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Journal of Biomechanics 41 (2008) 1995–2002

Nonlinear mechanical property of tracheal cartilage:A theoretical and experimental study

Zhongzhao Tenga,�, Ignacio Ochoab, Zhiyong Lic, Yihan Lind,Jose F. Rodriguezb, Jose A. Beab, Manuel Doblareb

aDepartment of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USAbAragon Institute of Engineering Research (I3A), University of Zaragoza, Zaragoza, Spain

cDepartments of Engineering & Radiology, University of Cambridge, Cambridge, UKdDepartment of Mechanics and Engineering Science, Fudan University, Shanghai, China

Accepted 25 March 2008

Abstract

Background: Despite being the stiffest airway of the bronchial tree, the trachea undergoes significant deformation due to intrathoracic

pressure during breathing. The mechanical properties of the trachea affect the flow in the airway and may contribute to the biological

function of the lung.

Method: A Fung-type strain energy density function was used to investigate the nonlinear mechanical behavior of tracheal cartilage. A

bending test on pig tracheal cartilage was performed and a mathematical model for analyzing the deformation of tracheal cartilage was

developed. The constants included in the strain energy density function were determined by fitting the experimental data.

Result: The experimental data show that tracheal cartilage is a nonlinear material displaying higher strength in compression than in

tension. When the compression forces varied from �0.02 to �0.03N and from �0.03 to �0.04N, the deformation ratios were

11.0372.18% and 7.2771.59%, respectively. Both were much smaller than the deformation ratios (20.0174.49%) under tension forces

of 0.02 to 0.01N. The Fung-type strain energy density function can capture this nonlinear behavior very well, whilst the linear

stress–strain relation cannot. It underestimates the stability of trachea by exaggerating the displacement in compression. This study may

improve our understanding of the nonlinear behavior of tracheal cartilage and it may be useful for the future study on tracheal collapse

behavior under physiological and pathological conditions.

Published by Elsevier Ltd.

Keywords: Trachea; Cartilage; Collapse; Mechanical property; Nonlinear

1. Introduction

The trachea is composed of a series of rings withhorseshoe-shaped cartilages and mucosal membrane link-ing the airway with the esophagus. It is the only passageconnecting the upper airway and lung, for air to flow inand out. Therefore, an understanding of the mechanicalbehavior of trachea is very important in studying the bio-function of the whole respiratory system. Despite being thestiffest structure of the bronchial tree, the trachea under-goes significant deformation during breathing. Changes in

the mechanical properties of the trachea may contribute toaltered lung function in obstructive lung diseases. Forexample, weakening of the wall structures of large airwayshas been found in chronic obstructive pulmonary disease(Baier et al., 1981), and altered cartilage geometry andelasticity appear to play a role in tracheomalacia (Lomas-ney et al., 1989; Newth et al., 1990; Baroni et al., 2005;Murgu and Colt, 2006). In these diseases, tracheal collapseis often observed with the flexible tracheal cartilagechanging its curvature and the mucosal membrane drop-ping into the cavity causing obstruction. The ticklingsensation of the membrane touching the tracheal lininggenerates coughing, eventually resulting in ventilatory flowlimitation (Penn et al., 1988; Deoras et al., 1991) and

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clinical distress. The analysis of tracheal collapse is also ofgreat importance for many clinical treatments, such asTotal Liquid Ventilation (Costantino and Fiore, 2001). Itrequires in-depth knowledge of the mechanical propertiesof the trachea in order to obtain the compliance of theairway.

To date, the mechanical properties of trachea are notfully understood. In previous experimental work, thetracheal cartilage was treated as a linear elastic material(Lambert et al., 1991a; Rains et al., 1992). In reportedsimulations using either analytical models (Holzhauser andLambert, 2001; Wang et al., 2006) or numerical simulations(Begis et al., 1988; Costantino et al., 2004), the trachealcartilage was similarly considered to be linearly elastic.However, tracheal cartilage, like most other biological softtissues, e.g. muscle, tendon and blood vessel, is likely tohave nonlinear mechanical properties.

In this paper, cartilages of tracheal rings from adult pigswere used to make the material test. Fung-type strainenergy density function (Fung, 1990) was used to charac-terize cartilaginous mechanical properties. The governingequations describing the deformation of the cartilage werebased on the physics of a shell (Kresch and Noordegraaf,1972; Lambert et al., 1991a; Wang et al., 2006). By fittingthe experimental data, the material constants included inthe strain energy density function were determined.

