Download pdf - FSI Analysis of a Healthy and a Stenotic Human Trachea Under Impedance-Based Boundary Conditions

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M. Malvèe-mail: [email protected]

A. Pérez del PalomarGroup 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. ChandraInstitute for Complex Engineered Systems,

Carnegie Mellon University,1205 Hamburg Hall, 5000 Forbes Avenue,

Pittsburgh, PA 15213

J. L. López-VillalobosDepartment of Thoracic Surgery,

Hospital Virgen del Rocío,Avenida de Manuel Siurot s/n,

41013 Seville, Spain

A. MenaGroup of Structural Mechanics and Materials



E-50018 Zaragoza, Spain;Centro de Investigación Biomédica en Red en


50018 Zaragoza, Spain

E. A. FinolInstitute for Complex Engineered Systems ,

Carnegie Mellon University,1205 Hamburg Hall, 5000 Forbes Avenue,

Pittsburgh, PA 15213

A. GinelDepartment of Thoracic Surgery,

Hospital Virgen del Rocío,Avenida de Manuel Siurot s/n,

41013 Seville, Spain

M. DoblaréGroup of Structural Mechanics and Materials



50018 Zaragoza, Spain;Centro de Investigación Biomédica en Red en


E-50018 Zaragoza, Spain

FSI Analysis of a Healthy and aStenotic Human Trachea UnderImpedance-Based BoundaryConditionsIn this work, a fluid-solid interaction (FSI) analysis of a healthy and a stenotic humantrachea was studied to evaluate flow patterns, wall stresses, and deformations underphysiological and pathological conditions. The two analyzed tracheal geometries, whichinclude the first bifurcation after the carina, were obtained from computed tomographyimages of healthy and diseased patients, respectively. A finite element-based commercialsoftware code was used to perform the simulations. The tracheal wall was modeled as afiber reinforced hyperelastic solid material in which the anisotropy due to the orientationof the fibers was introduced. Impedance-based pressure waveforms were computed usinga method developed for the cardiovascular system, where the resistance of the respiratorysystem was calculated taking into account the entire bronchial tree, modeled as binaryfractal network. Intratracheal flow patterns and tracheal wall deformation were analyzedunder different scenarios. The simulations show the possibility of predicting, with FSIcomputations, flow and wall behavior for healthy and pathological tracheas. The com-putational modeling procedure presented herein can be a useful tool capable of evaluat-ing quantities that cannot be assessed in vivo, such as wall stresses, pressure drop, andflow patterns, and to derive parameters that could help clinical decisions and improvesurgical outcomes. �DOI: 10.1115/1.4003130�

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received June 7, 2010; final manuscript
eceived November 16, 2010; accepted manuscript posted November 29, 2010; published online January 5, 2011. Assoc. Editor: Dalin Tang.
ournal of Biomechanical Engineering FEBRUARY 2011, Vol. 133 / 021001-1Copyright © 2011 by ASME

aded 02 Feb 2011 to 128.2.87.101. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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IntroductionThe trachea begins immediately below the larynx and runs

own the center of the front part of the neck, ending behind thepper part of the sternum. Here, it bifurcates in the left and rightain bronchus that enters the lung cavities. The trachea forms the

runk of an upside-down tree and is flexible, so that the head andeck may twist and bend during the process of breathing �1�. Theain components that constitute the trachea are fibrous and elastic

issues and smooth muscle, which runs longitudinally and poste-iorly to the trachea, with about 20 rings of cartilage, which helpshe trachea to keep it open during extreme movement of the neck2�. The main role of the tracheal cartilaginous structures is toeep the windpipe open despite the interthoracic pressure duringreathing, sneezing, or coughing �3�. Under these conditions,mong others, the smooth muscle collapses to regulate the air flowy dynamically changing the tracheal diameter. This contractionnd the transmural pressure generate bending and tensile stressesn the cartilage.

Although a clear understanding of the respiration process hasot yet been reached and the influence of the implantation of arosthesis is very important from a medical point of view as wells challenging from the computational aspect, a few studies havenalyzed the behavior of the tracheal walls under different venti-ation conditions. Most numerical analyses of the respiratory sys-em have been focused purely on computational fluid dynamicsCFD�, neglecting the interaction with the wall airways. In par-icular, most works dealt with the airflow patterns using idealized4,5� or approximated airways geometries �6–8� while only fewtudies were based on accurate airway geometries developed fromomputed tomography �CT� or magnetic resonance imagesMRIs� �9–14�. While these studies, following the CFD approach,ailed to incorporate constitutive material modeling of the trachealall �6–8,15–17�, recent studies in lower airway geometries have

uccessfully presented FSI results using simplified models/eometries for single bifurcations �18–21�. Wall and Rabczuk �22�nalyzed the behavior of a CT-based respiratory system through aSI analysis during breathing and mechanical ventilation evensing a simplified model for the composing material. Moreover,oombua and Pidaparti �23�, following a FSI approach, studied

he stress distributions in a two-bifurcation tracheal model andompared the relative differences between an isotropic and anrthotropic material model of the airways. Regarding the consti-utive behavior of the tracheal walls, there is in fact a large dis-ersion of the mechanical properties of the different composingissues and only a few studies have analyzed their mechanicalehavior for humans �24–26�. Abnormal airways deformation andlteration of airway mechanical properties were previously re-orted as a potential cause of tissue damage and associated tohronic complications, as reported by Spitzer et al. �27�. In mostf the works on this topic, the isolated tracheal cartilage wasonsidered as a linearly elastic isotropic material �28�. In many ofhe previous studies, the human tracheal smooth muscle dealt withts plasticity, stiffness, and extensibility and the influence of theemperature on force-velocity relationships. Begis et al. �29� de-eloped a structural finite element model of an adult human tra-heal ring, accounting for its morphometric features and mechani-al properties. More recently, Costantino et al. �28� analyzed theollapsibility of the trachea under different pressure conditions byomparing the numerical results with experimental tests. Regard-ng tracheal pathologies, Brouns et al. �30� provided a CFD nu-

