M. Malvè
e-mail: [email protected]
A. Pérez del Palomar
Group of Structural Mechanics and Materials
Modeling,
Aragón Institute of Engineering Research (I3A),
Universidad de Zaragoza,
C/María de Luna s/n,
E-50018 Zaragoza, Spain;
Centro de Investigación Biomédica en Red en
Bioingeniería, Biomateriales y Nanomedicina ,
C/Poeta Mariano Esquillor s/n,
50018 Zaragoza, Spain
S. Chandra
Institute for Complex Engineered Systems,
Carnegie Mellon University,
1205 Hamburg Hall, 5000 Forbes Avenue,
Pittsburgh, PA 15213
J. L. López-Villalobos
Department of Thoracic Surgery,
Hospital Virgen del Rocío,
Avenida de Manuel Siurot s/n,
41013 Seville, Spain
A. Mena
Group of Structural Mechanics and Materials
Modeling,
Aragón Institute of Engineering Research (I3A),
Universidad de Zaragoza,
C/María de Luna s/n,
E-50018 Zaragoza, Spain;
Centro de Investigación Biomédica en Red en
Bioingeniería, Biomateriales y Nanomedicina ,
C/Poeta Mariano Esquillor s/n,
50018 Zaragoza, Spain
E. A. Finol
Institute for Complex Engineered Systems ,
Carnegie Mellon University,
1205 Hamburg Hall, 5000 Forbes Avenue,
Pittsburgh, PA 15213
FSI Analysis of a Healthy and a
Stenotic Human Trachea Under
Impedance-Based Boundary
Conditions
In this work, a fluid-solid interaction (FSI) analysis of a healthy and a stenotic human
trachea was studied to evaluate flow patterns, wall stresses, and deformations under
physiological and pathological conditions. The two analyzed tracheal geometries, which
include the first bifurcation after the carina, were obtained from computed tomography
images of healthy and diseased patients, respectively. A finite element-based commercial
software code was used to perform the simulations. The tracheal wall was modeled as a
fiber reinforced hyperelastic solid material in which the anisotropy due to the orientation
of the fibers was introduced. Impedance-based pressure waveforms were computed using
a method developed for the cardiovascular system, where the resistance of the respiratory
system was calculated taking into account the entire bronchial tree, modeled as binary
fractal network. Intratracheal flow patterns and tracheal wall deformation were analyzed
under different scenarios. The simulations show the possibility of predicting, with FSI
computations, flow and wall behavior for healthy and pathological tracheas. The computational modeling procedure presented herein can be a useful tool capable of evaluating quantities that cannot be assessed in vivo, such as wall stresses, pressure drop, and
flow patterns, and to derive parameters that could help clinical decisions and improve
surgical outcomes. 关DOI: 10.1115/1.4003130兴
A. Ginel
Department of Thoracic Surgery,
Hospital Virgen del Rocío,
Avenida de Manuel Siurot s/n,
41013 Seville, Spain
M. Doblaré
Group of Structural Mechanics and Materials
Modeling,
Aragón Institute of Engineering Research (I3A),
Universidad de Zaragoza,
C/María de Luna s/n,
50018 Zaragoza, Spain;
Centro de Investigación Biomédica en Red en
Bioingeniería, Biomateriales y Nanomedicina ,
C/Poeta Mariano Esquillor s/n,
E-50018 Zaragoza, Spain
Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received June 7, 2010; final manuscript
received November 16, 2010; accepted manuscript posted November 29, 2010; published online January 5, 2011. Assoc. Editor: Dalin Tang.
Journal of Biomechanical Engineering
Copyright © 2011 by ASME
FEBRUARY 2011, Vol. 133 / 021001-1
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1
Introduction
The trachea begins immediately below the larynx and runs
down the center of the front part of the neck, ending behind the
upper part of the sternum. Here, it bifurcates in the left and right
main bronchus that enters the lung cavities. The trachea forms the
trunk of an upside-down tree and is flexible, so that the head and
neck may twist and bend during the process of breathing 关1兴. The
main components that constitute the trachea are fibrous and elastic
tissues and smooth muscle, which runs longitudinally and posteriorly to the trachea, with about 20 rings of cartilage, which helps
the trachea to keep it open during extreme movement of the neck
关2兴. The main role of the tracheal cartilaginous structures is to
keep the windpipe open despite the interthoracic pressure during
breathing, sneezing, or coughing 关3兴. Under these conditions,
among others, the smooth muscle collapses to regulate the air flow
by dynamically changing the tracheal diameter. This contraction
and the transmural pressure generate bending and tensile stresses
in the cartilage.
Although a clear understanding of the respiration process has
not yet been reached and the influence of the implantation of a
prosthesis is very important from a medical point of view as well
as challenging from the computational aspect, a few studies have
analyzed the behavior of the tracheal walls under different ventilation conditions. Most numerical analyses of the respiratory system have been focused purely on computational fluid dynamics
共CFD兲, neglecting the interaction with the wall airways. In particular, most works dealt with the airflow patterns using idealized
关4,5兴 or approximated airways geometries 关6–8兴 while only few
studies were based on accurate airway geometries developed from
computed tomography 共CT兲 or magnetic resonance images
共MRIs兲 关9–14兴. While these studies, following the CFD approach,
failed to incorporate constitutive material modeling of the tracheal
wall 关6–8,15–17兴, recent studies in lower airway geometries have
successfully presented FSI results using simplified models/
geometries for single bifurcations 关18–21兴. Wall and Rabczuk 关22兴
analyzed the behavior of a CT-based respiratory system through a
FSI analysis during breathing and mechanical ventilation even
using a simplified model for the composing material. Moreover,
Koombua and Pidaparti 关23兴, following a FSI approach, studied
the stress distributions in a two-bifurcation tracheal model and
compared the relative differences between an isotropic and an
orthotropic material model of the airways. Regarding the constitutive behavior of the tracheal walls, there is in fact a large dispersion of the mechanical properties of the different composing
tissues and only a few studies have analyzed their mechanical
behavior for humans 关24–26兴. Abnormal airways deformation and
alteration of airway mechanical properties were previously reported as a potential cause of tissue damage and associated to
chronic complications, as reported by Spitzer et al. 关27兴. In most
of the works on this topic, the isolated tracheal cartilage was
considered as a linearly elastic isotropic material 关28兴. In many of
the previous studies, the human tracheal smooth muscle dealt with
its plasticity, stiffness, and extensibility and the influence of the
temperature on force-velocity relationships. Begis et al. 关29兴 developed a structural finite element model of an adult human tracheal ring, accounting for its morphometric features and mechanical properties. More recently, Costantino et al. 关28兴 analyzed the
collapsibility of the trachea under different pressure conditions by
comparing the numerical results with experimental tests. Regarding tracheal pathologies, Brouns et al. 关30兴 provided a CFD numerical model of a healthy trachea without any bifurcation, in
which different stages of artificial stenosis were imposed, with the
aim to establish a relationship between local pressure drop and
velocity depending on the degree of stenosis. Sera et al. 关31兴 investigated the mechanism of wheeze generation with experimental
tests conducted on a rigid and an extensible realistic CT-based
tracheostenotic model. Cebral and Summers 关32兴 showed how virtual bronchoscopy can be used to perform aerodynamic calculations in anatomically realistic models.
