented particle of 0.5 μm in diameter, and for F_γ at γ_crit, and E = 20 kPa. A: deformed shape and stress and strain distributions for ν = 0.5. Stress field of σ_z (left) (blue = higher compressive stress); Stress components σ_z, σ_r, σ_θ, and effective stress, σ_v, along vertical axis of symmetry (middle); Strain components and effective strain along a vertical axis of symmetry (right). B: deformation and strain and stress distributions for ν = 0.3. Strain field of ε_r = ε_θ (left) (blue = higher compressive strain); Stress components and effective stress along a vertical axis of symmetry (middle). Strain components and effective strain along a vertical axis of symmetry (right).

The effects of E and v on stress and strain distributions in the tissue along the z-axis are shown in Fig. 11, A and B, respectively. An increase in E from 10 to 30 kPa (Fig. 11A) results in an appreciable increase in the magnitude of σ_z, σ_r ≈ σ_θ, and σ_v right beneath the particle (z < 0.4 μm) and decrease in ε_z, ε_r ≈ ε_θ, and ε_eff, especially for z >0.1 μm. The increase in stress with an increase in E appears to be much larger than the decrease in strains due to nonlinear nature of the Hertz problem. A decrease of v from 0.5 to 0.1 (Fig. 11B) changes not only the magnitude, but also the shape and the spatial distribution of the stress and strain. The most significant deviation caused by a change in v is seen in the shape of the distributions of σ_v and ε_eff, showing the strong effects of Poisson ratio on tissue shear deformation.

Fig. 11.

Effects of elastic modulus and Poisson's ratio: the stress and strain distribution along vertical axis of symmetry for an indented particle of 0.5 μm in diameter. The indentation force corresponds to F_γ at γ_crit. A: effect of Young's elastic modulus, E, for Poisson ratio v = 0.5 on axial stress, σ_z, a radial/hoop stress, σ_r, and von Mises stress, σ_θ (left); and on axial strain, ε_z, a radial/hoop strain, ε_r ≈ ε_θ, and effective strain, ε_eff (right). B: effect of Poisson ratio, v, for E = 20 kPa on stresses σ_z, σ_r, ≈ σ_θ, and σ_v (left); and on strains ε_z, ε_r ≈ ε_θ and ε_eff (right).

The difference in Poisson ratio (e.g., v = 0.5 vs. 0.3) results in quite different distributions in stress and strain fields (Fig. 12, left and right, respectively) not only along the z-axis (Fig. 11B). These distributions are contrasted for each component of stress and strain: on the left half for v = 0.5 and on the right half for v = 0.3. In some cases, both stress and strain change the sign in areas close to particle-tissue interface when v changes from 0.5 to 0.3, while the change in u_max is insignificant. These small differences in u_max are hardly visible as a small shift of the particle-tissue interface at the vertical axis of symmetry (Fig. 12). In contrast, the magnitude and spatial stress and strain distributions are significantly affected by the same change in v. These altered stress and strain distributions could be important for mechanotransduction and biological response of the epithelial cells to indentation of the particles in the alveolar tissue.

Fig. 12.

Stress and strain fields of the components of the stress and strain tensors under an indented particle (0.5 μm in diameter) into alveolar tissue for F_γ at γ_crit, E = 20 kPa, and Poisson ratios of v = 0.5 (left halves) and v = 0.3 (right halves).

DISCUSSION

The principal findings of this study are as follows: 1) an inhaled particle deposited on the alveolar surface can be forced toward the epithelium by surface tension forces, F_γ, to produce an indentation on alveolar septal wall. The degree of indentation can be primarily estimated from a force balance between F_γ and tissue elastic resistance to deformation, F_el. 2) The nonlinear behavior of F_γ with respect to the degree of indentation is determined by geometric factors, such as the size and shape of a particle, the thickness of hypophase, and the interfacial tensions acting on the three-phase line on the surface of the particle. 3) The magnitude of F_γ is strongly dependent on a relationship between surface tension γ and contact angle θ, represented graphically as Zisman's plot (10, 68), with F_γ usually having the maximum at γ_crit. 4) The degree of indentation is related inversely (nonlinearly) to the tissue elastic modulus, E, while it modestly (also nonlinearly) depends on the tissue Poisson ratio, v. 5) Stress and strain fields generated beneath the indented particle show complex patterns. These patterns are self-similar for different E, but they are quite different for different Poisson ratios, v. On the other hand, the magnitudes of the stresses and strains strongly depend on E, but only moderately on v.