erical model of a healthy trachea without any bifurcation, inhich different stages of artificial stenosis were imposed, with the

im to establish a relationship between local pressure drop andelocity depending on the degree of stenosis. Sera et al. �31� in-estigated the mechanism of wheeze generation with experimentalests conducted on a rigid and an extensible realistic CT-basedracheostenotic model. Cebral and Summers �32� showed how vir-ual bronchoscopy can be used to perform aerodynamic calcula-
ions in anatomically realistic models.
21001-2 / Vol. 133, FEBRUARY 2011

aded 02 Feb 2011 to 128.2.87.101. Redistribution subject to ASME

For both CFD and FSI simulations, the choice of the boundaryconditions is critical. Most studies used velocity or flow condi-tions at the entrance of the trachea as inlet boundary conditionsand usually utilize approximations such as time dependent resis-tance �8,22�, zero-pressure �9–11�, or simply outflow conditions�8,9,30� as outlet boundary conditions. Recently, Wall and co-workers proposed a structured tree for modeling the human lungsand analyzing the respiration and mechanical ventilation underimpedance conditions �33–35�. Finally, Elad et al. �36� proposed anonlinear lumped-parameter model to study the dependency ofairflow distribution in asymmetric bronchial bifurcations on struc-tural and physiological parameters. In this study, we analyzed thetracheal wall stresses and the flow patterns through a study ofhealthy and stenotic human tracheas during inhalation using a FSIapproach. In this analysis, we gave special emphasis to the soliddomain in order to predict stresses and displacements under physi-ological conditions. The aim of the this work is to create a func-tional method to simulate and classify the tracheal wall stress andstrain. Understanding the mechanical behavior of the central air-ways may allow for instance the investigation of their response tothe mechanical ventilatio-associated pressure loads, which is im-portant for clinical applications and for stenting technique. In thissense, our study can be considered as a step toward building aframework that could aide surgical decisions in future. Moreover,we tackle the challenging aspect of the physiological boundaryconditions by modeling the resistance of the entire respiratorysystem through impedance-based boundary conditions. This ap-proach allows the computation of physiological pressures that arenot measureable in vivo starting from patient specific spirometry.Computed pressures waveforms are essential for the evaluation ofstresses and strains during breathing and coughing, especially fordiseased and stented tracheas. The method followed, developed byOlufsen �37� and later extended by Steele et al. �38�, has beenapplied to the cardiovascular systems �39–43� and to the pulmo-nary system �33–35�.

2 Materials and Methods

2.1 Fluid-Airflow. Thoracic CT scans were taken from a 70year old healthy and a 56 year old diseased man and the imageswere then segmented and post-processed to create the fluid model.An initial graphics exchange specification �IGES� file of the com-putational fluid model of the internal tracheal cavity was createdfrom these segmented images. The model was then meshed withunstructured tetrahedral-based elements using the commercialsoftware FEMAP �Siemens PLM Software, Plano, TX� �see Fig. 1�.The fluid was assumed as Newtonian ��F=1.225 kg /m3, �

−5

(a) (b)

healthy stenosis

2

11

2

Fig. 1 Computational grids of the „a… healthy and „b… stenotictracheas with the respective computational fluid „1… and solid„2… grids

=1.83�10 kg /m s� and incompressible under unsteady flow

Transactions of the ASME

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onditions �6�. Flow was assumed turbulent for the analyzed casesince the Reynolds numbers, based on the median tracheal sec-ion, at peak velocity, were in the turbulent regime �Rehealthy30,000, Restenosis=10,000�. To ensure that the results were insen-

itive to the computational grid size, a sensitivity study was car-ied out. Four different meshes of about 100,000, 200,000,50,000, and 300,000 elements were tested for the healthy andtenotic tracheas, studying the inspiratory peak flow to ensure thathe results at the highest Reynolds number were grid independent6�. Comparison between four grid sizes revealed that the velocityrofiles for the grid of 200,000, 250,000, and 300,000 elementsere found almost coincident �see Fig. 2, section 1�. However, theelocity profile near the wall was found to be smoother for thesehree meshes with respect to the coarser mesh. This indicates thathe higher number of grid cells within the wall region can capturehe flow field in more detail than that for the coarser grid size. Theifference between coarser and finer grids is also noticeable as their flows downstream in the left bronchus �see Fig. 2, section 2�.s finer grid increases the computing time considerably, we se-

100000

200000

250000

300000

v[m

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diameter [mm]

diameter [mm]

100000200000

250000

300000

v[m

/s]

00

5

5

10

10

15

15

201

2 00 5 10 15 20

5

10

15

20

a)

1

2

20

Fig. 2 Axial velocity profiles through selected orthogonalinspiration

ected the grids �healthy and stenotic trachea� with a total number

ournal 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 theturbulent nature of the considered tracheal flow, which is basicallytransitional rather than fully turbulent, and taking into account thelimitation of the use of the K-� model for its modeling, as docu-mented in Refs. �44,45� among others, the standard K-� model�46� was used to modify the air viscosity. However, it is not clearif Reynolds averaged Navier-Stokes �RANS�-based turbulencemodels are a good choice or not. Due to the recirculatory flow andturbulent eddies caused by the laryngeal jet, Gemci et al. �47�, forinstance, used a large eddy simulation �LES� turbulence model intheir study. In particular, in the Navier–Stokes equations, the vis-cosity � is substituted by �0=�+�t, where �t is the turbulentviscosity, which is computed by Eq. �1�,

