quantification of tympanic membrane elasticity parameters

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Quantification of tympanic membrane elasticity parameters from in situ pointindentation measurements: Validation and preliminary study

Jef Aernouts *, Joris A.M. Soons, Joris J.J. DirckxLaboratory of Biomedical Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 July 2009Received in revised form 4 September 2009Accepted 17 September 2009Available online xxxx

Keywords:Tympanic membrane elasticityPoint indentationInverse modeling

0378-5955/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.heares.2009.09.007

* Corresponding author. Tel.: +32 32653435.E-mail address: [email protected] (J. AernoutsURL: http://www.ua.ac.be/bimef (J. Aernouts).

Please cite this article in press as: Aernouts, J., ements: Validation and preliminary study. Hear.

Correct quantitative parameters to describe tympanic membrane elasticity are an important input forrealistic modeling of middle ear mechanics. In the past, several attempts have been made to determinetympanic membrane elasticity from tensile experiments on cut-out strips. The strains and stresses insuch experiments may be far out of the physiologically relevant range and the elasticity parametersare only partially determined.

We developed a setup to determine tympanic membrane elasticity in situ, using a combination of pointmicro-indentation and Moiré profilometry. The measuring method was tested on latex phantom modelsof the tympanic membrane, and our results show that the correct parameters can be determined. Theseparameters were calculated by finite element simulation of the indentation experiment and parameteroptimization routines.

When the apparatus was used for rabbit tympanic membranes, Moiré profilometry showed that thereis no measurable displacement of the manubrium during the small indentations. This result greatly sim-plifies boundary conditions, as we may regard both the annulus and the manubrium as fixed withouthaving to rely on fixation interventions. The technique allows us to determine linear elastic materialparameters of a tympanic membrane in situ. In this way our method takes into account the complexgeometry of the membrane, and parameters are obtained in a physiologically relevant range of strain.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Finite element modeling is a powerful method for the investiga-tion of middle ear mechanics, (Eiber, 1999; Elkhouri et al., 2006;Koike et al., 2002; Sun et al., 2002), studying the effect of middleear pathologies, (Gan et al., 2006) and predicting the behaviourof middle ear prostheses (Eiber et al., 2006; Schimanski et al.,2006). The technique allows to take into account the complexshape and boundary conditions of the middle ear structure. Cur-rent finite element models are mainly restricted to acousticalsound pressures and low acoustical frequencies, and good datafor the mechanical properties of the tympanic membrane are stilllacking. It is known, however, that tympanic membrane elasticityhas a significant influence on the resulting output (Elkhouriet al., 2006). Furthermore, when one wants to extend finite ele-ment models to the quasi-static regime (Dirckx and Decraemer,2001), effects such as nonlinearity and viscoelasticity of the tym-panic membrane can play an important role. On the other hand,when one considers dynamical behaviour, the strain rate depen-dent tympanic membrane stiffness should be taken into account.

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).

t al. Quantification of tympanicRes. (2009), doi:10.1016/j.hear

Mechanical properties of the human tympanic membrane werefirst measured by Békésy in 1960 (Békésy, 1960). He applied aknown force on a cut-out strip and measured the resulting dis-placement. Assuming the sample to be a uniform, isotropic, linearelastic beam with a thickness of 50 lm, he calculated a Young’smodulus of 20 MPa. Kirikae measured the ‘dynamical’ stiffness ofa strip of human tympanic membrane in 1960 (Kirikae, 1960).From the system’s resonance frequency, he calculated a Young’smodulus of 40 MPa based on a thickness of 75 lm. In 1980, Decra-emer et al. showed a nonlinear stress–strain curve for human tym-panic membrane (Decraemer et al., 1980). In the large straincondition, starting from �8%, a Young’s modulus of 23 MPa wasproposed.

