Articles in PresS. J Appl Physiol (July 15, 2010). doi:10.1152/japplphysiol.00209.2010
Particle-induced indentation of the alveolar epithelium caused by surface
tension forces
S. Mijailovich1, M. Kojic1,2, A. Tsuda1
1) School of Public Health, Harvard University, Boston, USA.
2) Department of Nanomendicine and Biomedical Engineering,
University of Texas Medical Center at Houston, USA
Running head: Surface tension-induced tissue deformation by particles
July 8th, 2010
Corresponding authors:
Srboljub M. Mijailovich, Ph.D.
Molecular and Integrative Physiological Sciences
Department of Environmental Health
Harvard School of Public Health
665 Huntington Avenue
Bldg. I, Room 1010D
Boston, MA 02115
Tel: 1-617-432-4814
Fax: 1-617-432-3468
Email: [email protected]
Akira Tsuda, PhD
Molecular and Integrative Physiological Sciences
Department of Environmental Health
Harvard School of Public Health
665 Huntington Avenue
Boston MA 02115
Tel: 617 432 0127
Fax: 617-432-3468
Email: [email protected]
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Copyright © 2010 by the American Physiological Society.
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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 mechanotrasduction and mechanosensing pathways, as
well as potentially enhancing particle uptake by the cells.
Key words: Zismann’s plot, three-phase interfacial lines, ultra fine particles, contact angle,
aerosols, mechanotransduction
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Interaction between inhaled particles and the lung is of great interest in lung physiology
(13). It is likely that it triggers 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 trans-epithelial 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, 24,
57)]; 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.
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.
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INTRODUCTION
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 liquidair 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, Fγ , which pushes the particle toward the tissue, and
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resisting tissue elastic force, Fel , 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 umax . Both forces can be expressed by complex nonlinear functions of umax ;
Fγ includes geometric parameters, as described below, while Fel depends on the tissue
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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 we considered in this study (51).
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For submicron particles buoyancy force or particle weight are more than two orders of magnitude smaller than
either Fel or Fγ . For example the weight for typical 1 μm particle with density of 2-3 g/cm3 is about 200 times
smaller than the smallest equilibrium forces Fel = Fγ (calculated for the highest elastic modulus of alveolar wall
tissue and the largest thickness of hypophase). The effect of gravity decreases with third power with respect to the
particle and for particles <1 μm this effect is negligible, i.e. spatial orientation of alveolar surface with respect to
gravity vector affects very little the degree of particle indentation. However, the asymptotic solution for capillary
rise (shown below) contains a term that weakly depends on gravity. Therefore inclusion of gravity in these
calculations is necessary but overall impact of this term on the equilibrium indentation of the particle is small.
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The surface tension acting on a particle
Because the alveolar surface is covered by a liquid layer, a particle forms a particle-airliquid 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),
γ PA = γ PL + γ cosθ
(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 .
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 cos θ and γ , spanning a representative physiologic range is shown in Fig.
3. A similar relationship between cos 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”, cos θ is equal to 1 at low surface tensions
(i.e. γ < γ cr ), denoting “complete wetting”, while for higher surface tension ( γ > γ cr ), cos θ
decreases linearly with increasing The nonlinear effects due the line tension are negligible2 over
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.
2
The effect of line tension on the contact angle, θ , can be large for the small droplets in the flat surface especially
for θ ~ 0 (37; 38). However, for a small sphere in a liquid layer, as we considered here, this effect is negligible,
because the critical three phase contact radius, rc is of order or smaller than an average distance between the solid
and liquid molecules,
θi
δ mol . The rc
is calculated from modified Young equation cos θ i = cos(θ ) − τ (γ
ro )
is contact angle modulated by the line tension, τ , and
ro is radius of the three phase line. For small
sphere partially submerged in liquid layer the term τ (γ ro ) is approximately equal to − sin θ i yielding to
where
rc / δ = sin(θ ) /[cos(θ ) − cos(θ − π ) . For θ < 65° the critical radius rc / δ mol < 1 and it is below the lower
limit of experimental resolution in contact angle measurements (37; 38) and therefore effect of line tension on
negligible.
5
θ
is
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Knowing the relationship between θ and γ , we are now in a position to estimate the normal
surface tension force Fγ acting on the particle (47; 48; 51),
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Fγ = 2πRγ sin φ sin(θ + φ )
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 φ = 1 −
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δ hypo
R
−
zo umax
−
R
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.
The a priori unknown values of φ , z o , and u max are determined iteratively, following the
computational steps as follows: (i) for a given φ , Fγ is calculated from Eq. 2, and z o is
calculated from the solution of the Young-Laplace equation (46) (see Eq. 4, below); (ii) for
known Fγ , u max is calculated form the analytical solution (see Eq. 14, below) of the Hertz
problem (21); and (iii) 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 raise and menisci profile around small spherical particles
At the surface of 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 umax are determined. The capillary rise, z o , is obtained from
the hydrostatic Young-Laplace equation for the meniscus (46):
1
1 
(4)
Δp = ρgz − γ  +  = 0
 R1 R2 
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 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
3
ρ = ρ L − ρ A , where ρ L
ρ L >> ρ A , we assume ρ ≈ ρ L .
Strictly speaking, the density in Eq. 4 should be
respectively. However, because
6
and
ρ A are densities of liquid and air,
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(2)
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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
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(1 + z′ )
=
2
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z′′
r (1 + z′′2 )2
and R2 =
z′
1
(5)
2
where z ′ = dz dr and z′′ = d 2 z dr . The boundary conditions are defined at the three-phase line
and in the far field: (1) at radius ro = R sin ϕ , 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:
z ′ = tan β o
z′ = 0
at r = ro
as r → ∞
(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.