Surface Tension Force

The surface tension force acting on a particle, F_γ, pushes the particle toward the alveolar epithelial cell surface. This force can be calculated as a function of the surface tension, γ, contact (or wetting) angle, θ, and the length of the three-phase interface line. The contact (or wetting) angle θ, can be obtained from a balance of interfacial tensions, γ_PL, γ_PA, and γ_AL ≈ γ [see Fig. 2 and Eq. 1]. Because the surface energy of a given particle is fixed as a result of the constant excess energy at the surface of a particle compared with the bulk, the associated interfacial tensions, γ_PL and γ_PA, are also constant. Therefore, for a given particle, θ essentially depends solely on the strength of surface tension, γ. The relationship between γ and cos(θ) is characterized by two distinctly different linear regimes (Fig. 3): 1) for γ < γ_crit, where θ = 0, denoting complete wetting; and 2) for γ > γ_crit, where θ > 0, denoting partial wetting. It should be noted that this relationship holds for any particle; only the value of γ_crit (= γ_PA − γ_PL) is different, and this value depends on the particle surface energy (19). Gehr's group [e.g., Schürch et al. (51) and Gehr et al. (12)] studied latex (polystyrene) and PMMA particles. From their data, we determined γ_crit = 27.26, and 22.25 dyn/cm for latex and PMMA particles, respectively. The fact that these two different particles have the different values of γ_crit is consistent with the observation that polystyrene particles are less hydrophilic compared with PMMA (51). This is because the surface free energy of polystyrene is about 33 erg/cm² while the surface free energy of PMMA is ∼40 erg/cm² (19).

The air-liquid interfacial tension, γ, varies during breathing, while particle interfacial tensions γ_PL and γ_PA are constant throughout a breathing cycle because they solely depend on the surface characteristics of the particles. When the alveoli expand during inhalation, the alveolar walls stretch and the alveolar surface area increases. The increase in the area increases surface tension, γ, due to a reduction in concentration of lung surfactant at the air-liquid interface. Under normal breathing conditions of healthy subjects, surface tension γ is typically well below 30 dyn/cm², and at the end of the exhalation, the alveolar surface tension can even reach values close to zero (50, 51). This indicates that the physiologically relevant range of values of γ is likely below γ_crit for the most of common particles; thus, an inhaled particle most likely interacts with alveolar liquid phase in a perfect wetting condition. For instance, relatively hydrophilic inhaled particles, such as dust particles, pollen, spores, or even hydrophobic⁹ particles, such as Teflon particles (15), may be able to submerge below the air-liquid surface during the exhalation because the surface tension can reach values close to zero. Thus, even the particles that usually float on the water surface can be submerged into lung lining layers at low surface tension and indented into the alveolar epithelial cells. On the other hand, the surface tension, γ, in some pathological cases with dysfunctional surfactant may rise up to 50 dyn/cm² (51). In this case, a particle would not be completely wetted; hence, the contact angle θ needs to be obtained using the γ vs. cos(θ) relationship described in Zisman's plot (Fig. 3) to obtain the equilibrium force F_γ and u_max.

The position of the three-phase line (on the surface of the sphere, Fig. 5), defined by angle ϕ, is determined by Eq. 3; it depends on thickness of hypophase δ_hypo, capillary rise z_o, maximum depth of particle indentation u_max, and particle size. The effect of the reduction of δ_hypo on an increase in u_max is demonstrated in Figs. 6 and 8. This behavior cannot be explained straightforwardly because the capillary rise, z_o, depends intrinsically on angles θ and ϕ. Thus the magnitude of z_o is directly associated with the position and length of the three-phase line (Fig. 5). Because z_o is essential for assessing F_γ and u_max, and, in turn, u_max is also intrinsically linked to the position of the three-phase line, the indentation u_max is modulated, in complex fashion, by both δ_hypo and the particle size.