2. Materials and methods

2.1. Material test

A local slaughterhouse provided three pigs’ tracheas. Ten rings above

the tracheal bifurcation from two trachea (five from each trachea) were

used for the material test (see Fig. 1(A)) and labeled T1_Bi and T2_Bi

(i ¼ 1,y,5), respectively. The samples from the third one were used for

other purpose, which will be described later. All soft tissues, including

mucosa and trachealis muscles, were removed and a hole with a diameter

of 1.5mm was made on each cartilage tip (4mm away from the ends). To

allow the cartilage tip to rotate freely without vertical sliding during

testing, two cone-shaped screw heads and a steel bar with a diameter of

1mm were used to fix each tip (see Fig. 1(B)). Finally, the specimen was

mounted on an Instron Micro Tester 5548 (Instron Corporation, USA) to

perform tension and compression tests (Fig. 2), where these situations

have to be understood as the conditions under which the force opens the

cartilage gap (F in Fig. 1(A) is positive) and closes the gap (F in Fig. 1(A)

is negative), respectively.

After preconditioning 5 times, the test was performed with a

displacement ranging from �10 to 5mm under a constant velocity of

10mm/min at room temperature. The humidity was maintained by an

atomizer. There was no detectable dehydration according to the weights

before and after test. After the test, the cartilaginous width (ac) and

thickness (bc) were measured at seven different positions along the

circumference. These were used to determine the semimajor and

semiminor axes of the ellipse describing the cross-section of cartilage

(Fig. 1(C)). Following this, the specimen was immersed in saline solution

to record its shape with a digital camera for identifying the cartilage shape.

To study the contribution of soft connective tissue to the bending

behavior of cartilage, five rings (labeled as T3_Li, (i ¼ 1,y,5)) next to the

larynx from the third trachea were used. The test was made with soft

connective tissue first and then the test was repeated after removal of the

soft tissue. Before performing the test with soft tissue, a radial cut was

made at the middle point of the trachealis muscle connecting the two free

tips. Then the test was performed following the same steps described

above. After that, the soft connective tissue was carefully removed with

fine surgical scissors and the test was repeated.

2.2. Mathematical model

The following analysis of tracheal cartilage deformation is based on the

assumptions: (1) A cartilage ring is considered as a curved beam of

elliptical shape. The inner and outer walls of the cartilage can be

approximated as ellipses and their middle line defines the cartilage shape.

The positions of both cartilage tips are determined manually according to

the shape of the specimen. Therefore, the cartilage shape is not considered

to be symmetric as in previous models (Lambert et al., 1991a; Holzhauser

and Lambert, 2001; Costantino et al., 2004; Wang et al., 2006); (2) The

cross-section of the cartilage is also assumed to be elliptical (Fig. 1(C)); (3)

The cartilage is treated as an incompressible, hyper-elastic and transver-

sely isotropic material by assuming fibers reinforcing along the circumfer-

ential direction; and (4) The cartilage length is assumed to be constant

during deformation.

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Fig. 1. (A) Schematic drawing of bending tests on cartilage. Both the tips

are assumed to be fixed by hinges. F is the force applied on the tip and D is

the distance between both tips. (B) The amplification of handling the

cartilage at tips O and A aiming at eliminating the slide along the steel bar

and allowing free rotation. (C) The cross-section of the cartilage is

approximated as an ellipse (The length of semimajor axis is ac and that of

semiminor axis is bc).

Fig. 2. The mechanical test was performed with Instron Micro Tester

(5548) with 0.0001N and 0.001mm precisions in force and displacement.

An atomizer was used to maintain the tissue humidity and no detectable

weight loss was found during the test.

Z. Teng et al. / Journal of Biomechanics 41 (2008) 1995–20021996

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The free-body diagram for differential arc-length, ds, with a constant

cross-section is shown in Fig. 3. T and Q are the stretch force and shear

force, respectively; M is the bending moment; y is the angle between the

tangent to wall and the x-axis; and s denotes the arc-length of the

cartilage. The requirements of static equilibrium lead to the following

governing equations:

dT

ds�Q

dyds¼ 0 (1)

dQ

dsþ T

dyds¼ 0 (2)

dM

dsþQ ¼ 0 (3)

The boundary conditions are:

TA ¼ �F cosðyAÞ (4a)

QA ¼ F sinðyAÞ (4b)

and

M ¼ 0 at both tips (4c)

in which F is the vertical force applied on the tip A, TA and QA are the

stretch and shear forces induced by F at tip A, and yA is the tangent angle

at tip A.