�t = ��FK

��1�

100000

200000

250000

300000

diameter [mm]

v[m

/s]

v[m

/s]

diameter [mm]

100000200000

250000300000

00

10

12

8

2

4

6

2 4 6 8 10 12 14

00

2

2

4

4

6

6

8

8

10

12

10 12 142

1

b)

1

2

heal sections for different mesh sizes at peak flow during
trac
where � is a constant of the model, expressed as a function of the

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eynolds number �46�. The variable � is the specific dissipationate of turbulence. This is related to the so-called kinetic energynd rate of dissipation of the turbulence �K and �, respectively�nd is defined as follows:

� ��

K�2�

ith

K =1

2v� · v�, � =

�

�F��v�� v�� 3�

here v� is the fluctuating velocity. K and � are governed by Eqs.4� and �5�,

��ya�K��t

+ � · �ya��vK − qK�� = yaGK �4�

��ya��t

+ � · �ya��v� − q�� = yaG� �5�

here a is 0 for two- and three-dimensional flows and 1 for axi-ymmetric flows. Other corresponding terms are defined as fol-ows:

qK = ��0 +�t

�K� � K �6�

q� = ��0 +�t

�� 7�

GK = 2�tD2 − �K�K� + B �8�

G� =�

K�2��tD

2 − ��K� + �B� �9�

here �, ��, �K, ��, �K, ��, �, and � are model constants.hese values, input to the finite element solver at the beginning of

he simulation, are displayed in Table 1. A complete description ofhis turbulence model may be found in Ref. �46�.

2.2 Solid-Tracheal Wall. In order to determine the real ge-metry of the trachea and to distinguish between the muscularembrane and the cartilage rings, we proceeded to a nonauto-atic segmentation of the CT scans using MIMICS©. Each tra-

heal constitutive part could be detected through its different den-ity. Finally, with the commercial software ABAQUS

©, a fullexahedral mesh of about 60,000 elements for the healthy tracheand of about 90,000 elements for the stenotic trachea was gener-ted. In Fig. 1, these grids are shown. The stenosis, which is aigid fibrous cap surrounding the internal tracheal wall, due to itsrregular and asymmetric geometry was meshed with about0,000 tetrahedral elements. The properties of the different tissues

Table 1 Parameters used to d

� �� K ��

1 0.555 0.09 0.075

Table 2 Parameters used to define the consbehavior of cartilage, muscular membrane, an

Material parametersC1

�kPa�K1

�kPa�

Cartilage 577.7 —Muscular membrane 0.87 0.154Stenosis 1.74 0.308

21001-4 / Vol. 133, FEBRUARY 2011


of the trachea were analyzed and numerically modeled throughdifferent experimental tests described in previous studies �24,48�.Here, we will provide only a short explanation. Human tracheaswere obtained through the autopsy from two subjects �aged 79–82years� and subjected to different tensile tests to obtain the materialmodels. The cartilaginous rings were modeled as isotropic mate-rial �24,48�. The muscular membrane presented two orthogonalfamilies of smooth muscle cells, one running mainly longitudi-nally and the other transversely. Therefore, for its behavior, theanisotropy coming from the fiber orientations was introduced inthe material model. For cartilage, since there is no preferentialorientation, a neo-Hookean model with strain density energy func-

tion �SEDF�, =C1�I1−3�, was used to fit the experimental re-sults. Regarding the smooth muscle, and taking into account thatthe histology showed two orthogonal fiber families, the HolzapfelSEDF �49� for two families of fibers was used,

= C1�I1 − 3� +K1

2K2�exp�K2�I41 − 1�2� − 1 +

K3

2K4�exp�K4�I42

− 1�2� − 1 +1

D�J − 1�2

where C1 is the material constant related to the ground substance,Ki�0 are the parameters that identify the exponential behaviordue to the presence of two families of fibers, and D is identifiedwith the tissue incompressibility volumetric modulus. The invari-ants Iij are defined as

I1 = tr C I2 = 12 ��tr C�2 − tr�C2��

I41 = a0 · Ca0 I42 = b0 · Cb0

where a0 is a unitary vector defining the orientation of the firstfamily of fibers, b0 is the direction of the second family both in

the reference configurations, and C is the modified right Green

strain tensor defined as C=J1/3C being C=FTF. In Table 2, asummary of the material constants used for the different tissues isshown. For further explanations, see Refs. �24,48�. Not much in-formation is available about the constitutive properties of the tra-cheal stenotic tissue. For this reason, this fibrous tissue was mod-eled as a hyperelastic material, in which we assumed an increasedstiffness of about 2 times with respect to the muscular membrane�see Table 2�. In Fig. 3, the different tracheal constitutive tissuesare separately shown for the healthy �a� and stenotic �b� models.

2.3 Numerical Method and Boundary Conditions. To fa-cilitate the fluid dynamic model, in order to apply uniform veloc-ity field as inflow boundary condition, a five-inlet extension wasadded to each model. In this way, a fully developed airflow isassessed through the trachea. Moreover, to take into account theeffect of the laryngeal jet on the tracheal flow �50–52�, the tra-

e the turbulence K-� models

�K ��

2 2 1 0.712

tive models that characterize the mechanicaltenosis

K2

K3�kPa� K4

D�kPa�

— — — 484034.15 0.34 13.9 8.10868.3 0.68 27.8 17.4

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heal inlet extension was biased toward the rear wall �11,50�.utlet conditions for tracheal bifurcations are critical and influ-

nce the computational solution. While some studies used to setero-pressure condition at the outlets of the respective truncatedeneration �7–12�, the pressure at each outlet must be known ariori, reflecting the effect of the remaining truncated generationt the alveolar level �47�. An alternative approach could be to sethe flow ratios in each tracheal branch �5,8�. Luo and Liu �11�ested both zero-pressure and outflow conditions, finding that out-ow conditions forced the air flow rate to be equal on each outlet.emci et al. �47� modeled an almost complete anatomical replicaf the pulmonary tree down to the 17th generation where theymposed a constant outflow condition p=0.

