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1Department of Mechanical Engineering, Technion, Haifa 32000, Israel; and 2Physiology Program, Harvard School of Public Health, Boston, Massachusetts 02115
Submitted 23 August 2002 ; accepted in final form 11 March 2003
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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alveolus expansion; lungs; chaos
Until lately, it has been widely accepted that gravitational settling and deposition of fine particles deep inside the lung can reasonably be predicted, assuming that a simple Poiseuille flow can represent the flow field inside the acinus, ignoring completely the complexity of the flow that exists inside the alveoli. A mathematical model describing particle motion under gravity in a horizontal, long, straight pipe was originally given by Fuch (7) and Pich (20). It was later modified by Wang (30) and Hyder (12), who addressed the cases of particle motion inside inclining and declining pipes and randomly oriented pipes, respectively.
More recently, modifications to the foregoing "classic" model of the acinus were suggested, arguing that the alveolated structure of acinar ducts is likely to play a significant role in the acinar fluid mechanics and, consequently, in the transport of fine particles (25, 26). A simplified alveolar duct model included a rigid axisymmetric duct surrounded by toroidal pockets that opened to a central thoroughfare. The flow field was numerically solved, and trajectories of micrometer-sized particles were studied (26). The main finding of this initial investigation was that gas streamlines in the central channel were somewhat curved due to the presence of side-walled alveoli, and this curvature of the streamlines, coupled with the orientation of the duct relative to gravity, may cause the particles to enter the alveolus and deposit there. These findings were later confirmed by others (5, 6) in similar alveolated duct computer models.
Subsequently, the effect of an additional kinematic parameter, the rhythmical motion of the alveolar walls, on the flow was investigated. Sophisticated models of acinar fluid mechanics, representing time-dependent low Reynolds number flows in cyclically expanding and contracting alveolated duct structure, were developed (4, 8, 11, 24, 27–29). Both the theoretical and experimental analyses have demonstrated that fluid path lines in the acinus are highly complex. The forgoing complex flow patterns result in chaotic trajectories if 1) asynchrony exists between ductal and expansion flows (8, 28), or 2) inertial effects are considered (11, 27). These findings are of utmost importance and are significantly different from past investigations of alveolated rigid thoroughfares that resulted in periodic fluid path lines, which repeated themselves with every breathing cycle (5, 6, 25, 26). These findings also imply that models that fail to incorporate alveolar expansion and contraction may lead to incorrect predictions for the deposition of fine aerosols in the pulmonary acinus.
Two mathematical models of acinar flow in a rhythmically expanding and contracting alveolated duct structure were developed: 1) the acinar ductal flow model (11, 27), including an axisymmetric, multiple-alveolated expandable duct model with a closed end, and 2) the single-alveolus model (8), consisting of a fully three-dimensional (3D) cyclically expanding and contracting hemispherical alveolus subjected to a shear flow passing over the alveolar mouth. The former model (11, 27) is most suitable to study the bulk kinematic interaction between the central channel flow and alveolar entering and exiting flows and the associated aerosol behavior (e.g., deposition efficiency). The latter model (8), we believe, captures essential features of the 3D time-dependent flow field inside alveoli and may be used to study the detailed motion of aerosol particles that penetrate the alveoli, their duration of stay inside the alveoli, and their deposition locations on the alveoli walls.
The objective of this study was to investigate the influence of the flow
field induced by the rhythmical expansion and contraction of alveolar walls on
the gravitational motion of fine aerosol particles, by using the 3D,
single-alveolus model (8). The
results suggest that 1) the rhythmical expansion and contraction of
the alveolar walls and the associated fluid flow patterns have a major role in
determining the trajectories of particles settling under gravity. Fine
particles in the size range of 0.5 
1 µm in diameter are particularly
sensitive to the detailed patterns of recirculating alveolar flows, resulting
in highly nonuniform deposition distributions on the alveolar walls.
2) The relative magnitude of alveolar recirculation flow to ductal
shear flow passing over the alveolar opening (discussed in detail below) plays
a significant role in predicting particle behavior inside the alveolus.
Because this ratio varies along the acinar tree, the process of gravitational
deposition at the entrance of the acinus may differ from that at the periphery
of the acinus. 3) Even a small gravitational drift of aerosol
particles from the cyclic gas streamlines can cause a substantial convective
mixing. This gravity-induced convective mixing enhances deposition on the
alveolar walls. The new model reported here demonstrates that the existence of
alveolar recirculation is a key factor in determining the gravitational
deposition in rhythmically expanding and contracting alveoli.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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Dp 2.5 µm) and mass mp of unit
density (= 1 g/cm3) suspended in the foregoing oscillating flow
field. To make this paper self-contained, we recapitulate briefly the
underlying assumptions of the flow model and the key characteristics of the
flow field (8). This is
followed by the detailed methods that are employed to track the motion of a
suspended particle under the effect of gravitational forces. Alveolar Flow Model
Our model views a single-alveolus configuration as a hemispherical cavity1 attached at its rim to a flat plane (Fig. 1). The flow passing through the alveolar duct near the alveolus is approximated by a simple oscillatory shear flow over the flat plane, far upstream or downstream from the semispherical cavity. The plane and the attached cavity perform an oscillatory, self-similar expansion and contraction movement. Assuming that the flow field is governed by the creeping flow equations, superposition of the following two flow fields is allowed: 1) the flow induced by the self-similar expansion and contraction of the alveolus with zero downstream flow inside the adjacent airway, and 2) the flow induced by shear flow over a hemispherical rigid cavity with a vanishing velocity at the boundaries. Moreover, due to the quasi-steadiness of Stokes flows, the time variable can be viewed as a parameter that enters the problem via time-dependent boundary conditions. Thus two generic problems are addressed: the flow field vH induced by a unit surface radial velocity for a unit radius hemisphere (8), and the flow field vP induced by a unit shear flow over a unit hemispherical cavity (21). The solution representations for vH and vP are provided in APPENDIX A.
