Jessica M. Oakes
Steven Day
Department of Mechanical Engineering,
Rochester Institute of Technology,
Rochester, NY 14623
Steven J. Weinstein
Department of Chemical Engineering,
Rochester Institute of Technology,
Rochester, NY
Risa J. Robinson1
Department of Mechanical Engineering,
Rochester Institute of Technology,
Rochester, NY
e-mail: [email protected]
Flow Field Analysis in Expanding
Healthy and Emphysematous
Alveolar Models Using Particle
Image Velocimetry
Particulates that deposit in the acinus region of the lung have the potential to migrate
through the alveolar wall and into the blood stream. However, the fluid mechanics governing particle transport to the alveolar wall are not well understood. Many physiological conditions are suspected to influence particle deposition including morphometry of
the acinus, expansion and contraction of the alveolar walls, lung heterogeneities, and
breathing patterns. Some studies suggest that the recirculation zones trap aerosol particles and enhance particle deposition by increasing their residence time in the region.
However, particle trapping could also hinder aerosol particle deposition by moving the
aerosol particle further from the wall. Studies that suggest such flow behavior have not
been completed on realistic, nonsymmetric, three-dimensional, expanding alveolated geometry using realistic breathing curves. Furthermore, little attention has been paid to
emphysemic geometries and how pathophysiological alterations effect deposition. In this
study, fluid flow was examined in three-dimensional, expanding, healthy, and emphysemic
alveolar sac model geometries using particle image velocimetry under realistic breathing
conditions. Penetration depth of the tidal air was determined from the experimental fluid
pathlines. Aerosol particle deposition was estimated by simple superposition of Brownian
diffusion and sedimentation on the convected particle displacement for particles diameters of 100–750 nm. This study (1) confirmed that recirculation does not exist in the most
distal alveolar regions of the lung under normal breathing conditions, (2) concluded that
air entering the alveolar sac is convected closer to the alveolar wall in healthy compared
with emphysematous lungs, and (3) demonstrated that particle deposition is smaller in
emphysematous compared with healthy lungs. 关DOI: 10.1115/1.4000870兴
Keywords: alveolar sac, emphysema, particle image velocimetry, compliant model,
deposition
1
Introduction
6
In the United States approximately 35⫻ 10 Americans have
some form of chronic lung disease, which leads to about 400,000
deaths per year 关1兴. To predict particulate delivery to the lung
whether it is in the form of aerosol medication or airborne pathogens, it is crucial to understand the physiological conditions that
affect fluid flow and particulate behavior in the pulmonary region.
Since disease alters the structure of the lung, it is expected that
airflow and mass transport differ in diseased compared with
healthy lungs. With emphysema, there is a decrease in the alveolar
wall compliance and surface area accompanied by an increase in
lung volume. The mechanisms by which these changes influence
flow field and deposition in emphysemic patients are not fully
understood. Kohlhäufl et al. 关2兴 found an increase in ventilation
asymmetry in emphysema, which was suspected to alter particle
deposition. Sweeney et al. 关3兴 measured lower particle deposition
in emphysemic compared with healthy hamsters. Sturm and Hofmann 关4兴 found a decrease in alveolar particle deposition in emphysema compared with normal using multiple generation stochastic models with two-dimensional rigid geometries. However,
1
Corresponding author.
Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received February 4, 2009; final
manuscript received October 2, 2009; accepted manuscript posted December 22,
2009; published online January 29, 2010. Assoc. Editor: David A. Steinman.
Journal of Biomechanical Engineering
no work has been done to examine the fluid flow mechanisms that
may be responsible for the decreased particle deposition in emphysema.
It is widely accepted that the predominant mechanisms by
which particles are transported in the pulmonary region is by sedimentation and diffusion since slow moving velocities in the pulmonary region render impaction negligible. However, this understanding stems primarily from analyses of multiple bifurcating
tubes with nonmoving boundaries. It is possible that expanding
alveolar walls provide additional convective motion, or that flow
around alveolar septa induces recirculation zones that could aid in
mixing or particle trapping. Animal lung casting experiments have
suggested that mixing is present in the respiratory zone 关5兴 and
inhaled bolus dispersion experiments have shown that mixing influences particle deposition 关6–9兴. Several studies have been done
to examine the potential effect of these structural and physiological mechanisms on particle transport. Tsuda et al. 关10兴 in a numerical model of torus geometry and Karl et al. 关11兴 in an experimental model of square shaped alveoli, showed that fluid flow is
significantly affected by certain features of the alveolus; recirculation occurs with a large alveolus depth 共D兲 to mouth diameter
共MD兲 ratio 共see Fig. 1共b兲兲. Sznitman et al. 关12兴 in a threedimensional numerical model of an alveolus on a duct, Tippe and
Tsuda 关13兴 in a experimental alveolated torus model, and Tsuda
et al. 关10兴 in a numerical torus geometry, showed that the flow
becomes irreversible with a large flow rate ratio 共ratio of the flow
in the duct to the flow into the alveolus兲. Haber et al. 关14兴 and
Sznitman et al. 关15兴 showed in a numerical model that recircula-
Copyright © 2010 by ASME
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sight into the mechanisms that govern mixing and how alterations
in these mechanisms affect particle penetration and deposition in
diseased lungs.