021001-2 / Vol. 133, FEBRUARY 2011
1
1
2
2
healthy
stenosis
(b)
(a)
Fig. 1 Computational grids of the „a… healthy and „b… stenotic
tracheas with the respective computational fluid „1… and solid
„2… grids
For both CFD and FSI simulations, the choice of the boundary
conditions is critical. Most studies used velocity or flow conditions at the entrance of the trachea as inlet boundary conditions
and usually utilize approximations such as time dependent resistance 关8,22兴, zero-pressure 关9–11兴, or simply outflow conditions
关8,9,30兴 as outlet boundary conditions. Recently, Wall and coworkers proposed a structured tree for modeling the human lungs
and analyzing the respiration and mechanical ventilation under
impedance conditions 关33–35兴. Finally, Elad et al. 关36兴 proposed a
nonlinear lumped-parameter model to study the dependency of
airflow distribution in asymmetric bronchial bifurcations on structural and physiological parameters. In this study, we analyzed the
tracheal wall stresses and the flow patterns through a study of
healthy and stenotic human tracheas during inhalation using a FSI
approach. In this analysis, we gave special emphasis to the solid
domain in order to predict stresses and displacements under physiological conditions. The aim of the this work is to create a functional method to simulate and classify the tracheal wall stress and
strain. Understanding the mechanical behavior of the central airways may allow for instance the investigation of their response to
the mechanical ventilatio-associated pressure loads, which is important for clinical applications and for stenting technique. In this
sense, our study can be considered as a step toward building a
framework that could aide surgical decisions in future. Moreover,
we tackle the challenging aspect of the physiological boundary
conditions by modeling the resistance of the entire respiratory
system through impedance-based boundary conditions. This approach allows the computation of physiological pressures that are
not measureable in vivo starting from patient specific spirometry.
Computed pressures waveforms are essential for the evaluation of
stresses and strains during breathing and coughing, especially for
diseased and stented tracheas. The method followed, developed by
Olufsen 关37兴 and later extended by Steele et al. 关38兴, has been
applied to the cardiovascular systems 关39–43兴 and to the pulmonary system 关33–35兴.
2
Materials and Methods
2.1 Fluid-Airflow. Thoracic CT scans were taken from a 70
year old healthy and a 56 year old diseased man and the images
were then segmented and post-processed to create the fluid model.
An initial graphics exchange specification 共IGES兲 file of the computational fluid model of the internal tracheal cavity was created
from these segmented images. The model was then meshed with
unstructured tetrahedral-based elements using the commercial
software FEMAP 共Siemens PLM Software, Plano, TX兲 共see Fig. 1兲.
The fluid was assumed as Newtonian 共␳F = 1.225 kg/ m3, ␮
= 1.83⫻ 10−5 kg/ m s兲 and incompressible under unsteady flow
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1
1
2
2
a)
b)
20
1
1
12
10
15
v [m/s]
v [m/s]
8
10
100000
6
5
250000
0
5
10
200000
250000
2
300000
0
100000
4
200000
15
diameter [mm]
300000
0
0
20
2
4
6
8
10
12
14
10
12
14
diameter [mm]
12
20
10
15
v [m/s]
v [m/s]
8
10
100000
200000
250000
5
0
0
5
10
diameter [mm]
100000
200000
250000
300000
4
2
300000
2
6
15
20
2
0
0
2
4
6
8
diameter [mm]
Fig. 2 Axial velocity profiles through selected orthogonal tracheal sections for different mesh sizes at peak flow during
inspiration
conditions 关6兴. Flow was assumed turbulent for the analyzed cases
since the Reynolds numbers, based on the median tracheal section, at peak velocity, were in the turbulent regime 共Rehealthy
= 30,000, Restenosis = 10,000兲. To ensure that the results were insensitive to the computational grid size, a sensitivity study was carried out. Four different meshes of about 100,000, 200,000,
250,000, and 300,000 elements were tested for the healthy and
stenotic tracheas, studying the inspiratory peak flow to ensure that
the results at the highest Reynolds number were grid independent
关6兴. Comparison between four grid sizes revealed that the velocity
profiles for the grid of 200,000, 250,000, and 300,000 elements
were found almost coincident 共see Fig. 2, section 1兲. However, the
velocity profile near the wall was found to be smoother for these
three meshes with respect to the coarser mesh. This indicates that
the higher number of grid cells within the wall region can capture
the flow field in more detail than that for the coarser grid size. The
difference between coarser and finer grids is also noticeable as the
air flows downstream in the left bronchus 共see Fig. 2, section 2兲.
As finer grid increases the computing time considerably, we selected the grids 共healthy and stenotic trachea兲 with a total number
Journal of Biomechanical Engineering
of tetrahedral elements of about 200,000 for further simulations.