In this study, we follow the approach used in hemodymanics bylufsen et al. �37,53� and extended by Steele et al. �38�: Starting

rom patient-specific spirometries performed by the two patients,e modeled the entire respiratory system as a fractal network inhich we computed the impedance-based pressure waveformsriven by the procedure explained in the next section. The spirom-try, used as an input of the impedance computation, is a forcedreathing test that consists of enforcing the respiration, blowingff air after a long deep inspiration in order to measure the pul-onary flow, possible bronchial obstructions and, in particular, for

he diseased patient, to get the percentage of pulmonary capacitycquired after stent implantation. At the tracheal internal wall,hich represents the fluid-solid interface, a typical no-slip condi-

ion was applied. Finally, as in previous studies �22�, the move-ent of the solid mesh was restrained by fixing the tracheal top

urface. The coupled fluid and structure models were solved with

ig. 3 Finite element model of the „a… healthy and „b… stenoticracheas with their respective constitutive parts separatelylotted „A denotes anterior part and P denotes posterior part…

he commercial package ADINA �v.8.5, ADINA R&D Inc.�. The



governing equations were solved with the finite element method�FEM�. The fluid domain employs special flow-condition-based-interpolation �FCBI� tetrahedral elements �46�. We used the for-mulation with large displacements and small strains in the FSIcomputation available in ADINA. As an iteration, we used the fullNewton method while a sparse matrix solver based on the Gauss-ian elimination is used for solving the equation system �46�.

3 The Impedance-Based Method

3.1 Airways Fractal Network. In this work, we modeled therespiratory system as a structured fractal network in which wepredicted physiological airflow and pressure profiles. The fractalnetwork is modeled adapting the 1D approach used in the cardio-vascular system to the pulmonary system. In this case, the airwaysare modeled as compliant segments using the linearized 1D con-tinuity and momentum equation �see Refs. �37,38,43��,

�A

�t+

�q

�x= 0 �10�

where p is the pressure, A is the bronchial cross sectional area,and q the flow rate in x-direction, and

�ux

�t+ ux

�ux

�x+ ur

�ux

�r+

1

�

�p

�x=

�

r

�

�r�r

�ux

�r� �11�

where ux is the velocity in x-direction, t is the time, p is thepressure, � is the density, � is the viscosity, and r is the bronchialradius. In this 1D model, we predicted impedance and pressurewaveforms that were applied to the 3D tracheal model as outflowconditions. Equations �10� and �11�, as explained in Sec. 3, werecoupled into an expression for the impedance Z using the Wom-ersley solution of an oscillatory flow in a cylindrical tube.

The binary representation of airways can be considered as aconsiderable approximation of the human lung complexity sincethe respiratory system is defined with the help of general rules thatlimits the precision of the tree. Weibel �4� presented a similarapproach and he incorporated morphometric measurements in hismodel. He created a symmetrically branching tree with defineddiameters and length for each branch. A systematic recursive mor-phometric system was also developed by Horsfield and Wolden-berg �54�. Modeling human airways as a dichotomously branchingtree based on morphometric data was previously reported byRaabe et al. �55�. In that study, in order to keep track of theposition of the airways, a system of binary identification numberwas created. Each daughter branch was designated either a minoror major daughter depending on its diameter. These data wereused in many studies to describe and analyze the human lungmorphometry. However, such a model was not coupled to compu-tational model to predict airflow and lung mechanical strain andstresses. Based on these data, Yeh et al. �56� developed a modelthat identified single pathways without connectivity to representthe lung. They provided a dichotomous model for the rat, whichincluded lengths, diameter, and airway number of 23 generations.More recently, based on the Raabe and Yeh data, Latourelle et al.�57� presented a computational approach of a branching airwaytree that allows for one-to-one mapping of the airway anatomywhen predicting the overall lung mechanical properties. Finally,Comerford et al. �33,34� presented a symmetric and an asymmet-ric impedance-based structured tree for lungs simulations in whichthe method of Olufsen was applied.

In this paper, we presented a model in which two different partscan be distinguished: the trachea and the right and left bronchi. Inthe main bronchi, a relationship between flow and pressure repre-sents the outflow boundary conditions for the trachea. In bothtrachea and bronchi, the air flow and pressure were calculatedusing the incompressible axisymmetric Navier–Stokes equationsfor a Newtonian fluid. A fractal network for each main bronchuswas built prior to the impedance computation based on the radius
of each respective bronchus. Once these two binary trees are cre-
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ted, two pressure waveforms are computed using the methodeveloped by Olufsen et al. �37,53�. These pressures are laterpplied at the outlets of each FSI model.

The bronchial network was modeled as a binary asymmetric-tructured tree in which each branch was approximated by atraight compliant segment. The airways network resulted in aeries of bifurcations composed by a series of parent and daughterronchi, as shown in Fig. 4�a�. Each parent bronchus bifurcates inwo daughter bronchi following a scaling guided by the asymme-ry factors � and � of the root parent rroot according to Eqs. �12�nd �13�,

ri,j = �i� j−irroot, 0 i j �12�

�r0�d1= ��r0�pa �r0�d2

= ��r0�pa �13�

The structured tree continues branching until the radius of anyronchial segment is less than given minimum values rmin, as fornstance the alveolar radius �10–100 �m� where we assumedero impedance. Although we know that during inspiration andxpiration, the alveolar pressure sinusoidally varies between1 cm H2O and +1 cm H2O �1�, the assumption of zero imped-nce at the alveolar radius is clearly more appropriate than thessumption of zero-pressure applied in other works at intermedi-te generations �7,9–12�.