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The flow v inside the alveolus, which combines the effects of
expanding and contracting alveolus and the shear-induced flow generated by the
airflow in the adjacent duct, is
 | (1) |
(t), and for the oscillating
shear-flow in the adjacent airway, R(t)G(t), where
G(t) is the instantaneous shear rate at the airway wall. It is often
treated (albeit not exact) that R(t) = R0 [1 +
cos(
t)] and, consequently,
(t) = -R0
sin(t), where is the breathing
frequency, R0 is the mean radius of the alveolus, and
R0 is the expansion amplitude. The time protocol
R(t)G(t) = -R0G0
sin(t + 
) is assumed to possess an identical
and a small phase difference (
10°) that was physiologically
observed by Miki et al. (19)
[see Tsuda et al. (28) for
further discussion]. The value of G0 depends on the breathing
volumetric flow and the alveolus location down the acinar tree. The ratio
between the amplitudes of shear and expansion flows is given by
 | (2) |
= 0
 | (3) |
was
computed (see APPENDIX B) and plotted as a function of Weibel's
airway generation number (Fig.
2). The value of at the first few generations (e.g., 16
19th) from the entrance of the acinus is >1,000 and remains >100
for most of the acinar generations, suggesting that the ductal shear flow
plays an important role in determining alveolar flow. In previous studies
(8,
11,
27), we found that a
sufficiently strong shear flow passing by the alveolar opening induces
vortexes inside the alveolus, and the presence of alveolar recirculation flow
in an expanding and contracting alveolus indicates the existence of a
stagnation saddle point in the alveolar flow field.
Figure 5, A and
B, from Haber et al.
(8) depicts streamline maps of
such cases ( = 400 and 200, respectively, with = 0). Within
very few generations from the distal end of the acinar tree, the value of
is of the order of 10. This suggests that the effects of alveolar wall
motion are dominant and that alveolar flow is largely radial with no
recirculating flows [see Fig.
5D from Haber et al.
(8) for = 20 and
= 0].
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Particle Motion
The differential equation governing the motion of a spherical particle
subjected to the gravity field g is
 | (4) |
is a unit
vector along g, and FD and FBr are
the respective drag and stochastic Brownian forces exerted on the particle.
Henceforth, we focus on the combined deterministic effects of convection and
sedimentation and neglect the stochastic Brownian forces (see the rationale of
our approach in the DISCUSSION section).
The drag force exerted on a spherical particle suspended in a Stokesian
flow field v is given as
(10)
 | (5) |
 | (6) |
 | (7) |
 | (8) |
1
g/cm3, the maximum value of the Stokes number (St) that
determines the magnitude of the lefthand side term in Eq. 7
is
 |
 |
p
 |
p0,
p0,
p0). An additional three
parameters pertain to the kinematics of the problem, namely, the relative
amplitude between the shear velocity and the expansion velocity , the
phase lag between the flows , and the dimensionless amplitude of
alveolus expansion . A single
parameter,2
H, accounts for particle dynamics and is determined by the magnitude
of the gravity field, the breathing frequency, the alveolus size, the fluid
viscosity, and the particle size and density. Two parameters are required to
define the direction of the gravity field with respect to the
axes of symmetry of the alveolus and that of the adjacent duct (the z
and y directions, respectively). The last parameter 
=
Dp/R0 that describes the ratio between
sizes of the particle and the alveolus is required in determining
gravitational deposition. Namely, integration of Eq. 9 is terminated
when the distance between the particle center and the alveolus wall is equal
to its radius.
An analytic solution of the highly nonlinear set of three scalar first-order differential equations (Eq. 9) is a formidable task. A numerical approach is required that hinges on the data provided by Pozrikidis (21) and Haber et al. (8) for the vector fields vP and vH, respectively. A detailed numerical procedure is given in APPENDIX C.
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RESULTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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. The magnitude and direction of the
aerodynamic force, on the other hand, depend heavily on the local flow
patterns and physical characteristics (size, shape) of the particles. As
described in METHODS, there are a total of 10 independent
dimensionless parameters affecting the motion of spherical particles. In this
report, the investigation is focused on the three important parameters
, , and that represent particle size, alveolus expansion
amplitude, and alveolar location along the acinar tree, respectively. The
effect of the gravity orientation on particle deposition awaits future
investigation. Also, the lag angle is mostly kept zero in this
report. Effects of Alveolar Expansion
The effect of alveolar expansion on the behavior of aerosol particles may be best illustrated by tracking the motion of a particle in an alveolus whose opening is facing downward (Fig. 3). In the absence of wall motion (i.e., a rigid-walled alveolus; Fig. 3A), there is no upward air stream entering the alveolus, and particles cannot be convected into the alveolus. Furthermore, because the gravity force points downward, particles moving downstream shall simply bypass the alveolus. On the other hand, in case the alveolus performs a rhythmical expansion and contraction motion (Fig. 3B), an upward convection flow exists during inhalation. Thus particles can enter the alveolus, provided the aerodynamic drag force exerted on a particle moving upward can counterbalance the downward gravity force.
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To demonstrate this delicate balance of forces in detail, the trajectory of a 0.5-µm particle initially placed near the alveolus rim is plotted in Fig. 3C. The trajectory of a massless (i.e., fluid) particle is also plotted for comparison. In both cases, the particles are passively introduced near the proximal corner of the alveolar opening [position 0 (P0)] at the beginning of inspiration (t = 0). The fluid particle (shown as a dotted curve) enters the alveolus during inspiration (0 < t < T/2) and exits during expiration (T/2 < t < T), where T is the breathing period. Its trajectories during inspiration and expiration overlap; namely, the fluid particle undergoes a perfect kinematically reversible motion. The trajectory of an aerosol particle (shown as a solid curve) is, however, quite different. Starting at the same location (P0), it deviates from the path line of a fluid particle. During early inspiration (0 < t < T/3), the aerosol particle moves across the alveolar opening. At the time t = T/3 approximately, it approaches the distal corner of the alveolar opening (P1). The particle then continues moving upward into the alveolus against gravity and rotates along the expanding walls during the rest of the inspiration time (T/3 < t < T/2). The speed of aerosol movement is small during this period, because the tangential airflow velocities are typically small near the wall. At the end of inspiration (t = T/2), this aerosol particle is located deep inside the alveolus (P2). As the flow starts to reverse (i.e., early expiration), the velocity of airflow is nearly zero. During this time, therefore, gravity dominates the particle motion. Consequently, the particle starts to take an expiratory path, which is below the path taken during the late inspiration. This new path brings the particle to a region at the middle of the alveolus, where airflow is recirculating. Here, the particle rotates five and one-half times in the alveolus during the expiration period. At the end of the cycle (t = T), the particle remains suspended near the center of the alveolus (P3). In subsequent cycles (not shown), this particle continues to rotate in the alveolus, and, finally after a few cycles, the motion of the particle reaches steady state (rotating counterclockwise 8 times during inspiration and clockwise 8 times during expiration in this example). This particle never leaves and remains suspended inside the alveolus, a manifestation that a delicate balance between the gravitational force and the effects of alveolar recirculation flow has been achieved.