Substituting Eq. 5 into Eq. 4, defining the capillary constant c 2 = 2 γ / 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
non-dimensional form (47; 48):
y′′
(1 + y′2 )
3
2
+
y′
1
x(1 + y′2 ) 2
− 2y = 0
(7)
where y′ = dy dx , and y′′ = d 2 y dx 2 The boundary conditions (Eq. 6) can also be transformed
into a nondimentional form:
y′ = tan β o
y′ = 0
at x = xo
as x → ∞
(8)
where xo = ro / 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 Eqs. 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 << sin β (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 tan β = y ′ is a slope of the air-liquid
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R1
3
2
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 are 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 / dβ = − x / tan β
dy / dβ = − x
The solution of Eq. 9a for the known initial condition β = β o at x = xo , is:
x = − xo sin β o / sin β
(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 o + ( x o sin β o ) ln
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(9a)
(9b)
tan β o / 2
tan β / 2
(11)
The only remaining unknown is the value of the dimensionless capillary raise yo = 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 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 in ordered to precisely determine
yo and, therefore, the position of the three-phase line (via Eq. 3). One convenient method for
approximately determine yo the shooting method6 (47). Using this method, the value of yo is
determined by matching the above outer solution, which satisfies boundary conditions at three
phase line, with so called “the inner solution” which satisfies the boundary condition in the far
4
Multiplying Eq. 7 by x provides the term 2 xy , which is of much smaller magnitude than sin β for small
y ′ = tan β << 1 the second term in Eq. 7 can be approximated as
particles. For
1
2
y ' (1 + y′2 ) ≈ tan β ≈ sin β . Because xy << sin β the third term which includes gravity effect can be
neglected and the first term is of approximately equal magnitude as the second term. Because y rapidly decays
with x , for large x the product xy → 0 .
5
For example, the predicted values of x and
y from Eqs. (10) and (11) as a function of β are within 2% for x =
2.0 (i.e. r > 2 mm) at β = 0.5 ° compared with the tabulated values of Huh and Scriven (24). For larger values of
r > 2 mm the error increases and the solution diverges for large values of x .
6
The method of solving the boundary-value problem which involves transforming it into an initial-value problem is
called the shooting method, because this technique "shoots" from a point where one of the initial values is a guess to
another point where the effect of that guess may be judged owing to the known conditions at the final point of the
calculations.
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field (29). This solution yields to an approximate analytical expression of yo , in the limit of the small
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Bond number ε B2 = 2ro2 / c 2 → 0 7 as


2 2
− λ
y o = x o sin( β o ) ln
 x o (1 + cos β o )

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where λ is the Euler constant equal to 0.57721.
Deformation in alveolar tissue caused by particle indentation
Fγ (umax ) = Fel (umax )
(13)
The force Fel 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,
1.5
Fel =
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(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:
r
q(r ) = qo 1 −  
a
where a is the radius of contact,
2
(15)
3 (1 − ν ) 2
Fel R
4 E
and qo is the maximum contact pressure,
a=3
3
qo =
2π
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4 ⋅ E ⋅ R 0.5 ⋅ umax
3(1 − ν 2 )
3
(16)
16 Fr E 2
Fel R .
9 (1 −ν ) 2
(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
The limit of ε B → 0 is equivalent to one in which there is no effect of the hydrostatic pressure in the liquid. In
other words, the pressure difference across the interface is zero (33) which is valid for submicron particles.
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Under equilibrium conditions, umax (schematically shown in Fig. 4) is calculated from a force
balance between Fγ and Fel under quasi-static conditions,
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(12)
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The alveolar tissue, and especially epithelial cell layer, is very soft; thus, the particle
indentation could be larger than 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, Fplunge , with respect
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to the maximum Hertzian force FHertz , max = Fel (a = R) , is
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Fplunge = Fγ − FHertz , max = Fγ −
4 ER 2
3 (1 − ν 2 )
(18)
The linear increase in Fplunge for the force exceeding FHertz , max , i.e. for umax > uHertz ,max , is given as
2 ER
(u max − u Hertz ,max )
(19)
(1 − ν 2 )
and therefore umax is also linearly related to the total force Fγ > FHertz , max . Finally, the pressure
distribution can be approximated by using the solution for the flat indenter:
q
(20)
q plunge (r ) = o , plunge
2
r
1−  
a
and the reference pressure (at the axis of symmetry) is
F
qo , plunge = plunge2
(21)
2πR
F plunge =
The total pressure distribution for Fγ > FHertz , max is equal to the sum of q(r ) and q plunge (r ) .
For known normal traction (pressure) distribution imposed by 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:
1
(σ zz − σ rr ) 2 + (σ rr − σ θθ ) 2 + (σ θθ − σ zz ) 2 + 6(σ zr2 + σ r2θ + σ θ2 z ) 
συ =
(22)
2
Similarly, the effective strain is defined as:
2
2
2
2
eeff =
ε zz + ε rr + ε θθ + 12 (ε 2zr + ε 2rθ + ε θ2 z ) .
(23)
3
[
]
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335
deviates from the spherical, leading to the solution error. This will be analyzed in detail in the
Discussion section.