In this study, we present the first data that use realistic nonsymmetric healthy and emphysemic geometry with compliant walls
and realistic breathing conditions. For both geometries, morphometric data 关21–25兴 were used to create large scale physical models of the alveolar sac. The fluid flow was characterized using
particle image velocimetry 共PIV兲. The velocity fields were analyzed to determine whether or not recirculation exits and to quantify the difference in particle penetration depth due to convection
in the healthy and emphysemic models. Further analysis was completed to estimate, by simple superposition of convective motion
with sedimentation and diffusive motions, the difference in deposition on the alveolar walls between healthy and emphysematous
lungs.
2
Fig. 1 Alveolar sac model geometries for healthy side „a… and
top „b… view and emphysema side „c… and top „d… view. Measurements are shown in Table 1.
tion decreases as the generation number increases and is nonexistent in the alveolar sac. Cinkotai 关16兴 in an experimental alveolar sac model and Davidson and Fitz-Gerald 关17兴 in an analytical
solution of Stokes’s equations for expanding sphere showed that
flow in an alveolar sac is reversible and radial in the direction of
the wall motion. Choi and Chong 关18兴 added idealized nonexpanding alveolus geometry to a whole lung trumpet model and
reported the effect of tidal volume and flow rate on particle deposition. Yue et al. 关19兴 and Otani et al. 关20兴 looked at particle
deposition in an experimental pulsating balloon at a scale larger
than in vivo; however both flow and particle conditions were not
scaled by nondimensional analysis, which is necessary for the
results to appropriately represent in vivo conditions. The studies
mentioned above incorporated simplified or idealistic alveolar geometry. It is expected that more realistic geometric models of
healthy and diseased alveolar sacs would provide additional in-
Methods
2.1 Model Fabrication. Alveolar sac models were created to
represent healthy and emphysematic lungs at functional residual
capacity 共see Fig. 1兲 using the data available in literature 共see
Table 1兲. The healthy and emphysematous models’ effective airway diameters 共EAD兲 were within the standard deviation reported
by Kohlhäufl et al. 关23兴. Figure 2 shows the D/MD ratios 共D
= Alveolus depth, MD= Alveolus mouth diameter兲 of the models
compared with literature. The healthy alveolar sac D/MD ratio
was maintained at 0.8, to keep it within the range reported by
Mercer et al. 关24兴 and Klingele and Staub 关22兴 by reducing the
average number of alveoli from 17 as reported in Weibel 1964
关25兴 to 13. Although no D/MD ratio could be found in open literature for emphysema, the smaller ratio of 0.4 is consistent with
the pathology of the disease state in which the septa break down
and several alveoli merge into a single sac. The alveolar sac models were scaled to 75 times in vivo dimensions and manufactured
for flow visualization. Rapid prototyping was used to create a four
part hollow model, which was used to construct a solid aluminum
cast model. The hollow compliant alveolar sac models were created by dipping the aluminum cast into the molten compliant material 共Ultraflex, Douglas and Sturgess Inc., CA兲. Ultraflex is a
hyperelastic material and there is currently no models to describe
its behavior. The material properties are currently being studied.
2.2 Fluid Scaling. In order to properly represent in vivo flow
physics in the scaled up experimental model, the dominant forces
present in vivo were maintained. These dominant forces were determined by scaling the Navier–Stokes equations 共see Appendix
A兲, which resulted in two nondimensional parameters that govern
the fluid regime, Reynolds and Womersley numbers. The Reynolds number was calculated from Re= 8EVi f / D␯, where E is the
percent expansion, Vi is the initial volume of the alveolar sac, f is
Table 1 In vivo alveolar sac model dimensions compared with morphometric literature: Experimental models were scaled 75 times in vivo size listed in this table
Published
dimension
Duct diameter 共␮m兲
Alveolus radius 共␮m兲
Alveolus depth 共␮m兲
Sac length 共␮m兲
Number of alveoli
D/MD
a
EAD 共␮m兲—healthy
EAD 共␮m兲—healthy
EAD 共␮m兲—emphysema
a
250
140
230
750
17
0.8–1.2
250
280⫾ 50
630⫾ 200
Article
Haefeli-Bluer and Weibel 关21兴
Weibel 关25兴
Weibel 关25兴
Haefeli-Bluer and Weibel 关21兴
Weibel 关25兴
Klingele and Staub 关22兴
and Mercer et al. 关24兴
Haefeli-Bluer and Weibel 关21兴
Kohlhäufl et al. 关23兴
Kohlhäufl et al. 关23兴
Healthy
dimension
Emphysemic
dimension
250
122
165
748
13
640–870
250
140
108
748
17
340–450
312–554
312–554
553–589
Alveolus radius was scaled to 50% TLC 关31兴.