The velocity profiles obtained from the grids of about 200,000,
250,000, and 300,000 show a difference of only 2–3%. Due to the
turbulent nature of the considered tracheal flow, which is basically
transitional rather than fully turbulent, and taking into account the
limitation of the use of the K-␧ model for its modeling, as documented in Refs. 关44,45兴 among others, the standard K-␻ model
关46兴 was used to modify the air viscosity. However, it is not clear
if Reynolds averaged Navier-Stokes 共RANS兲-based turbulence
models are a good choice or not. Due to the recirculatory flow and
turbulent eddies caused by the laryngeal jet, Gemci et al. 关47兴, for
instance, used a large eddy simulation 共LES兲 turbulence model in
their study. In particular, in the Navier–Stokes equations, the viscosity ␮ is substituted by ␮0 = ␮ + ␮t, where ␮t is the turbulent
viscosity, which is computed by Eq. 共1兲,
K
␮t = ␣␳F
共1兲
␻
where ␣ is a constant of the model, expressed as a function of the
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Table 1 Parameters used to define the turbulence K-␻ models
␣
␣␻
␤K
␤␻
␴K
␴␻
␴␪
␤␪
1
0.555
0.09
0.075
2
2
1
0.712
Reynolds number 关46兴. The variable ␻ is the specific dissipation
rate of turbulence. This is related to the so-called kinetic energy
and rate of dissipation of the turbulence 共K and ␧, respectively兲
and is defined as follows:
␻⬃
␧
K
共2兲
␧=
␮
共ⵜv⬘兲 丢 共ⵜv⬘兲
␳F
共3兲
with
1
K = v ⬘ · v ⬘,
2
where v⬘ is the fluctuating velocity. K and ␻ are governed by Eqs.
共4兲 and 共5兲,
共⳵ y a␳K兲
+ ⵜ · 关y a共␳vK − qK兲兴 = y aGK
⳵t
共4兲
共⳵ y a␳␻兲
+ ⵜ · 关y a共␳v␻ − q␻兲兴 = y aG␻
⳵t
共5兲
where a is 0 for two- and three-dimensional flows and 1 for axisymmetric flows. Other corresponding terms are defined as follows:
冉
冉
冊
冊
␮t
qK = ␮0 +
ⵜK
␴K
共6兲
␮t
ⵜ␻
␴␻
共7兲
q␻ = ␮0 +
G K = 2 ␮ tD 2 − ␤ K␳ K ␻ + B
G␻ =
of the trachea were analyzed and numerically modeled through
different experimental tests described in previous studies 关24,48兴.
Here, we will provide only a short explanation. Human tracheas
were obtained through the autopsy from two subjects 共aged 79–82
years兲 and subjected to different tensile tests to obtain the material
models. The cartilaginous rings were modeled as isotropic material 关24,48兴. The muscular membrane presented two orthogonal
families of smooth muscle cells, one running mainly longitudinally and the other transversely. Therefore, for its behavior, the
anisotropy coming from the fiber orientations was introduced in
the material model. For cartilage, since there is no preferential
orientation, a neo-Hookean model with strain density energy function 共SEDF兲, ⌿ = C1共Ī1 − 3兲, was used to fit the experimental results. Regarding the smooth muscle, and taking into account that
the histology showed two orthogonal fiber families, the Holzapfel
SEDF 关49兴 for two families of fibers was used,
− 1兲2兴 − 1其 +
共9兲
where ␣, ␣␻, ␤K, ␤␻, ␴K, ␴␻, ␴␪, and ␤␪ are model constants.
These values, input to the finite element solver at the beginning of
the simulation, are displayed in Table 1. A complete description of
this turbulence model may be found in Ref. 关46兴.
2.2 Solid-Tracheal Wall. In order to determine the real geometry of the trachea and to distinguish between the muscular
membrane and the cartilage rings, we proceeded to a nonautomatic segmentation of the CT scans using MIMICS©. Each tracheal constitutive part could be detected through its different density. Finally, with the commercial software ABAQUS©, a full
hexahedral mesh of about 60,000 elements for the healthy trachea
and of about 90,000 elements for the stenotic trachea was generated. In Fig. 1, these grids are shown. The stenosis, which is a
rigid fibrous cap surrounding the internal tracheal wall, due to its
irregular and asymmetric geometry was meshed with about
40,000 tetrahedral elements. The properties of the different tissues
1
共J − 1兲2
D
where C1 is the material constant related to the ground substance,
Ki ⬎ 0 are the parameters that identify the exponential behavior
due to the presence of two families of fibers, and D is identified
with the tissue incompressibility volumetric modulus. The invariants Iij are defined as
Ī1 = tr C̄
共8兲
␻
共2␣␻␮tD2 − ␤␻␳K␻ + ␤␪B兲
K
K1
K3
兵exp关K2共Ī41 − 1兲2兴 − 1其 +
兵exp关K4共Ī42
2K2
2K4
⌿ = C1共Ī1 − 3兲 +
1
Ī2 = 2 关共tr C̄兲2 − tr共C̄2兲兴
Ī41 = a0 · C̄a0
Ī42 = b0 · C̄b0
0
where a is a unitary vector defining the orientation of the first
family of fibers, b0 is the direction of the second family both in
the reference configurations, and C̄ is the modified right Green
strain tensor defined as C̄ = J1/3C being C = FTF. In Table 2, a
summary of the material constants used for the different tissues is
shown. For further explanations, see Refs. 关24,48兴. Not much information is available about the constitutive properties of the tracheal stenotic tissue. For this reason, this fibrous tissue was modeled as a hyperelastic material, in which we assumed an increased
stiffness of about 2 times with respect to the muscular membrane
共see Table 2兲. In Fig. 3, the different tracheal constitutive tissues
are separately shown for the healthy 共a兲 and stenotic 共b兲 models.
2.3 Numerical Method and Boundary Conditions. To facilitate the fluid dynamic model, in order to apply uniform velocity field as inflow boundary condition, a five-inlet extension was
added to each model. In this way, a fully developed airflow is
assessed through the trachea. Moreover, to take into account the
effect of the laryngeal jet on the tracheal flow 关50–52兴, the tra-
Table 2 Parameters used to define the constitutive models that characterize the mechanical
behavior of cartilage, muscular membrane, and stenosis
Material parameters
C1
共kPa兲
K1
共kPa兲
K2
K3
共kPa兲
K4
D
共kPa兲
Cartilage
Muscular membrane
Stenosis
577.7
0.87
1.74
—
0.154
0.308
—
34.15
68.3
—
0.34
0.68
—
13.9
27.8
4840
8.108
17.4
021001-4 / Vol. 133, FEBRUARY 2011
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governing equations were solved with the finite element method
共FEM兲. The fluid domain employs special flow-condition-basedinterpolation 共FCBI兲 tetrahedral elements 关46兴. We used the formulation with large displacements and small strains in the FSI
computation available in ADINA. As an iteration, we used the full
Newton method while a sparse matrix solver based on the Gaussian elimination is used for solving the equation system 关46兴.