The stress–strain relationship of cartilage can be described by Fung-

type strain energy density function, which was initially developed for

modeling arteries (Fung, 1990)

W ¼uc

2½expðOÞ � 1� (5)

in which O ¼ b1E2x þ b2E2

Z þ b3E2z þ 2b4ExEZ þ 2b5ExEz þ 2b6EZEz and

uc, b1,y,b6 are material constants, Ex, EZ and Ez are the Green strains in

the local x, Z and z directions, respectively (Fig. 1(C)), related with the

stretch ratios by

Ei ¼12ðl2i � 1Þ ði ¼ x; Z; zÞ

The stress–strain relationship is then derived as

sij ¼qaj

qbm

qai

qbn

qW

qEnm

ði; j;m; n ¼ x; Z; zÞ (6)

in which ai and bi(i ¼ x, Z, z) are local coordinates in the deformed and

undeformed configurations, respectively. The model based on Eq. (5) is

named as ‘nonlinear model’ to distinguish it from the one based on a linear

stress–strain relation (named as ‘linear model’), which will be discussed

later. When the incompressibility and transverse isotropy are taken into

account, we have the following equations:

lx ¼ lZ ¼1ffiffiffiffiffilz

p (7)

Therefore, O in (Eq. (5)) can be simplified as

O ¼ b�1E2x þ b3E2

z þ 2b�5ExEz (8)

Considering the real shape of cartilage, which is a long and thin structure,

the stretch ratio is assumed varying linearly along the thickness:

lz ¼ 1� DkZ (9)

where Dk ¼ kðsÞ � k0ðsÞ, k (defined as k ¼ dy/ds) and k0 are the

curvatures in deformed and initial configurations, respectively. The

bending moment equals the weighted area integral of the stress in

the cross-section area, C,

M ¼ �

ZC

Zsz dC (10)

By considering the stress–strain relation (Eq. (6)) and the ellipse shape of

the cross-section, the above equation can be rewritten as

M ¼ �ucacb

2c

2

Z 2p

0

f1ðDktÞff2ðDktÞ þ c1ðDkÞgc2ðDkÞdt (11)

in which

f1 ¼ sinðtÞcos2ðtÞ½1� bcDk sinðtÞ�2

f2 ¼ b3½ð1� bcDk sinðtÞÞ2 � 1�

c1 ¼ b�51

1� bcDk sinðtÞ� 1

� �

and

c2 ¼ exp1

4b�1

1

1� bcDk sinðtÞ� 1

� �2((

þ b3½ð1� bcDk sinðtÞÞ2 � 1�2 þ 2b�51

1� bcDk sinðtÞ� 1

� �

�½ð1� bcDk sinðtÞÞ2 � 1�

))

It is inconvenient to use Eq. (11) directly, due to numerical integration

in the regression procedure. Therefore, c1 and c2 are both expanded into

Taylor series in terms of bcDk at 0 and the first three items are retained.

Finally the relation between moment and curvature is

M ¼ ucðI1B1Dkþ I3B3Dk3 þ I5B5Dk5 þ I7B7Dk7Þ (12)

in which

I1 ¼pacb

3c

8; I3 ¼

pacb5c

64; I5 ¼

5pacb7c

256; I7 ¼

7pacb9c

1024

and

B1 ¼ 2b3 � b�5

B3 ¼ 8b23 þ 4b�25 þ 2b�1b3 � b�1b�5 � 12b3b�5 þ 16b3

B5 ¼ 18b23 � b�25 � 3b�1b3 þ b�1b�5 � 5b3b�5 � 2b�5

B7 ¼ 4b23 þ 2b�25 � 2b�1b3 þ b�1b�5 � 6b3b�5

Cartilage is unlikely to have a uniform stiffness along its circumference

(Lambert et al., 1991a). The weakest parts locate at both tips and become

stiffer while approaching the middle point, which are described by

choosing the periodic functions for uc given below

uc ¼ C1 1� C2 cos2 ps

sA

� �� �(13)

in which sA is the length of cartilage and C1 is material constant. The value

for the parameter C2 can be chosen between 0 and 1 to control the

magnitude of the stiffness.

