Aerosol Science and Technology 37: 510–527 (2003)
c 2003 American Association for Aerosol Research
°
Published by Taylor and Francis
0278-6826/03/$12.00 + .00
DOI: 10.1080/02786820390126376
Application of Cloud Dynamics to Dosimetry of Cigarette
Smoke Particles in the Lungs
David M. Broday1 and Risa Robinson2
1
2
Faculty of Agricultural Engineering, Technion, Israel Institute of Technology, Haifa, Israel
Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, New York, USA
Clinical data suggest a relationship between in vivo deposition
patterns of cigarette smoke particles and the occurrence of tumors
in the lung. Traditional dosimetry models fail to predict the preferential proximal deposition of cigarette smoke in the human airways, which resembles deposition of aerosol with a larger mass
median aerodynamic diameter (MMAD) than that representative
of cigarette smoke. Previous work has shown that accounting for
the so-called cloud effect leads to enhanced proximal deposition
and to better agreement with clinical and experimental data. This
work presents an improved model of transport and deposition of
cigarette smoke in the airways of smokers, accounting for possible particle-particle interactions (cloud effect) and their effect on
the mobility of individual particles and on the deposition profile.
Brinkman’s effective medium approach is used for modeling the
flow through and around the cloud, with the cloud’s permeability
changing according to the cloud’s solid volume fraction.
Although the weakest of all interparticle hydrodynamic interactions is considered, it significantly alters the deposition pattern
along the respiratory tract, both alone and simultaneously with
other synergistic processes (coagulation, hygroscopic growth) that
dynamically modify the particle size distribution. Model results
compare favorably with clinical data available on CSP deposition
in the lungs and indicate that a combination of cloud behavior,
hygroscopic growth, and coagulation may explain the preferential proximal deposition of smoke particles in the tracheobronchial
region.
INTRODUCTION
Tobacco smoke has long been recognized as a major cause of
death and disease, responsible for an estimated 434,000 deaths
per year in the US (EPA 1993). It is known as a Group A carcinogen in humans, and is associated with a major risk factor for
Received 26 March 2001; accepted 8 January 2003.
This research was supported in part by the fund for the promotion
of research at the Technion and by Philip Morris Incorporated.
Address correspondence to David M. Broday, Faculty of Agricultural Engineering, Technion, Israel Institute of Technology, Haifa,
32000, Israel. E-mail: [email protected]
510
heart disease; eye, nose, and throat irritation; and various respiratory chronic disease and pulmonary disorders. In particular,
tobacco smoke is known to cause lung cancer (Hoffmann and
Hoffmann 1995) and to increase the severity and frequency of
asthma episodes in children exposed to environmental tobacco
smoke. Health symptoms associated with tobacco smoke occur
because of active smoking and as a result of secondary exposure, usually termed secondhand smoking (SHS) or exposure to
environmental tobacco smoke (ETS). The latter exposure route
involves contact with a mixture of smoke given off by the burning end of a cigarette and smoke exhaled by smokers.
Cancer etiology relating to inhaled cigarette smoke requires
understanding of the motion and deposition patterns of cigarette
smoke particles (CSP) in the human respiratory tract, as well as
its postdeposition fate. Exposure to cigarette smoke is known to
decrease the ability of the lungs to clear inhaled matter (Churg
et al. 1992, 1998; Keeling et al. 1993; Finch et al. 1998). Although chronic CSP clearance failure may be an important cause
of tumor initiation, leading to induction of lung cancer, this work
does not address this issue.
Previous mechanistic deposition models were generally unable to accurately predict the deposition of CSP in the human
respiratory tract. In this work we present a new description of
the dynamics of cigarette smoke in the airways of smokers, implemented within an improved deposition model. Predictions
of the model agree favorably with clinical and measured CSP
deposition data.
Clinical Studies
Clinical studies (see reviews by Ellett and Nelson 1985;
Martonen et al. 1987; Yang et al. 1989) established that bronchogenic carcinomas are preferentially spread in proximal airways in the lungs of smokers, whereas adenocarcinomas arise
more often in the periphery of the lung. Almost all cancers
proximal of the trachea are squamous cell carcinomas. These
findings indicate that histologic-type tumors are nonuniformly
distributed in the lungs. Data compiled in these and other studies
511
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
suggest that cancer (with no type distinction) is distributed
with the following approximate frequency: ∼13% in the orallaryngeal cavity, <1% in the trachea, 6–28% in the main
bronchi, 49–68% in the lobar bronchi, and 10–13% in the periphery including the segmental bronchi. The nature of the large
variability in bronchial tumor statistics is unknown to the authors, but may represent, in part, different detection methods. In
comparison, in vivo measurements of cigarette smoke deposition
in living subjects range from 22 to 89% (see review by Robinson
and Yu 2001), depending on the cigarette brand and reflecting intersubject physiological and morphological variability. Based on
analyses of clearance rates using radioactive tracers, estimates
of regional deposition of CSP in living subjects include 11–23%
deposition in the extrathoracic airways, 46–63% deposition in
the tracheobronchial (TB) airways, and 26–35% deposition in
the pulmonary (P) region (Pritchard and Black 1984; Hicks et al.
1986). In general, these findings suggest a relationship between
CSP deposition patterns and the occurrence of tumors in specific
regions of the respiratory tract (RT) (Robinson and Yu 2001).
Deposition Data
The clinical (indirect) evidence is supported by measurements of regional and local CSP deposition in replica casts of the
first few generations of the upper human airways (Ermala and
Holsti 1955; Martonen et al. 1987). Overall, CSP were found to
collect preferentially in specific tracheobronchial regions of the
human airways. In particular, hot spots of CSP deposition were
found at the carinal ridges, within bifurcation zones, and along
the posterior surfaces of downstream tubular airway segments
(Churg and Vedal 1996). These findings are consistent with local secondary flow patterns (Martonen 1992; Hofmann 1996).
However, such a deposition profile resembles deposition of an
aerosol with a much larger mass median aerodynamic diameter (MMAD) than that representative of cigarette smoke. This
suggests that cigarette smoke particles deposit as if they have a
much larger effective size.
Deposition Models
A puff of cigarette smoke consists of approximately 70%
ambient air, 17% gaseous species originated from combustion
and thermal degradation, 8% particulate matter (PM), and 5%
miscellaneous vapor components (Jenkins et al. 1979). The particulate matter in fresh CSP has a polydisperse size distribution,
with properties as specified in Table 1 (Robinson 1998). In particular, the miniscule solid-to-gas volume fraction (Martonen
1992) indicates that the average interparticle spacing is very
large relative to the particle size.
Since cigarette smoke is mostly air, previous studies usually
neglected the interparticle forces, calculating particle motion
and deposition based on a single particle mechanics. This approach results in about 22–53% total deposition, primarily in
the pulmonary region (Yu and Diu 1983; Task Group on Lung
Dynamics 1966; Muller et al. 1990), which is inconsistent with
Table 1
Typical properties of fresh cigarette smoke (compiled from
Robinson 1998)
Number concentration
Density of the solid fraction
Solid-to-gas volume fraction
Count mean diameter (CMD)
Mass mean diameter (MMD)
Mass mean aerodynamic
diameter (MMAD)
Geometric standard deviation
(GSD)
Coagulation rate
2–7 × 109 cm−3
0.98–2.47 g cm−3
<10−4
0.2–0.35 µm
0.25–0.35 µm
0.35–0.55 µm
1.2–1.64
4.8–23.8 × 10−10 cm3 /s
the clinical and experimental studies cited above. These studies
considered steady breathing with normal tidal volumes and stable monodisperse particles (i.e., PM of constant size). Yet CSP
is known to coagulate (Keith 1982; Robinson and Yu 1998) and
to be slightly hygroscopic (McCusker et al. 1981; Kousaka et al.