021008-2 / Vol. 132, FEBRUARY 2010
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Fig. 2 Depth „D… to MD ratio of alveolar sac models and those reported by Klingele and
Staub †22‡ and Mercer et al. †24‡
the breathing frequency, D is the diameter of the duct, and ␯ is the
kinematic viscosity. For normal breathing the average in vivo
Reynolds number for the alveolar sacs is approximately 4.85
⫻ 10−3 and the in vivo Womersley number is approximately
0.037. Based on the scaling analysis it was shown that pressure
and viscous forces will govern the fluid behavior in the alveolar
sac. Therefore experimental conditions will represent in vivo conditions for low Re and Wo.
The experimental Reynolds number Re= 4Q / ␲␯D, where Q is
the volumetric flow rate, was varied for the healthy and emphysemic alveolar sacs in accordance with the tidal breathing curve
共variation in flow rate with time兲 and therefore ranged between
approximately 0 and 0.42. The Womersley number was 0.25 for
the healthy alveolar sac and 0.21 for the emphysemic alveolar sac.
Because these values are less than unity, the pressure and viscous
forces will dominate the flow equations 共see Appendix A兲, yielding experimental streamlines and velocity vectors that will represent in vivo conditions.
2.3 Breathing Curve. The expansion and contraction of the
alveolar sacs in our experiments was controlled by a realistic unsteady breathing curve. Using a spirometer, the general form of
the flow rate curve was acquired from a 21 year old healthy female volunteer during resting breathing conditions. A two part
polynomial function was fitted to the flow rate to remove noise
induced during measurement 共see Fig. 3兲. Assuming the same
transient flow rate behavior throughout each airway branch and
acinus of the lung 共i.e., homogeneous ventilation兲, the spirometer
data were scaled for the alveolar sac models by applying a desired
Fig. 3 Experimental unsteady flow rate curve used to simulate
breathing conditions in healthy alveolar sac model. Curve is
based on a change in volume of 13.29 ml and a breathing cycle
of 8 s. Emphysema flow rate graph is similar with a change in
volume of 18.15 ml and a breathing cycle of 11 s. Solid line is
flow rate measured from the spirometer. The dashed line is the
two piece polynomial curve fitted to the spirometer data.
Journal of Biomechanical Engineering
maximum expansion of 30% change in volume of each model,
using an initial volume of 13.3 ml and 18.2 ml for the healthy and
emphysemic models. To keep the flow rate the same in each
model, the breathing period was scaled to 8 s and 11 s for the
healthy and emphysemic models, respectively. Keeping the flow
rate constant allowed us to isolate the effects of diseased geometry
from the influence of breathing conditions.
2.4 Experimental Setup. The main experimental components
included a laser, camera, data acquisition, and control system,
pressure expansion chamber, and compliant alveolar sac model
共see Fig. 4兲. The compliant model was attached inside a glycerin
filled pressure chamber. Glycerin 共kinematic viscosity ␯ for pure
glycerin is 11.26 cm2 / s兲 was used for the experimental fluid because it has an index of refraction 共1.487⫾ 0.002兲 that matched
the material 共1.481⫾ 0.002兲 used for the compliant alveolar sac
models. The glycerin inside the alveolar model was seeded with
fluorescent 8 ␮m diameter polymer microspheres with a density
of 1.05 g / cm3, peak excitation wavelength of 542 and peak emission wavelength of 612 nm 共Duke Scientific Corp, CA兲. A force
balance on the particles, involving buoyancy, drag, and gravitational forces was used to estimate that the particles migrate at a
rate of 5.3⫻ 10−7 cm/ s upward. This corresponds to a negligible
Fig. 4 Schematic of the experimental setup
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displacement of 0.063 ␮m from the streamlines in 12 s, the time
scale of our experiment. The alveolar sac models were expanded
and contracted drawing a negative pressure on the glycerin inside
the chamber. Expansion of the model allowed the seeded glycerin
in the model inlet tube to be drawn into the sac. The fluorescent
microspheres were illuminated by a laser sheet generation 共1 mm
thick and 48 mm wide兲 connected to a 20 W copper vapor laser,
which produced two wavelengths, 510.6 nm and 578.2 nm.