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Fig. 3. Free-body diagram for the analysis of the mechanics of deformed

cartilage. The thick line is a small section of cartilage arc, on which the

stretch and shear forces (T and Q) and bending moment (M) are shown.

Z. Teng et al. / Journal of Biomechanics 41 (2008) 1995–2002 1997

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By considering Eq. (12) and the governing equations (1)–(3), we finally

get the equation in terms of Dk

d3Dkds3¼ f

d2Dkds2

;dDkds

;Dk; k0

� �(14)

with the following boundary conditions:

DkðsAÞ ¼ 0 (15a)

dDkds

����s¼sA

¼ �QA

I1B1ucðsAÞ(15b)

d2Dkds2

����s¼sA

¼ �1

ucðsAÞI1B1

2QA

ucðsAÞ

duc

ds

����s¼sA

þ TAkðsAÞ

" #(15c)

Dkð0Þ ¼ 0 (15d)

The four-order Runge–Kutta method (Lambert and Lambert, 1991b) was

used to solve Eq. (14) with boundary conditions of (15a)–(15c). The

shooting method was introduced to determine yA included in TA and QA

in order to meet the condition of (15d). After solving the governing

equations, the displacement of tip A, DD, can be achieved with

y ¼Z s

0

ðDkþ k0Þdsþ y0 (16a)

and

DD ¼

Z sA

0

cos yds�D0 (16b)

in which y0 is the value of y at tip O and D0 is the initial distance between

two tips. Therefore, the parameters included in Eqs. (5) and (13) can be

determined by fitting the experimental data of F�DD.

3. Results

The image of the cartilage was processed using thesoftware developed by the authors with Matlab 6.5(The MathWorks, Inc.) to isolate the inner and outer walls(the points in Fig. 4), which were approximated byconcentric ellipses (the thin lines in Fig. 4). The middle

line of the two ellipses was used to describe the cartilageshape (the thick line in Fig. 4) with ends marked by twoblack points (black round points in Fig. 4).The ellipse can be used to describe the cartilage shape

with the regression line almost exactly passing through allpoints of the inner wall or outer wall. The asymmetry isobvious since the two black points in Fig. 4 are not in thesame horizontal line. Although it has been thought thatsolving the asymmetric problem may not improve theresults significantly (Holzhauser and Lambert, 2001), westill lack a quantitative study on this aspect.

3.1. The soft connective tissue contribution to bending

The inner wall of the tracheal cartilage is covered bysubmucosa and the free tips are connected by trachealismuscle and other connective tissues cover its outer part. Tostudy the contribution from soft tissues on the bendingbehavior of the cartilage, the F�DD curves with andwithout soft tissue are both shown in Fig. 5 (data are fromthe third trachea). The two F�DD curves of each specimenfit very well. This shows that soft tissue has little impact onthe bending behavior of tracheal cartilage. Therefore, inthe following sections, only the experimental data from thespecimen without the soft tissue are represented.

3.2. The regression result of force vs. displacement (F�DD)

The experimental data from 10 tracheal cartilageswithout soft tissue are shown in Fig. 6. The tendency ofthe curve indicates the nonlinearity of cartilage displayinghigher strength in compression than in tension. For theconvenience of analysis, we define dF1�F2

as the deforma-tion ratio, which equals:

dF1�F2¼

DDF1�F2

D0100%

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4

9

14

-6

-1

Inner wall

Outer wall

Cartilage shape

mm

-16

-11

-16 -11 -6 4 9 14-1mm

Fig. 4. Both the inner (triangle points) and outer (square points) walls of

cartilage can be well approximated by ellipses (thin lines). The cartilage

shape is described by the middle line (thick line) of the ellipses with both

ends marked by two round black points (data are from sample T1_B1).

5.5

1.5

3.5T3_L1S T3_L1

T3_L2S T3_L2

T3_L3S T3_L3T3_L4S T3_L4

-4.5

-2.5

-0.5 T3_L5S T3_L5

ΔD

(mm)

-10.5

-8.5

-6.5

-0.06 -0.04 -0.02 0 0.02

F (N)

Fig. 5. F�DD curves from the test of the specimens with and without the

soft tissue. The line is the experimental curve with the soft tissues (the

legend is with postfix ‘S’) and the points are the corresponding data from

the specimen without the soft tissue.