3
The Impedance-Based Method
3.1 Airways Fractal Network. In this work, we modeled the
respiratory system as a structured fractal network in which we
predicted physiological airflow and pressure profiles. The fractal
network is modeled adapting the 1D approach used in the cardiovascular system to the pulmonary system. In this case, the airways
are modeled as compliant segments using the linearized 1D continuity and momentum equation 共see Refs. 关37,38,43兴兲,
⳵A ⳵q
+
=0
⳵t ⳵x
共10兲
where p is the pressure, A is the bronchial cross sectional area,
and q the flow rate in x-direction, and
冉 冊
⳵ ux
⳵ ux 1 ⳵ p ␯ ⳵ ⳵ ux
⳵ ux
+ ur
+
=
r
+ ux
⳵x
⳵r ␳ ⳵x r ⳵r ⳵r
⳵t
Fig. 3 Finite element model of the „a… healthy and „b… stenotic
tracheas with their respective constitutive parts separately
plotted „A denotes anterior part and P denotes posterior part…
cheal inlet extension was biased toward the rear wall 关11,50兴.
Outlet conditions for tracheal bifurcations are critical and influence the computational solution. While some studies used to set
zero-pressure condition at the outlets of the respective truncated
generation 关7–12兴, the pressure at each outlet must be known a
priori, reflecting the effect of the remaining truncated generation
at the alveolar level 关47兴. An alternative approach could be to set
the flow ratios in each tracheal branch 关5,8兴. Luo and Liu 关11兴
tested both zero-pressure and outflow conditions, finding that outflow conditions forced the air flow rate to be equal on each outlet.
Gemci et al. 关47兴 modeled an almost complete anatomical replica
of the pulmonary tree down to the 17th generation where they
imposed a constant outflow condition p = 0.
In this study, we follow the approach used in hemodymanics by
Olufsen et al. 关37,53兴 and extended by Steele et al. 关38兴: Starting
from patient-specific spirometries performed by the two patients,
we modeled the entire respiratory system as a fractal network in
which we computed the impedance-based pressure waveforms
driven by the procedure explained in the next section. The spirometry, used as an input of the impedance computation, is a forced
breathing test that consists of enforcing the respiration, blowing
off air after a long deep inspiration in order to measure the pulmonary flow, possible bronchial obstructions and, in particular, for
the diseased patient, to get the percentage of pulmonary capacity
acquired after stent implantation. At the tracheal internal wall,
which represents the fluid-solid interface, a typical no-slip condition was applied. Finally, as in previous studies 关22兴, the movement of the solid mesh was restrained by fixing the tracheal top
surface. The coupled fluid and structure models were solved with
the commercial package ADINA 共v.8.5, ADINA R&D Inc.兲. The
Journal of Biomechanical Engineering
共11兲
where ux is the velocity in x-direction, t is the time, p is the
pressure, ␳ is the density, ␯ is the viscosity, and r is the bronchial
radius. In this 1D model, we predicted impedance and pressure
waveforms that were applied to the 3D tracheal model as outflow
conditions. Equations 共10兲 and 共11兲, as explained in Sec. 3, were
coupled into an expression for the impedance Z using the Womersley solution of an oscillatory flow in a cylindrical tube.
The binary representation of airways can be considered as a
considerable approximation of the human lung complexity since
the respiratory system is defined with the help of general rules that
limits the precision of the tree. Weibel 关4兴 presented a similar
approach and he incorporated morphometric measurements in his
model. He created a symmetrically branching tree with defined
diameters and length for each branch. A systematic recursive morphometric system was also developed by Horsfield and Woldenberg 关54兴. Modeling human airways as a dichotomously branching
tree based on morphometric data was previously reported by
Raabe et al. 关55兴. In that study, in order to keep track of the
position of the airways, a system of binary identification number
was created. Each daughter branch was designated either a minor
or major daughter depending on its diameter. These data were
used in many studies to describe and analyze the human lung
morphometry. However, such a model was not coupled to computational model to predict airflow and lung mechanical strain and
stresses. Based on these data, Yeh et al. 关56兴 developed a model
that identified single pathways without connectivity to represent
the lung. They provided a dichotomous model for the rat, which
included lengths, diameter, and airway number of 23 generations.
More recently, based on the Raabe and Yeh data, Latourelle et al.
关57兴 presented a computational approach of a branching airway
tree that allows for one-to-one mapping of the airway anatomy
when predicting the overall lung mechanical properties. Finally,
Comerford et al. 关33,34兴 presented a symmetric and an asymmetric impedance-based structured tree for lungs simulations in which
the method of Olufsen was applied.
In this paper, we presented a model in which two different parts
can be distinguished: the trachea and the right and left bronchi. In
the main bronchi, a relationship between flow and pressure represents the outflow boundary conditions for the trachea. In both
trachea and bronchi, the air flow and pressure were calculated
using the incompressible axisymmetric Navier–Stokes equations
for a Newtonian fluid. A fractal network for each main bronchus
was built prior to the impedance computation based on the radius
of each respective bronchus. Once these two binary trees are creFEBRUARY 2011, Vol. 133 / 021001-5
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ated, two pressure waveforms are computed using the method
developed by Olufsen et al. 关37,53兴. These pressures are later
applied at the outlets of each FSI model.
The bronchial network was modeled as a binary asymmetricstructured tree in which each branch was approximated by a
straight compliant segment. The airways network resulted in a
series of bifurcations composed by a series of parent and daughter
bronchi, as shown in Fig. 4共a兲. Each parent bronchus bifurcates in
two daughter bronchi following a scaling guided by the asymmetry factors ␣ and ␤ of the root parent rroot according to Eqs. 共12兲
and 共13兲,
ri,j = ␣i␤ j−irroot,
共r0兲d1 = ␣共r0兲pa
0ⱕiⱕj
共12兲
共r0兲d2 = ␤共r0兲pa
共13兲
The structured tree continues branching until the radius of any
bronchial segment is less than given minimum values rmin, as for
instance the alveolar radius 共⬇10– 100 ␮m兲 where we assumed
zero impedance. Although we know that during inspiration and
expiration, the alveolar pressure sinusoidally varies between
−1 cm H2O and +1 cm H2O 关1兴, the assumption of zero impedance at the alveolar radius is clearly more appropriate than the
assumption of zero-pressure applied in other works at intermediate generations 关7,9–12兴.