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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., Fγ < Fγ crit ) an increase in surface tension increases the particle
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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 Zismann’s plot (Fig. 3), where cos θ 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 ( ∝ γ sin β o )
sharply decreases. Thus, according to Eq. 4, the decrease in Fγ is caused by a larger decrease in
sin β o compared to smaller increase in the product of γ and the length of the three phase line,
2πR sin φ . In other words, the effects of an increase in θ on Fγ is strong so that even a large
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decrease in φ (i.e., a large increase in the length of the three line phase line), cannot compensate
a large decrease of β o and z o (Fig. 6A). Since Fγ is a strong function of these geometric factors
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(see Eq. 2), an increase of γ above γ crit would result in decreasing Fγ and consequently a
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decrease in the degree of particle indentation (Fig. 6A). In this analysis the thickness of
hypophase δ hypo is taken to be constant.
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particle indentation increases (Fig. 6B). In this case, an increase in Fγ is caused by a decrease in
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φ as a consequence of larger capillary rise, zo , for thinner hypophase. Since smaller φ
increases the length of the three phase line and in addition increases magnitude of βo = φ − π (i.e.
RESULTS
Degree of particle indentation into alveolar tissue by surface tension forces
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the vertical component of the surface tension force), the increase in Fγ is amplified by these two
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synergetic effects. The cumulative effect, therefore, increases Fγ and the depth of particle
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indentation, u max .
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indentation u max , the equilibrium indentation, umax,eq , is determined iteratively, and is represented
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graphically as a cross-over point of the Fel −u max and Fγ −u max curves. Here, the Fγ −u max
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relationship is obtained from Eqs. 2 and 3, while the Fel −u max relationship is derived from the
Herzian contact (Eqs. 14 and 19). In Figs. 7 and 8A,B the equilibrium indentation, umax,eq , is
denoted as diamond symbols at the crossover points.
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Due to the fact that both Fel and Fγ are nonlinear functions of the maximum particle
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When δ hypo decreases (at constant γ and θ ), Fγ increases and consequently the degree of
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In Fig. 7, u max is plotted vs. Fγ and Fel 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, umax,eq , increases with increasing γ
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for a fixed E or with decreasing E for a fixed γ . When γ > γ crit (Fig. 7B), however, the
situation is different: umax,eq decreases with increasing γ for a fixed E and with increasing E
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for a fixed γ . As we discussed above, the difference between these cases can be explained due
to the difference in the γ vs. cos θ 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 Fγ nearly
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linearly decreases with the degree of the particle submerging. 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 Fγ decreases to 0. Thus, when Fγ = 0 at umax =
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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 Fγ vs. .u max relationship
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increases (in absolute value) much faster with increasing γ , leading to a progressively lower
values of u max when Fγ = 0. In this last case, increasing θ (with increasing γ ) cause β o to
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decrease to zero (see Fig. 6A), and consequently Fγ =0 occurs at φ < 180°. This, in turn,
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indicates that the particle is only partially submerged, even with no resistance form tissue. For
simplicity in further text and Figures other than 7, and 8A,B the equilibrium indentation, umax,eq ,
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is denoted as umax .
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which is demonstrated in Fig. 8A for θ =0 and γ crit . Decrease in δ hypo increases both Fγ and
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the degree of particle indentation. This is because the decrease of δ hypo (i.e., thinner hypophase)
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leads to a combined effect of an increase of the length of 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 affects of the thickness of hypophase are summarized in Fig. 8C. Both, the
equilibrium force and equilibrium indentation (denoted here as umax instead umax,eq for
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simplicity), decrease approximately linearly with increasing δ hypo in the range of δ hypo from 0.05
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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 rigid sphere with elastic medium (Eq. 14).
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,
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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, Fγ = Fel , for a
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decrease of ν from 0.5 to 0.1 is roughly equivalent to 5 kPa decrease in E . This change in
Fel −u max relationship only slightly increases the particle indentation, showing that the overall
effect of ν is modest. In fact, the quantitative effect of Poisson’s ratio on the equilibrium force
is about 16%, for four fold increase in ν and increase is more pronounced at higher values of ν ,
while decrease in displacements is only 8% (Fig. 8D).
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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 Fγ = Fel , and u max , all
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dramatically decrease. Also for smaller particles, the effect of E variation on u max is
dramatically reduced (Fig. 9A upper panel), while the effect of δ hypo variation is amplified (Fig.
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9B upper panel). 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 = 0.25μm) at
E =10 kPa and δ hypo = 0.1 μm (Fig. 9A lower panel). Interestingly, a much larger effect of
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decreased particle size on change in u max R is caused by dencrease of hypophase thickness.
Although the smallest particle causes much smaller indentation, the decrease of δ hypo causes the
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largest relative imbedding of smaller comparing to larger particles (Fig. 9B lower panel). In
contrast, the smallest particles tested ( R = 0.125μm) with thicker δ hypo (= 0.20 μm) does not
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show any indentation when γ >45dyn/cm.
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
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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 ν =
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
8
Although it is widely accepted that Poisson’s ratios of cells and tissues are close to 0.5 (e.g. Fukaya et al. (11),
Ofek et al. (43)), the local Poisson’s ratio can be much lower as observed in cartilage (22; 30; 58) due to local
compressibility of the cell cytoskeleton immerged in cytosol.
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Stress-strain fields
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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 υ , 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 ν from 0.5 to 0.3
affects the stress and strain distributions in vicinity of indentation surface of the particle. For
ν = 0.5, both σ z and σ r (≈ σ θ ) start from the same value (~25kPa), whereas for ν = 0.3, σ z and
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σ r (≈ σ θ ) start from the different values. Similarly, for ν = 0.5, both strains ε z and ε r (≈ ε θ )
start from zero at the particle-tissue interface, whereas for ν = 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 )
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switches the sign (from negative to positive, i.e. from compression to extension) with the
distance from the particle surface in case of ν = 0.3. These large changes in the distributions of
stresses and strains result in minimal change in Fel = Fγ and u max . The stress field for ν =0.5
The effects of E and ν on stress and strain distributions in the tissue along the z-axis are
shown in Figs. 11 A and 11B, 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 υ right beneath the
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particle ( z <0.4μm) and decrease in ε z , ε r ≈ ε θ , and ε eff , especially for z >0.1 μm. The increase
in stress with increase in E appears to be much larger than the decrease in strains due to
nonlinear nature of the Hertz problem. A decrease of ν 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 ν is seen in the shape of the distributions of σ υ
and ε eff , showing the strong effects of Poisson ratio on tissue shear deformation.