In 2005, Fay et al. estimated the elastic modulus of human tym-panic membrane based on the elasticity of the collagen fibers andfiber density (Fay et al., 2005). Their data suggested an elastic mod-ulus in the range 100—300 MPa, significant higher than previousvalues. More recently, new experimental observations of theYoung’s modulus of human tympanic membrane were publishedby Cheng et al. (2007). They conducted a uniaxial tensile test andobtained a stress–strain curve which was in good agreement withthat of Decraemer et al. Describing the nonlinearity using the first-order Ogden model, they found l1 ¼ 0:46 MPa and a1 ¼ 26:76. In

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addition, preliminary stress relaxation tests were performed tostudy the viscoelastic behaviour of the tissue. Two years later,the same research group characterized more detailed the visco-elastic behaviour by in-plane and out-of-plane nanoindentationsand found Young’s relaxation moduli (Huang et al., 2008; Daphala-purkar et al., 2009). Their results show an in-plane modulus chang-ing by � 10% from 1 to 100 s with steady-state values of 37:8 and25:73 MPa respectively for different samples. The out-of-planemodulus seemed to reduce by 50% from 1 to 100 s with a stea-dy-state value varying from 2 to 15 MPa over the surface. Theseexperiments show a wide variety and it is not straightforwardhow to interpret the different moduli. Regardless Kirikae’s mea-surement, all previous mentioned publications report quasi-staticbehaviour of the tympanic membrane. Very recently, Luo et al.measured the Young’s modulus at high strain rates using a minia-ture split Hopkinson tension bar (Luo et al., 2009). Due to the vis-coelastic properties of the membrane, they found that the Young’smodulus increases with the increase in strain rate. However, theyonly measured up till 2 kHz.

It should also be mentioned that in addition to human tympanicmembrane data, a lot of experiments have been performed on tym-panic membranes of laboratory animals as well.

It is clear that accurate data for the mechanical properties of thetympanic membrane, both for human and animal samples, cannotbe determined using uniaxial tensile tests. At first, when one con-siders the tympanic membrane to be linear elastic, which is rea-sonable in the acoustical regime, a Young’s modulus can only beassigned in a physiologically irrelevant strain range starting from�8%. Moreover, small and long strips with uniform thickness ofthe tympanic membrane need to be cut to perform uniaxial tensiletests. In general, however, tympanic membrane thickness varieswith a factor 4 between periphery and the center (Kuypers et al.,2005), making it impossible to cut long uniform samples.

Another parameter that might have an influence on themechanical behaviour of the tympanic membrane is the presenceof prestrain. Békésy and Kirikae are the only ones who investigatedthis characteristic, but their results depended on the type of spe-cies and were not unambiguous (Decraemer and Funnell, 2008). Fi-nally, the pars flaccida has never been examined and propertieslike anisotropy, inhomogeneity, nonlinearity and viscoelasticityare only poorly studied. These are mechanical properties thatmay become important in more detailed examinations.

In order to obtain data in more realistic circumstances, wedeveloped a setup to determine tympanic membrane elasticityin situ. The measurement method consists of doing a point inden-tation perpendicular on the membrane surface; measuring theindentation depth, resulting force and three-dimensional shapedata; simulating the experiment with a finite element model andadapting the model to fit the measurements using optimizationprocedures. In the first part of this paper, the method of measuringand model optimization is validated on a scaled phantom model ofthe tympanic membrane with known elasticity parameters. In thesecond part, preliminary results from one rabbit tympanic mem-brane experiment are shown.

Fig. 1. Schematic drawing of the point indentation setup: (1) translation androtation stage, (2) phantom model: ‘manubrium shaped rod’ pushed in circularrubber membrane, (3) needle connected to a load cell, (4) stepper motor and (5)LVDT.

2. Materials and methods

2.1. Phantom model

In the validation experiment, we used rubber from medicalgloves, which mainly consists of natural latex. The thickness is0:18� 0:02 mm and the material is isotropic and homogeneous.Like other rubber-like materials, natural latex exhibits very largestrains with strongly nonlinear stress–strain behaviour. For thisreason, the rubber can be described as a hyperelastic material. A

Please cite this article in press as: Aernouts, J., et al. Quantification of tympanicments: Validation and preliminary study. Hear. Res. (2009), doi:10.1016/j.hear

hyperelastic material is typically characterized by a strain energydensity function W. A well known constitutive law for rubber-likematerials is the Mooney–Rivlin law (Treloar, 1975):