1982; Hicks et al. 1986; Muller et al. 1990; Robinson and Yu
1999). In addition, smokers have a breathing pattern different
from normal breathing, with about twice the tidal volume (Hinds
et al. 1983). Yet accounting for these aspects in deposition models, alone and in combination, could not explain the enhanced
deposition of CSP in the upper bronchial airways (Robinson and
Yu 2001).
Cloud Effect
The predicted preferential pulmonary CSP deposition is attributed to the smallness of individual particles, the deposition
of which is governed by diffusion. Indeed, theoretical considerations suggest that due to their small size, CSP penetrate into
distal regions of the respiratory tract and deposit mostly in the
pulmonary airways. A plausible explanation for the discrepancy
between theoretical deposition results and clinical and laboratory observations may be attributed to particle-particle interactions, sometimes referred to as the cloud or colligative effect
(Martonen 1992; Martonen and Musante 2000; Robinson and
Yu 2001). Due to screening, particle-particle interactions have
the effect of decreasing the drag on some particles, thereby increasing their mobility. This modifies the particle relaxation time
and, hence, the parameters that govern particle deposition, and
may lead to enhanced deposition in the upper tracheobronchial
(TB) region. In fact, cloud behavior has been observed (Slack
1963a,b), and it was suggested that CSP entering the respiratory
tract may indeed have characteristics consistent with cloud behavior (Ingebrethsen 1989; Phalen et al. 1994). Possible mechanisms by which clouds of cigarette smoke can form in the human
respiratory tract were discussed by Martonen (1992). Interactions among cloud members make cloud mechanics far more
complex than that of individual particles.
In general there are two types of swarm interactions: hydrodynamic and thermodynamic. While hydrodynamic interactions
512
D. M. BRODAY AND R. ROBINSON
require intervening of fluid between the particles, thermodynamic interactions occur as a result of contact between particles or due to the presence of a conservative potential field.
Thermodynamic interactions such as volume exclusion, elastoplastic collisions, and electrostatic forces are characterized by
strong albeit short-range forces. Hydrodynamic interactions, on
the other hand, are relatively weak (with the exception of lubrication forces) but can pertain at a distance. This work considers
the weakest among the interparticle hydrodynamic interactions,
which nevertheless will be shown to significantly alter the deposition pattern along the respiratory tract.
CLOUD MODEL
Cloud Definition
An aerosol cloud is defined as a region of relatively high
aerosol concentration in a much larger region of clean air (Hinds
1999). More precisely, it is a swarm of fine particles distributed
throughout an identifiable connected volume (albeit with fuzzy
boundaries due to a negligible surface tension) within a much
larger expanse of particle-free suspending fluid (air) (Schaflinger
and Machu 1999; Machu et al. 2001). Thus, the aerosol cloud
represents an effective continuum of excess mass upon which
gravity is pulling. In dilute clouds (and droplet suspensions) the
pseudo fluid is Newtonian, with a viscosity essentially the same
as that of the pure fluid (Martonen 1992).
Consider a cloud of n identical spherical solid particles, each
with a diameter 2a and density ρ p . For simplicity, the cloud is
assumed to be a spherical enclosure of diameter 2b. The number
concentration, c, of the particles in the cloud is
c=
n
,
4/3πb3
[1]
and the volume fraction, φ, of the solid phase is
φ = 4/3πa 3 c = n
µ ¶3
a
= n R −3 ,
b
[2]
R being the size ratio of the cloud and the monodisperse individual particles constituting it. The density of the cloud, ρc , can
be expressed as
ρc = φρ p + (1 − φ)ρ f ,
[3]
where ρ p is the density of the solid phase (0.98–2.47 times the
density of water at standard conditions) and ρ f is the density of
the gas phase (∼1.31 mg cm−3 ) (Martonen 1992).
The settling velocity (the equilibrium velocity attained by a
freely moving particle settling slowly under gravity in a viscous
fluid) of infinitely diluted fine solid spherical particles in an
unbounded viscous domain, v p∞ , is
v p∞ =
2a 2 (ρ p − ρ f )g
Cc (a).
9µ
[4]
Here g is the gravity acceleration, µ is the medium (air) viscosity,
and Cc is Cunningham’s slip correction factor. The latter corrects
the drag on fine particles for which the continuum regime does
not apply, and which may experience slip velocity. For spheres,
the Cunningham’s correction factor is (Hinds 1999)
Cc (a) = 1 + Kn(1.17 + 0.525 exp{−0.78/Kn}),
0.1 < Kn < 10,
[5]
where Kn = λ/a is the Knudsen number, measuring the ratio of
the mean free path of air at given conditions, λ, to the particle
radius. Equation (4) is used to normalize the settling velocities
of particle clouds in the models discussed hereinafter.
The derivation of Equation (4) is based on the assumption
that the relative particle-air velocity can be described as a creeping flow. Namely, that Re p = v p∞ a/ν ¿ 1, where Re p is the
particle Reynolds number and ν is the kinematic viscosity of air.
A finite swarm of particles may take a meaningful macroscopic
identity as a cloud. To be able to describe hydrodynamic interactions within clouds using expressions pertinent to creeping flows
requires a more stringent condition, which can be expressed as
2(ρc − ρ f )gb3
= φ R 3 Re p /Cc (a)
9µν
= nRe p /Cc (a) < 1,
Rec = vc b/ν =
[6]
where Rec is the cloud Reynolds number. For CSP, Re p is of
the order of O(10−8 –10−6 ). If indeed cloud motion occurs in
the respiratory tract, the cloud size cannot exceed that of the
glottis aperture, d ∼ O(1) mm. For the maximum CSP concentration reported, c = 3×109 cm−3 , the number of particles in the
cloud, n, can not exceed O(106 ), hence Rec ≤ O(1). Thus, the
disturbance to the local flow induced by the motion of a swarm
of fine particles confined to a finite volume can be described
in most cases by expressions relevant to low Reynolds number
hydrodynamics.
If a swarm of particles indeed moves as an effective cloud, the
particles constituting it will each move with a common apparent
velocity. Since the velocity of cloud members is larger than the
velocity of any individual particle had it been moving as an
isolated particle, it is important to know whether the motion of
the particles is affected by their own inertia. This issue will be
addressed in the Deposition Model section. For now, the inertia
of individual cloud particles is measured by means of the Stokes
number, St,
St ≡
m p Mp
,
dgen /U
[7]
where m p is the particle’s mass, M p is its mobility, dgen is the
airway diameter, and U is the average air velocity in the airway. The denominator in Equation (7) represents a characteristic time the particle stays within an airway, used to normalize
the particle’s relaxation time (the numerator). For isolated CSP
513
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
particles, St ¿ 1. When cloud motion prevails, the particle mobility is altered and, hence, St may be of the order of unity. It will
be shown (see Equation (28)) that for a given particle size the
larger the cloud the larger the particle inertia, resulting from the
increased relative (settling) velocity between the particle and
the air. Namely, particle inertia is affected by the presence of
nearby like particles via the alteration they induce to its (self)
mobility.