Particle Deposition in a Rigid vs. Cyclically Expanding and Contracting Alveolus
Although the present model is not designed for quantitative analysis (e.g.,
a determination of exact deposition concentration per alveolus), it can be
used to compare the relative difference in wall deposition concentrations
between a rigid-walled alveolus and a cyclically expanding and contracting
alveolus. We simulate the behavior of an aerosol bolus (of width 40 µm,
consisting of 0.5-µm particles) approaching an alveolus in the case of
cyclically expanding and contracting vs. rigid-walled models. In both cases,
the alveolus is placed horizontally with its mouth facing upward, and the
ductal shear flow condition is kept constant at = 400 (the effects of
are discussed below). The movement of the bolus is monitored during
one breathing cycle, and the number of particles deposited inside the alveolus
is counted (Table 1). Results
show that, under equivalent ductal shear flow conditions, deposition in a
cyclically expanding and contracting alveolus ( = 0.05 and 0.1) is
substantially higher (an increase of 134 and 112%, respectively) than that
observed in a stationary alveolus ( = 0). [Note that the parameter
(0.1, 0.05, and 0) indicates the extent of the alveolar volume
expansion (30, 15, and 0%, respectively).] This clearly demonstrates the
dominant role alveolar wall motion can play in enhancing particle deposition
inside the alveoli.
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Effects of Particle Size
For a constant mass density of 1 g/cm3, the trajectories of
DP = 0.5, 1, and 2.5 µm are obtained
(Fig. 4A). The
alveolus is horizontally placed with its mouth facing upward, and the flow
conditions are set at = 400. The particles are introduced at the
beginning of inspiration (t = 0) in the duct near the proximal corner
of the alveolar opening (P0). For a small particle (0.5 µm; see
Fig. 4A,
top), whose motion is strongly influenced by the local flow field,
the flow patterns near P0 are critical in determining the fate of the
particle. When the initial position P0 of a particle is slightly moved from
the alveolar opening toward the strong ductal thoroughfare airstreams, the
particle is likely to be convected downstream without entering the alveolus
(trajectory shown as a green line). On the other hand, when the position P0 is
close enough to the alveolar opening, the particle (shown in red) follows the
airflow entering the alveolus (shown as a dashed line) and simultaneously
settling due to gravity (0 < t < 0.2T). Inside the
alveolus, the behavior of the particle depends on the extent of instantaneous
alveolar recirculation flow. When the recirculation flow is strong (e.g.,
0.1T < t < 0.4T), the particle follows the
recirculation path, but, when the strength of the airflow begins to diminish
as the end of inspiration approaches (e.g., 0.4T < t <
0.5T), the particle trajectory starts to deviate from the fluid path
line and drifts toward the alveolar wall due to gravity. During expiration,
the particle moves back near the wall following the expiratory airflow (shown
in blue) and continues to drift downward. Finally, the particle is deposited
on the wall during the expiration period at t = 0.7T. The
deposition process during expiration is augmented by the facts that airflow
near the walls is generally weak and that the walls are contracting (i.e.,
moving toward the particle). It is interesting to point out that the direction
of the particle's vertical drift due to gravity is opposite to that shown in
Fig. 3C in which the
particle moves away from the walls toward the alveolar center where the
airflow is more intense.
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The trajectory of a 0.5-µm particle initially placed off the midplane of
the alveolus (x 
0) is very complex
(Fig. 4B). During the
inspiration period of the first cycle, the particle enters the alveolus and
drifts appreciably toward the sidewall. This is due to the influence of
substantial sideway airflow in the alveolus during inspiration
(8). During expiration, it
moves back toward the center of the alveolus while rotating one and one-half
times. At the end of the first cycle, the particle remains suspended in the
air deep in the alveolus. When the second inspiration starts, the particle
moves again toward the sidewall, and, this time, this sideway motion soon
causes the particle to be deposited.
For slightly larger particles (1 µm; see Fig. 4A, middle), the balance between the gravity and aerodynamic forces again plays an important role, determining whether the particles would enter the alveolus and be deposited. Despite the fact that 1-µm particles are eight times heavier than 0.5-µm particles (of equal density), their trajectories do not follow the gravity vector. If these particles enter the alveolus, they follow the alveolar recirculation paths and approach close to the walls, where the airflow velocity is nearly zero relative to the wall velocity, and finally deposition occurs (shown as short vertical turns of solid lines in Fig. 4A, middle). Most of the deposition sites are on the proximal side of the alveolar walls and occur during the end of inspiration (discussed below in detail).
For the largest particles (2.5 µm; see Fig. 4A, bottom), during the early stages of inspiration, the velocity of ductal airflow is still small, and the particles move appreciably downward due to gravity, entering the alveolus. Once the particles are inside the alveolus, they tend to follow the alveolar recirculation flow, especially during peak inspiration, but, due to their heavy weight, these 2.5-µm particles continue to settle downward and soon are deposited on the alveolar wall.
Deposition Site
The distribution of deposition sites inside the alveolus is determined for
particles of different sizes (Fig.