W ¼XN

iþj¼1

Cij I1 � 3ð Þi I2 � 3ð Þj þ 12

K ln Jð Þ; ð1Þ

with N the order of the model, I1 and I2 the strain invariants of thedeviatoric part of the Cauchy–Green deformation tensor, Cij theMooney–Rivlin constants, K the bulk modulus and J the determi-nant of the deformation gradient which gives the volume ratio.The invariants are given as:

I1 ¼ k21 þ k2

2 þ k23; ð2Þ

I2 ¼1k2

1

þ 1k2

2

þ 1k2

3

; ð3Þ

with ki ði ¼ 1;2;3Þ the principal stretches. It is common to assumethat rubber materials are incompressible when the material is notsubjected to large hydrostatic loadings, so that the last term in Eq.(1) will be neglected (Bradley et al., 2001). In this study, we willonly consider the first-order Mooney–Rivlin equation ðN ¼ 1Þ. Inthis case, Eq. (1) becomes:

W ¼ C10 I1 � 3ð Þ þ C01 I2 � 3ð Þ: ð4Þ

This low order strain energy function is described by two con-stants: C10 and C01.

Since the latex rubber is perfectly homogeneous in all its phys-ical properties, a uniaxial tensile test was carried out which resultsin accurate determination of the first-order Mooney–Rivlinparameters.

The phantom model of the tympanic membrane was created bypushing a ‘manubrium shaped rod’ in a flat circular rubber mem-brane which was constrained at its circumference. In this way,the typical conical tympanic membrane shape was obtained. Thediameter of the phantom model was 50 mm and the height was16 mm, approximately eight times the size of a human tympanicmembrane. The phantom model was placed on a translation– and rotation stage, a schematic drawing is shown in Fig. 1. Inden-tations in and out in a direction perpendicular to the surface mem-brane were carried out using a stepper motor with indentationdepths up till 2 mm. The needle had a cylindrical ending with adiameter of 1:7 mm and indentations were carried out slowlyð_r ¼ 0:125 mm s�1Þ, meaning that quasi-static behaviour was stud-ied. The points of indentation were chosen in the undermost part



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of the ‘cone’ (the inferior part of the ‘tympanic membrane’) be-cause in the tympanic membrane case, it showed to be experimen-tally impossible to do indentations in the superior part. By doingthe indentation perpendicular to the surface and not too close tothe boundary, slippage between the needle and the rubber wasavoided. The resulting force was measured with a load cell (Senso-tec Model 31). The exact indentation depth was assessed with anLVDT (HBM KWS3071). All signal processing was done via A/D con-version in Matlab (using NI DAQPad-6015).

Three-dimensional shape data were measured with a projectionLCD-Moiré profilometer (Buytaert and Dirckx, 2008; Dirckx andDecraemer, 1989), developed in our laboratory. The device is ableto perform topographic measurements with a height resolutionof 25 lm on a matrix of 1392� 1040 pixels. The Moiré shape mea-surement was used to locate the exact point of indentation and todetermine if the membrane is locally at right angle with the inden-tation needle.

The numerical simulations were performed with the finite ele-ment code FEBio, which is specifically designed for biomechanicalapplications. The rubber membrane was modeled as a first-orderMooney–Rivlin material, triangular incompressible shell elementswere used to create a non-uniform mesh, with increased meshdensity in the contact areas. The border of the membrane wasx; y; z constrained, with x; y and z the axes of a Cartesian coordinatesystem with the x; y plane parallel to the ‘annulus plane’. In thefirst model steps, the ‘manubrium shaped’ rigid body was movedvertically into the flat membrane using a combination of rigid –and sliding contact. In the following model steps, the perpendicu-lar needle indentation was modeled as a rigid body translationwith frictionless sliding contact.

In order to find the first-order Mooney–Rivlin parameters, weoptimized the average square force difference defined as

errorforce ¼1N

XN

j¼1

FexpðqjÞ � FmodðqjÞ� �2

; ð5Þ

in which N is the number of evaluated points, qj the indentationdepth, FexpðqjÞ the experimental force and FmodðqjÞ the simulatedforce. The optimization was carried out with a surrogate modelingtoolbox (Gorissen et al., 2009; Jones et al., 1998). Using this routine,firstly one has to specify the input domain. Then, a first surrogatemodel calculated from 24 homogeneously distributedðC10;C01; errorforceÞ data points is built. Afterwards, the model is re-fined on the basis of a gradient-optimum based sample selection inorder to localize very accurately the optimum values.