Cloud Settling
Solid Sphere Model. Previous studies addressing the hydrodynamic colligative effect considered the particle cloud to be a
solid sphere with air flowing around it (cf. Hinds 1999). Thus,
the settling velocity was given in terms of the cloud’s diameter
and density. After accounting for buoyancy, the settling velocity
relative to Equation (4) is
vsolid sphere =
φ R2
.
Cc (a)
[8]
Note that since the size of the cloud is large, Cc (b) ∼
= 1 and is
neglected in Equation (8). In the unlikely event that Rec > 1,
the motion of the cloud will be non-Stokesian. The cloud’s
settling velocity, vc , in such cases (normalized with respect to
Equation (4)) is approximately (Frielander 1977; Hinds 1999)
s
φR n
¢
vc = ¡
,
[9]
1 + 0.25Re2/3
Cc (a)
c
since the drag coefficient is C D = 24/Rec (1 + 0.25Re2/3
c ). This
expression is considered valid for Rec < 1000, with a maximum
of 7% error (Frielander 1977).
Fluid Sphere Model. Alternatively, the cloud can be considered a spherical fluid domain of distinct density and viscosity,
unmixed with the surrounding air (Martonen 1992). In contrast
to the solid sphere model, this model involves nonvanishing
shear stresses on the cloud boundary, which induces internal circulation. The density of the fluid region is given by Equation (3);
with respect to the bulk viscosity µ, its viscosity µc is modified
to first order according to Einstein’s formula, µc = (1+2.5φ)µ,
and the force acting on the fluid sphere follows the Hadamard–
Rybcznski drag (Happel and Brenner 1973)
Ffluid sphere = 6πµbU
3µc + 2µ
.
3µc + 3µ
[10]
In cases when φ ¿ 1, µc ∼
= µ and the factor (i.e., the reduced
drag) in Equation (10) equals 5/6. It turns out, therefore, that due
to less dissipation experienced when a fluid particle rather than
a solid particle travels within a viscous fluid, the normalized
settling velocity of the cloud is 20% larger than that given in
Equation (8),
vfluid sphere =
6φ R 2
.
5Cc (a)
[11]
The fluid sphere model predicts that small clouds of diameter
ranging from 5 to 20 microns (about 10–100 times larger than
the CMD of cigarette smoke particles) deposit with a very high
deposition efficiency, thus contributing to the ∼99% deposition
efficiency predicted in the tracheobronchial airways (Martonen
and Musante 2000). This prediction is inconsistent with experimental evidence (Pritchard and Black 1984; Hicks et al. 1986),
although it agrees with the integrated spatial distribution of malignant tumors in the upper bronchial airways (Martonen 1992).
Both the solid sphere and the fluid sphere cloud models use
a simplified approach to describe the colligative effect. Jointly,
they consider homogeneous impermeable media and permit only
bimodal motion. Namely, individual particles can move either
as isolated particles in an ideal viscous flow, being unaffected by
the presence of nearby particles, or collectively as a cloud. The
cloud is modeled as an impermeable domain with the air passing
around it, and its size is the dominant physical parameter.
Clearly there are cases where the surrounding air does flow
through the cloud and mixes with it, dispersing cloud particles, reducing the cloud’s initial concentration, and deforming
its shape. The solid sphere and the fluid sphere models do not
account for these processes, since they consider interactions between the cloud and the particle-free fluid surrounding it only
via the action of normal and shear stresses on its rigid boundary.
Furthermore, they do not account for any interactions among the
particles within the cloud.
Reflection Method. For weakly interacting, freely moving
particles in a dilute cloud, the settling velocity of individual
particles can be calculated by the reflections method. Summation
of all the pairwise hydrodynamic interactions for a given particle
array yields (Happel and Brenner 1973)
vreflection =
1
.
1 + kφ 1/3
[12]
The constant k ranges between 1.3 and 1.9, depending on the
randomness (or orderliness) of the structure of the particle array.
For example, for a cubic lattice arrangement k ∼ 1.6.
To describe the dynamics of many-particle systems one needs
to consider hydrodynamic and thermodynamic interactions, as
described above. Unlike thermodynamic interactions, pairwise
hydrodynamic interactions between any two particles are modified by the presence of a third nearby particle. Mathematically,
the reduced hydrodynamic resistance (per particle) can be described by a modified friction tensor, which replaces the Stokes’
friction tensor for an isolated sphere. The modified drag on any
particle can be thought of as an infinite sum of reflections of
the disturbances induced by all the particles in the cloud, which
alter the local flow field around any particle.
Swarm Model. The classical approach to describe the motion of a particle cloud within a viscous fluid is by the cell model
(Happel and Brenner 1973), also known as the swarm model
(Neal et al. 1973). In this model the cloud is represented by a
single typical particle, surrounded by air that fills a hypothetical
514
D. M. BRODAY AND R. ROBINSON
spherical fluid domain of diameter 2b∗ . In terms of the actual
cloud size, 2b, and the number of particles per cloud, n, b∗ can
be expressed as b∗ = bn −1/3 . Using Happel’s well-known formula for the drag on a solid sphere contained in a shear-free cell
and implementing Equation (2), the setting velocity is
vswarm =
6 − 9φ 1/3 + 9φ 5/3 − 6φ 2
.
6 + 4φ 5/3
[13]
Note the canceling out of the Cunningham correction factor in
Equations (12) and (13), because these cloud models are based
on the motion of a single representative particle. Equation (13)
describes the motion of each particle within the cloud, accounting for the crowding effect only in terms of the effect of the finite
interparticle spacing on the particle mobility. It is intrinsically
assumed that the cloud is monodisperse and that all particles
share a similar motion. Therefore, there is no relative motion
among cloud particles, i.e., the cloud moves like a rigid, albeit
porous, body.
One of the main critiques of the swarm model refers to the
fact that boundary conditions are fulfilled on a spherical enclosure and, therefore, an aggregate of such cells does not fill the
space but rather leaves “holes.” Another problem relates to the
boundary conditions imposed on each cell’s boundary, causing
it to be detached from neighboring cells.
Stokesian Dynamics Simulations. The direct way to account for hydrodynamic interactions among cloud particles is by
utilizing Stokesian dynamics (Brady and Bosis 1998). Stokesian
dynamics tracks the motion of numerous particles in viscous liquid when thermal motions are overdamped and, hence, random
forces are neglected. Recently, Machu et al. (2001) used an even
cruder representation of the hydrodynamic interactions among
the particles. In their model each particle settles essentially
in isolation, with the Stokes settling velocity in Equation (4)
relative to the local velocity. The velocity field is obtained by
superimposing Stokeslet disturbance fields induced by all the
particles, each being regarded as itself settling in isolation relative to its own local flow. At this level of approximation the
particles appear like point forces and their size does not enter
the calculation directly. The justification for this is that under the
influence of a conservative force field, the collective far-field
effects rather than the nearest-neighbor interactions dominate
the motion of individual particles in a swarm (Brenner 1999).
For a sufficiently low volume fraction (0.05 < φ < 0.07) of
non-neutrally buoyant particles in a quiescent fluid, the agreement between experimental data and theoretical predictions is
remarkable (Machu et al. 2001). These findings support viewing the swarm as a continuum pseudofluid, like in the Brinkman
model.