Particle Indentation and Tissue Resistance to Deformation

For a particle larger than δ_hypo, the surface tension force, F_γ, pushes the particle into the alveolar wall, which results in deforming the tissue. Because F_γ depends on γ and the magnitude of u_max, and conversely u_max depends on F_el = F_γ and alveolar wall tissue elasticity, the interplay between these factors defines the F_γ-u_max-γ relationship. The solution of this implicit relationship is found iteratively as described in Surface Tension Acting on a Particle. The overall results showed that the maximum values of F_γ and u_max are usually observed in our calculations at γ_crit (Fig. 9).

The particle size has diverse effects on the particle indentation. The plot of the raw value of indentation, u_max, vs. particle size clearly shows that smaller particles generally make smaller indentations due to weaker indentation force (Fig. 9, top). This tendency can be seen by varying the thickness of the hypophase and elasticity, E, of alveolar wall tissue (Fig. 9). However, the plot of indentation normalized by particle size, u_max/R (Fig. 9, bottom), shows that u_max/R depends very little on particle size and points out more complex behavior. For example, within the physiological range of E, u_max/R is the largest for a particle of ∼0.5 μm in diameter. However, in thin hypophase, u_max/R displays the largest relative imbedding for the smallest particle (but larger than δ_hypo). This is somewhat consistent with the experimental observation by Schürch and colleagues (18), who showed that smaller particles submerge more rapidly in the liquid layer than larger particles. The fact that smaller particles make indentations more rapidly may partially explain why smaller (submicron/ultrafine) particles are taken up by the underlining epithelial cells at a greater rate than larger (>1 μm) particles (17). As soon as particles land on the alveolar surface, they encounter both epithelial cells and alveolar macrophages. Thus, these two types of cells compete for engulfing the particles. The fact that surface force enhances particle indentation into epithelial cells may potentially diminish (or change) the rate of macrophage-mediated clearance of smaller particles. In addition, for small particles (R = 0.125 μm), there is a significant dependency of u_max/R on thickness of the hypophase (Fig. 9B, bottom). This suggests that the thickness of the hypophase plays an important role in indentation of nanoparticles into underling tissues (42). Also, the reduction in thickness of hypophase may contribute to slowing macrophage-mediated clearance on nanoparticles.

The force normalized by product Eu_maxR, showed much smaller variation with respect to large changes in Young's elastic modulus and particle size (Table 1). In general, increase in Young's elastic modulus only modestly decreased the normalized force. Similarly, the large change in particle size changed the normalized force by at most 10%. The effect of a four-fold change in the thickness of the hypophase was also significantly reduced, having the normalized force in the range between 1 and 2.

Model Assumptions and Simplifications

In this study, we used the semianalytic approach to elucidate basic physics involved in particle indentation when an inhaled particle deposited on the alveolar septal surface interacts with alveolar tissue. As the process is complex because of the interplay of many different variables, we had to adopt several assumptions and simplifications to keep the analysis simple and tractable. A point-by-point critical evaluation of the assumptions is given below.

Particle shape.

We studied spherical, smoothly surfaced, and well-characterized (e.g., latex or PMMA) particles. The geometric simplification and the knowledge of particle surface characteristics were necessary to employ our semianalytic approach, which focuses on elucidating basic physics operating at the three-phase interfacial lines, as well as tissue resistive forces in the vicinity of particle-tissue interface. Although real ambient particles are certainly more complex in their shapes, our approach—surface microscopic geometry and the surface characteristics—albeit highly simplified, can serve as the basis for further detailed analysis.

Capillary rise.