In this study, we followed the approach used by Steele et al.
关38兴, dividing the entire bronchial network in three different levels, to mimic the respiratory system structure. For each level, the
parameters describing the fractal tree were varied. The minimum
radius was set at the alveolar level 共10 ␮m兲 where, as discussed,
we assumed zero impedance. In the Table 3, the starting radii of
the considered trachea models used as the input of the fractal tree
scaling are shown. The asymmetry and area ratios of the bronchial
network 关57兴 were defined as 关37兴
␩=
共r0兲d21 + 共r0兲d22
共r0兲2pa
␥=
冉 冊
r0d2
r0d1
2
共14兲
where r0d1 and r0d2 are the radii of the daughter bronchi and rpa is
the radius of the parent bronchus, while ␩, the area ratio, and ␥,
the asymmetry ratio, are related to each other through the expression 关37兴
␩=
1+␥
共1 + ␥␰/2兲2/␰
共15兲
␥ is known also as asymmetry index and describes the relative
relationship between the daughter bronchi. The exponent ␰ is
known in the cardiovascular literature as the power exponent and
comes from the power law 关59兴,
␰
␰
共r0兲␰pa = 共r0兲d1
+ 共r0兲d2
共16兲
This relation is valid for a range of flows 关37兴. Several studies
show that ␰ varies between 2 and 3 关38,59,60兴. The power exponent ␰ = 3 is optimal for laminar flows while ␰ = 2.33 is optimal for
turbulent flows 关37兴.
Assuming that rd1 ⱕ rd2 关57兴, ␥ is between 0 and 1. The value of
␥ widely varies from humans to dogs and rodents as documented
by Latourelle et al. 关57兴. Based on the morphometry of the models
described in Refs. 关4,54,61兴, in this work, we chose to vary the
value of asymmetry index with ␥ = 0.85, 0.9, and 0.95 共see Table
4兲. The value of ␩ in Eq. 共15兲 describes the branching relationship
across bifurcations between the radius of the parent bronchi rpa
and the radii of the daughter bronchi rdi, i = 1 , 2. Using the mentioned values of ␥ and the power-exponent ␰ = 2.33, we calculated
the values of ␩ for each respective level 共see Table 4兲. The scaling
parameters are obtained from the following expressions 共see Table
4兲:
␣ = 共1 + ␥␰/2兲−1/␰
␤ = ␣冑␥
021001-6 / Vol. 133, FEBRUARY 2011
共17兲
Finally, the length of a given branch L can be related to the
radius r0 of each branch segment through the parameter lrr
= L / r0 关38,53兴. Morphometric and anatomic data for humans and
rodents are described in Refs. 关4,6,55,58,62兴; these works showed
that this parameter rapidly varies from the trachea to the bronchioles, until the alveolar level 共from around 12 to 2 关1,4兴兲. Based on
these studies, we finally fixed a constant average value of lrr = 6
关1,4,54,61兴.
4
Impedance Recursive Computation
Impedance was computed in a recursive manner starting from
the terminal branch 关38,53兴 of the structured tree and was used as
the outlet boundary condition for the tracheal left and right bronchi. The details of the recursive calculation are given elsewhere
关37,38,53兴. Input impedance was evaluated at the beginning of
each airway daughter z = 0 as a function of the impedance at the
end of the airway daughter z = L:
Z共0, ␻兲 =
ig−1 sin共␻L/c兲 + Z共L, ␻兲cos共␻L/c兲
cos共␻L/c兲 + igZ共L, ␻兲sin共␻L/c兲
共18兲
where L is the bronchial length, c = 冑s0共1 − FJ兲 / 共␳FC兲 is the wavepropagation velocity, g = 冑CA0K / ␳F, and
Z共0,0兲 = lim Z共0, ␻兲 =
␻→0
8␮lrr
␲r30
+ Z共L,0兲,
FJ =
2J1共w0兲
w0J0共w0兲
共19兲
J0共x兲 and J1共x兲 being the zeroth and first order Bessel functions
with w0 = i3w, w2 = ␳Fr20␻ / ␮, i is the complex unity, and w is the
Womersley number. The compliance C is approximately given by
C = 3A0r0 / 2Eh, where A0 is the cross sectional area and r0 is the
bronchus radius corresponding to section A0, while E is the Young
modulus, whose value is 3.33 MPa according to the experimental
study of Trabelsi et al. 关48兴. In Fig. 4, the spirometry 共b兲 with the
corresponding left and right bronchus pressure waveforms 共c兲
used in the computations are sketched. It has to be noted that in
this figure, inspiratory flows are assumed as positive while expiratory flows are assumed as negative.
5
Results and Discussion
The airflow inside the trachea is analyzed at peak inspiration
flow through bidimensional streamlines projected on selected tracheal longitudinal sections. Velocity distributions are plotted for
each section to give a complete description of the flow patterns in
the trachea during inhalation. The air flow, which is oscillatory, is
governed by the Womersley number. This characteristic number
represents a nondimensional frequency in oscillatory flows defined by Eq. 共20兲,
w=
D
2
冑
2␲ f
␯
共20兲
with the forced breathing frequency f, the kinematic air viscosity
␯, and the diameter D of the trachea. The Womersley number for
the two patient-specific models is showed in Table 5. Due to the
high flow rates obtained from the patient-specific spirometry
共these are forced breathing rather than normal breathing兲, w is
higher than what is reported in other studies 关6,13,61兴. Noteworthy is that although the two spirometries have different frequencies, w was nearly identical for both models. With regard to the
airway mechanics, wall stresses and displacements are analyzed at
peak inspiration to show the function of the different tracheal
tissues and the overall behavior of the trachea during inspiration.
5.1 Flow Patterns in the Healthy Trachea. The flow inside
the healthy trachea is shown during inhalation with its respective
pressure distribution in Fig. 5. During the inspiration, the trachea
shows a strong primary axial flow resulting in a flattened velocity
profile 共in the order of 10 m/s兲 due to the turbulence 共see Fig.