The motion of the fluid was captured by a monochromatic MotionPro X3 Plus camera 共Redlake Inc., FL兲, which had a 1280
⫻ 1024 pixel2 resolution, for a frame rate up to 2000 fps. Each
pixel on the array was 12 ␮m in length and width. A frame rate of
100 fps and a shutter speed of 4996 ␮s were used for the experimentation. The magnification of the physical object by the sensor
array was 0.23 times. Even though the magnification was small,
the particle image size was between 2 pixels and 3 pixels because
of diffraction limited optics. The glycerin/microsphere mixture
was created to allow for approximately 32 pixels per each cross
correlation interrogation region.
2.5 PIV Analysis. The fluid displacement vectors were calculated by cross correlation of the light intensity within small interrogation regions in successive images with the use of the TSI
product, Insight 3G 共TSI Inc., MN兲. A fast Fourier transform
based 关26兴 multiple step recursive Nyquist grid cross correlation
method was used to allow for good detection of displacement
vectors at high spatial resolution 共32 pixels2兲. The initial displacement vectors used an interrogation size of 64 pixels2 with a
50% window overlap and were used to determine the best overlap
for the interrogation region of 32 pixels2. To allow for subpixel
accuracy in determination of the location of the cross correlation
peak a Gaussian curve fit was used 关27兴. Only measurements with
a signal to noise ratio, defined as the ratio between the highest and
second highest correlation peak, greater than 2 were accepted. The
pixel size of 2–3 pixels and 32 pixel2 interrogation region has
been shown to have a displacement error less than 0.1 pixels 关27兴
for the cross correlation analysis.
Tecplot 360 共Tecplot Inc., WA兲 was used to visualize the flow
field. Streamlines plotted tangent to the velocity vector were used
to analyze the instantaneous flow. Lagrangian pathlines 共location
of the particle at each point in time兲 were used to visualize the
flow over a breathing cycle. Details of the pathline analysis and
error estimation are in Appendix B. We showed that the PIV
analysis error may be reduced by increasing the time separating
frames ⌬t but there is a limit to how large ⌬t can be; if ⌬t was too
large, the PIV particles may leave the interrogation region, and
there would be an increased PIV analysis error e.
The alveolar sac was photographed at the middle cross-section,
which allowed for analysis of the duct and several alveoli. The
model was divided vertically in two sections since the laser light
sheet was not large enough to illuminate the entire model. Streamlines and pathlines were plotted to evaluate the presence of recirculation and reversible flow. The differences between healthy and
emphysemic models were analyzed relative to the measurement
error 共Eq. 共B3兲兲.
2.6 Transport and Deposition Approximations. Fluid pathlines were used to evaluate the difference in bulk fluid motion and
particle transport between the healthy and emphysemic models.
Specifically, penetration depth for the bulk fluid and aerosol particles was estimated as well as the number of breaths to achieve
100% particle deposition efficiency. These analyses were completed on pathlines scaled to in vivo lengths 共divided by 75兲 so
that particle motion calculations would be relevant to an alveolar
sac in vivo.
One measure of the penetration depth utilized was the distance
that the bulk fluid 共inhaled tidal air兲 travels into the expanding
alveolar sac during inhalation due to the expansion of the alveolar
walls. The difference in tidal air penetration depth was estimated
021008-4 / Vol. 132, FEBRUARY 2010
by comparing the average pathline lengths at the end of inhalation
for the healthy and emphysemic models.
Another metric of penetration depth utilized was the remaining
distance that an aerosol particle must travel to reach the alveolar
wall, after being carried into the alveolar sac by the bulk fluid
motion of the tidal air. As a first approximation, it was assumed
that the aerosol particles traveled along the tidal air pathlines with
minimum drift flux by diffusion or sedimentation, until the end of
inhalation. The particle size range having a 20% drift flux was
calculated to quantify the validity of this assumption. The shortest
distance of travel 共in any direction兲 to reach the wall from the
location at the end of inhalation was measured for particles traveling on each pathline. We assumed reversible flow and therefore
particles were assumed to not defer from tidal flow during exhalation. The distances to reach the wall for the healthy and emphysemic models were averaged and compared.
To determine the relative significance of tidal air penetration
depth on aerosol particle deposition, the time for an aerosol particle to travel from the inhaled position to the alveolar wall was
estimated by superimposing a calculated diffusive and gravitational settling distance 关28兴 on top of the initial penetration depth.