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The difference in Poisson ratio (e.g., ν = 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 left half for ν =0.5
and on right half for ν =0.3. In some cases both stress and strain change the sign in areas close to
particle-tissue interface when ν 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 particletissue 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 ν . 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.
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(Fig. 10A, left) shows spherical iso- σ z distribution which decay fast with distance from the
particle-tissue interface (Fig. 10A, left). The strain field for ν =0.3 shows a complex pattern of
strain sign change; for example ε r (≈ ε θ ) starts from a minimum negative value of ~ -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).
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The principal findings of this study are as follows: 1) an inhaled particle deposited on the
alveolar surface can be forced towards the epithelium by surface tension forces, Fγ , to produce
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an indentation on alveolar septal wall. The degree of indentation can be primarily estimated
from a force balance between Fγ and tissue elastic resistance to deformation, Fel . 2) The
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nonlinear behavior of Fγ with respect to the degree of indentation is determined by geometric
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factors, such as the size and shape of a particle, the thickness of hypophase, as well as by
interfacial tensions acting on the three-phase line on the surface of the particle. 3) The
magnitude of Fγ is strongly dependent on a relationship between surface tension γ and contact
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angle θ , represented graphically as Zismann’s plot (10; 68), with Fγ usually having the
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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, ν .
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,
ν . On the other hand, the magnitudes of the stresses and strains strongly depend on E , but only
moderately on ν .
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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 due to the constant
excess energy at the surface of a particle compared to 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): i) for γ < γ crit where θ =0,
denoting complete wetting; and ii) 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); Gehr et al. (12)) studied latex (polystyrene) and polymethylmethacrylate (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 to
PMMA (51). This is because the surface free energy of polystyrene is about 33 erg/cm2 while the
surface free energy of PMMA is approximately 40 erg/cm2 (19).
DISCUSSION
The surface tension force acting on a particle, Fγ , pushes the particle toward the alveolar
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Surface Tension Force
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 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 dyne/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 perfectly 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 dyne/cm2 (51). In this case, a
particle would not be completely wetted; hence, the contact angle θ need to be obtained using
the γ vs. cos(θ ) relationship described in Zismann’s plot (Fig. 3) in order to obtain the
equilibrium force Fγ and u max .
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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 0 ,
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 0 , depends intrinsically on angles θ and φ . Thus
the magnitude of z 0 directly associated with the position and length of the three phase line (Fig.
5). Because z 0 is essential for assessing Fγ and u max , and, in turn, u max is also intrinsically
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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.
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alveolar wall which results in deforming the tissue. Because Fγ depends on γ and the
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magnitude of u max , and conversely u max depends on Fel = Fγ and alveolar wall tissue elasticity,
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the interplay between these factors defines the Fγ - u max - γ relationship. The solution of this
Particle indentation and tissue resistance to deformation
For a particle larger than δ hypo , the surface tension force, Fγ , pushes the particle into the
9
The particles with low γ crit are considered as hydrophobic. These particles have large θ at air-water interface if
the surfactant is not present.
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implicit relationship is found iteratively as described above. The overall results showed that the
maximum values of Fγ and u max are usually observed in our calculations at γ crit (Fig. 9).
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The particle size has diverse effects on the particle indentation. The plot of the raw value
of indentation, umax, versus particle size clearly shows that smaller particles generally make
smaller indentations due to weaker indentation force (Fig. 9 upper panels). 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 lower panels)
shows that u max R depends very little on particle size and points at 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 grater rate than larger (>1μm) 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 lower panel). This suggests that the thickness of the hypophase plays an
important role in indentation of nanoparticles into underling tissues (42). As well, the reduction
in thickness of hypophase may contribute to slowing macrophage-mediated clearance on
nanoparticles.
The force normalized by Eu max R , 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, large change in
particle size changed the normalized force by at most 10%. The effect of 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 semi-analytic approach to elucidate basic physics involved in
particle indentation when an inhaled particle deposited on the alveolar septal surface interacts
with alveolar tissue. Since the process is complex due to the interplay of many different variables
(discussed above), we had to adapt several assumptions and simplifications to keep the analysis
simple and tractable. A point-by-point critical evaluation of the assumptions is given below.
Particles shape: We studied spherical, smoothly surfaced, and well-characterized (e.g.,
latex or PMMA) particles. These geometric simplification and the knowledge of particle surface
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characteristics were necessary to employ our semi-analytic 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, surface microscopic geometry and the surface characteristics, our
approach, albeit highly simplified, can serve as the basis for further detailed analysis.
Capillary rise: Based on the boundary condition at the three phase line and the angle β ,
the capillary rise z 0 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 here of the main interest.
James (29) analyzed multiple approaches with increasing complexity for determining capillary
rise, zo . He claims that the prediction of yo = zo / c from Eq. (12) is accurate when the parameter
c / ro >10. The smallest c / ro 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 zo , and in 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 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 ( 2r / c) ≡ K o (εr / ro ) . 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
lungs the alveolar walls are of finite dimensions for which the boundary conditions in far field
may not be important except for the mathematical clarity.