5). About 1,000 particles of three different sizes (0.5, 1, and
2.5 µm) are equally spaced over the alveolar opening (z = 0) at
t = 0. The simulation is performed at = 400 for a few cycles;
for each cycle, the number of particles deposited on the alveolar walls and
their deposition site are recorded. For 0.5-µm particles, 47% of the
particles enter the alveolus and deposit during the first cycle (41% during
mid-to-late inspiration, 6% during early expiration). The deposition
distribution of 0.5-µm particles on the alveolar walls is highly
nonuniform, with a peak on the proximal side of the alveolar wall at about
one-third of the way from the rim to the bottom (see pink symbols in
Fig. 5; the deposition peak
further moves closer to the rim as increases, as discussed later). All
of the medium-sized particles (1.0 µm in diameter) deposit during the first
cycle, mostly at the peak inspiration (92%), whereas few particles (8%)
deposit at peak expiration. The deposition distribution of 1.0-µm particles
is also nonuniform, with a peak on the proximal side of the alveolus at about
midway from the rim to the bottom (see cyan symbols in
Fig. 5). It is the area that
the particles reach convectively after following the recirculation airflow
during inspiration. As expected, the larger the particles are, the quicker the
deposition occurs. For 2.5-µm particles, almost all of the particles (99%)
deposit during the early inspiration period of the first cycle. These large
particles are mainly deposited at the bottom region of the alveolus. The
deposition distribution deep in the alveolus is nearly uniform, with a slight
elevation on the proximal side (see yellow symbols in
Fig. 5).
Gravitational Deposition Along the Acinar Tree (The Effects of
)
As described in METHODS, the structure of alveolar flow is
determined by the combined effects of the alveolar recirculation flow induced
by ductal shear flow (QD) and alveolar entering and exiting radial
flow (QA) induced by the rhythmic motion of the alveolar walls.
Because the relative magnitude of QD to QA rapidly
decreases as the tidal air moves deeper into the acinus, the velocity profile
of the ductal flow entering the alveolus and the velocity field inside the
alveolus are expected to change along the acinar tree, a change represented by
the parameter . Consequently, we expect that the parameter has
a significant influence on 1) the efficiency of particles entering
the alveolar cavity and 2) the gravitational deposition process of
particles inside the alveolus. We test this idea by simulating the motion of
0.5-µm particles near and inside the alveolus for various values of
.
A sheet of equally spaced (1,000) particles is placed over the
alveolar opening at t = 0, and the number of particles entering the
alveolus 
, expressed as an "alveolar-entering efficiency," is
examined for different values of (3,000, 800, 400, 200, 100, and 10)
(Fig. 6A). For high
values of (say, = 3,000, which corresponds to the alveolar
flow near the entrance to the acinus, see
Fig. 2), a relatively small
number of particles enter the alveolar cavity, especially from the proximal
side of the alveolar opening, and are deposited on the proximal side of the
alveolar walls near the rim. As the value of decreases (corresponding
to the alveolar flow deeper into the acinus, see
Fig. 2), the increases.
At 100, all of the particles released at the alveolar surface
enter and are deposited. For comparison, a similar simulation is performed for
the case of a rigid-walled alveolus, with the latter showing that the number
of particles entering a rigid-walled alveolus is much smaller than that for an
expanding alveolus for equivalent values of (see rigid
in Fig. 6A).
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Approximately 6,500 particles, equally spaced, are also placed inside the
alveolus, and their behavior is monitored for different values of . For
high values of (e.g., 3,000, 1,000), deposition occurs mainly during
the early phase of inspiration (Fig.
7). The number density distribution of deposited particles is
decidedly nonuniform, with a very high density on the proximal side close to
the alveolar rim (Fig.
6B, top). For lower values of , but still
higher than 100 (a range that corresponds to the majority of the acinar
generations, except the very distal ones), the results are basically similar
to those of = 3,000, namely, deposition occurs mainly during the first
half of the inspiration period (Fig.
7) and preferentially on the proximal side of the alveolar wall
surface (Fig. 6B,
middle) further down from the rim. Remarkably, the deposition process
is different in the case of = 10 (corresponding to the flow conditions
occurring at the very distal alveoli). In this case, deposition is almost
temporally invariant throughout two breathing cycles
(Fig. 7), and deposition sites
are spread almost uniformly on the alveolar surface
(Fig. 6B,
bottom), in sharp contrast to the cases of larger values
(Figs. 6B,
top and middle).
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Convective Mixing Induced by Gravity
To further demonstrate the effects of alveolar flow patterns on
gravitational deposition, the behavior of a cloud of aerosols compared with
that of massless (i.e., fluid) particles is investigated. The flow parameters
are kept constant, assuming the following values, = 0.1, = 400,
and = 0° (i.e., no phase lag exists between the expansion and the
tidal shear flows). A small square-shaped cloud of massless particles (that
exemplifies particle behavior in a gravitation-free environment) is introduced
at the middle of the alveolus symmetry plane, x = 0
(Fig. 8A), and its
behavior is monitored through one breathing cycle. During inspiration, the
initially square-shaped cloud is stretched and progressively deforms to an arc
shape (pink symbols for T/4; gray symbols for T/2). During
expiration, however, this large arc shrinks into a smaller arc (blue symbols
for 3T/4), with exactly the same shape as the one obtained at
t = T/4, and finally it returns to the original square shape
as the expiration process terminates (green symbols for t =
T). As expected, the path of each massless particle is perfectly
reversible under the simulated flow conditions ( = 0° and Reynolds
number = 0). In other words, during expiration, each particle faithfully
retraces the path it outlined during inspiration, and, as a result, the
square-shaped cloud is restored after a full cycle.
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In the simulation illustrated in Fig.
8B, a small square-shaped cloud of 0.5-µm particles
(shown in black) is introduced at the middle of the alveolus at t =
0, and its behavior in a gravity field is monitored through a period of three
cycles. Up until the end of the first inspiration period, the cloud of
0.5-µm particles behaves somewhat similar to that of the massless
particles, namely, the square-shaped cloud possesses an elliptical shape at
t = T/2 (shown in gray). However, by the end of the first
cycle, the cloud of aerosol particles possesses a spiral shape (shown in
green), instead of returning to its original square shape, demonstrating that
the trajectory of each particle is not kinematically reversible. Irreversible
spreading continues during the second cycle, and the initial square shape is
no longer recognizable by the end of the second cycle (shown in red). By the
end of the third cycle, all the particles are deposited on the walls (shown as
blue dots). Note that the extent of convective mixing of 0.5-µm particles
is nearly equivalent or slightly higher than the extent of mixing induced by a
phase difference between ductal flow oscillation and cyclic alveolar wall
motion (Fig. 8C,
= 20°).