Brinkman’s Model. For a solid volume fraction less than
∼30%, it was shown (Durlofsky and Brady 1989; Sangani and
Mo 1994) that the drag on and the diffusivity of a test particle, obtained by Stokesian dynamics, are in excellent agreement
with those obtained by the much more simplified Brinkman’s
effective medium approximation. Since the volume fraction in
cigarette smoke is orders of magnitude lower than 30%, it seems
appropriate to apply Brinkman’s model to describe the cloud
colligative effect. Properties derived by applying Brinkman’s
model to porous media with volume fractions not exceeding
30%, such as the medium permeability, its hydrodynamic resistance, and its heat conductivity, are all in excellent agreement with those derived by the rigorous numerical calculations
involving averaging over numerous initial stochastic configurations (Sangani and Yao 1988a,b). Furthermore, predictions
obtained using Brinkman’s effective medium approach agree
favorably with experimental data on homogeneous porous materials (Jackson and James 1986). Hence, the motion of CSP
clouds, where individual particles are supposed to weakly interact with each other, will be described here in terms of a sparse
porous medium using Brinkman’s equation.
Brinkman’s effective medium accounts for hydrodynamic interactions attributed to the presence of a less mobile (quasistationary) particle fraction within the cloud. In essence, since
CSP clouds consist of a distribution of particles with respect
to size, and since the settling velocity is highly size dependent,
at a “local” scale some particles can be regarded as relatively
immobile compared to others. In such “fixed” clouds (Hinds
1999) the less mobile particles move uniformly without a relative motion between each other and, hence, the cloud structure
remains constant. The presence of a (quasi-) immobile fraction
significantly slows the local-scale (relative) motion of particles
from the mobile fraction (Dodd et al. 1995; Broday 2000). This
so-called Brinkman’s screening dominates the hydrodynamic
resistance acting on the mobile fraction, due to the smallness of
its characteristic length scale.
Brinkman’s equation is a modification of Stokes’ equation,
including an additional term that accounts for the hydrodynamic
resistance induced by the quasi-immobile fraction,
µ
∇ p̃ = − ũ + µ∇ 2 ũ,
k
[14]
where p is pressure, u is the particle velocity, and k is the
medium permeability. Two relevant problems can be solved
using Equation (14). The first problem involves the motion of a
solid spherical particle within a finite random array of like particles that consist a porous structure immersed in a viscous fluid.
In fact, this configuration is very close to the original problem,
presented by Brinkman (1947) and later reworked by Howells
(1974), describing the motion of a solid particle in an unbounded
swarm of similar particles. The infiniteness of the array of singularities is of negligible significance for particles far away from
the (fuzzy) boundary of the cloud, since the size ratio of the
cloud to individual particles is very large, R À 1, and the effect
of the boundary decays very fast (James and Davis 2001). Under such conditions, the normalized settling velocity, v p , of such
particles is
v p = F p−1 = (1 + α + α 2 /9)−1 ,
[15]
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
where F p is the excess drag on the particle over the Stokesian
drag and α ≡ a 2 /k is the reciprocal of the normalized permeability, also called (the square of) Brinkman’s screening length.
It can be shown (cf. Broday 2000) that the relation between α
and φ is
φ=
2α 2
.
9(1 + α + α 2 /3)
[16]
The other case relevant to this work is the motion of the cloud
as a whole in a relatively particle-free viscous domain. Consider
a spherical cloud composed of monodisperse fine particles and
assume that the cloud does not deform and that it moves in an unbounded viscous domain (the hydrodynamic resistance induced
by proximate walls will be discussed in the next section). In our
terminology, the drag Fc on a cloud moving in an unbounded
viscous domain relative to the Stokesian drag on one of the solid
particles consisting it is (Brinkman 1947; Neal et al. 1973; Davis
and Stone 1993)
Fc = R
£
¤
R)
2(α R)2 1 − tanh(α
αR
£
¤.
R)
2(α R)2 + 3 1 − tanh(α
αR
[17]
The normalized settling velocity of the cloud by this model is
vc =
φ R3
.
Fc Cc (a)
515
unity. (4) For large φs, the settling velocity of a “Brinkman
cloud” asymptotes the settling velocity calculated by the fluid
sphere model, the latter accounting for an inner circulation;
(5) Brinkman’s effective medium approach provides a simple
yet powerful model to describe cloud settling at arbitrary solid
fraction (up to φ ∼ 0.3). In practice two regimes are identified,
for φ ≤ φcr particles tend to settle individually, being affected
slightly by the presence of neighboring particles (the crowding
effect). For φ > φcr particles tend to settle as a cloud with a permeability that depends on φ. The transition between the single
solid particle behavior and the rigid impermeable cloud regime
is continuous, and its smoothness depends on the size of the
particles constituting the cloud (via Cc (a)).
Judging by the surprisingly good asymptotic behavior of the
results of Brinkman’s cloud model, the assumption about the relative stillness of some cloud particles seems to introduce no difficulty. Yet, it is noteworthy that the internal circulation observed
within regions of different viscosities (droplets, bubbles) and/or
permeabilities (suspension droplets, clouds) cannot be described
by Brinkman’s cloud model. Hence, while dilute clouds experience air flowing through them, curved streamlines that bypass
the dense region and a diminishing flow through it are evident for
less-permeable areas (Broday 2000). The results in Figures 1a
and 1b and the data in Table 1 suggest that for CSP, α is of the
order of 10−2 .
[18]
An expression similar to Equation (18) was obtained by Arachi
(1999).
Model Comparison
Figures 1a and 1b depict the cloud’s settling velocity calculated by the different models presented above, with Cunningham’s slip factor calculated for a = 0.15 µm (half the CMD of
CSP, see Table 1). The difference in the settling velocities predicted for small φs by the two Brinkman models, Equations (15)
and (18), corresponds exactly to the reciprocal of Cc (a). This
term appears in Equation (18) as a result of normalization by
the settling velocity of a typical particle, which is much smaller
than the cloud (R À 1). On the other hand, Cc (a) is missing
from Equation (15), since the latter deals with a single particle identical in size to the particle for which Equation (4) was
derived.
From Figures 1a and 1b it is evident that: (1) the fluid sphere
model is not applicable for very dilute clouds, since it does not
match the settling of individual particles consisting the cloud;
(2) the swarm model underpredicts the settling velocity of particles in dilute clouds (see also Broday 2000); (3) for low φs, the
cloud settling velocity in Equation (18) approaches the settling
velocity of individual CSP in Brinkman media (Equation (15)),
which itself is asymptotic to the Stokes settling velocity.
However, due to the normalization, at low φs vc appears in
Figure 1 within the Cunningham’s slip correction factor from
MOTION OF CSP CLOUDS IN THE HUMAN AIRWAYS
During their motion along the respiratory tract, CSP clouds
may undergo macroscopic deformation. Simultaneously, the fine
particles constituting the clouds may experience coagulation,
hygroscopic growth, and deposition on airway surfaces. While
hygroscopic growth results in an increase in the cloud’s solid
volume fraction, coagulation and deposition processes tend to
decrease the number concentration of persistent CSP, weakening
the cloud effect in subsequent airways. Furthermore, as will be
discussed hereinafter, recent experimental findings suggest that
the dynamics of suspension drops (drops containing a swarm of
fine particles) is identical to that of liquid drops (a connected
volume of homogenous liquid, distinguished by some thermophysical properties from an expanse of outer fluid) with a negligible surface tension (Schaflinger and Machu 1999; Machu et al.
2001). In addressing the motion of CSP in the human airways we
will conceptually extend this analogy to particle clouds, accounting for the relevant parameter domain. The motion of particle
clouds may be subjected to further effects, such as the proximity
to the airway walls. Both particle deformation and particle–wall
interactions affect the drag acting on the cloud, and may induce
its drift across the streamlines towards or away from the walls.