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Quasi-static analysis: The surface tension γ and the thickness of hypophase, δ hypo , are
known to vary during 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 Fγ and Fel . Using this approach, the indentation during breathing cycle
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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
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calculations, time is considered only as a parameter. This analysis takes partially in 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 visco-plastic dissipation effects in
alveolar wall tissues (11; 40) and the dynamic changes in γ and δ hypo during breathing cycle
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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.
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Hertzian contact problem: To describe the mechanics of particle indentation into 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,
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
semi-analytical approach is the simple and quick way to analyze large number of possible
scenarios and quantitative assessment of the effect of the variation of 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 (CSK). 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 above. 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
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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, interstitum, 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 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.
Acknowledgments
We gratefully acknowledge Drs. P. Gehr and M. Geiser for useful discussions, Dr. D.
Stamenović for critical review, A. Marinkovic for graphical design and Dr. A. Perin for
proofreading of this article. This work was supported by Grants NIH R01 AR048776 (SMM),
National Heart, Lung, and Blood Institute HL054885 (AT), HL070542 (AT), HL074022 (AT),
and Mijailovich Family Foundation (SMM). This work was also supported (for M. K.) by NASA
NNJ06HE06A and State of Texas, Emerging Technology Fund; and Ministry of Science and
Technological Development of Serbia (Grant OI-144028)
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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 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 infinitesimally small area and integrating it over the whole area of
the contact between the particle and alveolar tissue. Thus, from known Boussinesq formulae for
displacements and stresses a convolution integral can be constructed (Fig. A1), and the
components of the stress tensor can be determined at any point of the deformed semi-solid.
Calculation of the stress field in semi-infinite solid. The stress components at a point Q
with coordinates δ and z of a semi-infinite solid per unit of normal force acting on the plane
boundary surface in point S (see Fig A1) are (2; 56)
5
3 3 2
2 −2
(
)
σˆ nBouss
=
z
δ
+
z
(A1a)
z
2π
5
1
1 
z 2
1
2 −2 
2
2
2 −2 
σˆ nBouss
=
(A1b)
(1 − 2ν )  2 − 2 (δ + z )  − 3δ z (δ + z ) 
r
2π 
δ
δ


3
1
1 
z 2
 1
2 −2 
2
2 −2 
σˆ nBouss
=
(A1c)
(1 − 2ν ) − 2 + 2 (δ + z )  + z (δ + z ) 
θ
δ
2π 
 δ


5
3
2
2
2 −2
(A1d)
τˆnBouss
=
δ
z
δ
+
z
rn z
2π
(A1e)
τˆnBouss
= τˆnBouss
=0
r nθ
θ nz
(
)
Here δ is the distance between the infinitesimal force dP = q(ξ ) ξ dξ dα 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):
δ (r , ξ ,α ) =
(ξ cosα − r )2 + (ξ sin α )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. 8 and 11) over the entire circular area of contact of
radius, a , is calculated from the following convolution integral:
σ z (r , z ) = −
a
2π
ξ α σˆ
Bouss
nz
[(δ (r , ξ , α ), z )] q(ξ ) ξ dξ dα
=0 =0
21
(A3)
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767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
804
805
The calculation of stresses σ r and σ θ is more complicated because the Boussinesq
806
and σ̂ nBouss
act in the plane of integration, thus both components contribute
stresses σ̂ nBouss
r
θ
807
808
simultaneously for the either σ r and σ θ (Fig. A1). 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
809
, σˆ nBouss
and τˆnBouss
using Mohr circle (56) to obtain the component of the stresses,
force) σ̂ nBouss
r
θ
r nθ
810
811
812
σˆ rBouss* and σˆθBouss* , 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:
σˆ rBouss* = 12 (σˆ nBouss + σˆ nBouss ) + 12 (σˆ nBouss − σˆ nBouss ) cos(2 (ϕ + π ))
(A4a)
813
821
=
1
2
(σˆ
r
θ
Bouss
nr
+ σˆ
Bouss
nθ
) − (σˆ
1
2
r
Bouss
nr
θ
− σˆ
Bouss
nθ
) cos(2 (ϕ + π ))
(A4b)
where ϕ = atan [ξ sin α /(ξ cos α − 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 ). Because τˆrBouss
= 0 , it is
θ
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 the equations A4a,b. It is
interesting that the equations 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 :
σ r (r , z ) ≡ σ θ (r , z ) =
a
2π
  σˆ
ξ α
Bouss *
r
[(δ (r ,ξ ,α ), z )] q(ξ ) ξ dξ dα
(A5)
=0 =0
822
823
824
825
The only nonzero shear stress, τ rz , is obtained from the integral of the projection of the
vector τˆnBouss
which rotates with α (Fig. A1), is:
rnz
τ rz (r , z ) =
a
2π
 τˆ
ξ α
Bouss
nr n z
[(δ (r , ξ ,α ), z )]cos(ϕ − π / 2) q(ξ ) ξ dξ dα
(A6)
=0 =0
826
827
828
829
830
831
832
833
834
Calculation of the strain field in semi-infinite solid. Using the stress-strain relationship of
a linear elastic solid in cylindrical coordinates (56) the strains can be calculated from known
stresses at the position (r,z) as:
1
(A7a)
ε r (r , z ) = [σ r − ν (σ θ + σ z )]
E
1
(A7b)
ε θ (r , z ) = [σ θ − ν (σ r + σ z )]
E
1
(A7c)
ε z (r , z ) = [σ z − ν (σ θ + σ r )]
E
1 +ν
(A7d)
ε rz (r , z ) =
τ rz
E
ε rθ (r , z ) = ε θ z (r , z ) = 0
(A7e)
835
22
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814
815
816
817
818
819
820
σˆ
Bouss *
θ
836
837
838
839
840
841
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 zaxis, 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 direction,