Again, the particle size range examined was limited to particles
having less than 20% drift flux off the inhalation pathline. Specifically, the analysis was limited to particle sizes whose migration
from the tidal air pathline during inhalation 共due to combination
sedimentation or diffusion兲 would be less than 20% of the measured inhalation pathline lengths for a 2 s inhalation. This particle
size range was calculated for a combination of sedimentation and
diffusion 关28兴 using the settling velocity of a unit density sphere in
air and the diffusion coefficient 共D = KTCc / 3 ␮d, where K is the
Boltzman constant, T = 98.6 deg F, Cc is the Cunningham slip
correction factor, d is particle diameter兲, respectively. At the end
of inhalation, particles will diffuse in the direction of the concentration gradient. The travel distance is different for each pathline
and for each direction from that pathline to the wall. The pathline
that brought the particles furthest from the alveolar wall at the end
of inhalation, and the longest direction from this point to the alveolar wall was chosen for this analysis; so that, 100% deposition
efficiency could be implied if particles on the chosen pathline
were able to travel the additional distance to reach the wall in the
given amount of time. The differences between healthy and emphysemic distances were evaluated relative to a reasonable range
of inhalation times.
The particles that do not have enough time to reach the alveolar
wall during the first breath may migrate by diffusion or sedimentation to another pathline. With each inhalation and exhalation
cycle, the sequence of convective and drift motion 共which actually
occur simultaneously, although described here as sequential for
illustrative purposes兲 moves the particle closer to the wall. To
estimate the number of breaths required for particles to travel the
distance in this idealized manner, we assumed reversible tidal air
flow and zero particle drift flux during exhalation. The maximum
number of 3 s breaths required for 100% deposition efficiency was
estimated and compared for each model.
3
Results
Observation of the seeded flow over several breathing cycles in
multiple image planes indicated that there were no obvious recirculation regions in either the healthy 共H兲 or emphysemic 共E兲
model, even during high flow rate portions of the breathing cycle.
Furthermore, it was confirmed that the flow was reversible.
Streamlines and pathlines calculations further confirm that the
flow was behaving reversibly. Figure 5 shows a plot of five
streamlines in the top left alveolus, of the H model at maximum
flow rate 共7.13 ml/s at 1.43 s兲. These streamlines confirmed that
there were no low pressure zones around the alveolar walls that
would result in recirculation. Similar streamline plots were created for the E model 共not shown兲, indicating no low pressure
zones. Figure 6 shows the experimental velocity magnitude vector
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Fig. 5 Streamlines in the top left healthy alveolus model at
maximum flow rate „1.2 s into the breathing cycle…
plots for maximum flow rate for H 共a兲 and E 共b兲 for the top
portion of the models only. The velocity in the lower region 共not
shown兲 is close to zero. It was observed that throughout the inhalation portion of the breathing cycle, the direction of the vectors
was consistent. The magnitude of the vectors is proportional to the
flow rate at the inlet and can be related to in vivo velocity magnitude by multiplying by the ratio of the in vivo flow rate to the
experimental flow rate. Figure 7 shows a plot of eleven pathlines
for one complete breathing cycle of the H and E models. The
pathlines show that the fluid particles trace the same path during
inhalation and exhalation with an estimated uncertainty of 0.195
mm, indicating reversible flow for both E and H models.
The fluid pathlines were further evaluated to estimate the difference in penetration depth of the bulk fluid 共inhaled tidal air兲
between the H and E models. Pathlines were longer in the E
model compared with the H model for all but pathline 11. However, for most pathlines the difference was very small 共see Table
2兲. Tidal air in the E model penetrated on average only 9% further
共range of 0–46%兲 than the H model, even though the volume
intake was 37% larger in the E model compared with the H model
共both were expanded 30% of their initial volume兲.
The remaining distance that an aerosol particle must travel to
reach the alveolar wall, after being carried into the alveolar sac by
the bulk fluid motion of the tidal air is also shown in Table 2.
Assuming the particles have not drifted off the streamline by
gravitational or diffusive drift flux, inhaled aerosol particles would
need to travel on average 71 ␮m 共71%兲 further to reach the alveolar wall in the E model compared with the H model. The
increase in distance ranged from 35 ␮m 共43%兲 to 138 ␮m
共96%兲.