On the basis of the boundary condition at the three-phase line and the angle β_o, the capillary rise z_o can be assessed reasonably accurately by using an approximate formula (Eq. 12) developed by Rapacchietta and Neumann (47). In particular, it is known that this solution is sufficiently accurate for the small particles (<1 μm), which, indeed, are of main interest here. James (29) analyzed multiple approaches with increasing complexity for determining capillary rise, z_o. He claims that the prediction of y_o = z_o/c (from Eq. 12) is accurate when the parameter c/r_o > 10. The smallest c/r_o considered here is larger than 200 for the size of particles and the range of physiological surface tensions. Taken together, this paper was primarily concerned with sufficiently accurate calculations of z_o, and to a lesser degree, regional profiles of the interface proximate to a submicron particle.

If of interest for further mathematical clarity, the profile of the entire interface may be obtained from the solution of the Laplace's equation, in which the boundary conditions are simultaneously satisfied at the three-phase line and in the far field. Alternatively, the profile of the entire interface can be calculated using an approximate equation based on an additive composite expansion of the outer and inner solutions (60). In the latter approach, the approximate analytical solution is uniformly valid as ε_B → 0 and involves Bessel function K_o($\sqrt{2}$rc)≡K_o(ε_Br/r_o). The limit ε_B → 0 signifies that the effect of gravity is small. However, this solution is of little practical interest for submicron particles because in real geometry of the lungs, the alveolar walls are of finite dimensions, for which the boundary conditions in the far field may not be important, except for the mathematical clarity.

Quasi-static analysis.

The surface tension γ and the thickness of hypophase, δ_hypo, are known to vary during the breathing cycle. For simplicity, we performed a quasi-static analysis, in which we assessed the degree of particle indentation from a force balance between the instantaneous values of F_γ and F_el. Using this approach, we found that the indentation during breathing cycle can also be evaluated by the above quasi-static approach using the instantaneous values for γ, δ_hypo, and alveolar wall material characteristics as the model parameters. In these stepwise calculations, time is considered only as a parameter. This analysis takes partially into account the hysteresis of the surface tension during expansion and contraction of the alveolar surface (64). However, the truly dynamic nature of the process, such as viscoplastic dissipation effects in alveolar wall tissues (11, 40) and the dynamic changes in γ and δ_hypo during breathing cycle cannot be fully accounted by the quasi-static analysis. Nevertheless, our quasi-static analysis captures the important basic features of the particle indentation process, consistent with experimental data of Gehr and colleagues (12, 16, 18, 51, 52), permitting quick and effective analysis of the effect of the variation of the numerous model parameters.

Hertzian contact problem.

To describe the mechanics of particle indentation into the alveolar wall, we used the simplest solution of the contact problem—the Hertzian contact (21, 34). This simplification was necessary because the analysis described in this study involves variation of a large number of parameters with a complex geometry at the particle-tissue contact and, therefore, a large number of simulations. The Hertzian solution is accurate for small indentations, say, u_max/R < 0.2, where the pressure distribution in the tissue beneath a particle remains approximately spherical. For larger indentations, on the other hand, this analysis may yield an error that manifests as a deviation of indentation pressure distribution shape from the spherical one. Note that the error also depends on the Poisson ratio. We estimate that the error would be of the order of 30% when u_max/R ≈ 1. In that case, a numerical approach, such as a finite element analysis, would be appropriate to obtain more accurate solution.

Tissue stress-strain field.

Tissue deformation and stress and strain fields in the tissue induced by the particle indentation were calculated by convolution integrals of the contact pressure distributions using the Boussinesq analytical solution (2, 56). These integrals are obtained numerically. Since the Boussinesq solution is valid and accurate for a semi-infinite solid, our results (Figs. 10–12) are reasonably accurate in the proximity of the particle-tissue interface. We also limited our solution to the frictionless contact. The advantage of using this semianalytical approach is the simple and quick way to analyze large number of possible scenarios and quantitative assessment of the effect of the variation of a large number of the model parameters on the system behavior.

Despite many restricting assumptions and simplifications, the present approach conveniently provided a fast and effective tool for elucidating effects of model parameter variation on the degree of particle indentation (sensitivity analysis). This provides a valuable contribution to unlocking the principal mechanisms driving the indentation of particles into the alveolar wall by the surface tension forces.