Z. Teng et al. / Journal of Biomechanics 41 (2008) 1995–20021998

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in which DDF1�F2denotes the change of displacement when

the applied force varies from F1 to F2. When the loadingstep is the same, the deformation under compression ismuch smaller than under tension (see Table 1 for details).The deformation ratio is 50% less when the force variesfrom �0.02 to �0.03N than when the force varies from0.02 to 0.01 or 0.01N to 0. In compression, an increaseof loading leads to a smaller deformation ratio. Thisphenomenon reveals the ability of trachea to resistcollapse.

The continuous lines in Fig. 6 are those corresponding toregression curves. For each specimen, the predicted linealmost exactly passes through all the experimental datapoints with the coefficient of determination R2 very close to1 (see Table 2). Thus the nonlinear model can represent thenonlinearity of tracheal cartilage very well. The material

constants included in the function are listed in Table 2. It isworth pointing out that the final regression result does notrely on the initial guessed value. When we increased ordecreased the initial input value 3 times, there was nosignificant change in the final results.If the cartilage is considered as a linear material by using

s ¼ Ee (s is stress, e is strain and E is the Young modulus)instead of Eq. (5), we obtain the linear relation between M

and Dk as M ¼ EIDk, in which I is the inertia. The modelhere is named ‘linear model’. Similarly, the uniformity ofstiffness was also considered. Fig. 7 gives the regressionresult with experimental data from T1_B1. The regressionresult is poor, which implies the linear model cannotcapture the nonlinear property of tracheal cartilage. Whenthe force is lower than �0.02N, the displacement predictedby the linear model is exaggerated. That is, the linear model

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6

0

2

4 T1_B1 T1_B2 T1_B3 T1_B4 T1_B5

T2_B1 T2_B2 T2_B3 T2_B4 T2_B5

ΔD

(mm)

8

-6

-4

-2

F (N)

-12

-10

-8

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01

Fig. 6. Experimental data of F�DD and the corresponding regression

curves.

Table 1

Deformation ratios in different loading ranges (n ¼ 10)

d0.02–0.01 (%) d0.01–0 (%) d0 to �0.01(%) d�0.02 to �0.03 (%) d�0.02 to �0.03 (%) d�.03 to �0.04 (%)

20.0174.49 26.4276.34 24.7975.92 17.2273.51 11.0372.18 7.2771.59

Table 2

Determined material constants and the coefficient of determination

Specimen C1(Mpa) C2

b�1 b3b�5 R2

T1_B1 77.466 0.742 24,700.087 0.674 0.570 0.9994

T1_B2 48.172 0.597 35,882.782 0.909 0.541 0.9993

T1_B3 60.991 0.711 44,705.771 0.710 0.470 0.9987

T1_B4 48.858 0.547 53,232.077 0.915 0.518 0.9992

T1_B5 46.555 0.506 71,916.896 0.945 0.382 0.9988

T2_B1 66.789 0.702 37,765.427 0.805 0.640 0.9997

T2_B2 57.584 0.513 43,547.089 1.060 0.444 0.9995

T2_B3 49.793 0.547 35,267.997 1.468 0.457 0.9996

T2_B4 65.015 0.351 41,749.746 1.222 0.478 0.9989

T2_B5 57.929 0.427 61,119.663 1.156 0.422 0.9994

Mean7SD 57.91579.9606 0.56470.126 44,988.754713,772.563 0.98670.245 0.49270.076

1

3

5T2_B1

Regression curve

ΔD

(mm)

-9

-7

-5

-3

-1

F (N)

-13

-11

-0.035 -0.025 -0.015 -0.005 0.005

Fig. 7. Regression result with a linear constitutive law (C1 ¼ 32.480Mpa,

C2 ¼ 0.5).

Z. Teng et al. / Journal of Biomechanics 41 (2008) 1995–2002 1999

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underestimates the ability of cartilage to resist collapse.When the regression with linear model was carried out, C2

was fixed to be 0.5. Without this, the regressed result of C2

would be close to 1, which was thought to be irrational andthe regression was similarly poor.

4. Discussion

This study is the first attempt to model the nonlinearmechanical properties of tracheal cartilage. The force–displacement (F�DD) curve in the material test shows thenonlinear behavior of tracheal cartilage. Rings both closeto the larynx (Fig. 5) and above the tracheal bifurcation(Fig. 6) display this behavior. The mathematical modelusing Fung-type strain energy density function has beendeveloped to fit the experimental data. This model can beused to describe the material properties of trachealcartilage. This study may provide insight into the nonlinearbehavior of tracheal cartilage. It may be useful for futurestudy on tracheal collapse behavior under physiologicaland pathological conditions.