Wall Effect
In general, particles approaching bounding walls experience
excess hydrodynamic drag as a result of the increased shear
in the wall proximity. Due to the linearity of both Stokes’ and
516
D. M. BRODAY AND R. ROBINSON
(a)
(b)
Figure 1. Variation of the cloud settling velocity with the solid volume fraction according to different models (a), and variation
of the Brinkman’s cloud settling velocity with the cloud size (b). All models consider monodisperse fine particles with a diameter
of 2a = 0.3 µm.
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
Brinkman’s equations, the extra drag imposed by the wall can be
accounted for by a factor in the force expression. The linearity
also implies that no lift (i.e., normal) force acts on spherical
particles that translate slowly along a flat wall (Goldman et al.
1967a,b; Pozrikidis 1990).
With respect to the Laplace-like Stokes equation, there are
fewer separable solutions in orthogonal coordinate systems
for the Helmholtz-like Brinkman equation. In particular, the
Brinkman equation is inseparable in a bispherical coordinate
system (Kim and Russel 1985), the natural coordinate system
to describe particle-particle and particle-wall interactions. Indeed, this explains the lack of analytical expressions for wall effects in Brinkman’s flows. For example, the motion of a porous
sphere towards a flat boundary was studied only recently (Davis
2001), whereas translation of a porous sphere proximate to a
solid boundary has not been studied yet to the best of our knowledge. For the latter case, a possible approximation may involve
using expressions for the excess hydrodynamic drag induced by
adjacent walls on solid or liquid particles (Happel and Brenner
1973). The excess drag on a solid sphere that translates slowly
in a viscous fluid at a constant separation H from a flat wall,
FW,s , is (Happel and Brenner 1973).
·
FW,s
µ 4 ¶¸−1
9a
a
a3
= 1−
+O
.
+
16H
8H 3
H4
[19]
517
The authors are unaware of any similar expression for nonsolid
particles. According to Clift et al. (1978), wall effects significantly alter drop motion if the ratio of the drop to tube radii is
greater than 0.1. Applying this criterion to cloud motion as well,
CSP clouds moving concentrically in the upper tracheobronchial
airways may experience non-negligible excess hydrodynamic
resistance resulting from a limited particle-wall spacing. The
comparable case of a solid sphere translating parallel to a flat
wall in a Brinkman effective medium (i.e., motion of individual particles within a bounded swarm) was recently worked out
numerically (Broday 2002). Depending on the separation between the particle and the wall and on the medium permeability,
the excess drag is depicted in Figure 2. In general, as H/a decreases the wall effect becomes more appreciable, modifying
the particle mobility, increasing the residence times within airway segments, and consequently enhancing particle deposition
by diffusion and gravitational settling. Accounting for the radial
position of the cloud is beyond the scope of this work. Hence,
in the following simulations concentric motion of the cloud in
the airways is assumed (see also the Cloud Deformation section
of this article).
Particle Migration
In flows with non-negligible shear, fine particles that either
lead ahead or lag behind the bulk flow (e.g., as a result of gravity)
Figure 2. Excess drag on a solid sphere that translates in Brinkman and Stokes media parallel to a solid wall. Solid lines are the
numerical results (Broday 2002), dashed line is the analytical approximation (Equation (19)).
518
D. M. BRODAY AND R. ROBINSON
may experience redistribution as the result of drifting across
the streamlines (Broday et al. 1998). In part, the migration results from hydrodynamic interactions between deformable or
nonspherical particles and non-reversible inertial effects in the
presence of bounding walls. Specifically, isolated drops and bubbles moving in shear flows and characterized by finite Re p tend
to migrate toward or away from the bounding wall (Goldsmith
and Mason 1962) and to accumulate in an annular region about
halfway between the tube centerline and the wall. This behavior resembles that of spherical particles (Happel and Brenner
1973) and fibers (Broday 1996; Broday et al. 1998), where particles migrate across the streamlines even when the fluid inertia
is negligible. On the other hand, particle-particle interactions
in a swarm tend to spread out droplets and fine solid particles
and to disperse them more homogenously in the cross section
(Han et al. 1999). The smallness of CSP and their negligible
inertia suggest that effects caused by particle migration across
the streamlines within the airways are marginal.
Cloud Deformation
The degree to which particles deform is an outcome of a
competition between shear forces and capillary forces, the latter
tending to restore an interfacial surface corresponding to minimum free energy. The larger the interfacial surface tension, the
larger the tendency of the particle to have a spherical shape (Koh
and Leal 1989, 1990; Pozrikidis 1990). For deformable particles,
the ratio of the gravity force to the interfacial force that opposes
deformation during settling in a quiescent fluid is called the Bond
number, Bo = 1ρgb2 /σ , where σ is the surface tension and 1ρ
is the density difference between the particle and the surrounding
fluid. When deformable particles move in a shear flow, the force
ratio is termed the capillary number, Ca = µU/σ , representing the relative magnitude of the shear force to the interfacial
force. The ratio of the Bond number to the capillary number
corresponds to the relative magnitude of the settling velocity
compared to the bulk velocity. Usually, when shear prevails
Ca À Bo. When either the Bond or the capillary numbers are
large, the surface tension cannot oppose deformation, and deformable particles tend to be nonspherical.
The deformation process is highly nonlinear because the
shape evolves as a result of shear stresses that are determined
by the local disturbance to the flow, caused by the motion of
the particle itself. In shear flows, droplets and bubbles extend
longitudinally and become elongated (prolate) spheroids that
eventually disintegrate into a number of small spherical entities (Brenner 1999). To minimize shear stresses, stretched out
droplets tend to align with the flow and situate at the centerline (Koh and Leal 1989, 1990). In our simulations we therefore
consider concentric cloud motion.
Recently, Schaflinger and Machu (1999) studied the settling
of suspension drops in quiescent fluid. Throughout the experiment, a distinct interface between the suspension and the clear
fluid surrounding it was visible. Internal circulation was observed within the drop, as well as an evolutionary sequence of
deformations that exhibited copious similarities to the deformation process of miscible drops in viscous fluid (Kojima et al.
1984). Following Kojima’s work, an apparent interfacial tension
between the pure fluid and the suspension drop was estimated
(σ ∼ 3.e-5 Nm−1 for φ = 0.04–0.08).1 The finite interfacial
tension is consistent with observations that a spherical blob settling at finite Rec in an unbounded fluid do not experience shape
changes (Nitsche and Batchelor 1997). The envelope of closed
streamlines surrounding the blob seems to lie slightly inside
of the fuzzy spherical boundary of the drop, with particles lying outside of that envelope being swept to the rear, forming
a tail. These observations are reminiscent of the Hadamard–
Rybchinsky fluid-sphere model of immiscible fluids, for which
the drag corresponds to that calculated by Brinkman’s model
for a dense, quasi-impermeable, core (Figure 1). For thinner regions (dilute clouds) mixing due to entrainment of clear fluid
(air) is expected, resulting from streamlines that cross the cloud
“boundary.”