denoted as u and w, respectively, we only need the following Boussinesq strains per unit force:
842
εˆnBouss =
843
εˆnBouss =
θ
uˆnBouss
r
(A8a)
r
∂wˆ nBouss
z
844
(A8b)
∂z
Substituting appropriate Eqs. A1 into Eqs. A7b,c provides the analytical formulae for εˆnBouss and
845
εˆnBouss from which ŵ can be obtained directly as:
847
848
849
850
851
θ
z
=
wˆ nBouss
z
(1 − 2ν )(1 +ν )  z (r 2 + z 2 )−

2π E r
1
2
−1+
(
1
r2z r2 + z2
1 − 2ν
)
− 32


(A9a)
and û after integration as (56):
=
uˆnBouss
r
(
1 
(1 +ν )z 2 r 2 + z 2
2π E 
)
− 32
(
)(
+ 2 1 −ν 2 r 2 + z 2
)
− 12


(A9b)
The following convolution integrals provide the displacements u and w at any radius r
and depth z :
w (r , z ) =
a
2π
  wˆ
ξ α
Bouss
nz
[(δ (r ,ξ ,α ), z )] cos(ϕ + π ) q(ξ ) ξ dξ dα
(A10a)
=0 =0
852
u (r , z ) =
a
2π
  uˆ
ξ α
Bouss
nr
[(δ (r , ξ ,α ), z )] q(ξ ) ξ dξ dα
=0 =0
853
854
855
23
(A10b)
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846
z
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Watanabe, N, Horie, S, Michael, GJ, Keir, S, Spina, D, Page, CP, & Priestley, JV.
Immunohistochemical co-localization of transient receptor potential vanilloid (trpv)1 and
sensory neuropeptides in the guinea-pig respiratory system. Neuroscience 141(3): 15331543, 2006.
Wilson, TA. Surface tension-surface area curves calculated from pressure-volume loops.
J Appl Physiol 53(6): 1512-1520, 1982.
Wirtz, HR, & Dobbs, LG. Calcium mobilization and exocytosis after one mechanical
stretch of lung epithelial cells. Science 250(4985): 1266-1269, 1990.
Yacobi, NR, Malmstadt, N, Fazlollahi, F, Demaio, L, Marchelletta, R, HammAlvarez, SF, Borok, Z, Kim, KJ, & Crandall, ED. Mechanisms of alveolar epithelial
translocation of a defined population of nanoparticles. Am J Respir Cell Mol Biol, 2009.
Yu, J. Airway mechanosensors. Respir Physiol Neurobiol 148(3): 217-243, 2005.
Zisman, WA. Relation of equilibrium contact angle to liquid and solid constitution. In
RF Gould (Ed.), Contact angle, wettability, and adhesion. Washington DC: American
Chemical Society, 1964.
27
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991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
Table Legend
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.
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28
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
Fig. 4 A schematic view of particle-hypophase-tissue interactions. At the equilibrium the
surface tension force Fγ is balanced with the elastic tissue restoring force Fel , and this
1055
1056
1057
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, zo . The angle φ defines the angular position of the
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
point where three surfaces meet with respect to the vertical axis, 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 .
Figure Legends
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 submersed. 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,
osmiophilic lining layer material. Bars= 2 (A and D), 0.5 (B), or 5 um (C). From Geiser et al.
(18), by permission.
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 = 0 at lower
γ crit = 22.25 dyn/cm.
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: The 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 zo , where zo 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: A 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
29
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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.
1071
1072
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 Fγ .
1073
1074
1075
1076
1077
1078
Fig. 6. Air-liquid and liquid-solid interfaces after particle indentation by surface tension forces.
A: The 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: The 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.
1087
maximum indentation, umax,eq are obtained at the intersection of the lines for Fγ (u max ) the fixed
1088
1089
1090
surface tension, γ , and those for Fel (u max ) fixed elastic moduli, E and ν =0.5. The particle
diameter is 0.5 μm. The diamond symbols denote points of static equilibrium and hence values of
umax,eq for that particular pair of forces. The increase of Young’s elastic modulus, E , always
1091
increases the equilibrium force and decreases umax,eq . A: For γ < γ crit both the equilibrium force
1092
and umax,eq increase with increase of γ , while B: for γ > γ crit both the equilibrium force and
1093
umax,eq decrease.
1094
1095
1096
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 affects of δ hypo on
1097
the equilibrium force, Fγ = Fel , and indentation umax,eq (C and D). The diamond symbols denote
1098
the equilibrium force, Fγ = Fel , and indentation umax,eq . A: The force and indentation cross-over
1099
1100
1101
points for three thicknesses of hypophase (0.05, 0.1 and 0.2 μm), three Young’s elastic muduli
( E = 30, 20, and 10 kPa), and ν = 0.5. The particle diameter is 0.5 μm. The decrease of the
thickness of the hypophase increases both Fγ and umax,eq . B: The force and indentation cross-
1102
1103
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 Poison’s ratio increases Fγ and decreases u max , having
1104
1105
similar quantitative effect as modest increase in Young’s elastic modulus. Note: for simplicity,
u max here and in all following figures denotes equilibrium indentation umax,eq . C: The effect of
1106
the thickness of hypophase three Young’s elastic muduli E , and ν = 0.5 on Fγ and u max . D:
1107
The effect of the Poisson’s ratio for the range of ν = 0.1-0.5 and for E = 20 kPa on Fγ and
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, Fγ = Fel , and the corresponding
30
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1079
1080
1081
1082
1083
1084
1085
1086
1108
u max . In (C) and (D) Fγ = Fel vs. δ hypo is denoted by solid lines and the symbols indicate which
1109
1110
1111
1112
1113
1114
1115
1116
Young modulus was used; the indentation u max vs. ν is denoted by dashed lines.