The analysis of time for an aerosol particle to travel from the
inhaled position to the alveolar wall was limited to particles
within the 20% drift flux range 共100–750 nm兲. Since particles
above and below this range, would have migration distances
greater than 20% of the inhalation pathlines by sedimentation and
diffusion, respectively. Pathline 6 was chosen for the H model and
Fig. 6 Experimental velocity magnitude at high inlet flow rate „7.13 ml/s and 7.09 ml/s for H and E, respectively… for „a… H
at 1.43 s and „b… E at 1.95 s for the top half of the model
Fig. 7 Pathlines starting at duct entrance in healthy „a… and emphysemic „b… alveolar
sac models plotted over one complete breathing cycle
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Table 2 In vivo tidal pathline lengths for tidal air entering the top of the model and the minimum remaining distance „in any
direction… a particle must travel to reach the alveolar wall after inhalation
Pathline length after inhalationa
Pathline
Healthy 共H兲 model
␮m
Emphysema 共E兲
⌴m
1
2
3
4
5
6
7
8
9
10
11
141
200
266
301
327
333
328
307
273
223
149
205
263
305
327
340
341
331
310
273
227
129
Average
Shortest distance from end of the pathline to alveolar walla
Increase for
E model compared
with H model
␮m
%
64
63
39
27
13
8
3
3
0
4
−20
46⫾ 4
31⫾ 3
15⫾ 2
9⫾2
4⫾2
2⫾2
1⫾2
1⫾2
0⫾2
2⫾2
−13⫾ 3
Healthy 共H兲 model
␮m
Emphysema 共E兲
model
␮m
81
51
65
111
143
189
191
145
109
68
106
116
135
173
223
281
313
257
205
150
106
83
9⫾4
Average
Increase for
E model compared
with H model
␮m
%
35
84
108
112
138
124
66
61
41
39
−23
43⫾ 6
164⫾ 10
165⫾ 8
102⫾ 5
96⫾ 4
66⫾ 3
35⫾ 3
42⫾ 4
38⫾ 5
57⫾ 8
−22⫾ 5
71⫾ 5
a
Experimentally derived distances were scaled to obtain in vivo lengths 共divided by 75兲. The error was based on the 0.195 mm experimental error converted to 2.60 ␮m
representing the in vivo length scales.
pathline 5 was chosen for the E model since particles traveling on
this pathline would be further from the wall at the end of inhalation 共325 ␮m for the H model and 425 ␮m for the E model兲 than
particles traveling on any other pathline. The calculations predicted that none of the particle sizes would migrate to the alveolar
wall during a 3 s inhalation 共see Fig. 8兲. Furthermore, an additional breath hold of 5 s, for a total residence time of 8 s was not
sufficient time. Figure 8 also shows the number of 3 s breaths that
would be required for particles in the 20% maximum drift flux
range to reach the wall with 100% deposition efficiency. A 750 nm
particle, which is dominated by sedimentation in the alveolar sac,
will reach the wall in at most five breaths for H and seven breaths
for E. A larger discrepancy between H and E was found for the
smallest particles in the size range considered, which are diffusion
dominated while traveling in the alveolar sac. At most 23 breaths
for H and 42 breaths for E were required to achieve 100% deposition of 100 nm particles.
4
Discussion
The alveolar sac model geometry used in this study represents
incremental improvement over the experimental models that have
already been presented in the literature. Prior to this work, no
results were available to accurately assess the flow field in both a
healthy and emphysema alveolar sac. This study provides further
convincing evidence that recirculation is not likely present in the
alveolar sacs of the human lung. Although unique geometric models analyzed in this study, the conclusion that flow is reversible is
consistent with Sznitman et al. 关12兴, Tsuda et al. 关10兴, Tippe and
Tsuda 关13兴, Haber et al. 关14兴, Sznitman et al. 关15兴, Cinkotai 关16兴,
and Davidson and Fitz-Gerald 关17兴, when one considers similar
generation numbers, flow rate ratios, and D/MD ratios, respectively. Inconsistencies between our results and reports of recirculation over rigid walls from Karl et al. 关11兴, for similar flow rate
and D/MD ratios, can be explained by the nature of the flow when
Fig. 8 Distance traveled by aerosol particles for a given number of 3 s
inhalations, compared with the maximum distance required to reach the
alveolar wall „using pathline 6 for H and pathline 5 for E… for combination
sedimentation and diffusion †28‡
021008-6 / Vol. 132, FEBRUARY 2010
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induced by the slowly expanding alveolar walls. Negative pressure created by the wall motion draws the fluid toward the alveolar wall in an orderly fashion. By contrast, flow over a rigid backward facing step, creates separation points, low pressure zones,
and induces recirculation.
Further evidence that acinus morphometry and wall motion
play an important role in particle transport through convective
motion was found when analyzing particle penetration into the
alveolar sac. Even though the E model tidal air penetrated the
alveolar sac deeper than the H model, the difference was very
small, and would not likely enhance particle deposition because
the E model did not bring the aerosol particles closer to the wall
compared with the H model. The larger travel distance required
for aerosol particles to reach the wall after inhalation, for the E
model compared with H model can be attributed to a larger effective airway diameters and slower moving fluid in the E model.
Specifically, the velocity gradient between the alveolar wall and
the center of the sac was larger for the healthy model compared
with the emphysemic model, so that more particles were traveling
slower, and therefore not as deep into the emphysemic model
compared with the healthy model for the same breathing period.