DISCUSSION
In previous models of alveolated ducts with rigid walls (5, 6, 25, 26), the acinar flow was typically treated as a flow consisting of two distinct flow regions. In the acinar ductal region, an axial flow streams by the alveolar openings without entering the alveoli, whereas, inside the alveoli, the flow rotates slowly without convectively communicating with the ductal region. Thus, according to these models, there is no convective transport of particles between acinar channels and the surrounding alveoli, and fine-sized aerosol particles that do not undergo Brownian motion can only enter the alveolus due to gravity. In the present study, we demonstrate that the inclusion of an additional factor, the rhythmical motion of the alveolar walls, fundamentally changes the acinar flow patterns and substantially alters the fate of particles moving inside the alveoli. Associated with tidal breathing, each alveolus expands and contracts about 12 times per minute; that is, the alveolar volume changes, necessarily inducing airflow between the duct and alveolus. This entering and exiting alveolar convective flow may carry aerosol particles into and outside the alveolus. The effects of alveolar convective airflow on small aerosol particles (e.g., 0.5 µm) are likely to be critical in the study of aerosol deposition in the pulmonary acinus, yet no existing model has dealt with this effect. The present work, therefore, represents the first systematic study focusing on this important issue.
It should be noted that our present analysis neglects the stochastic Brownian force exerted on a moving particle, whose influence relative to the gravity force, especially on small particles 0.5 µm in diameter, is still in dispute (e.g., Ref. 13) and might prove to be potentially important. Nevertheless, the exclusion of diffusion is necessary here, because the objective of this study is to perform, as a first step, a mechanistic analysis that would elucidate the fundamental physics of gravitational sedimentation coupled with particle convection in the complex, rhythmically expanding and contracting, chaotic alveolar flow, with the latter being so fundamentally different from the steady Poiseuille flow applied in many past investigations. Thorough investigation on the combined effects of Brownian motion and alveolar chaotic flow on small particles, as well as simultaneous effects of Brownian motion and gravitational sedimentation in complex alveolar flows, is left as future research topics.
Alveolar Flow
The basic premise of the present model is that alveolar flow can be
considered as creeping (inertialess) flow. We argue that this assumption is
justifiable, at least as a first-order approximation, because the Reynolds
number of airflow in the acinus is generally much smaller than unity. This
also means that the "Reynolds number" is no longer relevant for
inertialess flows; we must introduce a more appropriate parameter to
characterize the alveolar flows. As our laboratory has been emphasizing in
recent investigations (8,
11,
27), we proposed that
, the ratio between QA and QD, multiplied by a
constant and a local geometric factor
(RD/R0)3,
should be the new important
parameter3 that
characterizes alveolar flow. Because both QA and QD are
proportional to a product of tidal volume (VT) and , both
VT and are canceled out from the
QA-to-QD ratio, and, therefore, is solely a
function of acinar tree geometry. Our calculation (see APPENDIX B)
shows that starts as a small number (10-4) at the
entrance of the acinus (16 18th generations), exponentially increases as
the tidal air moves deeper into the acinus, and becomes an order of unity at
the most distal part of the acinus. In this paper, we use (=
1/), the reciprocal of , to express the relative importance
between QD and QA.
In recent studies (8,
11,
27), we demonstrated that the
inclusion of the cyclic motion of alveolar walls (represented by nonzero
in the present study) fundamentally changes acinar flow patterns. The
detailed alveolar flow patterns, especially the presence (or absence) of
recirculation flow inside the alveolus, depend largely on , which
represents the relative strength of the axial thoroughfare ductal flow and the
lateral alveolar entering and exiting flow. The larger the is
(corresponding to the flow conditions in more proximal acinar generations),
the more likely that alveolar recirculation flows occur and the larger the
alveolar space the recirculating region occupies (see
Fig. 5 in Ref.
8). On the other hand, when
is small (e.g., <10; corresponding to the flow conditions in the
postdistal region in the acinar tree), the alveolus hardly possesses a
recirculating region, and the alveolar flow is largely radial (see
Fig. 5 in Ref.
8). It should be noted that the
presence (or absence) of alveolar recirculation is paramount in understanding
the behavior of fine-sized particles in the alveolus, because introduction of
a small phase lag can make the alveolar flow chaotic
(8,
27).
In what follows, we focus our discussion on the effects of rhythmical
motion of the alveolar walls on gravitational particle deposition inside the
alveolus based on these two parameters, and , separately.
Effects of (Wall Motion)
The parameter denotes the lung expansion
strain.4 In the
analysis varying from 0 (i.e., rigid walls) to 0.1 while keeping fixed
the acinar ductal flow conditions, we compare the behavior of particles in an
alveolus with moving walls ( > 0) to that of rigid ones ( = 0)
(Table 1). The effect of wall
motion is dramatic; the alveolar deposition efficiency can be more than
doubled when the alveolar wall moves even to a very small degree (i.e.,
= 0.05). This suggests that previous predictions based on the alveolar models
with rigid walls (5,
6,
25,
26) might have substantially
underestimated particle deposition inside the alveolus.
Effects of (Alveolus Location Along the Acinar Tree)
Varying and keeping at a fixed physiologically relevant
value ( = 0.1), we study the effects of alveolus location along the
acinar tree on particle deposition. A strong dependence of alveolar entering
efficiency, of particle trajectories, and of particle deposition sites within
the alveolus on can be observed (Figs.