In this study, we followed the approach used by Steele et al.38�, dividing the entire bronchial network in three different lev-ls, to mimic the respiratory system structure. For each level, thearameters describing the fractal tree were varied. The minimumadius was set at the alveolar level �10 �m� where, as discussed,e assumed zero impedance. In the Table 3, the starting radii of

he considered trachea models used as the input of the fractal treecaling are shown. The asymmetry and area ratios of the bronchialetwork �57� were defined as �37�

� =�r0�d1

2 + �r0�d2

2

�r0�pa2 � = � r0d2

r0d1

�2

�14�

here r0d1and r0d2

are the radii of the daughter bronchi and rpa ishe radius of the parent bronchus, while �, the area ratio, and �,he asymmetry ratio, are related to each other through the expres-ion �37�

� =1 + �

�1 + ��/2�2/� �15�

is known also as asymmetry index and describes the relativeelationship between the daughter bronchi. The exponent � isnown in the cardiovascular literature as the power exponent andomes from the power law �59�,

�r0�pa� = �r0�d1

� + �r0�d2� �16�

his relation is valid for a range of flows �37�. Several studieshow that � varies between 2 and 3 �38,59,60�. The power expo-ent �=3 is optimal for laminar flows while �=2.33 is optimal forurbulent flows �37�.

Assuming that rd1 rd2 �57�, � is between 0 and 1. The value ofwidely varies from humans to dogs and rodents as documented

y Latourelle et al. �57�. Based on the morphometry of the modelsescribed in Refs. �4,54,61�, in this work, we chose to vary thealue of asymmetry index with �=0.85, 0.9, and 0.95 �see Table�. The value of � in Eq. �15� describes the branching relationshipcross bifurcations between the radius of the parent bronchi rpand the radii of the daughter bronchi rdi, i=1,2. Using the men-ioned values of � and the power-exponent �=2.33, we calculatedhe values of � for each respective level �see Table 4�. The scalingarameters are obtained from the following expressions �see Table�:

�/2 −1/� �
� = �1 + � � � = � � �17�
21001-6 / Vol. 133, FEBRUARY 2011


Finally, the length of a given branch L can be related to theradius r0 of each branch segment through the parameter lrr=L /r0 �38,53�. Morphometric and anatomic data for humans androdents are described in Refs. �4,6,55,58,62�; these works showedthat this parameter rapidly varies from the trachea to the bronchi-oles, until the alveolar level �from around 12 to 2 �1,4��. Based onthese studies, we finally fixed a constant average value of lrr=6�1,4,54,61�.

4 Impedance Recursive ComputationImpedance was computed in a recursive manner starting from

the terminal branch �38,53� of the structured tree and was used asthe outlet boundary condition for the tracheal left and right bron-chi. The details of the recursive calculation are given elsewhere�37,38,53�. Input impedance was evaluated at the beginning ofeach airway daughter z=0 as a function of the impedance at theend of the airway daughter z=L:

Z�0,�� =ig−1 sin��L/c� + Z�L,��cos��L/c�

cos��L/c� + igZ�L,��sin��L/c��18�

where L is the bronchial length, c=�s0�1−FJ� / ��FC� is the wave-propagation velocity, g=�CA0K /�F, and

Z�0,0� = lim�→0

Z�0,�� =8�lrr

�r03 + Z�L,0�, FJ =

2J1�w0�w0J0�w0�

�19�

J0�x� and J1�x� being the zeroth and first order Bessel functionswith w0= i3w, w2=�Fr0

2� /�, i is the complex unity, and w is theWomersley number. The compliance C is approximately given byC=3A0r0 /2Eh, where A0 is the cross sectional area and r0 is thebronchus radius corresponding to section A0, while E is the Youngmodulus, whose value is 3.33 MPa according to the experimentalstudy of Trabelsi et al. �48�. In Fig. 4, the spirometry �b� with thecorresponding left and right bronchus pressure waveforms �c�used in the computations are sketched. It has to be noted that inthis figure, inspiratory flows are assumed as positive while expi-ratory flows are assumed as negative.

5 Results and DiscussionThe airflow inside the trachea is analyzed at peak inspiration

flow through bidimensional streamlines projected on selected tra-cheal longitudinal sections. Velocity distributions are plotted foreach section to give a complete description of the flow patterns inthe trachea during inhalation. The air flow, which is oscillatory, isgoverned by the Womersley number. This characteristic numberrepresents a nondimensional frequency in oscillatory flows de-fined by Eq. �20�,

w =D

2�2�f

��20�

with the forced breathing frequency f , the kinematic air viscosity�, and the diameter D of the trachea. The Womersley number forthe two patient-specific models is showed in Table 5. Due to thehigh flow rates obtained from the patient-specific spirometry�these are forced breathing rather than normal breathing�, w ishigher than what is reported in other studies �6,13,61�. Notewor-thy is that although the two spirometries have different frequen-cies, w was nearly identical for both models. With regard to theairway mechanics, wall stresses and displacements are analyzed atpeak inspiration to show the function of the different trachealtissues and the overall behavior of the trachea during inspiration.

5.1 Flow Patterns in the Healthy Trachea. The flow insidethe healthy trachea is shown during inhalation with its respectivepressure distribution in Fig. 5. During the inspiration, the tracheashows a strong primary axial flow resulting in a flattened velocity
profile �in the order of 10 m/s� due to the turbulence �see Fig.