Physiological Implication and Future Direction

Potential physiological consequences of our findings are as follows. First, the particle indentation may trigger mechanotransduction pathways, either by directly deforming epithelial cells (3, 5, 6, 8, 20, 26–28, 35, 44, 45, 49, 55, 61, 62), physically insulting cell surface molecules (e.g., 4, 14, 54, 57, 65), and by remodeling of the intracellular cytoskeleton. The activation of these pathways may alter cellular biochemistry (1, 7, 23, 36, 41) and thereby the normal cell function. For instance, we have shown previously that physical contact between particles and cell surface adhesion molecules under cyclic (tidal) motion results in a profound secretion of proinflammatory cytokine (39, 59). Second, particle indentation increases the contact area between the particle and cell surface. This may trigger biochemical pathways directly and enhance the pathogenic response to toxic and allergenic particles, as well as particle internalization (17, 53, 66). Enhanced particle uptake by epithelial cells may indirectly alter the rate of particle clearance from the lung periphery, as discussed in Particle Indentation and Tissue Resistance to Deformation. Third, particle indentation may trigger signals, which activate afferent nerve fibers. Innervations of alveolar septa with sensory neurons have been visualized by us (31) and others (9, 25, 63). Because the afferent fibers can locate very close to the alveolar surface, particle-induced unphysiological stresses and strains in the proximity of afferents may mechanically stimulate afferent fibers and trigger neuronal responses. Similar mechanosensing phenomena have been recently studied in many biological models (67).

Regarding further model refinements, there are many additional details and factors, such as realistic shape and surface characteristics of (ambient) particles, viscoelastic nature of alveolar tissue characteristics, and dynamic changes of the surface tension imposed by breathing, all of which can be considered for a more realistic model analysis. Among all model parameters, the consideration of realistic structure and composition of the alveolar wall might be the most important. In the current model, we treated the alveolar wall as a whole, using a range of effective Young's modulus and Poisson's ratio measured by Fukaya et al. (11). However, the alveolar walls, in reality, are made of several different components with different mechanical properties, such as soft epithelial, endothelial, and other cellular components, relatively stiff collagen-based basement membrane, as well as other residual connective tissues. Furthermore, the alveolar walls are typically formed in a thin multiple-layered structure (i.e., epithelial cells, basement membrane, interstitium, endothelial cells). In addition, this detailed overall analysis could also include the lateral alveolar boundaries [e.g., other particles (32), alveolar shape, and airway bifurcations] and the effect of gravity, which, for particles significantly larger than 1 μm, becomes progressively important. Although the inclusion of these additional factors in the model would certainly make the analysis more complicated, the consideration of those factors might be necessary when one aims to more precisely model the specific effects of the particle indention on biological consequences.

Summary

We have developed a mathematical model to study mechanisms of the particle indentation into alveolar tissue. The analysis reveals that these mechanisms are centered on a mechanical balance between surface tension forces and tissue elastic forces; the former push the particle against the alveolar epithelial surface; the latter resist alveolar tissue deformation. The model describes in detail how various factors are involved in the indentation process. The quantitative model predictions can be used for understanding of mechanisms associated with mechanotransducting pathways triggered by indentation of the alveolar septa. For simplicity and mathematical transparency, several idealizations were employed in the model. Nevertheless, the model is capable of capturing principal features of the particle indentation process and can be used as the basis for the further detailed analysis.

GRANTS

This work was supported by Grants National Institutes of Health R01 AR048776 (to S. M. Mijailovich), National Heart, Lung, and Blood Institute HL054885 (to A. Tsuda), HL070542 (to A. Tsuda), HL074022 (to A. Tsuda), and Mijailovich Family Foundation. This work was also supported by National Aeronautics and Space Administration Grant NNJ06HE06A and State of Texas, Emerging Technology Fund and the Ministry of Science and Technological Development of Serbia (Grant OI-144028) (to M. Kojic).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

APPENDIX

Calculation of the Displacement Vector and the Components of the Stress and Strain Tensors