From a theoretical perspective, the existence of a macroscopic
interfacial tension is expected because of the tendency of a cluster of particles (molecules) to lower its collective potential energy by agglomeration, thus forming a lattice in which inner
particles are completely surrounded (Nitsche and Schaflinger
2001). This naturally leads to the standard concept of surface
energy, i.e., the work required to increase the number of particles (molecules) residing in an incompletely surrounded state
at the surface. This idea can be extended to include interfacial
tension-like phenomena even when a clear physical boundary
between two domains (i.e., the cloud and the surrounding air)
does not exist. This approach was utilized indirectly for simulating the settling and stretching of a swarm of particles (blob)
within a bounded domain, accounting for particle-wall interactions (Brenner 1999). If we assume that the interface of a
particle cloud (suspension drop) has no mechanical properties
other than a pseudosurface tension, continuity of velocity and
tangential stresses on it must exist (Batchelor 1967). This conforms with the present description of the cloud in terms of an
effective medium and with the boundary conditions applied to
Equation (14) in obtaining Equation (17) (Brinkman 1947; Neal
et al. 1973; Davis and Stone 1993).
The deformation process described by Schaflinger and
Machu (1999) and Machu et al. (2001) was observed for suspension droplets settling in viscous liquid under the following conditions: a = 50 µm, R = 80, 1ρ/ρl = 1.32, φ = 0.05–0.075,
and a kinematic viscosity 6.6 times higher than that of air. In contrast, the CSP cloud is characterized by CMD of about 0.3 µm,
1ρ/ρg ∼ 750–1800, and φ ≈ 10−4 . Thus, it is not clear whether
the findings cited above are relevant to cigarette smoke. Rigorously, the requirements are Rec ¿ 1 and Bo À 1. While the first
condition usually holds for CSP clouds with b ≤ O (1 mm),
1 In a subsequent study, however, Machu et al. (2001) presented simulations showing that the deformation of suspension drops may result solely from
Stokesian hydrodynamics, without accounting for interfacial forces.
519
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
the second condition normally requires clouds of size larger
than about 5 mm, introducing a contradicting requirement.
Yet the importance of the analogy, if it holds, is in the ability to
learn about the deformation of CSP clouds within the respiratory
tract from studies conducted on the motion of deformable drops
in tubes and pipes.
The relaxation time for cloud deformation is τ ∼ bµ/σ ∼
R ∗ 10−7 s. For CSP clouds it is expected to be of the order
of O(10−6 − 10−3 s), much shorter than the residence time
in any bronchial generation. The shape distortion is such that
the excess velocity of the particle relative to the air is reduced.
Namely, for a given particle mass (volume) the settling velocity is higher for particles with larger interfacial tension. Since
we expect very small (if any) interfacial tension for the “miscible” clouds, deformations will be appreciable. According to
Bernoulli’s law, cloud deformation will take place up to an equilibrium state, at which point the difference in dynamic pressure
between the fore and aft portions of the cloud is balanced by
the capillary pressure. Consequently, the cloud extends in the
flow direction to an order of magnitude of l ∼ O(2σ/ρU 2 ),
while its cross sectional area is of an order of magnitude of
S ∼ O(V /l), V being the cloud volume. Cloud fragmentation
is expected for extremely elongated or distorted clouds, possibly in bifurcation zones where flow reversal and separation may
prevail.
Deposition Model
General. Deposition of CSP in the human respiratory tract
is determined for an adult smoker using the improved trumpet model presented previously by Robinson and Yu (2001),
based on Weibel’s model A. In this model CSP size distribution and concentration are updated at each generation, accounting for changes in hygroscopic growth, polydisperse coagulation that results from kinematic effects (turbulence, laminar
shear, differential settling), Brownian motion, particle charge,
and deposition due to the combined mechanisms of gravitational settling, inertial impaction, diffusion, and electrostatic image forces. These processes are implemented within a transport
and deposition model that solves the one-dimensional aerosol
general dynamic equation after it is integrated over the crosssectional area of the respiratory airways to yield average concentrations per unit length. It is assumed that concentration changes
due to deposition and growth are independent. The model, which
has been evaluated against experimental data for stable and nonstable aerosols, was modified here to account for the new cloud
effect module.
The model allows for an accurate account of the smoker’s
breathing pattern, dividing the total 1870 ml tidal volume into
a bolus of 54 ml—the puff—followed by clean air (Robinson
and Yu 2001). Flow rates are based on a 3 s inhale, 1 s breath hold,
and 3 s exhale (Hinds et al. 1983) and a uniformly expanding
lung in which only the airways’ diameters change. Uniform velocity profiles and negligible axial diffusion are assumed. The
initial CSP size distribution is taken from Keith (1982), which
measured the properties of fresh smoke aged for less than 0.05 s
(CMD of 0.25 µm, 1.3 GSD). The particle initial charge distribution follows data compiled by Robinson and Yu (2001).
Since CSP are characterized by independent size and charge
distributions, a three-dimensional array that is updated at each
generation, representing discretizeation to 0.1 µm and 1 elementary charge, was used to track the concentration of CSP as
they move through the lung.
Cloud Behavior. At each generation the calculated CSP volume fraction is used to determine the particle-in-a-swarm and
the swarm-of-particles normalized settling velocities, which are
Equations (15) and (18), respectively. Particles are assumed to
settle in a manner that minimizes energy dissipation (and therefore drag), and thus the model that yields the largest settling velocity (i.e., individual particles versus a cloud) is used. This criterion is similar to the one utilized when the solid sphere (Hinds
1999; Robinson and Yu 2001) or the fluid sphere (Martonen
1992; Martonen and Musante 2000) cloud models were considered. Yet, as seen previously, the transition between the two
regimes is smoother for the Brinkman cloud model.
All deposition mechanisms depend to a great extent, via the
specific deposition parameters, on the particle mobility M p .
Similarly, the colligative effect affects the deposition process
by modifying the particle mobility and changing the deposition
efficiencies accordingly. When only hydrodynamic interactions
are accounted for, M p is given by
Mp =
v
fp
or
Mp =
vCc (a)
,
f c /n
[20]
where f p is the force acting on individual particles and f c is the
collective force acting on the cloud as a whole. From Equations
(15) and (17), f p = 6π µav F p /Cc (a) and f c = 6π µav Fc , thus
the mobility of individual particles is
Mp =
 C (a)
c


 6π µa F
for single particle motion,
p

φ R 3 Cc (a)


6π µa Fc
[21]
for cloud motion.
Unlike previous work (cf. Martonen and Musante 2000;
Robinson and Yu 2001), we do not consider CSP to move within
large rigid particle clouds while “losing” their individual properties. Rather, here CSP behave independently according to their
own set of attributes, while simultaneously subjected to reduced
drag (increased mobility) resulting from the disturbance to the
flow introduced by the presence of numerous similar particles
in their vicinity. In essence, CSP move as if contained within
clouds while approaching the airway walls, but are susceptible
to deposition as individual particles.
Diffusion. Diffusion is the most significant deposition
mechanism for fine particles that are subjected to and affected
by random impaction of air molecules. According to Einstein’s
theorem, the diffusion coefficient D is related to the particle
520
D. M. BRODAY AND R. ROBINSON
mobility via
D = kT M p ,
[22]
where kT is the particle thermal energy. The colligative effect on
diffusion of individual particles is accounted for by modifying
the diffusion coefficient of individual CSP using the modified
mobility given in Equation (21). The expression for deposition
by diffusion implemented in the model follows the derivation of
Ingham (1975), with the diffusion parameter expressed as
1=
DL
,
2
U dgen
[23]
where L and dgen are the length and diameter of airways in the
current generation, respectively, and U is the mean air velocity
within the airways.
Inertial Impaction. Deposition by inertial impaction occurs
at locations characterized by sharp streamline curvature, where
the particle persistence promotes deposition on posterior airway
walls. The efficiency of inertial deposition is calculated using
expressions derived by Zhang et al. (1997), with the cloud effect
introduced into the calculation via the definition of the Stokes
number (Equation (7)),
St =
2a 2 ρ p U Cc (a) φ R 3
.