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root
(a)
α
α
β
2
αβ
α
2
α2β
αβ
4
α3 β
2
αβ
3
2
α
β
αβ
α3 β
3
αβ
2
β
α 2β
αβ2
2 2
αβ
unit bifurcation
daughter 1
a
i
a b
i+j (j+1)−(i+1)
b
j
parent
i (j+1)−1
a b
daughter 2
right bronchus
left bronchus
v [m/s]
Flow [l/s]
5
0
20
60
10
40
0
−10
right bronchus
left bronchus
20
p [cmH20]
10
0
−20
−5
−20
−10
0
2
4
6
time [s]
8
10
−30
0
12
2
−40
2
4
6
time [s]
8
10
12
−60
0
10
2
4
6
time [s]
0
−1
−5
−2
0
−10
0
12
right bronchus
left bronchus
4
p [cmH20]
v [m/s]
Flow [l/s]
5
0
10
8
6
1
8
2
0
−2
−4
2
4
time [s]
6
8
(b)
2
4
time [s]
(c)
6
8
−6
0
1
2
3
4
time [s]
5
6
7
(d)
Fig. 4 „a… Structured tree „adapted from Olufsen †37‡…, „b… airflow, „c… inlet velocity, and „d… computed pressure
waveforms „healthy trachea up, stenotic trachea bottom…
Journal of Biomechanical Engineering
FEBRUARY 2011, Vol. 133 / 021001-7
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Table 3 Radii of the CT-based trachea model used as input of
the fractal tree and impedance recursive computation
Table 5 Forced breathing frequencies and corresponding
Reynolds and Womersley numbers
Trachea
Tracheal
diameter 共m兲
Right bronch.
diameter 共m兲
Left bronch.
diameter 共m兲
Trachea
Frequency
共Hz兲
Re
w
Healthy
Stenotic
0.023
0.018
0.016
0.015
0.0125
0.0095
Healthy
Stenotic
0.097
0.159
30,000
10,000
2.32
2.33
6共b兲, section 1兲. Up to the bifurcation the flow is nearly parabolic,
while after the flow divider, the stream has split as the fluid moves
toward the daughter tubes. The high axial flow intensity 共8 m/s兲
reduces the effect of the geometrical bending curvature so that a
fully developed Dean-flow pattern is not present, as documented
in Refs. 关9–11兴. Also, and as found in other studies 关8,22,63,64兴,
the right main bronchus shows a typical M-shaped velocity profile, which is not fully developed probably due to the limited
length of bronchus and its particular geometry 共Fig. 6共b兲, section
4兲; this typical distribution is normally associated to a high Reynolds number as documented in Ref. 关22兴 among others. For the
same reason, in the left main bronchus the air flow possesses a
high axial component, whose velocity is in the order of 6 m/s.
Also here the typical Dean-flow pattern, found by Refs.
关11,22,65兴, cannot be observed 共see Fig. 6共a兲, section 3兲. The reason of these differences on the secondary flows can be explained
considering the higher inlet velocity we used in comparison with
other works. For the present computations we used in fact a forced
respiration extracted from a patient specific spirometry, while
other studies 关9–11兴 on CFD with rigid wall and Wall and Rabczuk 关22兴 on the FSI approach used velocity data extracted from
normal breathing. Finally, the velocity distribution in the left main
bronchus is slightly skewed toward the outer wall 共see Fig. 6共b兲,
section 3兲.
5.2 Wall Stresses and Deformations in the Healthy
Trachea. Results corresponding to the wall deformation during
the different phases of the inspiration are shown in Figs. 7共a兲 and
7共b兲, corresponding to the maximum logarithmic strain in the circumferential direction. This illustrates the radial deformation of
the tracheal walls as a function of their initial configuration. The
highest deformation 共10% at t = 2 s兲 appeared in the muscular
membrane, which increases the section of the trachea along its
longitudinal axis during inhalation and decreases it during expiration contributing to the pressure-balance inside the lungs, as documented in Ref. 关2兴. Besides, the deformation of tracheal rings was
smaller than that of the muscular membrane. In Fig. 7共c兲, the
amplified deformed shape of the tracheal rings at inspiration is
shown. The cartilage rings opening, as shown in the figure, allows
muscular membrane dilatation. In particular, in the figure, an increase of the cartilage expansion during inspiration from 0 s to 2
s and a reduction of this from 2 s to 4 s can be seen, at the end of
which the expiratory process begins. This expansion is basically
due to the positive pulmonary pressure 关1,2兴. In Fig. 9, the principal stresses are shown. The maximum registered value is 3.3
KPa. At inspiration, the muscular membrane is in tension. The
muscular deflection generates tensile stresses in the cartilage and
modulates the airway diameter as reported also in Ref. 关3兴. If only
the cartilaginous rings of the trachea are shown, it can be seen
how they absorbed most of the load, as shown in Fig. 7.
5.3 Flow Patterns in the Stenotic Trachea. In the diseased
trachea, strong modifications of the local airway flow patterns
were found due to the elevated pressure drop caused by the stenosis, as shown in Fig. 5共b兲, where the pressure distribution of the
two analyzed models are compared. The pressure drop between
the trachea entrance and the stenosis is around 8 cm H2O, which
is higher with respect to values found in literature. This is basically due to the stenosis degree of the considered model being
more severe than that in other works 关30兴. Another reason could
be the nature of the computation. In the present study, the tracheal
wall are in fact extensible while other works used a CFD approach. These different approaches could lead to different fluid
field, as demonstrated by Ref. 关31兴. The deformation of the trachea at the stenotic constriction caused in fact an increase of the
longitudinal vortex intensity, which progressively blocks the air
flowing in the trachea, generating an additional increase of the
pressure drop before and after the constriction 共see Fig. 5共a兲兲. The
pressure distribution of Fig. 5共b兲 shows almost the same pressure
values on both bronchi. This is basically due to the computed
pressure waveforms 共see Fig. 4兲, which are very similar for the
left and the right bronchi. This can be clearly seen also in the
healthy trachea pressure distribution 共see Fig. 5共a兲兲. Furthermore,
the considered models only show one bifurcation. The flow in the
daughter tubes is influenced by the absence of more bronchial
generations while the pressure distribution is surely affected by
this fact. For these reasons, perhaps the impedance approach
could have a stronger impact than what we found for a onegeneration model in more complex models considering many
bronchial generations with different radii.