While performing the mechanical test, the authors foundthe tracheal cartilage displayed a nearly linear trend ofF�DD in tension. This may explain why it has been widelyconsidered as a linear elastic material. To study itsbehavior under tension, a test was performed with adisplacement ranging from 0 to 10mm. The F�DD curvewas a nearly straight line (see the data points in Fig. 8),which was well captured by the linear model (see thecontinuous lines in Fig. 8). The determined material

constant is listed in Table 3. Therefore, the linear modelis a good approximation if the tension is the only interest,for instance, in the case of studying the mechanicalbehavior of the airway when balloon expansion surgery iscarried out to cure a stenosis. Here, the authors would liketo explain that the linear model, treating the cartilage asa linear elastic material, does not yield a linear relationof F and DD.For each material, its mechanical property must be

determined by the inherent micro-components and struc-ture. Therefore, a closer view of the structure will provideinsight into the nonlinear mechanical behavior of trachealcartilage. Tracheal cartilage is a type of hyaline cartilage,characterized by small aggregations of chondrocytesembedded in an amorphous matrix of ground substancereinforced by collagen fibers designated as collagen type II(Young et al., 2006). The surrounding layer is called theperichondrium, composed of collagen fibers and spindle-shaped cells that resemble fibroblasts. The collagen fibers inthe thin layer close to the lumen are oriented eithercircumferentially (along the cartilage length) or long-itudinally (along the axis of the airway). In the deeperlayer, which is much thicker in the cartilage, collagen fibersare oriented less regularly with a circumferential tendency(Roberts et al., 1998). It has been demonstrated thatcollagen type II is a molecule with highly nonlinearmechanical properties (Sun et al., 2004). These may explainthe nonlinear mechanical properties of tracheal cartilage.Since a larger proportion of fibers are oriented in thecircumferential direction than longitudinally, the cartilageis treated as a transverse isotropic material.The nonlinear model was developed by considering

the incompressibility and transverse isotropy (Eq. (7)).Thus O in Eq. (5) has been simplified as Eq. (8). Therefore,it seems that we may not be able to derive all of thematerial constants, and this may limit the result’s applica-tion in 3D numerical simulations. However by comparingEqs. (5) and (8), it is clear that the simplification is basedon the relations among the constants, b�1 ¼ b1 þ b2 þ 2b4,b1 ¼ b2 ¼ 2b4, b�5 ¼ b5 þ b6 and b5 ¼ b6. According tothese relations, the constants in the general formation(Eq. (5)) can be easily obtained from the data listed inTable 2.From the experimental data, the tracheal cartilage close

to the bifurcation is quite stiff. The value of C1, a mainparameter reflecting the stiffness, listed in Tables 2 and 3 isquite high. This C1 value is around 3 times larger than theYoung modulus for humans (Rains et al., 1992) and thoseadopted by Begis et al. (1988) and Costantino et al. (2004).

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8

10

4

6

T1_B1

ΔD

(mm)

0

2

0 0.005 0.01 0.015 0.02 0.025

T1_B4

T2_B2 T2_B4

T2_B1

T1_B3

T2_B5

F (N)

T1_B2

T1_B5

T2_B3

Fig. 8. F�DD experimental data points in tension and the corresponding

regression curve using the linear model.

Table 3

Material constants from the linear model (DD range from 0 to 10mm)

T1_B1 T1_B2 T1_B3 T1_B4 T1_B5 T2_B1 T2_B2 T2_B3 T2_B4 T2_B5 Mean7SD

C1(Mpa) 29.644 31.633 30.111 34.371 36.287 30.891 48.981 56.479 63.543 54.146 41.609712.842

C20.617 0.603 0.722 0.580 0.537 0.601 0.455 0.321 0.218 0.329 0.49870.162

Z. Teng et al. / Journal of Biomechanics 41 (2008) 1995–20022000

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It implies the mechanical property of tracheal cartilagemight be species specific. From the result of C2 in Tables 2and 3, it is reasonable to set this parameter to be 0.5 as wasdone by Holzhauser and Lambert (2001). This implies thatthe stiffness at the tip is half that of the middle point.However, according to the measurement at differentpositions in this study, the thickness at the middle couldbe more than 3 times of the value of the position close tothe tip (data not shown here). It is well known that a thinmaterial is weak. Therefore, it is not clear whether theinherent smaller stiffness or the relative thinner makes thepart close to the tip softer. Future study to investigate thisposition-dependent variation of the mechanical propertiesmay be useful.