9µdgen
Fc
[24]
Note that when particles move independently of each other
(i.e., when no cloud motion occurs) the Stokes number is only
marginally affected by the presence of nearby particles, since
F p is close to unity (Equation (15) and Figure 1).
Gravitational Settling. Deposition by gravitational settling
is implemented in the model using the expressions derived by
Yu (1978) and Pich (1972), with the settling parameter being
ε=
3gLv sin θ
,
4dgen U
[25]
and θ being a typical generation-dependent airway orientation.
The settling velocity v is either v p (Equation (15)) for single particle motion, or vc (Equation (18)) when cloud motion prevails.
Electrostatic Precipitation. Deposition due to electrostatic
image forces is important if the number of charges per particle
is greater than ∼30 (Chan and Yu 1982; Yu 1985), with the
electrostatic precipitation parameter being
ζ =
e2 z 2 L Cc (a)
,
2 u
96π 2 ε0 µ a dgen
[26]
where ε0 is the dielectric constant of vacuum, e is the elementary
charge, and z is the number of charges per particle. Normally,
electrostatic deposition is negligible for CSP, since even after
coagulation there are usually less than 10 charges per particle
(Robinson and Yu 2001).
Cloud Breakup. The total number of CSP in any subsequent
(i + 1) generation can be calculated knowing the deposition efficiency in the previous (ith) generation and the details of the
coagulation and hygroscopic growth processes in that generation. Yet, the fate of the clouds is unclear. Due to deformation,
clouds may change size and increase in number. Furthermore,
particle-free air residing in the airways may mix with the puff,
diluting it and increasing its effective volume. Lacking experimental data, we account for these processes in a rather simplistic
way, changing the cloud size deterministically along the respiratory pathway while keeping the total puff volume constant.
Our results show (see below) that for realistic conditions, cloud
behavior occurs only in the very proximal tracheobronchial airways, where mixing with resident air is negligible. This is in contrast with the previous predictions of Robinson and Yu (2001),
where cloud motion was estimated in distal generations as deep
as the 21st Weibel’s generation.
Description of the mechanistic processes involved in cloud
breakup is beyond the scope of this study. We thus assume here
that the relationship between the size of the newly formed cloud
and that of the parent cloud is
·
bi+1
dgen,i+1
= bi
dgen,i
¸k
,
[27]
where k = 0, 1, 2. In particular, for k = 0 the size of the cloud remains constant throughout the lung (see Martonen and
Musante 2000). By using a nonzero value for k the cloud size
changes along the respiratory pathway. Thus, k = 1 represents
a size change such that the ratio of the cloud to the airway
diameters is constant (Robinson and Yu 2001), while k = 2
represents a constant ratio of the cloud to the airway crosssectional area. Clearly, in all cases conservation of particle mass
is ensured.
Model Results
Deposition of CSP in the human respiratory tract was calculated using three model variants. The simple (S) variant considers a stable polydisperse aerosol and does not account for
particle-particle interactions. The cloud model variant (L) considers stable polydisperse aerosol, accounting for the colligative
effects as described previously. The base case L-variant comprises a cloud with an initial diameter of 4000 µm and size that
changes along the lung pathway according to Equation (27) with
k = 1. A cloud diameter of 4000 µm, roughly the size of the
glottis, represents an upper bound cloud size expected in the
human respiratory tract. The third variant (LHC), in addition to
colligative behavior, further includes the effect of hygroscopicity
and coagulation on the evolution of the particle size distribution.
Effect of Initial Cloud Size. Figure 3 depicts model predictions of CSP deposition profile in the lung for different initial
cloud sizes. In contrast to the solely peripheral deposition of
noninteracting individual CSP particles (model S), the increase
in cloud size is associated with increase in proximal deposition
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
521
Figure 3. Effect of initial cloud size on the deposition profile of CSP in the lungs, CMD = 0.25 µm, GSD = 1.3, k = 1.
(in the trachea, main bronchi, and lobar bronchi) and a simultaneous decrease in pulmonary deposition (Table 2). As expected,
the colligative effect is pronounced in the upper thoracic airways
and ceases to exist shortly after. This is unlike Robinson and Yu’s
(2001) results, where cloud motion prevailed in the pulmonary
region (21st generation) as well.
Modeled previously as fluid or solid spheres (Martonen and
Musante 2000; Robinson and Yu 2001), CSP clouds as small as
40 µm in diameter already showed very high deposition (70–
99%) with proximal preference. Here, CSP clouds of initial size
smaller than 1000 µm show negligible cloud behavior, having
deposition profile similar to that predicted for noninteracting
Table 2
Regional and total CSP deposition for different initial cloud
diameter 2b0 and cloud size change parameter k = 1
Cloud size
Total
(%)
No cloud effect
39
(model S)
43
2b0 = 1000 µm
46
2b0 = 2000 µm
55
2b0 = 3000 µm
62
2b0 = 4000 µm
In vivo measurements 32–89
Tracheobronchial Pulmonary
(%)
(%)
4
35
5
15
29
40
46–63
38
31
26
22
26–35
particles (model S). Being essentially unaffected by sedimentation, which usually dominates deposition in the lower thoracic
airways, deposition of CSP exhibits a double peak profile. Owing
to the swarm effect, particles tend to deposit in the upper airways, whereas particles that survive proximal deposition tend to
persist and deposit only in the deep pulmonary region. Indeed,
the model predicts a wide and flat deposition minimum in the
distal bronchial airways. Table 2 details the total and regional
deposition, where tracheobronchial deposition is defined as deposition in generations 0–16 of Weibel’s lung model A. These
results compare favorably with those obtained in clinical studies
(Ellett and Nelson 1985; Martonen et al. 1987; Yang et al. 1989),
and in in vivo (Pritchard and Black 1984; Hicks et al. 1986) and
replica casts (Ermala and Holsti 1955; Martonen et al. 1987)
measurements.
Effect of Cloud Size Decrease Rate. The effect of the degree by which the size of the cloud changes between successive
generations, implemented by means of the magnitude of k in
Equation (27), is depicted in Figures 4a and 4b. Figure 4a shows
deposition profiles of clouds with different size decrease rates,
all having initial sizes of 4000 µm. It is evident that the smaller
the decrease in size between succeeding generations, the larger
the deposition in the proximal tracheobronchial airways (see
also Table 3). Note that the initial cloud size implies that the
k = 0 case is aphysical, since it represents clouds that cannot
penetrate any airways distal of the fourth generation. Figure 4b
depicts the effect of cloud size decrease on deposition of CSP
522
D. M. BRODAY AND R. ROBINSON
(a)
(b)
Figure 4. Effect of the extent of cloud size decrease between successive generations on CSP deposition profile in the lungs,
CMD = 0.25 µm, GSD = 1.3. (a) Initial cloud size is 2b0 = 4000 µm, (b) final common cloud size is 5 µm.