During inhalation, the velocity profile before the stenosis is
almost axial 共around 10 m/s兲 while the highly reduced geometrical
cross section creates a dead fluid zone before the bifurcation 共see
Fig. 5兲 with a longitudinal vortex that influences the flow distribution after the flow divider. This velocity field is in agreement
with those found by Brouns et al. 关30兴. The asymmetry of the
stenosis causes the already mentioned high velocity jet that
crosses the constriction. The flow distribution on the right and left
main bronchi is shown in Fig. 6共c兲 on sections 5 and 6. Different
from that reported by Sera et al. 关31兴, the secondary flow is not
present probably because of the higher axial velocity entering
both bronchi 共see Fig. 6共d兲 on sections 5 and 6兲 with respect to
their work.
5.4 Wall Stresses and Deformations in the Stenotic
Trachea. The wall deformation of the different constitutive parts
of the stenotic trachea during inspiration is shown in Figs. 8共a兲
Table 4 Parameters used to describe the structured tree. The three-tiered tree is divided into
three levels as a function of the bronchial radius. Scaling parameters are varied along the tree.
Radius
共␮m兲
␣
␤
␰
␥
␩
200⬍ r
50⬍ r ⬍ 200
r ⬍ 50
0.772
0.7619
0.7521
0.7116
0.7228
0.7331
2.33
2.33
2.33
0.85
0.9
0.95
1.10
1.102
1.1031
Level
Trachea-bronchi
Bronchi-bronchioli
Bronchioli-alveoli
021001-8 / Vol. 133, FEBRUARY 2011
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5
20
2.5
10
0
0
20
3
10
−1
0
−5
a)
t = 0.1 s
0.0187
t=2s
10.5
0
0
0.008
8.4
b)
0
a)
0
b)
Fig. 5 Projected 2D streamlines „in m/s… on frontal and lateral
sections of the „a… healthy and „b… stenotic tracheas with respective pressure distributions „in cm H2O… at peak flow during inhalation
c)
t = 0.1 s
t=4s
t=2s
Fig. 7 Logarithmic circunferential strain of the „a… complete
model and of the „b… cartilage rings of the healthy trachea at
selected time points during inspiration. In „c…, the deformed
shape of the cartilage rings during inspiration is shown.
t = 0.1 s
a)
0.0175
t = 1.3 s
0
t = 0.1 s
b)
0.0105
0.315
0
t = 1.3 s
0.175
0
0
c)
t = 0.1 s
Fig. 6 Projected 2D streamlines „in m/s… at selected longitudinal sections of the „a… healthy and „c… stenotic tracheas with
„„b… and „d…… respective velocity distributions at peak flow during inhalation
Journal of Biomechanical Engineering
t = 1.3 s
t = 3.3 s
Fig. 8 Logarithmic circunferential strain „a… of the complete
model and „b… of the cartilage rings of the stenotic trachea at
selected time points during inspiration. In „c…, the deformed
shape of the cartilage rings during inspiration is shown.
FEBRUARY 2011, Vol. 133 / 021001-9
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a)
FSI
FSI
5
20
0
t= 2 s
3000
t= 1.3 s
t= 1.3 s
t= 2 s
A
A
0
CFD
CFD
5
20
0
0
b)
A
P
P
P
a)
−6000
b)
t= 1.3 s
Fig. 9 Comparison of the maximum principal stresses „in Pa…
„a… „left… of the complete model and „a… „right… of the cartilage
rings between healthy and stenotic trachea respectively, at
peak flow during inspiration. In „b… the stress distribution of the
stenotic region is shown.
and 8共b兲. As for the healthy case, the muscular membrane expands, increasing the section of the trachea 共in this case the maximum circumferential strain was around 37%兲 during inhalation.
At the bottom of Fig. 8共c兲, the amplified deformed shape of the
stenotic tracheal rings at inspiration is shown. The trachea shows
a relative small increase in volume to allow the air flowing into
the lungs 共from 0 s to 1.3 s兲. Comparing the deformation of the
cartilage rings between healthy and stenotic tracheas, it can be
seen that these open only partially with respect to the healthy
trachea 共Fig. 8共b兲兲. In Fig. 9 , the maximum principal stresses are
shown and compared with those of the healthy trachea. At inspiration, the muscular membrane is in tension and the maximum
stress is 3 KPa. Showing only the tracheal cartilaginous parts as in
Fig. 9共a兲, it can be seen how they open during inspiration while
the muscle expands 共compare with Fig. 8兲. However, in this case,
the deflections are smaller than those registered in the healthy
trachea due to the presence of the stenotic fibrous cap, which
absorbs most of the load and restrains the cartilage opening capacity. The maximum stress is not obtained at the cartilage rings
but, as expected, in the stenotic region, which increases the rigidity of the tracheal tube upstream of the constriction during inspiration and the downstream of it during exhalation. In addition, the
maximum stress is slightly smaller for the diseased trachea. This
is due to the different pressures applied to the models 共smaller for
the diseased trachea兲, which are subjected to the individual
spirometries of the two patients. The membrane muscle is not free
to dilate in the same way as shown in the healthy trachea 共see Fig.
7兲 where the maximum displacements were located along the tracheal axis, on the same muscle. On the contrary, the maximal
displacement for the pathological trachea is located in a smaller
area of the interface between muscle and cartilage. Finally, at the
bottom of Fig. 9, the maximum principle stresses of the stenotic
region are displayed. The maximum values on the interior surface
of the constriction are around 2 kPa. The highest values are shown
in the posterior side of the stenosis 共point P in Fig. 9兲 where the
fibrous stenotic cap is in contact with the cartilage rings. This
confirms the overload of the stenosis, which discharges the cartilage rings, preventing their opening/closing function. The maximal stress region on the cartilage is in fact registered at the interface with the stenosis 共Fig. 9 top-right and bottom兲, on the
021001-10 / Vol. 133, FEBRUARY 2011
Fig. 10 Comparison between FSI and CFD computations at
peak flow during inspiration: „a… healthy trachea and „b…
stenotic trachea
superior part of the trachea. On the contrary, the healthy trachea
shows its maximal values longitudinally on the central-lower part
of the cartilage, as shown in Fig. 9 top-left.