From the curves in Fig. 5, we can see that the impact ofsoft connective tissues on the bending behavior can bedisregarded. This may be because of the weakness of softtissue. The Young modulus of the tracheal mucosalmembrane of rabbit is 3.721Kpa (Wang et al., 2000) andthe value of guinea pig is 3.992Kpa (Wang et al., 2005).Our result confirms previous reports that the stiffness ofcartilage is much higher than that of the mucosalmembrane (Lambert et al., 1991a; Rains et al., 1992).

Some important issues relative to the experiment and theassumptions for the theoretical frame need to be explained.Regarding the experiment, the fixing method shown inFig. 1(B) is efficient to eliminate the slide along the steel barduring the test. Since the load cell is highly precise (with0.0001N precision), it is rational to assume that the slidewill lead to discontinuities in force (F) in the F�DD curve.Actually there was no discontinuity in the data curve whileperforming the test. Thus it is rational to model theconnection between the tip and testing machine as a hinge.This satisfies the boundary conditions (15a) and (15d). Theassumption of unchanged total length of cartilage isrelevant to the value of stiffness (in 10Mpa order) of thecartilage and the small force (less than 0.05N) appliedduring the test. It can be estimated that the stretch will bemuch less than 1% during the material test (the area ofcross-section ranges from 5.04 to 6.37mm2). Thus it isreasonable to assume the length keeps constant during thedeformation. In order to provide a solid validation for themathematical model and the solving procedure, a seriesimage of deformed shape of cartilage was recorded duringthe test. The inner wall and outer wall of the deformedshape were identified and were further approximated byconcentric skew ellipses. The middle points of these twoellipses described the deformed shape. The deformedshapes of the cartilage with different vertically appliedforce (F) and the corresponding predicted shapes areshown in Fig. 9. The represented line (the continuous linein Fig. 9) almost exactly passes through the shape points(points in Fig. 9). Thus, Fig. 9 provides support for themathematical model and solving procedure in this study.

Despite the good fit of the experimental data, somelimitations exist in the theoretical model. (1) The non-uniform thickness along the cartilage length is not

considered. Here the thickness is treated as uniform alongthe length. However, it has been observed that a localthinning leads to a different bending behavior (Lambertet al., 1991a); (2) According to the micro-structure ofcartilage (Roberts et al., 1998), tracheal cartilage should bean anisotropic material. In this paper, it is treated to betransversely isotropic by assuming the fibers are predomi-nately arranged in the circumferential direction. And thevariation of mechanical property along the thickness is nottaken into account; (3) The length of tracheal cartilage isassumed constant during the deformation. It might not betrue under some extreme conditions. For instance, when theforce (F) is very high and eventually closes the gap, the forcein the cartilage would induce an un-ignorable extension andmake the assumption of constant length incorrect; and (4)similarly, Eq. (7) might not be true when F is high andeventually closes the gap, which might lead to lxalZ.

5. Conclusion

Tracheal cartilage is a nonlinear material displayinghigher strength in compression than in tension. Thisbehavior can be captured by Fung-type strain energydensity function. In tension, the curve of F�DD displays anearly straight line, which can be predicted by the linearmodel. This indicates that the linear model is efficient in

ARTICLE IN PRESS

10 mm

F=-0.0260

F=-0.0150

F=0.0040

F=0.0080

F=0.0158

Fig. 9. Deformed shape of tracheal cartilage with a given force (data

points) and the corresponding represented curve by the nonlinear model

(the continuous line). The arrow denotes the force (F) applied on the

cartilage tip.

Z. Teng et al. / Journal of Biomechanics 41 (2008) 1995–2002 2001

Author's personal copy

modeling tension. Furthermore, soft tissue attached to thecartilage wall has little impact on the bending behavior oftracheal cartilage.

Conflict of interest statement

The authors declare that they have no competinginterests relative to the paper.

Acknowledgments

The study was supported by the Spanish Ministry ofScience and Technology through the research projectDPI2004-07410-C03-01, the Spanish Ministry of Healththrough the National Network IM3, the Aragon’s govern-ment through the research project PIP113/2005 and theJuan de la Cierva Program. These supports are gratefullyacknowledged. We thank Mr. Sohrab Virk (undergraduateat the Massachusetts Institute of Technology) for checkingthe manuscript.

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