Our experimental model geometry was based on average morphometric data from the literature, however, accuracy could be improved by considering the inhomogeneous nature of alveolar size
and shape. Furthermore, slightly larger D/MD ratios than used in
the current study have been reported, however it is unlikely that
the subtle changes in shape would create recirculation zones that
were not observed in the present model nor change the conclusion
from this study that that alveolar sac flow is reversible. A healthy
breathing curve was utilized in this study for both H and E models, in order to isolate the effect of morphology changes in emphysema 共smaller D/MD ratio and larger EAD兲 from pulmonary
function test changes. It is anticipated that differences between
healthy and emphysemic particle transport could be further distinguished by accounting for pathophysiological changes to the
breathing pattern as well.
The significance in the fluid flow results, specifically how close
the tidal fluid travels to the alveolar wall, was amplified by linking
the results to particle transport and deposition. Although a simple
superposition of convective displacement with diffusive displacement and sedimentation was employed, the results are appealing
in that they qualitatively agree with animal studies reporting lower
particle deposition in emphysemic compared with healthy hamsters 关3兴. Certainly, a more complex analysis in which the
convective-diffusion equation is solved in a transient flow field, is
the next step to better quantify the particle deposition differences
in healthy and emphysematous lungs.
5
Conclusion
Our experiments confirmed that irreversible flow is not present
in the alveolar sac of either healthy or emphysemic lungs. The
pathlines for particle motion in the healthy alveoli terminate
closer to the walls then in the emphysemic alveolar sac by approximately 71%. Consequently, the rate of particle diffusion and
sedimentation is postulated to be greater in healthy then emphysemic lungs, and this, in turn means particle deposition will occur
more quickly when lungs are healthy. This study illustrates the
influence of the increased presence of dead space in emphysema
on fluid flow into the alveolar sac. We have confirmed that recirculation does not occur in the alveolar sac, therefore future flow
and particle analysis should include alveoli in more proximal generations that do have irreversible flow behavior. We showed that
expanding walls is necessary in accurately representing acinar
flow, therefore the presence of moving walls in experimental and
theoretical models is required to adequately model acinar flow.
Finally, it is proposed that a more accurate lung deposition model
may be developed by incorporating a convective-diffusion model
for particle deposition coupled with a realistic moving alveolar
geometry.
Journal of Biomechanical Engineering
Acknowledgment
This work was supported by the American Cancer Society
共Contract No. RSG-05-021-01-CNE兲. The authors would like to
thank Nick Daniel, William Dapolito, Chris Salvato, and Matt
Waldron for the design and fabrication of the experimental apparatus and Jackie Russo for the design of the alveolar sac model.
Nomenclature
D ⫽ diameter
E ⫽ error of calculating the pathlines
H ⫽ location of the alveolar sac with respect to the
superior most lobe
P ⫽ pressure
Palv ⫽ pressure in the alveolar sac
Pel ⫽ elastic recoil pressure
Ppl ⫽ pleural pressure
Ps ⫽ pressure scale
Rm ⫽ mean radius of curvature
Re ⫽ Reynolds Number
T ⫽ membrane tension
U ⫽ average velocity at the entrance of the alveolar
sac
Wo ⫽ Womersely Number
d ⫽ interpolated displacement of the velocity vector
e ⫽ RMS PIV analysis error
f ⫽ frequency
g ⫽ gravity vector
k ⫽ numerical time step
t ⫽ time
tint ⫽ integration time
u ⫽ velocity vector
x ⫽ pathline position
z ⫽ local sac height
⌬t ⫽ time separating PIV picture frames
␥ ⫽ lung’s specific gravity
␮ ⫽ dynamic viscosity
␯ ⫽ kinematic viscosity
␳ ⫽ fluid density
␻ ⫽ angular frequency
Appendix A
Fluid scaling and nondimensional analysis was performed for
the alveolar sac. The momentum equation is given by
␳
冋
册
⳵u
+ u ⵜ u = − ⵜP + ␮ⵜ2u + ␳g
⳵t
共A1兲
where ␳ is the density of the fluid, u is the velocity vector, t is
time, P is the pressure, ␮ is the dynamic viscosity and g is the
gravitational vector. The effect of the gravitational term in Eq.
共A1兲 is to adjust the shape of the alveolar sac through the balance
of forces across the alveolar membrane. Specifically, if we assume
a simple membrane model for the alveolar septa, the balance of
forces across the alveolar membrane is given by
Pel = Palv − 关␥共H + z兲 + Ppl兴
共A2兲
where Palv is the pressure in the alveolar sac, the second term in
brackets represents the external forces on the alveolar sac, which
are comprised of the lung’s weight per unit area ␥共H + z兲, where ␥
is the lung’s specific gravity, H is the location of sac with respect
to the superior most lobe, and z is the local sac height and the
pleural pressure Ppl. Pel is the elastic recoil pressure of the alveolar septa and can be defined as the membrane tension T divided by
the local mean radius of curvature Rm 关29兴. It is clear from Eq.