5.5 FSI Versus CFD Comparison. Both FSI and CFD models were computed using the same fluid flow boundary conditions.
In Fig. 10, the results of rigid wall simulations of the healthy and
stenotic tracheas are shown with those obtained with the FSI approach. For the healthy trachea, the peak velocity at the median
section of the trachea is 12 m/s in the CFD solution, which is
higher than that obtained with FSI 共10 m/s兲. In addition, the velocity in the daughter airways is higher, even though the increment is less than 5% for both left and right bronchi. For the
diseased trachea, the flow patterns obtained at the constriction
yield a different longitudinal vortex at peak flow during inspiration 共see Fig. 10共b兲兲. The longitudinal vortex in the CFD solution
is larger and appears shifted downstream with respect to the FSI
solution. In addition, the stenotic velocity jet in the FSI model is
about 10% higher than that in the CFD model. Air flowing
through the constriction begins to roll up after the narrow section,
generating a strong recirculation area, as shown in Fig. 11, where
FSI
CFD
1000
0
−1000
Fig. 11 Comparison between FSI and CFD vorticity magnitude
„in s−1… at peak flow during inspiration „top: healthy trachea;
bottom: stenotic trachea „frontal/lateral section……
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5
2.5
0
Fig. 12 Contours of the turbulent kinetic energy in m2 / s2 „left:
healthy trachea; right: diseased trachea for each subfigure…
the vorticity magnitude is illustrated for each model, indicating
the severity of the flow conditions in the diseased trachea. While
for the healthy case no particular vortex structures are identifiable,
mainly due to the strong axial inflow velocity, a strong vortical
structure is visible in the stenotic region for both 共CFD and FSI兲
diseased tracheas. The vorticity is higher for the CFD solution
than for FSI, which agrees with the velocity field discrepancy
obtained in that region. Downstream of the constriction, the vorticity progressively loses its intensity and in the daughter airways
there are no secondary structures visible. The depicted maximal
vorticity values are higher than those reported in other studies
关66,67兴 because of the normal breathing inflow conditions used in
those studies compared with the forced respiration adopted in the
present work. Differences in the flow field for CFD and FSI simulations are also previously reported by Wall et al. 关22兴 for a
healthy trachea.
5.6 Turbulence Statistics. To gain insights into the turbulent
characteristics of the tracheal airflow, we analyzed the turbulent
kinetic energy 共TKE兲. Figure 12 shows the TKE contours corresponding to a median longitudinal section of both models. For the
healthy trachea 共Fig. 12, left兲, the kinetic energy exhibits homogeneity through the tracheal axis due to the aforementioned strong
axial velocity component 共slightly stronger for the CFD solution兲.
On the contrary, for the stenotic trachea, a higher TKE is observed
at the constriction. The CFD model exhibits a lower TKE as the
flow fiels illustrated in Fig. 10 showed lower velocities in the
stenotic region. Noteworthy is that while the scale in Fig. 12 is the
same for the stenotic and healthy tracheas, the TKE is proportionally higher for the diseased trachea, taking into account the
smaller flow rate used for CFD and FSI models of this pathology.
5.7 Limitations. Although this work contributes to the understanding of the response of a human trachea under impedancebased conditions and represents an improvement with respect to
studies done before, there are some limitations. In these patientspecific tracheas, due to the complexity of the complete geometry,
we have modeled only the first bifurcation and neglected the upper tracheal geometry. It is well known that the laryngeal flow
causes a jet that affects the tracheal airflow. Even if we count the
effect of this jet through skewed tracheal extensions, a complete
model including this part should be considered for the next studies. In the impedance computation, several geometrical approximations were taken. A study of the influence of geometrical parameters of the fractal network on the pressure waveforms 共as for
instance, ␣, ␤, and lrr兲 should be done. In addition, as already
mentioned, we assumed zero impedance at the alveolar level even
if we know that the alveolar pressure is not zero but varies from
negative to positive pressures 共with respect to the atmospheric
pressure兲 during breathing. Moreover, the presented FSI functional method should be extended and tested for a big number of
pathologies, considering also the inclusion of prosthesis. In this
sense, this is the first step of a future classifying work.
6
Conclusion
The effect of the unsteady airflow on a healthy and a stenotic
trachea was studied under impedance-based boundary conditions
Journal of Biomechanical Engineering
and turbulence modeling. While our models have a single bifurcation, we took into account the resistance of the respiratory system through the use of the impedance-based method obtaining
physiological flow features that qualitatively agree with previous
studies for both healthy and pathologic tracheas. The importance
of this work can be seen in the quantification of flow patterns and
wall stresses inside different airway geometries such as those with
stenotic and stented tracheas. Such numerical analysis could be
performed before a surgery in order to provide help in the surgical
technique and in determining the choice of stent type. In this
paper, we presented a computational approach with the aim of
constructing a diagnostic tool that in the future could predict the
tracheal stresses and deformations due to the influence of different
flow patterns originating from different geometries, pathological
situations, and endotracheal prosthesis implantations. In this way
it could be possible to extract physical variables not assessable in
vivo as local pressure drops, muscular deflections, and stresses.
Finally, with the aim of showing the importance of FSI simulation
in the study of the respiratory system, we compared the fluid fields
computed through the FSI and CFD approaches. The fluid structures postprocessed from these solutions, as found in previous
works, showed differences between results obtained with rigid and
deformable tracheal walls. This confirms that FSI is not only necessary for calculating stresses and strains inside the trachea but it
is also important to extract fluid features that are influenced by the
deformable walls.
Acknowledgment
This study was supported by the CIBER-BBN, an initiative
funded by the VI National R&D&i Plan 2008-2011, Iniciativa
Ingenio 2010, Consolider Program, CIBER Actions and financed
by the Instituto de Salud Carlos III with assistance from the European Regional Development Fund and by the Research Project
No. PI07/90023. The technical support of Plataform for Biological
Tissue Characterization of the Centro de Investigación Biomédica
en Red en Bioingeniería, Biomateriales y Nanomedicina 共CIBERBBN兲 is highly appreciated. We finally wish to thank Christine M.
Scotti 共W.L. Gore & Associates, Inc. Flagstaff, AZ兲.
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