The indentation of a spherical particle in a compliant solid causes symmetrical deformation with respect to the axis of revolution, i.e., the z-axis. In this case, all stress components are independent of the circumferential angle θ, all derivatives with respect to θ disappear, and the displacements in θ direction are equal to zero. This symmetry significantly simplifies the calculations of the stress, strain, and displacement fields. Assuming that the alveolar tissue is linearly elastic further simplifies the problem; thus the stress, strain, and displacement fields can be conveniently calculated using Boussinesq analytical solution (2) for a concentrated force acting on the boundary of a semi-infinite solid. According to the principle of superposition, the Boussinesq analytical solution can be applied on distributed loads by using infinitesimal force acting on an infinitesimally small area and integrating it over the whole area of the contact between the particle and alveolar tissue. Thus, from known Boussinesq formulas, for displacements and stresses, a convolution integral can be constructed (Fig. 13), and the components of the stress tensor can be determined at any point of the deformed semi-solid.

Fig. 13.

Construction of the convolution integral for displacements, stress and strains from analytical solution of concentrated force on boundary of a semi-infinite elastic solid integrated over the distributed contact load q(r) (Fig. 5). The Boussinesq solution for displacements, stress and strains for an infinitesimal force applied to an infinitesimal area, i.e., contact pressure, are integrated over the area of the contact between particle and elastic substrate. Integration domains are for radial direction (along ξ) from 0 to a, and for circumferential direction a is from 0 to 2π. For q(ξ) acting at point S on the surface, the Boussinesq solution provides functions for displacements, stresses, and strains at the point Q at the depth z, at radial distance δ rotated by angle ϕ relative to the zr plane from point P. The point P is a projection of point S to the plane on depth z, i.e., to the plane of integration. The directions of components of the Boussinesq displacement vector and stress and strain tensors are denoted as init vectors n_z, n_r, and n_θ. The calculation of radial and hoop components of displacements, stresses, and strains takes into account the Boussinesq solutions for all values of α.

Calculation of the stress field in semi-infinite solid.

The stress components at a point Q with coordinates β and z of a semi-infinite solid per unit of normal force acting on the plane boundary surface in point S (see Fig. 13) are (2, 56)

$${\widehat{\sigma }}_{n_z}^{Bouss}=\frac{3}{2\pi }{z}^3{\left({\delta }^2+z^2\right)}^{-\frac{5}{2}}$$ (A1a)

$${\widehat{\sigma }}_{n_r}^{Bouss}=\frac{1}{2\pi }\left\{\left(1-2v\right)\left[\frac{1}{{\delta }^2}-\frac{z}{{\delta }^2}{\left({\delta }^2+z^2\right)}^{-\frac{1}{2}}\right]-3{\delta }^{2}z{\left({\delta }^2+z^2\right)}^{-\frac{5}{2}}\right\}$$ (A1b)

$${\widehat{\sigma }}_{n_{\theta }}^{Bouss}=\frac{1}{2\pi }\left\{\left(1-2v\right)\left[-\frac{1}{{\delta }^2}-\frac{z}{{\delta }^2}{\left({\delta }^2+z^2\right)}^{-\frac{1}{2}}\right]+z{\left({\delta }^2+z^2\right)}^{-\frac{3}{2}}\right\}$$ (A1c)

$${\widehat{\tau }}_{n_r{n}_z}^{Bouss}=\frac{3}{2\pi }\delta z^2{\left({\delta }^2+z^2\right)}^{-\frac{5}{2}}$$ (A1d)

$${\widehat{\tau }}_{n_r{n}_{\theta }}^{Bouss}={\widehat{\tau }}_{n_{\theta }{n}_z}^{Bouss}=0$$ (A1e)

Here, δ is the distance between the infinitesimal force dP = q(ξ) ξdξdα at the position S (ξ,α) on the surface of the semi-infinite solid (i.e., at z = 0) and the point Q (r, α = 0, z) at which the stress σ_z is calculated (Fig. A1):

$$\delta \left(r,\xi ,\alpha \right)=\sqrt{{\left(\xi \cos \alpha -r\right)}^2+{\left(\xi \sin \alpha \right)}^2}$$ (A2)