共A2兲 that gravity has little effect on the local deformation of the
alveoli in vivo since z is much smaller than H. Apart from this,
gravity does not impart motion to the air in vivo, as can be demonstrated by the introduction of the stream functions into the
FEBRUARY 2010, Vol. 132 / 021008-7
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Navier–Stokes equations in which gravity is eliminated 关30兴.
Since our experiments involve using fluids whose properties are
the same on both sides of the sac, the boundary conditions across
the membrane are also gravity-independent. Thus, for both in vivo
conditions and model experiments, the effect of gravity on the
fluid motion can be neglected.
The remaining terms in Eq. 共A1兲 were scaled as follows: u
⬇ U, where U is the average velocity at the entrance of the sac
t ⬇ 1 / ␻, where ␻ = 2␲ f and f is the frequency 共Hz兲, ⵜ = 1 / D, P
⬇ Ps, where Ps is an arbitrary pressure scale. The Navier–Stokes
equation can then be written in nondimensional terms as
冋 册 冋 册
冋 册
␳␻D2 ⳵ uⴱ
␳DU
P sD ⴱ ⴱ
关uⴱⵜⴱuⴱ兴 = −
ⵜ P + ⵜⴱ2uⴱ
ⴱ +
␮
⳵t
␮
D␮
共A3兲
ⴱ
where the denotes the nondimensional variables. Three dimensionless parameters are found in the nondimensional equation.
The first parameter, Reynolds number, defined by Re= ␳DU / ␮
determines the role of the steady inertia forces relative to viscous
forces. The second parameter, B = ␳␻D2 / ␮, determines the role of
the nonsteady inertia forces. The Womersley number 关32兴 is defined as Wo= 共D / 2兲冑2␲ f / ␯, which is related to the nondimensional parameter B, B = 4共Wo兲2. The third nondimensional parameter is C = PsD / ␮U, which defines the resulting scale of Ps.
The Reynolds and Womersley numbers for in vivo alveolar sacs
are much less than unity, therefore, the left-hand side of the
Navier–Stokes equation is small compared with the viscous terms.
Since pressure is required to interact with dominant flow terms,
the pressure gradient term balances the viscous term and the parameter C is chosen to be 1, which sets the pressure scale as Ps
= ␮U / D. In the limit of small Wo and Re, then, the Navier–Stokes
equation thus reduces to
− ⵜⴱ Pⴱ + ⵜⴱ2uⴱ = 0
共A4兲
which when coupled with the continuity equation yields Stokes’
description for steady flow.
Appendix B
The Lagrangian pathline calculation and error analysis was performed. The position x at any numerical step k was calculated by
numerical integration in time according to
k max
xk = x0 +
兺 ut
k int
共B1兲
k=1
where x0 was the initial position, tint was the integration time, the
summation term is the total distance traveled and k may be related
to real time t by t = k ⫻ tint. Several integration steps occur for each
measurement file and each measurement file is only used when t
corresponds to the time of that particular measurement. For example, if the time separating the data files was 0.1 s and the tint
was 0.01 s there would be m = 10 integration steps for each vector
field. The velocity for each velocity vector solution file was defined as
uj =
dj + e
⌬t j
共B2兲
where d j was the interpolated displacement for the velocity vector
solution time, e was the assumed constant RMS 0.1 pixel PIV
error 关27兴 and ⌬t was the time separating the frames used for the
cross correlation, which ranged between 0.02 s and 0.2 s. The d
and ⌬t varied for each velocity vector solution file; the ⌬t was
chosen such that e would be much smaller than d but small
enough so that the PIV particles did not travel outside of the
interrogation region. The integration error was minimized by decreasing the tint until convergence was met in the pathline distance
021008-8 / Vol. 132, FEBRUARY 2010
calculation. The pathline calculation error, independent of integration error, was given by
k
Ei+1,j+1 =
m
兺 兺 ⌬t t
j=1 i=1
e
int
共B3兲
j
Equation 共B3兲 was utilized to access the error in pathline calculations. The average d was about 2.38 pixels, the average ⌬t was
0.13 s, the tint was 0.2 s, the e was 0.1 pixels, and the initial
displacement was 0 pixels. The time separating the velocity vector
files was 0.4 s, therefore, m was 2. The x2,1 and x3,1 were calculated to be 3.79 pixels each leading to the total displacement for
that vector file to be 7.57 pixels. The d would be 7.27 pixels if Eq.
共B1兲 was solved without e. By subtracting 7.27 from 7.57, the E
for each data file is 0.3 pixels. By multiplying the E by k = 20, the
total maximum error would be 6 pixels or 0.39 mm for each
pathline 共0.195 mm each for inhalation and exhalation兲.
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