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
Table 3
Regional and total deposition for different cloud size change
parameter, k
Cloud rule
No cloud effect
(model S)
k=1
k=2
Total
(%)
Tracheobronchial
(%)
Pulmonary
(%)
39
4
35
62
48
40
19
22
29
The initial diameter is 2b0 = 4000 µm.
that attain a 5 µm cloud size in the airways of the last generation of Weibel’s lung model. The difference between our results
for constant cloud size (k = 0) and those obtained by Martonen
and Musante (2000) are attributed to the different cloud models
used, and in particular to the way by which these models are
implemented and modify the particle mobility. Indeed, using
the fluid sphere cloud model and considering motion and deposition of rigid clouds, we reproduced Martonen and Musante’s
results, obtaining 99% total deposition while reducing drastically the deposition in the peripheral airways (TB 72.7%,
P 26.5%). In contrast, the small cloud size in the k = 0 case
(2b0 = 5 µm) and the moderate decrease in cloud size in the
k = 1 case (2b0 = 220 µm) reveal a negligible cloud behavior
according to our model, in agreement with the trend depicted
523
in Figure 3. In fact, the three lines in Figure 4b corresponding
to k = 0, k = 1, and “no cloud” (model S) are indistinguishable, representing total deposition of ∼39% (k = 0: TB 3.9%,
P 35.2%; k = 1: TB 4.1%, P 35.1%; “no cloud”: TB 3.9%,
P 35.3%). On the other hand, the k = 2 case (2b0 = 9640 µm)
does show significant cloud behavior, having total deposition of
67.5% (TB 50.4%, P 17.1%). Apart from being in better agreement with in vivo data (see Table 2), it is our belief that accounting for a permeable and deforming (size-changing) CSP cloud
represents a more realistic description of the physical process
occurring in the lungs.
Deposition by Mechanism. Figure 5 portrays the deposition
profile by mechanism. As expected, particle inertia and gravitational settling play the major role in promoting proximal deposition. Once the colligative effect (Equation (17)) ceases (due to
cloud disintegration) and the Stokes number represents the inertia of individual CSP, which is only slightly affected by the presence of nearby like particles, diffusion becomes the major deposition mechanism. The weak crowding effect (Equation (15))
increases diffusion slightly in the very distal pulmonary airways
(Figure 5), but does not change significantly the overall deposition by diffusion. Gravitational settling increases by an order of
magnitude, and its contribution to the total deposition becomes
comparable to that of diffusion. This is mainly attributed to the
decrease in the particle relaxation time, which promotes gravitational settling in the pulmonary region. Overall, predictions
of CSP deposition by the L model-variant show significant shift
Figure 5. Deposition profile by mechanism. CMD = 0.25 µm, GSD = 1.3, 2b0 = 4000 µm, k = 1.
524
D. M. BRODAY AND R. ROBINSON
(a)
(b)
Figure 6. Deposition profile of CSP with a dynamically changing size distribution. Initial parameters: CMD = 0.25 µm, GSD =
1.3, k = 1, (a) 2b0 = 4000 µm, (b) 2b0 = 7695 µm.
CLOUD DYNAMICS OF CIGARETTE SMOKE PARTICLES
Table 4
Deposition predictions for unstable and stable CSP behavior
Cloud model
No cloud effect
(model S)
Cloud behavior
(model L)
Cloud behavior,
hygroscopicity,
and coagulation
(model LHC)
Total
(%)
Tracheobronchial
(%)
Pulmonary
(%)
39
4
35
62
40
22
78
66
12
Cloud diameter 2b0 = 4000 µm, size change parameter k = 1.
from distal diffusional deposition (model S) to increased proximal deposition and suppressed deposition in the distal airway
generations (Table 4).
Effect of Dynamic Size Distribution. As CSP move along
the respiratory tract the fine particles undergo coagulation and
hygroscopic growth. The decreasing number concentration due
to coagulation and deposition reduces the coagulation rate and
weakens the colligative cloud effect, promoting single particle dynamics. On the other hand, hygroscopic growth promotes
cloud behavior by increasing the volume fraction. The effect of
the dynamic size distribution on the deposition profile is demonstrated in Figure 6a. Cloud behavior (model L with initial cloud
size of 4000 µm and k = 1) increases CSP deposition in the
TB region by 36% over the “no cloud” (S) model (Table 4).
Since fewer particles reach the lower airways, deposition in the
pulmonary region decreases. The net increase in the total deposition due to the cloud behavior (model L versus model S)
is 23%. Accounting for hygroscopicity and coagulation (model
LHC) further increases deposition in the TB region by another
26% over model L. Pulmonary deposition further decreases,
and the net deposition increases (Table 4). The increased deposition in the LHC model over model L is due mainly to the
combination of hygroscopic growth and cloud behavior. Hygroscopic growth of smoke particles occurs only in the trachea,
since the particles reach equilibrium size in <0.1 s (Robinson
and Yu 1998; Broday and Georgopoulos 2001). This has the
effect of changing the initial size distribution of CSP entering the respiratory tract. Thus, although hygroscopic growth
does not change the number concentration of CSP, the particle volume fraction increases from about 10−5 in model L to
∼10−4 in the combined LHC model. This increase in φ results
in a significant increase in the intensity of the cloud behavior, enhancing interparticle screening. The increased deposition
in the TB region is consistent with in vivo regional deposition
measurements in human subjects (Pritchard and Black 1984;
Hicks et al. 1986), and with in vitro regional deposition in
hollow cast models (Ermala and Holsti 1955; Martonen et al.
1987).
525
The different cloud model (“solid sphere”) employed by
Robinson and Yu (2001) results in a completely different deposition profile, as seen in Figure 6b. Although Robinson and Yu’s
cloud model increased the overall deposition in the lungs, most
of the CSP still deposited in the periphery. For example, while
Robinson and Yu (2001) reported CSP deposition of 16.5% (TB)
and 53.7% (P) for a CSP cloud with an initial size of 7695 µm
(their notation was dc = 1/2dx ), Figure 6b depicts 88.6% deposition in the tracheobronchial airways and <0.2% deposition
in the pulmonary region. It is noteworthy that these results are
referred to here for the mere reason of comparing the two models, since Robinson and Yu considered initial cloud sizes larger
than the glottis aperture, and thus unlikely.
CONCLUSIONS
Contrary to previous models, where bimodal cloud settling
was considered, Brinkman’s effective medium approach regards
the cloud as a porous medium and accounts for weak hydrodynamic interactions among the particles constituting it. The main
features of the model are that the viscosity of the medium is
taken identical to that of the surrounding air and the permeability of the cloud is finite and changes according to its solid volume
fraction. The finite permeability allows air to pass through the
cloud as well as around it, yet Brinkman’s model does not support the development of internal circulation. Only very dense
CSP clouds settle like fluid spheres, unmixing with the outer air.
Within dilute small clouds particles settle almost in isolation,
being only weakly affected by the presence of nearby similar
particles. Within denser and/or larger clouds, a non-negligible
hydrodynamic screening affects the particle mobility, altering
the deposition profile as a result of modifications introduced to
the deposition efficiencies of all three major deposition mechanisms. Model results compare favorably with data available on
CSP deposition in the human airways and in replica casts.
The model predicts that CSP deposition is significantly influenced by the details of cloud formation, deformation, and
disintegration. These processes are simulated rather simplistically at present. Much work is still needed to characterize these
processes in vivo, parameterize them, and study their effect on
the CSP deposition profile. Nonetheless, the model presented
here is superior over previous models because it examines the
effect of particle crowding on various processes (coagulation,
growth, and deposition) simultaneously, and because it implements particle-particle screening effects in a more realistic way.
In particular, the present cloud model shows a relatively smooth
transition between single particle and cloud behavior.
Model results indicate that a combination of cloud behavior,
hygroscopic growth, and coagulation may explain the preferential deposition of cigarette smoke particles in the TB region in
spite of their smallness.
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