The stress component σ_z at arbitrary point Q of the semi-infinite solid with coordinates r and z, produced by traction distribution (Eqs. 15 and 20) over the entire circular area of contact of radius a is calculated from the following convolution integral:

$${\sigma }_z\left(r,z\right)=-{\displaystyle \underset{\xi =0}{\overset{a}{\int }}{\displaystyle \underset{\alpha =0}{\overset{2\pi }{\int }}{\widehat{\sigma }}_{n_z}^{Bouss}\left[\left(\delta \left(r,\xi ,\alpha \right),z\right)\right]q\left(\xi \right)\xi d\xi d\alpha }}$$ (A3)

The calculation of stresses σ_r and σ_θ is more complicated because the Boussinesq stresses σ̂_nr^Bouss and σ̂_nθ^Bouss act in the plane of integration; thus, both components contribute simultaneously for both σ_r and σ_θ (Fig. 13). Therefore, the convolution integrals of σ_r and σ_θ at a point Q(r, α = 0, z) requires recalculation of the Boussinesq stresses (per unit force) σ̂_nr^Bouss, σ̂_n0^Bouss, and σ̂_nrnθ^Bouss using Mohr circle (56) to obtain the component of the stresses, σ̂_r^Bouss* and σ̂_θ^Bouss*, in directions r and θ at in the horizontal plane at height z. The Boussinesq stresses in r and θ directions from the Mohr circle are:

$${\widehat{\sigma }}_r^{Bouss}=\frac{1}{2\pi }\left({\widehat{\sigma }}_{n_r}^{Bouss}+{\widehat{\sigma }}_{n_{\theta }}^{Bouss}\right)+\frac{1}{2}\left({\widehat{\sigma }}_{n_r}^{Bouss}-{\widehat{\sigma }}_{n_{\theta }}^{Bouss}\right)\cos \left(2\left(\varphi +\pi \right)\right)$$ (A4a)

$${\widehat{\sigma }}_{\theta }^{Bouss}=\frac{1}{2\pi }\left({\widehat{\sigma }}_{n_r}^{Bouss}+{\widehat{\sigma }}_{n_{\theta }}^{Bouss}\right)-\frac{1}{2}\left({\widehat{\sigma }}_{n_r}^{Bouss}-{\widehat{\sigma }}_{n_{\theta }}^{Bouss}\right)\cos \left(2\left(\varphi +\pi \right)\right)$$ (A4b)

where ϕ = atan[ξsinα/(ξcosα − r)] is the angle between plane rz of the Boussinesq stresses and the plane in which the stress is calculated at point Q(r, α = 0, z). Becauseτ̂_rθ^Bouss = 0, it is omitted in Eqs. A4a, b. To calculate stress produced by the entire contact pressure distributed over the contact circular area with the radius a, we must integrate Eqs. A4a, b. It is interesting that Eqs. A4a and A4b represent vectors shifted by π/2 (56). Thus, after integration over the full circle, the convolution integral gives the resulting values for the stresses in r and θ directions, at any radius r and depth z:

$${\sigma }_r\left(r,z\right)\equiv {\sigma }_{\theta }\left(r,z\right)={\displaystyle \underset{\xi =0}{\overset{a}{\int }}{\displaystyle \underset{\alpha =0}{\overset{2\pi }{\int }}\times {\widehat{\sigma }}_r^{Bouss*}\left[\left(\delta \left(r,\xi ,\alpha \right),z\right)\right]q\left(\xi \right)\xi d\xi d\alpha }}$$ (A5)

The only nonzero shear stress, τ_rz, is obtained from the integral of the projection of the vector τ̂_nr,nz^Bouss, which rotates with α (Fig. 13), is

$${\tau }_{rz}\left(r,z\right)={\displaystyle \underset{\xi =0}{\overset{a}{\int }}{\displaystyle \underset{\alpha =0}{\overset{2\pi }{\int }}{\widehat{\tau }}_{n_r{n}_z}^{Bouss}\left[\left(\delta \left(r,\xi ,\alpha \right),z\right)\right]\cos \left(\varphi -\pi /2\right)q\left(\xi \right)\xi d\xi d\alpha }}$$ (A6)

Calculation of the strain field in semi-infinite solid.