6 and
7). For alveoli near the acinar
entrance (e.g., respiratory bronchioles) where ductal flow substantially
dominates the alveolar lateral flow, the value of is large, and the
strong ductal shear flow passing by the alveolar opening induces a large
recirculation inside the alveolus. Consequently, the number of particles
entering the alveolus mouth and moving inside the alveolus is largely
influenced by this alveolar recirculation flow, with the particles moving
quickly to the proximal side of the alveolar walls, where they are finally
deposited (Fig. 6B,
top, and Fig. 7). This
basic airflow pattern and deposition mechanism persist in most of the alveoli
along the acinar tree up until > 100. In this region of acinus,
aerosols are quickly deposited, mostly during an inspiration period,
consistent with experimental observation reported by Bennett and Smaldone
(1). For small values of
(<100) representing the alveolar flow conditions at the distal end
of the acinus, however, the deposition process and deposition patterns are
very different from the one described above. Deposition occurs continuously
(Fig. 7), and the pattern of
deposition inside the alveolus is largely uniform
(Fig. 6B,
bottom). This distinct difference of the = 10 from the
> 100 cases is most probably due to the fact that, for =
10, there is no recirculating flow, and the flow is largely radial inside the
alveolus. Thus the existence of alveolar recirculation and expansion is a key
factor in determining deposition processes and deposition patterns.
Effects of Particle Size
Based on results shown in Figs.
3,
4,
5,
6, aerosol particles considered
in this study (0.5 2.5 µm) may be categorized into two size groups:
submicron-sized particles (0.5 < DP 1 µm) and
micron-sized particles (1 < DP < 2.5 µm).
Particles in the former size range are highly sensitive to the detailed
instantaneous alveolar flow patterns, and, therefore, their trajectories are
determined as a result of the competition between gravitational and alveolar
aerodynamic forces. When is large (e.g., > 1,000), the
gravitational deposition of submicron particles (e.g., DP
= 0.5 µm) occurs preferentially near the proximal side of the alveolar rim
during early inspiration (Fig.
7). When is small (e.g., < 10), the submicron
particles are deposited rather uniformly at the bottom of the alveolus and
continuously throughout both the inspiration and the expiration periods during
a few breathing cycles (Fig.
7). These results suggest that the direction and magnitude of the
gravitational force and the alveolar flow patterns must be known to make it
possible to predict the characteristics of submicron particle deposition
within the alveoli (i.e., timing and site as a function of alveolus location
along the acinar tree). In contrast, larger particles (1 µm <
DP < 2.5 µm) are generally not significantly
influenced by the detailed alveolar flow patterns.
Convective Mixing and Enhanced Deposition Mechanisms
One of the most peculiar features of rhythmically expanding alveolar flow is that an alveolar flow with recirculation can be chaotic, enhancing mixing under the effect of small disturbances (8, 11, 24, 27). In previous studies, the effects of small nonzero Reynolds numbers (inertial effects) were addressed (11, 27), and the effects of a small-phase difference between ductal flow oscillation and cyclic alveolar wall motion were examined (8, 28). In the present study, we investigated gravitational effects on convective mixing by following particle trajectories (Fig. 8).
The terminal velocity of 0.5-µm particles with density of 1
g/cm3 in quiescent air is 10 µm/s. It means that, if
gravity were the only mechanism driving deposition, it would take 15 s
(or 3–5 breaths) for the particle to move across a rigid-walled
alveolus, 300 µm in diameter. This is a rather slow process. However,
particles that are subjected to the combined effects of gravity and the
complex flow existing inside the alveoli continuously cross streamlines, and
their trajectories become increasingly convoluted. Thus substantial convective
mixing occurs.
In a rhythmically expanding and contracting alveolus with characteristic recirculation flow, deposition is enhanced (Table 1). A possible qualitative explanation is as follows. Due to the expansion movement of the alveolar walls, the probability of aerosol particles crossing the alveolus mouth is enhanced [see Figs. 3, 4, and also Fig. 5 from Haber et al. (8) depicting streamlines that enter the alveolus]. After entering the alveolus, the circulating flow field carries the particles, conveying them closer to the alveolar walls. Near the walls, the tangential velocity is exceedingly small. Consequently, the particles would spend most of the time at the wall vicinity, increasing their probability of being deposited. Simultaneously, the particles are being pulled downward by gravity, a process that causes their final approach toward the alveolar wall. Note also that the deposition pattern can be highly nonuniform5 (Fig. 8B), a very different pattern than would be conceived for the case of a rigid alveolus.
Physiological Implication
Comprehensive knowledge of aerosol deposition in the lung is needed so that the adverse effects of environmental particulate pollution can be understood and various therapeutic strategies of aerosolized drug delivery can be examined. Thus a large number of experimental investigations, both in animals and with human subjects, have been performed. Despite such considerable research efforts, much of the experimental data are limited to measurements of total particle deposition, and accurate assessments of regional deposition are still technically difficult. To complement this deficiency in an experimental approach, many mathematical models of aerosol deposition in the lungs have been developed, and our current knowledge of regional deposition is largely based on the predictions of these models (e.g., Ref. 15). The goal of the present study is to make model predictions of acinar deposition more realistic and accurate by incorporating an important factor, the rhythmical motion of alveolar walls.
At the level of alveolus. The alveolar surface is vulnerable to
inhaled particles because the alveolar wall, in contrast to the
well-protected, ciliated surface of conducting airways, is made of an
ultrathin layer of epithelium to maximize the efficiency of gas exchange.
Furthermore, this wall is not merely a simple layer of type I epithelial
cells; it has its own structural complexity. For instance, smooth muscle is
preferentially localized to the alveolar entrance rings
(32), and sensory nerve
endings are situated at specific locations within the alveolar wall
(18). There is also a report
that alveolar type II cells tend to be localized in the corners of the alveoli
(35). The localization of
particle deposition patterns within the alveolus, therefore, may have
particular physiological implications. For instance, particles in the
submicron-sized range would preferentially deposit near the entrance rings of
the alveoli and especially in the proximal region of the acinus (i.e., high
value). Because the alveolar entrance rings are rich with stress
fibers and smooth muscle, and these components are important players for
maintaining structure and dynamics of the lung parenchyma, a high dose of
pollutants specifically concentrated on this region of the alveoli may result
in adverse effects on respiratory function.
At the level of acinus. In the case of horizontally placed
alveolus with its mouth upward, all of the particles entering the alveolus are
eventually deposited. Therefore, the alveolar entering efficiency
(i) shown in Fig.