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root

right bronchus

parent

daughter 2

daughter 1

left bronchus

unit bifurcation

a

a

b

b

a b

i (j+1)−1

i+j (j+1)−(i+1)

i j

0 2 4 6 8 10 12−30

−20

−10

0

10

20

time [s]

v[m

/s]

0 2 4 6 8 10 12−10

−5

0

5

10

time [s]

Flo

w[l/

s]

0 2 4 6 8−2

−1

0

1

2

time [s]

Flo

w[l/

s]

0 1 2 3 4 5 6 7−6

−4

−2

0

2

4

6

8

time [s]

p[c

mH

20

]

right bronchusleft bronchus

0 2 4 6 8−10

−5

0

5

10

time [s]

v[m

/s]

0 2 4 6 8 10 12−60

−40

−20

0

20

40

60

time [s]

p[c

mH

20]

right bronchusleft bronchus

(b) (c) (d)

(a)

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…
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aded 02 Feb 2011 to 128.2.87.101. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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�b�, section 1�. Up to the bifurcation the flow is nearly parabolic,hile after the flow divider, the stream has split as the fluid moves

oward the daughter tubes. The high axial flow intensity �8 m/s�educes the effect of the geometrical bending curvature so that aully developed Dean-flow pattern is not present, as documentedn Refs. �9–11�. Also, and as found in other studies �8,22,63,64�,he right main bronchus shows a typical M-shaped velocity pro-le, which is not fully developed probably due to the limited

ength of bronchus and its particular geometry �Fig. 6�b�, section�; this typical distribution is normally associated to a high Rey-olds number as documented in Ref. �22� among others. For theame reason, in the left main bronchus the air flow possesses aigh axial component, whose velocity is in the order of 6 m/s.lso here the typical Dean-flow pattern, found by Refs.

11,22,65�, cannot be observed �see Fig. 6�a�, section 3�. The rea-on of these differences on the secondary flows can be explainedonsidering the higher inlet velocity we used in comparison withther works. For the present computations we used in fact a forcedespiration extracted from a patient specific spirometry, whilether studies �9–11� on CFD with rigid wall and Wall and Rabc-uk �22� on the FSI approach used velocity data extracted fromormal breathing. Finally, the velocity distribution in the left mainronchus is slightly skewed toward the outer wall �see Fig. 6�b�,ection 3�.

5.2 Wall Stresses and Deformations in the Healthyrachea. Results corresponding to the wall deformation during

he different phases of the inspiration are shown in Figs. 7�a� and�b�, corresponding to the maximum logarithmic strain in the cir-umferential direction. This illustrates the radial deformation ofhe tracheal walls as a function of their initial configuration. Theighest deformation �10% at t=2 s� appeared in the muscularembrane, which increases the section of the trachea along its

ongitudinal axis during inhalation and decreases it during expira-ion contributing to the pressure-balance inside the lungs, as docu-

ented in Ref. �2�. Besides, the deformation of tracheal rings wasmaller than that of the muscular membrane. In Fig. 7�c�, themplified deformed shape of the tracheal rings at inspiration ishown. The cartilage rings opening, as shown in the figure, allowsuscular membrane dilatation. In particular, in the figure, an in-

rease of the cartilage expansion during inspiration from 0 s to 2and a reduction of this from 2 s to 4 s can be seen, at the end ofhich the expiratory process begins. This expansion is basicallyue to the positive pulmonary pressure �1,2�. In Fig. 9, the prin-ipal stresses are shown. The maximum registered value is 3.3Pa. At inspiration, the muscular membrane is in tension. Theuscular deflection generates tensile stresses in the cartilage andodulates the airway diameter as reported also in Ref. �3�. If only

able 3 Radii of the CT-based trachea model used as input ofhe fractal tree and impedance recursive computation

racheaTracheal

diameter �m�Right bronch.diameter �m�

Left bronch.diameter �m�

ealthy 0.023 0.016 0.0125tenotic 0.018 0.015 0.0095

Table 4 Parameters used to describe the struthree levels as a function of the bronchial radi

LevelRadius��m� �

Trachea-bronchi 200� r 0.772Bronchi-bronchioli 50� r�200 0.761Bronchioli-alveoli r�50 0.752

21001-8 / Vol. 133, FEBRUARY 2011


the cartilaginous rings of the trachea are shown, it can be seenhow they absorbed most of the load, as shown in Fig. 7.

5.3 Flow Patterns in the Stenotic Trachea. In the diseasedtrachea, strong modifications of the local airway flow patternswere found due to the elevated pressure drop caused by the steno-sis, as shown in Fig. 5�b�, where the pressure distribution of thetwo analyzed models are compared. The pressure drop betweenthe trachea entrance and the stenosis is around 8 cm H2O, whichis higher with respect to values found in literature. This is basi-cally due to the stenosis degree of the considered model beingmore severe than that in other works �30�. Another reason couldbe the nature of the computation. In the present study, the trachealwall are in fact extensible while other works used a CFD ap-proach. These different approaches could lead to different fluidfield, as demonstrated by Ref. �31�. The deformation of the tra-chea at the stenotic constriction caused in fact an increase of thelongitudinal vortex intensity, which progressively blocks the airflowing in the trachea, generating an additional increase of thepressure drop before and after the constriction �see Fig. 5�a��. Thepressure distribution of Fig. 5�b� shows almost the same pressurevalues on both bronchi. This is basically due to the computedpressure waveforms �see Fig. 4�, which are very similar for theleft and the right bronchi. This can be clearly seen also in thehealthy trachea pressure distribution �see Fig. 5�a��. Furthermore,the considered models only show one bifurcation. The flow in thedaughter tubes is influenced by the absence of more bronchialgenerations while the pressure distribution is surely affected bythis fact. For these reasons, perhaps the impedance approachcould have a stronger impact than what we found for a one-generation model in more complex models considering manybronchial generations with different radii.