2.2. Tympanic membrane

The animal model selected for the present study was the NewZealand white rabbit. Animals were sacrificed by injection of120 mg/kg natrium pentobarbital and temporal bones were re-moved. Openings were made in the bulla so that visual accesswas provided from both sides of the tympanic membrane, leavingthe tympanic membrane and entire middle ear ossicle chain intact.Total preparation time was always less than 1.5 h. The study wasperformed according to the regulations of the Ethical Committeefor Animal Experiments of the University of Antwerp.

The tympanic membrane was modeled as a linear isotropichomogeneous elastic material which is described with two inde-pendent elastic constants: the Young’s modulus E and Poisson’sratio m. The thickness of the rabbit tympanic membrane variessomewhat along the membrane, however, in this study the mem-brane was modeled with a uniform thickness. Since no data onrabbit tympanic membrane thickness are available, we used amean value of 12:5 lm measured for the cat tympanic membrane


with confocal microscopy (Kuypers et al., 2006). In forthcomingwork, we will add a measured thickness distribution to our model.

The prepared sample was glued on a small tube so it could beplaced on a translation and rotation stage in a similar way as inFig. 1. In a direction perpendicular to the membrane, the indenta-tion needle was moved back and forth from the lateral (externalear) side with a maximal indentation depth up till 400 lm usinga piezoelectric actuator (PI P-290) in combination with an LVDT(Solartrion) and a feed-back controller unit (PI E-509 E-507). Theneedle had a cylindrical ending with a diameter of 210 lm. Theindentation was carried out in steps of 40 lm with a step time of4 s, thus quasi-static behaviour was studied. The resulting forcewas measured with a strain-gage force transducer (SWEMA SG-3). Input and output signals were controlled with a computer viaA/D conversion in Matlab (using NI DAQPad-6015).

Three-dimensional shape data were measured from the medialside with our LCD-Moiré profilometer. Because tympanic mem-brane dimensions are significant smaller than those of the phan-tom model, the setup was adjusted resulting in a heightresolution of 15 lm. Since the tympanic membrane is almostnon-reflective, it was coated with a layer of white paint mixed withglycerin to avoid dehydration. Again, the Moiré data were used todetermine exact needle position and to verify perpendicular needlepositioning.

Moreover, a highly detailed non-uniform mesh was created onthe basis of the Moiré shape images. In the needle indentation areaand in the manubrium neighbourhood, mesh density was in-creased. Since in the rabbit case the pars flaccida can geometricallybe regarded as almost completely separated from the pars tensa(Fumagalli, 1949), the influence of the pars flaccida can be ignoredso that the membrane was modeled as a homogeneous material.The manubrium was modeled as a rigid body that is x; y; z con-strained because no measurable motion was observed duringindentation (see further). The border of the tympanic membranewas modeled as fully clamped, the perpendicular needle indenta-tion was modeled as a rigid body translation with frictionless slid-ing contact. Afterwards, in a similar way as described for thephantom model case, the errorforce was optimized in order to deter-mine linear elasticity parameters.

3. Results

3.1. Phantom model

The mean output of the uniaxial tensile test, carried out on threedifferent samples, was C10 ¼ ð43� 5Þ kPa and C01 ¼ ð159� 6Þ kPa.

In Fig. 2a, the output of a force-indentation experiment is plot-ted (line). The indentation was carried out slowly and only a smallhysteresis is seen. The associated finite element model is shown inFig. 2b, with the number of membrane shell elements equal to4337. In the first model steps, the ‘manubrium shaped rod’ rigidbody displacement was applied which results in the typical conicalshape. Next, the perpendicular point indentation was applied. Thecolor map represents the effective strain, which rises up to approx-imately 40% in the indentation area.