6A can be utilized as the "local alveolar
deposition probability" (although it does not mean that
i represents a deposition probability of an alveolus
in case a different gravity orientation, with respect to that of the alveolus
axis, is employed) to compute deposition patterns along the acinar tree with
rhythmically expanding and contracting alveolus compared with that of a
rigid-walled alveolus.
The number of particles that are deposited in an alveolus at generation
i is defined by a product of the local alveolar deposition
probability i and the number of particles available
in the gas phase for that alveolus. Accounting for the fact that the number of
particles in the gas phase decreases as the inhaled particles sample more and
more alveoli along the acinar tree, the average number of particles
ci that are deposited in a single alveolus at the
ith generation may be expressed as ci =
Ctotal·{(1 -
16)n16 (1 -
17)n17.... (1 -
i-1)ni-1 [1 - (1 -
i)ni]/2i-16
ni}, where Ctotal is the
total number of particles leaving a particular 15th generation duct and
entering the acinar ducts, generations 16–23, and
ni denotes the number of alveoli attached to a
single ith generation duct.
In Fig. 9, the ratio ci/Ctotal as a function of the acinar generation number i is illustrated for 0.5-µm particles. A comparison is made between the classical rigid-walled and the rhythmically moving-wall alveolar models. According to the former model, the mean alveolar deposition is practically zero in the first three acinar generations (i = 16, 17, 18), peaks at the next two generations (i = 19, 20), and finally reduces to zero deep in the acinus (i = 22, 23). In the more realistic rhythmically expanding and contracting model, on the other hand, the local alveolar deposition starts high at the entrance to the acinus (i = 16), monotonically decreases in the first few generations (i = 17, 18, 19), and becomes practically zero in the last four acinar generations (i = 20, 21, 22, 23). These results suggest that 1) predictions of deposition patterns along the acinar tree would unrealistically be compromised if one ignores the motion of alveolar walls, and 2) the present model with alveolar wall motion predicts that deposition of 0.5-µm particles occurs in the first half of acinar pathway, with a preferentially higher deposition at the entrance of the acinus [see also Saldiva et al. (22), which agrees with our second conclusion].
|
It is interesting to point out that the present model predicts almost no
deposition in the distal half of the acinus generation (i =
20–23), even though the distal local alveolar deposition probabilities
are much higher than those in the proximal half of the acinus. This is due to
the fact that, in the proximal region, the local alveolar deposition process,
despite the associated, relatively low deposition probabilities (0.1 <
i < 0.2, i = 16–19), consistently
and significantly reduces the number of particles available for deposition in
the distal region of the acinar tree. In the early 1980s, Weibel et al.
(33) suggested that a similar
phenomenon, called "screening," is likely to occur in the case of
O2 transport and recently demonstrated that "screening"
is significantly influenced by acinar geometry and O2 diffusion in
the gas phase relative to its permeation through the alveolar surfaces
(23). Based on the present
study, we also conclude that "screening" may occur in the case of
aerosol transport in the acinus, and it is largely influenced by alveolar wall
characteristics (shape and motion) and the associated alveolar flow. Note that
it is important to incorporate the concept of "screening and
unscreening" when one interprets the local deposition distribution along
the airway pathway from experimental measurements (e.g., Ref.
16,
17).
At the level of the tracheobronchial tree. As we described, much of our current knowledge of local deposition in the lung is based on mathematical model predictions. To the best of our knowledge, however, none of the currently available mathematical models (except ours) incorporates the effects of expansion and contraction of the lungs, one of the most crucial determinants of particle deposition in the acinus, on local deposition. Therefore, even though the total deposition predicted by the "classical" models, calculated as the sum of local deposition values predicted for each airway compartment, corresponds well to the total deposition measured experimentally, the distribution of particle deposition along the tracheobronchial tree predicted by these models could be far from the one occurring in reality. The fact that the inclusion of alveolar wall motion in the present model results in much higher local acinar deposition than that by the classical model suggests that, in reality, more (less) particles are deposited in the proximal alveolar ducts (the conducting airways) than predicted previously.
Summary
Gravitational deposition processes and deposition sites in a rhythmically expanding and contracting alveoli differ from conventional predictions made by classical models, which treat the acinar duct as a straight pipe with rigid walls, and thus approximate gravitational deposition of particles immersed in unidirectional Poiseuille-like simple flows. We conclude that gravitational deposition in the pulmonary acinus is much more complex than assumed previously, and it is essential to incorporate the cyclic motion of alveolar walls to predict gravitational trajectories and deposition of fine particles.