During inhalation, the velocity profile before the stenosis isalmost axial �around 10 m/s� while the highly reduced geometricalcross section creates a dead fluid zone before the bifurcation �seeFig. 5� with a longitudinal vortex that influences the flow distri-bution after the flow divider. This velocity field is in agreementwith those found by Brouns et al. �30�. The asymmetry of thestenosis causes the already mentioned high velocity jet thatcrosses the constriction. The flow distribution on the right and leftmain bronchi is shown in Fig. 6�c� on sections 5 and 6. Differentfrom that reported by Sera et al. �31�, the secondary flow is notpresent probably because of the higher axial velocity enteringboth bronchi �see Fig. 6�d� on sections 5 and 6� with respect totheir work.

5.4 Wall Stresses and Deformations in the StenoticTrachea. The wall deformation of the different constitutive partsof the stenotic trachea during inspiration is shown in Figs. 8�a�

red tree. The three-tiered tree is divided intoScaling parameters are varied along the tree.

� � � �

0.7116 2.33 0.85 1.100.7228 2.33 0.9 1.1020.7331 2.33 0.95 1.1031

Table 5 Forced breathing frequencies and correspondingReynolds and Womersley numbers

TracheaFrequency

�Hz� Re w

Healthy 0.097 30,000 2.32Stenotic 0.159 10,000 2.33

ctuus.

91



Fssing inhalation

Fn„

ing inhalation

Journal of Biomechanical Engineering

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0

8.40.008

0

c)

b)

a)10.50.0187

0 0

t = 0.1 s t = 4 st = 2 s

t = 0.1 s t = 2 s

Fig. 7 Logarithmic circunferential strain of the „a… completemodel and of the „b… cartilage rings of the healthy trachea atselected time points during inspiration. In „c…, the deformed
shape of the cartilage rings during inspiration is shown.
t = 1.3 st = 0.1 s

a)

b)

c)

t = 3.3 s

0.315

0

0.0175

0

0.0105

0

0.175

0

t = 1.3 st = 0.1 s

t = 1.3 st = 0.1 s

Fig. 8 Logarithmic circunferential strain „a… of the completemodel and „b… of the cartilage rings of the stenotic trachea atselected time points during inspiration. In „c…, the deformed

20

00

5

0

3

−5

b)a)

20

10 −1

102.5

ig. 5 Projected 2D streamlines „in m/s… on frontal and lateralections of the „a… healthy and „b… stenotic tracheas with re-pective pressure distributions „in cm H2O… at peak flow dur-

ig. 6 Projected 2D streamlines „in m/s… at selected longitudi-al sections of the „a… healthy and „c… stenotic tracheas with„b… and „d…… respective velocity distributions at peak flow dur-

shape of the cartilage rings during inspiration is shown.

FEBRUARY 2011, Vol. 133 / 021001-9


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nd 8�b�. As for the healthy case, the muscular membrane ex-ands, increasing the section of the trachea �in this case the maxi-um circumferential strain was around 37%� during inhalation.t the bottom of Fig. 8�c�, the amplified deformed shape of the

tenotic tracheal rings at inspiration is shown. The trachea showsrelative small increase in volume to allow the air flowing into

he lungs �from 0 s to 1.3 s�. Comparing the deformation of theartilage rings between healthy and stenotic tracheas, it can beeen that these open only partially with respect to the healthyrachea �Fig. 8�b��. In Fig. 9 , the maximum principal stresses arehown and compared with those of the healthy trachea. At inspi-ation, the muscular membrane is in tension and the maximumtress is 3 KPa. Showing only the tracheal cartilaginous parts as inig. 9�a�, it can be seen how they open during inspiration while

he muscle expands �compare with Fig. 8�. However, in this case,he deflections are smaller than those registered in the healthyrachea due to the presence of the stenotic fibrous cap, whichbsorbs most of the load and restrains the cartilage opening ca-acity. The maximum stress is not obtained at the cartilage ringsut, as expected, in the stenotic region, which increases the rigid-ty of the tracheal tube upstream of the constriction during inspi-ation and the downstream of it during exhalation. In addition, theaximum stress is slightly smaller for the diseased trachea. This

s due to the different pressures applied to the models �smaller forhe diseased trachea�, which are subjected to the individualpirometries of the two patients. The membrane muscle is not freeo dilate in the same way as shown in the healthy trachea �see Fig.� where the maximum displacements were located along the tra-heal axis, on the same muscle. On the contrary, the maximalisplacement for the pathological trachea is located in a smallerrea of the interface between muscle and cartilage. Finally, at theottom of Fig. 9, the maximum principle stresses of the stenoticegion are displayed. The maximum values on the interior surfacef the constriction are around 2 kPa. The highest values are shownn the posterior side of the stenosis �point P in Fig. 9� where thebrous stenotic cap is in contact with the cartilage rings. Thisonfirms the overload of the stenosis, which discharges the carti-age rings, preventing their opening/closing function. The maxi-

al stress region on the cartilage is in fact registered at the inter-

3000

−6000

A

P P

A

A

P

a)

b)

t= 2 s t= 2 st= 1.3 s t= 1.3 s

t= 1.3 s

ig. 9 Comparison of the maximum principal stresses „in Pa…a… „left… of the complete model and „a… „right… of the cartilageings between healthy and stenotic trachea respectively, ateak flow during inspiration. In „b… the stress distribution of thetenotic region is shown.

ace with the stenosis �Fig. 9 top-right and bottom�, on the

21001-10 / Vol. 133, FEBRUARY 2011


superior part of the trachea. On the contrary, the healthy tracheashows its maximal values longitudinally on the central-lower partof the cartilage, as shown in Fig. 9 top-left.