The resulting surrogate model is shown in Fig. 3. The evaluatedsamples are plotted as white points and the color map representsthe surrogate model fitted through the errorforce values. A rangeof minima lying on a linear curve can be seen (dashed line). Onecan see that as a consequence of the gradient-optimum criterionsample selection, more samples were chosen in the minimum areain comparison to elsewhere. The force-indentation output for threedifferent points of ðC10;C01Þ optimum values is plotted in Fig. 2a(different dots). According to this plot, we see that the first-orderMooney–Rivlin equation is able to describe the load curve quite



0 500 1000 1500 20000

20

40

60

80

100

120

140

160

indentation depth [µm]

forc

e [m

N]

experimental datamodel data: C10=20 kPa; C01=183 kPamodel data: C10=75 kPa; C01=130 kPamodel data: C10=130 kPa; C01=77 kPa

Fig. 2. Phantom model: (a) plot of experimental force-indentation data (line) and best models output (different dots). (b) FE model with applied ‘manubrium shaped’ rigidbody displacement and needle indentation. The effective strain in the point indentation area after indentation rises up to approximately 40%.

C_10 [Pa]

C_0

1 [P

a]

2 4 6 8 10 12 14

x 104

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

erro

r forc

e [mN

2 ]

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 3. Phantom model: contour plot with result of errorforce optimization after 119samples. The samples are plotted as white dots, the color map represents the fittedsurrogate model and the resulting range of local minima is highlighted with adashed line.

25 30 35 40 45 50

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

r [mm]

erro

r_z

[mm

]baselinemodel data: C10=31 kPa; C01=172 kPamodel data: C10=11 kPa; C01=191 kPamodel data: C10=51 kPa; C01=153 kPa

Fig. 4. Phantom model: plot of difference between experimental and modeldeformation data along a point indentation section line for different Mooney–Rivlinparameters. In the case of C10 ¼ 31 kPa and C01 ¼ 172 kPa, the difference error is thesmallest.

0 1 2 3 4 5−0.5

0

0.5

1

1.5

2

2.5

heig

ht [m

m]

section axis [mm]

without indentationwith indentation


ARTICLE IN PRESS

well. It is not possible, however, to extract a unique set of Mooney–Rivlin parameters only on the basis of a load curve optimization.Therefore, also a shape analysis was carried out in which the aver-age square difference between model – and experimental deforma-tion (conform Eq. (5)) along a point indentation section wasoptimized so that the final optima were filtered out of the rangeof force minima. In Fig. 4, the difference between model and exper-imental shape data is shown for different Mooney–Rivlin parame-ters. The final output after shape optimization was C10 ¼ 31 kPaand C01 ¼ 172 kPa. The final output for a second indentation exper-iment was C10 ¼ 33 kPa and C01 ¼ 163 kPa.

Fig. 5. Rabbit tympanic membrane (right): experimentally measured section plotsbefore and after indentation to show fixed behaviour of manubrium.

3.2. Tympanic membrane

Fig. 5 shows an experimentally measured section plot of tym-panic membrane deformation during indentation. This experimentshows no manubrium movement, so the manubrium can beregarded as fixed.


The finite element model of the perpendicular point indentationexperiment is shown in Fig. 6, with the number of membrane shellelements equal to 5988. The perpendicular point indentation re-



Fig. 6. Rabbit tympanic membrane (right): FE model with indentation: the effective strain in the point indentation area after indentation rises up to approximately 15%.

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sults in an effective strain of approximately 15% in the point inden-tation area and due to manubrium fixation, no deformation in theopposite membrane part can be seen.

In Fig. 7a, the associated force output of the experiment is plot-ted (dotted line). When doing slow indentations, only a small hys-teresis can be seen. Using the finite element model and theexperimental force-indentation data, the surrogate model was cal-culated. The input interval was set to E ¼ ½20;40�MPa andm ¼ ½0:2;0:49�, the resulting surrogate model is shown in Fig. 7b.The evaluated samples are plotted as white points and the colormap represents the surrogate model fitted through the errorforce

values. In total, 118 samples were evaluated to guarantee an accu-rate determination of the model.

Inspecting the resulting output, one can see that the optimumvalue for the Young’s modulus E in the Poisson’s ratio intervalm ¼ ½0:2;0:45� is practically constant: E ¼ 30:4 MPa. For higherPoisson’s ratio’s, the optimum value decreases. For a Poisson’s ratiom ¼ 0:49, towards the theoretical upper limit m ¼ 0:5 that repre-sents incompressibility, the optimum value is E ¼ 26:4 MPa. InFig. 7a, the force-indentation curve is plotted for the optimum val-ues with respectively m ¼ 0:2; 0:45 and 0:49. The simulated curvesshow good agreement with the experimental one. For all theseoptimum values, models show practically no difference in modeldeformation, so that a shape analysis is not under discussion.