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APPENDIX A |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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and
in the radial and axial
directions, 
and z, respectively, that depend on the two
cylindrical coordinates (,z). Thus, in a Cartesian coordinate
system, vH possesses the following
form6
 | (A1) |
is the polar angle. The 3D solution for vP was
numerically obtained (21)
utilizing the boundary element method. The numerical procedure was greatly
simplified because the velocity's analytic dependence on the polar angle
had been known a priori. Consequently, the velocity field depends on three
independent unknown functions of the coordinates (,z) to be
determined numerically. The solution possesses the general following form in
Cartesian coordinates7
 | (A2) |
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APPENDIX B |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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Assuming that an alveolus is represented by a hemisphere of radius
R0 (= 100 µm), the amplitude of a minute ventilatory
flow, QA, entering and exiting each alveolus may be expresses as
 | (B1) |
is the tidal maximum strain of alveolar radius. Ductal Flow
Using Weibel's counting
system8 of airway
generation, a flow, QDn, through an acinar duct
of the generation n may be expressed as follows. We consider that the
airflow entering the lung, Qao, is represented by a sum of flows,
which enters the expanding and contracting acini (i.e., alveoli and associated
acinar ducts). Because alveolar spaces are typically about three times larger
than the volume of ductal region
(9), Qao may be
expressed as
 | (B2) |
),
and the distribution of Nn is given as
N16
0.2 x106 (a rough estimate),
N17 = 0.6 x 106, N18
= 2.0 x 106, N19 = 6.0 x
106, N20 = 21.0 x 106,
N21 = 41.5 x 106, N22
= 84.0 x 106, and N23 = 143.0 x
106. Now, let us consider a flow through each acinar duct,
QDn, at generation n. It can be
approximated as a difference between the total flow entering the lung,
Qao, and the flows expanding the alveoli and the associated ducts
proximal to generation n
 | (B3) |
Relative Ratio of QD to QA
As we discussed in the text, acinar flow may be better characterized by a
dimensionless parameter, QDn/QA,
instead of the commonly used Reynolds number. Using Eqs. B1 and
B3, this ratio can be expressed
 | (B4) |
 | (B5) |
depends only on the geometrical configuration of
generation n down the acinar tree and is independent of lung
kinematics such as VT or . |
APPENDIX C |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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,
, and
(Eq. A2, see
APPENDIX A) can be tabulated at given discrete nodal points in the
,z plane. The vector field vH was analytically
obtained and can, in principle, be calculated for every time step at any point
in the ,z plane. However, it turns out to be more expedient to
compute the velocity components
and
(see Eq. A1 in
APPENDIX A) at the same discrete nodal points and employ a
two-dimensional linear interpolation method to obtain the velocity values at
any place in the field. The same interpolation method is required to obtain
vP everywhere in the field with a similar error in velocity
estimates. It is also advantageous to arrange the computed velocity data for
,
,
,
, and
in five matrices where the two
indexes of a matrix element correspond to a location in the ,z
plane and the element's value corresponds to the velocity component value at
that location. An additional layer of fictitious nodal points external to the
alveolus was added to enable interpolation near the alveolus wall. For the
components of vP, zeroes were added (in accordance with the
no-slip condition applied in this case), whereas, for the vH
components, the projection of a unity vector perpendicular to the wall was
added (according to the expansion boundary condition that applies at the wall
in this case). Matrix sizes of 100 x 100 were found to introduce large
errors in particle trajectories, and only when we reached 1,000 x 1,000
sized matrices was the error reduced to an acceptable value (see
Fig. 10A).
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The differential equation (Eq. 9) was solved by using Matlab's
ODE45 solver (2) with a
relative tolerance equal to 10-8 and absolute tolerance of
10-16 that resulted in an acceptable error value (see
Fig. 10B). Normally,
errors can be computed by comparing results that are obtained by progressively
refining mesh sizes, time steps, etc. In our case, however, we can also
exploit the kinematic reversibility property of the Stokes equations. For
example, if = 0 and H = 0, trajectories must reverse during
inhalation and exhalation, and the solution must be 2
periodic. Thus a
test case has been defined with initial conditions
(p0,
p0,
p0) = (0.2, -0.1, -0.8),
= 200, and Dp = 2 µm, and the mean and maximum
errors eav and emax, respectively,
that appear in Fig. 10 were
calculated by the respective L1 and
L
norms as follows
 |
i(tj)
=
i(tj)
-
1(tj),
and
i(tj)
is the particle normalized location that it possessed during the i
breathing cycle at time tj (modulus 2, the
dimensionless breathing period). In other words, eav
illustrates the mean deviation between particle trajectory within breathing
period i with that obtained after the first breathing period, during
m periods sampled at n fixed times during the breathing
periods. Figure 10 was
obtained for n = 50 and m = 20. Notice that, generally,
L1 norms yield larger values than the more commonly used
L2 Euclid norms and, as such, define a more stringent
accuracy test.
The solution process includes the following significant steps. The current
particle position in Cartesian coordinates is transformed into cylindrical
coordinates (,,z). The (,z) coordinates of
the particle center are normalized with respect to alveolus time-dependent
dimensionless radius 
. The functions
,
,
,
, and
at the normalized
(,z) plane are calculated utilizing the data matrices and a
linear interpolation procedure. The velocity components are calculated by
using Eqs. A1, A2, and 3. Matlab's ODE45 procedure is
applied on Eq. 9 to obtain the new real dimensionless position of the
particle center. The new position is utilized to examine whether the particle
has or has not penetrated the alveolus or the duct walls. If penetration has
occurred, a deposition site is registered, and the program terminates.
Otherwise, the program continues running, returning to the first step
described in the above. The program also terminates whenever the time exceeds
a prescribed number of breathing periods or whenever the particle leaves the
ductal volume adjacent to the alveolus and moves further up or down the
alveolar tree.
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DISCLOSURES |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1 The present paper focuses on hemispherical cavities, because flow fields
for general spherical caps are yet unsolved. Haber et al.
(8) provided a general solution
for various shapes of expanding spherical caps. Alas, Pozrikidis
(21) obtained a numerical
solution for shear flows over semispherical cavities only. Thus mathematical
investigation of the effect of various alveolar openings must be postponed
until the flow fields induced by shear flow over a nonhemispherical cavity are
addressed. 
2 It is interesting to note that many physical properties are lumped into a
single dimensionless parameter H. Thus, for instance, the effect of
lowering particle density can be achieved by an equivalent increase of
breathing rate.
3 In previous papers (8,
27), was expressed by
the ratio QA/QD.
4 The product of and alveolar radius R0 is the
amplitude of alveolar wall expansion.
5 Deposition occurs near the alveolar opening on both proximal and distal
sides of alveolar walls, vertically higher than the original position of
particle cloud [Fig.
8B; compare the vertical positions of blue dots
(deposition sites) to the black square (original position)].
6 Haber et al. (8) express the
solution in toroidal coordinates. However, a simple one-to-one correlation
exists [see Eq. 5 in Haber et al.
(8)] between the toroidal and
the cylindrical coordinates and the associated velocity components.
7 Pozrikidis (21) provides
explicit expressions for the velocity components in a cylindrical coordinate
system. The transformation to a Cartesian coordinate system is
straightforward.
8 Based on the Weibel model
(31), the acinus starts from
generation 16 at respiratory bronchiole and ends at generation
23 (alveolar sac).
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REFERENCES |
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TOPABSTRACTMATERIALS AND METHODSRESULTSAPPENDIX AAPPENDIX BAPPENDIX CDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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  C. Darquenne, L. Harrington, and G.K. Prisk Alveolar duct expansion greatly enhances aerosol deposition: a three-dimensional computational fluid dynamics study Phil Trans R Soc A, June 13, 2009; 367(1896): 2333 - 2346. [Abstract] [Full Text] [PDF]
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