Using the stress-strain relationship of a linear elastic solid in cylindrical coordinates (56), we can calculate the strains from known stresses at the position (r,z) as

$${\epsilon }_r\left(r,z\right)=\frac{1}{E}\left[{\sigma }_r-v\left({\sigma }_{\theta }+{\sigma }_z\right)\right]$$ (A7a)

$${\epsilon }_{\theta }\left(r,z\right)=\frac{1}{E}\left[{\sigma }_{\theta }-v\left({\sigma }_r+{\sigma }_z\right)\right]$$ (A7b)

$${\epsilon }_z\left(r,z\right)=\frac{1}{E}\left[{\sigma }_z-v\left({\sigma }_{\theta }+{\sigma }_r\right)\right]$$ (A7c)

$${\epsilon }_{rz}\left(r,z\right)=\frac{1+v}{E}{\tau }_{rz}$$ (A7d)

$${\epsilon }_{r\theta }\left(r,z\right)={\epsilon }_{\theta z}\left(r,z\right)=0$$ (A7e)

Calculation of the displacement field in semi-infinite solid.

Since the indentation of a spherical particle into a compliant solid causes symmetrical deformation with respect to the z-axis, the displacements in θ direction, as well as the derivatives with respect to θ are equal to zero, and the strains are related to the displacements by simple relationships (56). Because we are interested in the simplest way to determine nonzero displacements in z and r directions, denoted as u and w, respectively, we only need the following Boussinesq strains per unit force:

$${\widehat{\epsilon }}_{n_{\theta }}^{Bouss}=\frac{{\widehat{u}}_{n_r}^{Bouss}}{r}$$ (A8a)

$${\widehat{\epsilon }}_{n_z}^{Bouss}=\frac{\partial {\widehat{w}}_{n_z}^{Bouss}}{\partial z}$$ (A8b)

Substituting appropriate Eqs. A1 into Eqs. A7b, c provides the analytical formulae for ε̂_nθ^Bouss and ε̂_nz^Bouss from which ŵ can be obtained directly as:

$${\widehat{w}}_{n_z}^{Bouss}=\frac{\left(1-2v\right)\left(1+v\right)}{2\pi Er}\left[z{\left(r^2+z^2\right)}^{-\frac{1}{2}}-1+\frac{1}{\left(1-2v\right)}{r}^{2}z{\left(r^2+z^2\right)}^{-\frac{3}{2}}\right]$$ (A9a)

and û after integration as (56):

$${\widehat{u}}_{n_r}^{Bouss}=\frac{1}{2\pi E}\left[\left(1+v\right)z^2{\left(r^2+z^2\right)}^{-\frac{3}{2}}+2\left(1-v^2\right){\left(r^2+z^2\right)}^{-\frac{1}{2}}\right]$$ (A9b)

The following convolution integrals provide the displacements u and w at any radius r and depth z:

$$w\left(r,z\right)={\displaystyle \underset{\xi =0}{\overset{a}{\int }}{\displaystyle \underset{\alpha =0}{\overset{2\pi }{\int }}{\widehat{w}}_{n_z}^{Bouss}\left[\left(\delta \left(r,\xi ,\alpha \right),z\right)\right]\cos \left(\varphi +\pi \right)q\left(\xi \right)\xi d\xi d\alpha }}$$ (A10a)

$$u\left(r,z\right)={\displaystyle \underset{\xi =0}{\overset{a}{\int }}{\displaystyle \underset{\alpha =0}{\overset{2\pi }{\int }}{\widehat{u}}_{n_r}^{Bouss}\left[\left(\delta \left(r,\xi ,\alpha \right),z\right)\right]q\left(\xi \right)\xi d\xi d\alpha }}$$ (A10b)