5.5 FSI Versus CFD Comparison. Both FSI and CFD mod-els were computed using the same fluid flow boundary conditions.In Fig. 10, the results of rigid wall simulations of the healthy andstenotic tracheas are shown with those obtained with the FSI ap-proach. For the healthy trachea, the peak velocity at the mediansection of the trachea is 12 m/s in the CFD solution, which ishigher than that obtained with FSI �10 m/s�. In addition, the ve-locity in the daughter airways is higher, even though the incre-ment is less than 5% for both left and right bronchi. For thediseased trachea, the flow patterns obtained at the constrictionyield a different longitudinal vortex at peak flow during inspira-tion �see Fig. 10�b��. The longitudinal vortex in the CFD solutionis larger and appears shifted downstream with respect to the FSIsolution. In addition, the stenotic velocity jet in the FSI model isabout 10% higher than that in the CFD model. Air flowingthrough the constriction begins to roll up after the narrow section,generating a strong recirculation area, as shown in Fig. 11, where

0

5

FSI

a)

CFD

0

20

FSI

CFD

b)

20

00

5

Fig. 10 Comparison between FSI and CFD computations atpeak flow during inspiration: „a… healthy trachea and „b…stenotic trachea

FSI 1000CFD

−1000

0

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……


ttfmsdtottv�tplh

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he vorticity magnitude is illustrated for each model, indicatinghe severity of the flow conditions in the diseased trachea. Whileor the healthy case no particular vortex structures are identifiable,ainly due to the strong axial inflow velocity, a strong vortical

tructure is visible in the stenotic region for both �CFD and FSI�iseased tracheas. The vorticity is higher for the CFD solutionhan for FSI, which agrees with the velocity field discrepancybtained in that region. Downstream of the constriction, the vor-icity progressively loses its intensity and in the daughter airwayshere are no secondary structures visible. The depicted maximalorticity values are higher than those reported in other studies66,67� because of the normal breathing inflow conditions used inhose studies compared with the forced respiration adopted in theresent work. Differences in the flow field for CFD and FSI simu-ations are also previously reported by Wall et al. �22� for aealthy trachea.

5.6 Turbulence Statistics. To gain insights into the turbulentharacteristics of the tracheal airflow, we analyzed the turbulentinetic energy �TKE�. Figure 12 shows the TKE contours corre-ponding to a median longitudinal section of both models. For theealthy trachea �Fig. 12, left�, the kinetic energy exhibits homo-eneity through the tracheal axis due to the aforementioned strongxial velocity component �slightly stronger for the CFD solution�.n the contrary, for the stenotic trachea, a higher TKE is observed

t the constriction. The CFD model exhibits a lower TKE as theow fiels illustrated in Fig. 10 showed lower velocities in thetenotic region. Noteworthy is that while the scale in Fig. 12 is theame for the stenotic and healthy tracheas, the TKE is proportion-lly higher for the diseased trachea, taking into account themaller flow rate used for CFD and FSI models of this pathology.

5.7 Limitations. Although this work contributes to the under-tanding of the response of a human trachea under impedance-ased conditions and represents an improvement with respect totudies done before, there are some limitations. In these patient-pecific tracheas, due to the complexity of the complete geometry,e have modeled only the first bifurcation and neglected the up-er tracheal geometry. It is well known that the laryngeal flowauses a jet that affects the tracheal airflow. Even if we count theffect of this jet through skewed tracheal extensions, a completeodel including this part should be considered for the next stud-

es. In the impedance computation, several geometrical approxi-ations were taken. A study of the influence of geometrical pa-

ameters of the fractal network on the pressure waveforms �as fornstance, �, �, and lrr� should be done. In addition, as alreadyentioned, we assumed zero impedance at the alveolar level even

f we know that the alveolar pressure is not zero but varies fromegative to positive pressures �with respect to the atmosphericressure� during breathing. Moreover, the presented FSI func-ional method should be extended and tested for a big number ofathologies, considering also the inclusion of prosthesis. In thisense, this is the first step of a future classifying work.

ConclusionThe effect of the unsteady airflow on a healthy and a stenotic

2.5

0

5

ig. 12 Contours of the turbulent kinetic energy in m2/s2„left:

ealthy trachea; right: diseased trachea for each subfigure…

rachea was studied under impedance-based boundary conditions



and turbulence modeling. While our models have a single bifur-cation, we took into account the resistance of the respiratory sys-tem through the use of the impedance-based method obtainingphysiological flow features that qualitatively agree with previousstudies for both healthy and pathologic tracheas. The importanceof this work can be seen in the quantification of flow patterns andwall stresses inside different airway geometries such as those withstenotic and stented tracheas. Such numerical analysis could beperformed before a surgery in order to provide help in the surgicaltechnique and in determining the choice of stent type. In thispaper, we presented a computational approach with the aim ofconstructing a diagnostic tool that in the future could predict thetracheal stresses and deformations due to the influence of differentflow patterns originating from different geometries, pathologicalsituations, and endotracheal prosthesis implantations. In this wayit could be possible to extract physical variables not assessable invivo as local pressure drops, muscular deflections, and stresses.Finally, with the aim of showing the importance of FSI simulationin the study of the respiratory system, we compared the fluid fieldscomputed through the FSI and CFD approaches. The fluid struc-tures postprocessed from these solutions, as found in previousworks, showed differences between results obtained with rigid anddeformable tracheal walls. This confirms that FSI is not only nec-essary for calculating stresses and strains inside the trachea but itis also important to extract fluid features that are influenced by thedeformable walls.

AcknowledgmentThis study was supported by the CIBER-BBN, an initiative

funded by the VI National R&D&i Plan 2008-2011, IniciativaIngenio 2010, Consolider Program, CIBER Actions and financedby the Instituto de Salud Carlos III with assistance from the Eu-ropean Regional Development Fund and by the Research ProjectNo. PI07/90023. The technical support of Plataform for BiologicalTissue Characterization of the Centro de Investigación Biomédicaen Red en Bioingeniería, Biomateriales y Nanomedicina �CIBER-BBN� is highly appreciated. We finally wish to thank Christine M.Scotti �W.L. Gore & Associates, Inc. Flagstaff, AZ�.

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