0 50 100 150 200 250 300 350 400

0

2

4

6

8

10

12

14

16

indentation depth [µm]

forc

e [m

N]

experimental datamodel data: E=30.4 MPa; v=0.2model data: E=30.4 MPa; v=0.45model data: E=26.4 MPa; v=0.49

Fig. 7. Rabbit tympanic membrane (right): (a) plot of experimental force-indentation daoutput after 118 evaluated samples: the samples are plotted as white dots and the colo


4. Discussion

If large enough samples of a material are available, tensile testson a strip can be performed to determine elasticity parameters. Onsmall samples like the tympanic membrane, it becomes very diffi-cult to control the exact size and extension of a strip. In this case, itbecomes inevitable to perform in situ measurements because spec-imens are too small and the natural shape of the membranes arecomplex.

In this paper, a new approach is presented in which elasticityparameters are determined by inverse modeling of an in situ pointindentation experiment.

At first, the characterization method was validated on a latexrubber phantom model. Measuring the first-order Mooney–Rivlinelasticity parameters with a uniaxial tensile test, carried out onthree different samples, the output was C10 ¼ ð43� 5Þ kPa andC01 ¼ ð159� 6Þ kPa. Applying the point indentation method, theoutput of a first point indentation experiment was C10 ¼ 31 kPaand C01 ¼ 172 kPa. The output of a second experiment on a differ-ent location was C10 ¼ 33 kPa and C01 ¼ 163 kPa. One can see thatthere is a good agreement between the two approaches. However,there is approximately a 25% relative error for the C10 values. It isknown that the C10 parameter has a small influence on the stress–strain curve for strains smaller than 50%. Since effective strains in

E [Pa]

v

2 2.5 3 3.5 4x 107

0.2

0.25

0.3

0.35

0.4

0.45

erro

r forc

e [mN

2 ]

0

1

2

3

4

5

6

7

8

9

10

ta (line) and best models output (different dots). (b) Contour plot with optimizationr map represents the fitted surrogate model.




ARTICLE IN PRESS

the indentation experiments go only up to approximately 80% and,this apparently high relative error can be argued.

Having validated the method, a preliminary experiment on aright rabbit tympanic membrane was performed. Indentationswere carried out slowly, so that quasi-static behaviour was stud-ied. In this first approach, the tympanic membrane was consideredto be a linear isotropic homogeneous elastic material. Although weknow this is not the best choice, it allows a comparison with pre-vious studies.

From our Moiré shape measurements, we learned that themanubrium does not move under influence of a small point inden-tation on the tympanic membrane. This is a very important result,as it shows that no complex procedures are needed to fixate themanubrium in order to obtain well defined boundary conditions.

Since there was practically no difference in model deformationfor different force-indentation optimum values, we cannot filterout one optimum value for the Young’s modulus. In contrast, it willdepend on the choice of Poisson’s ratio. In most of the middle earfinite element models, a Poisson’s ratio in the range m ¼ ½0:3;0:4�is proposed (Elkhouri et al., 2006; Koike et al., 2002; Sun et al.,2002), with a corresponding optimum value of E ¼ 30:4 MPa. How-ever, it might be argued that the Poisson’s ratio should be close to0.5 because soft tissues are thought to be nearly incompressible. Inthis case, the corresponding optimum value is E ¼ 26:4 MPa. Inboth cases, the optimum values are close to those found in previ-ous studies.

In future work, more point indentation experiments will be per-formed and a measured tympanic membrane thickness distribu-tion will be added to the finite element models. Simultaneously,we will investigate the behaviour under dynamic load conditionsand the influence of nonlinearity, inhomogeneity, anisotropy, vis-coelasticity and the possible presence of prestress.

Acknowledgments

This work was supported by the Institute for the Promotion ofInnovation through Science and Technology in Flanders (IWT-Vlaanderen).

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quantification of tympanic membrane elasticity parameters

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