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
γ crit .
Fig.10. Analytically calculated tissue deformation, stress and strain fields under an indented
particle of 0.5 μm in diameter, and for Fγ 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 υ , 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 vertical
axis of symmetry (middle); Strain components and effective strain along a vertical axis of
symmetry (right).
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 Fγ at γ crit . A: The effect of Young’s elastic modulus, E , for Poisson ratio
1132
1133
ν =0.5 on axial stress, σ z , a radial/hoop stress, σ r ≈ σ θ , and Von Mises stress, σ υ (left); and on
axial strain, ε z , a radial/hoop strain, ε r ≈ ε θ , and effective strain, ε eff (right). B: The effect of
1134
1135
Poisson ratio, ν , for E =20 kPa on stresses σ z , σ r ≈ σ θ and σ υ (left); and on strains ε z ,
ε r ≈ ε θ and ε eff (right).
1136
1137
1138
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 Fγ at γ crit , E = 20 kPa, and
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
Poisson ratios of ν = 0.5 (left halves) and ν = 0.3 (right halves).
Fig.A1. 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 area of the contact between particle and elastic substrate. Integration domains
are: for radial direction (along ξ ) is from 0 to a , and for circumferential direction α is from 0
to 2π. For q(ξ ) acting at point S on the surface, the Boussinesq solution provides functions for
displacements, stress and strains at the point Q at the depth z , at radial distance δ rotated by
31
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1117
1118
1119
1120
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: The
effect of Young’s elastic modulus and particle size; B: The effect of the thickness of the
hypophase and particle size. Upper panels show absolute value of displacements. Lower panels
show displacements normalized to the particle size. All calculations are performed for Fγ at
1149
1150
1151
1152
1153
1154
1155
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. in 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 nθ . The
calculation of radial and hoop components of displacements, stresses and strains takes in account
the Boussinesq solutions for all values of α .
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on June 15, 2017
32
R
E
δhypo
umax
F
umax/R
F/E
F/(EumaxR)
γcrit
μm
kPa
μm
μm
nN
--
μm2
--
dyn/cm
10.0
20.0
30.0
0.10
0.10
0.10
0.1291
0.1154
0.1053
0.2915
0.4930
0.6444
1.0330
0.9235
0.8425
0.0292
0.0247
0.0215
1.8061
1.7084
1.6318
27.260
27.260
27.260
0.250
0.250
0.250
10.0
20.0
30.0
0.10
0.10
0.10
0.3044
0.2546
0.2229
1.4734
2.2834
2.8063
1.2174
1.0184
0.8916
0.1473
0.1142
0.0935
1.9365
1.7938
1.6787
27.260
27.260
27.260
0.500
0.500
0.500
10.0
20.0
30.0
0.10
0.10
0.10
0.5767
0.4533
0.3831
5.4672
7.6742
8.9437
1.1534
0.9067
0.7663
0.5467
0.3837
0.2981
1.8960
1.6928
1.5562
27.260
27.260
27.260
0.125
0.125
0.125
20.0
20.0
20.0
0.05
0.10
0.20
0.1501
0.1154
0.0420
0.7229
0.4930
0.1080
1.2008
0.9235
0.3356
0.0361
0.0247
0.0054
1.9264
1.7084
1.0286
27.260
27.260
27.260
0.250
0.250
0.250
20.0
20.0
20.0
0.05
0.10
0.20
0.2822
0.2546
0.1983
2.6519
2.2834
1.5704
1.1289
1.0184
0.7934
0.1326
0.1142
0.0785
1.8794
1.7938
1.5839
27.260
27.260
27.260
0.500
0.500
0.500
20.0
20.0
20.0
0.05
0.10
0.20
0.4740
0.4533
0.4113
8.2041
7.6742
6.6326
0.9480
0.9067
0.8227
0.4102
0.3837
0.3316
1.7308
1.6928
1.6126
27.260
27.260
27.260
Table 1
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0.125
0.125
0.125
Figures
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Fig. 1
1
JPA
Fig. 2
2
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Particle
T
Liquid
JPL
JAL
Air
Three Phase Line
1.0
0.9
Jcrt = 26.27
0.7
0.6
Polystyrene
PMMA
PMMA-DPPC
0.5
0.4
0
10
20
30
40
50
Surface Tension (dyne/cm)
Fig. 3
3
60
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cos(T)
0.8
Eo
umax
Fel
Fig. 4
4
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FJ
I
T
R
Ghypo
zo
Surfactant
Layer
umax
5
u(r)
Z(r)
ro
FJ
r
a
r
qo
q(r)
Fig. 5
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R1
The Three
Phase Line
r
z
B
z
A
r=R2
Er)
z (Pm)
A
Air-Liquid
Surface
ER
0.2
Jcrit= 27.26
o
T=0
ER
I
0.1
J = 50
T = 60
0.0
zo
Ghypo
o
I
-0.1
zo
Cell Surface
Ghypo
umax
umax
-0.2
z (Pm)
Eo
Eo
Air
zo
0.2
0.1
T=0
o
Jcrit
Liquid
zo
Ghypo
Ghypo
0.0
-0.1
Particle
Cell
-0.2
0.0
0.1
0.2
0.3
0.4
0.5
2.0
4.0
6.0
r (mm)
Fig. 6
6
umax
umax
8.0
10.0
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B
6
B
Jcrit = 27.26 dyne/cm
T=0
20
J
T
umax
E = 30 kPa
E
o
Fel
30 kPa
o
Jcrit =27.26
20 kPa
4
T>0
J = 40
20 kPa
10
J = 50
10 kPa
2
10 kPa
0
0.0
0.1
0.2
Indentation
0.3
0.4
umax (Pm)
0.0
0.1
0.2
Indentation
0.3
umax (Pm)
Fig. 7
7
0.4
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Force FJ, Fel (nN)
A
FJ
Ghypo
Ghypo = 0.05 Pm
B
E = 30 kPa
6
Jcrit =27.26 dyne/cm
T =0o
0.10
4
Ghypo = 100 nm
E = 30 kPa
Jcrit =27.26 dyne/cm
T =0o
20 kPa
0.5
20 kPa
Q = 0.3
Q = 0.1
10 kPa
2
10 kPa
0
0.1
0.2
Indentation
Force (nN) FJ=Fel
C
0.3
0.4
0.0
umax (Pm)
0.1
0.2
Indentation
0.3
0.4
umax (Pm)
D
4
0.3
3
0.2
2
E (kPa)
1
30
20
10
0
0.05
Q = 0.5
0.1
Ghypo = 0.1 Pm
Force
Max. Displ.
E
= 20 kPa
0.0
0.10
0.15
0.20
0.1
Hypophase Thickness Ghypo (Pm)
0.2
0.3
0.4
Poisson ratio Q
Indentation umax (Pm)
0.0
0.5
Fig. 8
8
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Force FJ, Fel (nN)
A
Hypophase Thickness
Indntation
umax/R
Norm. Indent.
R =0.500 Pm
A
0.5
T =0 T >0
B
T =0 T >0
R =0.500 Pm
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umax (Pm)
Elastic Modulus
0.6
0.4
R =0.250 Pm
0.3
R =0.250 Pm
0.2
R =0.125 Pm
0.1
Jcrit
0.0
Jcrit
R =0.125 Pm
1.2
1.0
0.8
Ghypo (Pm) R (Pm)
E (kPa) R (Pm)
0.6
10
20
30
10
20
30
10
20
30
0.4
0.2
0.0
0
10
20
0.125
0.125
0.125
0.250
0.250
0.250
0.500
0.500
0.500
30
Surface Tension
0.05
0.10
0.20
0.05
0.10
0.02
0.05
0.10
0.20
40
J
50
60
(dyne/cm)
0
10
20
0.125
0.125
0.125
0.250
0.250
0.250
0.500
0.500
0.500
30
Surface Tension
40
J
50
60
(dyne/cm)
Fig. 9
9
A
0
Stress
Q=0.5
0
0.0
-0.1
5 10 15 20 25
(Pm)
Z
10
-0.4 -0.2 0.0 0.2 0.4
Particle Alveolar Wall
Surface
Surface
Particle
Surface
Alveolar Wall
Surface
-0.2
-0.3
Strain
(kPa)
V r, V T
-0.4
Vz
VX
Hr, HT
Hz
Heff
-0.5
-0.6
-0.7
-0.9
kPa
-1.0
B
0.1
Q=0.3
Strain
Stress (kPa)
0
0.0
5 10 15 20 25
Alveolar Wall
Surface
-0.2
-0.3
Z (Pm)
0
-0.4
Particle Alveolar Wall
Surface
Surface
Particle
Surface
-0.1
-0.4 -0.2 0.0 0.2 0.4
Vz , V T
Vz
VX
Hz
Hr, HT
-0.5
-0.6
-0.7
-0.1
-0.8
-0.9
-1.0
Fig. 10
10
Heff
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-0.8
20
-0.6
0.0
-0.1
-0.3
25
0.6
20
15
10
5
E (kPa)
-0.2
0
Stress
5
0.1
Q = 0.5
10
20
30
VX(kPa)
Strain
10
Col 1 vs Col 4
Col 8 vs Col 11
Col 15 vs Col 18
0.4
0.2
0.0
0
0.0
0.2
0.4
0.6
Distance from Particle Tip
0.8
Hz
Strain
10
5
0.2
0.4
0.6
0.8
Distance from Particle Tip
11
0.5
0.3
0.1
0.3
Q
E = 20 kPa
0.5
0.3
0.1
20
15
10
5
0.2
0.1
0.0
-0.1
-0.2
0
-0.3
25
0.6
20
15
10
5
Col 1 vs Col 4
Col 8 vs Col 11
Col 15 vs Col 18
0.4
0.2
0.0
0.0
0.2
0.4
0.6
Distance from Particle Tip
Fig. 11
Q
-0.4
-0.6
25
0
0.0
E = 20 kPa
-0.2
HrHT
0.2
HrHT
10
20
30
15
VrVT(kPa)
E (kPa)
20
15
0
0.3
Stress
Q= 0.5
25
Eff. Strain Heff
VrVT(kPa)
Stress
VX(kPa)
-0.4
0.0
20
Strain
5
-0.2
Poisson Ratio
25
Eff. Strain Heff
10
Vz(kPa)
15
Stress
Hz
20
0
Stress
B
0.0
Strain
Stress
Elastic Modulus
25
0.8
0.0
0.2
0.4
0.6
Distance from Particle Tip
0.8
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Vz(kPa)
A
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Fig. 12
12
q([) S
0
Fig. A1
13
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nz
nT
nr
M
DQ
r
d[
P
[
dD
z
G
r
z