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Analysis of the Mechanical Properties of the Human Tympanic Membrane and Its Influence on the Dynamic Behaviour of the Human Hearing System

Analysis of the Mechanical Properties of the Human Tympanic Membrane and Its Influence on the... Hindawi Applied Bionics and Biomechanics Volume 2018, Article ID 1736957, 12 pages https://doi.org/10.1155/2018/1736957 Research Article Analysis of the Mechanical Properties of the Human Tympanic Membrane and Its Influence on the Dynamic Behaviour of the Human Hearing System 1 2 2 2 L. Caminos, J. Garcia-Manrique, A. Lima-Rodriguez, and A. Gonzalez-Herrera Departamento de Ingeniería Mecánica, Universidad Nacional Experimental del Táchira, San Cristobal, Venezuela Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, Malaga, Spain Correspondence should be addressed to A. Gonzalez-Herrera; agh@uma.es Received 18 January 2018; Revised 25 April 2018; Accepted 29 April 2018; Published 9 May 2018 Academic Editor: Thibault Lemaire Copyright © 2018 L. Caminos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The difficulty to estimate the mechanical properties of the tympanic membrane (TM) is a limitation to understand the sound transmission mechanism. In this paper, based on finite element calculations, the sensitivity of the human hearing system to these properties is evaluated. The parameters that define the bending stiffness properties of the membrane have been studied, specifically two key parameters: Young’s modulus of the tympanic membrane and the thickness of the eardrum. Additionally, it has been completed with the evaluation of the presence of an initial prestrain inside the TM. Modal analysis is used to study the qualitative characteristics of the TM comparing with vibration patterns obtained by holography. Higher-order modes are shown as a tool to identify these properties. The results show that different combinations of elastic properties and prestrain provide similar responses. The presence of prestrain at the membrane adds more uncertainty, and it is pointed out as a source for the lack of agreement of some previous TM elastic modulus estimations. 1. Introduction numerical simulation and limit the extension of the conclu- sions. This is particularly significant in the case of the TM. The function of the tympanic membrane (TM) in the sound A great deal of work has been devoted to estimate transmission process is easy to understand intuitively. The these properties, especially to the determination of the piston-like motion, which transfers air sound pressure wave TM elastic modulus (EM). Most of the results range from into the cochlea, has been clearly identified long time ago 20 to 40 MPa or are close to these values. They are based on tension tests on small samples [1, 2, 20, 21] and are [1, 2]. Nevertheless, at higher frequencies, the TM motion is not so simple and new characteristic patterns appear. subject to important uncertainties. Different indentation There have been many experimental and numerical techniques have been developed providing result in the studies to evaluate the behaviour of the TM. These pat- same range [22–25]. Nevertheless, recently, values on the terns have been widely studied experimentally since the range of 3 MPa have been reported [26], showing the dif- development of techniques as holography [3–5]. With this ficulty to obtain accurately this parameter separately from technique, the complex TM vibration patterns at high fre- the estimation of the thickness. quencies have been revealed. Fay et al. [27] suggested that these values could be under- The alternative is numerical simulation. Since the early estimated; they used composite laminate theory and evalu- works led by Funnell and Laszlo [6] and Funnell et al. [7, 8], ated their result comparing with the dynamic response of finite element (FE) models have been used to study the behav- the TM. They calculated the effect of the orientation of the iour of the system [9–19]. Nevertheless, uncertainties regard- fiber on the EM obtained on tension or bending tests when ing the accuracy of the material properties hinder the an isotropic assumption was made. They showed that the 2 Applied Bionics and Biomechanics elastic material behaviour and small displacement condition EM could be in a broad range depending of the angle of ori- entation of the fibers of the sample tested. They confirmed are assumed. Both models have been simplified and limited their finding by means of correlating experimental dynamic to the components necessary for the purpose of this study. wavelength pattern. These patterns were obtained from live The anatomic measures and functional properties were anesthetized cat but conclusions can be transferred to the based on published data. The geometrical model is divided human case. Fitting a mathematical model and by means of into three parts: TM, ossicular chain, and the system of liga- a parameter estimation procedure, they suggested that the ments, tendons, and joints (Figure 1(a)). Details of the geom- EM should be in the range 0.1 to 0.3 GPa. etry and property estimation are described in different A potential reason for this high stiffness value could be references [16, 33]. the presence of an unaccounted prestrain. It is well accepted The FE model was developed using ANSYS. TM was mod- the presence of active muscles on the middle ear system [28], elled with shell elements assuming uniform thickness (50 μm) but apart from hypothesized active tension effects inside the in order to simplify the analysis. The tympanic annulus is TM due to smooth muscles [29], only passive prestrain has modelled as a band (0.2 mm wide and 0.2 mm thickness) with been named by several authors [1, 2], but no quantitative the external border fixed. Solid elements are used for the ossic- estimation has been clearly made. Recently, Aernouts and ular chain, posterior incudal ligament, and incudomalleolar Dirckx [30] assumed an in situ strain for the elastic charac- and incudostapedial joints. Ligaments and tendons support- terization of the gerbil pars flaccida, but no other numerical ing malleus are modelled with beam elements. The stapedial model has included this effect. annular ligament is modelled with shell elements surrounding In the present paper, the parameters that define the bend- the stapedial footplate (0.05 mm wide and 0.05 mm thickness). ing stiffness properties of the membrane have been studied. Figure 1(a) shows the middle ear FE model, and the mechani- Mainly, the influence of two parameters has been studied: cal properties are summarized in Table 1. the Young’s modulus of the tympanic membrane and the The effect of the impedance of the fluid of the cochlea was thickness of the eardrum. Additionally, it has been completed modelled using a mechanical equivalent load consisting of a with the evaluation of the presence of an initial prestrain block of mass and dashpots according to [14]. This simplified inside the TM. model provides a good accuracy with a low computational Firstly, a harmonic analysis is made to evaluate the range cost. When modal analysis is applied to the TOS model, this of influence of these properties. Secondly, a modal analysis is equivalent load is removed to eliminate its own vibration used to check how the properties affect the dynamic modes, which are outside the scope of this study. response. We will demonstrate that variations in membrane Regarding the TM simplified model (Figure 1(b)), most parameters have small effects on the lower natural frequen- of the description previously stated is valid. The area corre- cies; the effects of thickness, EM, and prestrain on the higher sponding to the connection with the manubrium has also natural frequencies are larger and separable. been meshed with shell elements, but the mechanical proper- Rosowski et al. [4] used holography to observe the TM ties have been adapted to represent the inertial and stiffness vibration pattern in different species including human. They effect of the ossicular chain. The umbo was represented by established a qualitative description based on the pattern an area 0.6 mm wide with an EM equal to 14 GPa and the observed and related with the frequency range. They consid- density 1900 kg/m . In this way, the ossicular chain complex ered a simple pattern when only one maximum displacement vibration patterns—out of the scope of this part of the was observed (below 2 kHz in human), complex pattern study—are excluded from the model. Another important dif- when more than one maximum (from 2 to 8 kHz), and ference is that the membrane has been meshed with a higher ordered pattern when a high number of maximum appear number of elements as this model is intended to capture (above 8 kHz). This description will be used to evaluate the accurately high-frequency vibration pattern. response of the TM at different ranges of frequencies. Direct comparison of numerical and experimental results 2.2. Middle Ear Harmonic Response. As a starting point of is not easy to do [31, 32] as the acoustomechanic coupling this study, a sensitivity analysis has been done in order to must be solved; nevertheless, comparing numerical modes evaluate the influence of two key mechanical properties that and experimental vibration pattern leads to identify which strongly influence the response of the system: TM elastic of these material properties combinations are more suitable modulus and TM thickness. and suggest potential future working lines. First, the behaviour of the middle ear model will be shown. A harmonic analysis was conducted from 100 to 10,000 Hz. A uniform harmonic 80 dB stimulus pressure was applied to SPL 2. Materials and Methods the lateral side of the eardrum. The amplitude of umbo and 2.1. FEA Modelling Methodology. A brief summary of the stapedial footplate displacements versus frequency is shown main modelling methodology is presented in this section in Figure 2 compared with experimental measurements [37]. (see detail in [16]). It has been developed from precedent These experimental curves are the average data of 10 human works [6–15]. temporal bones with intact cochlea, using LDV with 80 dB sound pressure applied at the TM in similar condition to the Two models will be described. The first one includes all the elements of the middle ear, referred as tympanic-ossicular sys- present study. Rayleigh damping β = 0.0001 s is assumed. tem (TOS) model. The second one only includes the TM and With Table 1 parameters, the numerical result approximately the approximate effect of the manubrium. In both cases, linear predicts the experimental displacement. Applied Bionics and Biomechanics 3 Superior malleolar ligament Posterior incudal Malleus Incudomalleolar ligament joint Stapes Lateral malleolar Incus footplate ligament Stapedial annular ligament Eardrum (pars flaccida) Stapes Anterior malleolar ligament Stapedial tendon Tensor tympani Incudostapedial joint tendon Tympanic annulus Manubrium Eardrum Umbo (pars tensa) (a) (b) Figure 1: Finite element models: (a) tympanic-ossicular system (TOS) model and (b) tympanic membrane simplified model. Figures 3 and 4 show the influence of the EM and the thickness, respectively. The thick line without marks corre- Table 1: Mechanical properties used in middle ear components for sponds to the results obtained with the reference model finite element model. (Figure 2, Table 1). Values assigned to EM range from 3.2 MPa to 320 MPa, covering the extreme values obtained Density Young’s Poisson’s Component 3 2 from literature. (kg/m ) modulus (N/m ) ratio The TM EM presents a qualitatively similar influence in Eardrum both cases (Figure 3). Below 1000 Hz, the displacements 3 7 Pars tensa 1.2 × 10 [9] 3.2 × 10 [10] 0.3 [13] increased as the stiffness decreases. The peak value is at a 3 7 Pars flaccida 1.2 × 10 [9] 1 × 10 [13] 0.3 [13] lower frequency with a lesser stiffness. The displacement at 3 10 Malleus 1.9 × 10 [34] 1.41 × 10 [35] 0.3 [13] high frequency (above 2000 Hz) presents an opposite rela- 3 10 Incus 1.9 × 10 [34] 1.41 × 10 [35] 0.3 [13] tion; it increases slightly as the stiffness increases. 3 10 Stapes 1.9 × 10 [34] 1.41 × 10 [35] 0.3 [13] Regarding the thickness, its response is inversely propor- tional to the increase of stiffness and mass due to the incre- 1.2 × 10 Tympanic annulus 6 × 10 [13] 0.3 [13] ment of thickness (Figure 4). The influence is more (assumed) 3 9 significant at lower frequencies. The peak value close to Manubrium 1.0 × 10 [12] 4.7 × 10 [12] 0.3 [13] 1000 Hz is similar for all curves. Tensor tympanic 3 6 2.5 × 10 [12] 2.6 × 10 [12] 0.3 [13] These results are similar to those obtained by Gan et al. tendon [14] suggesting that an isotropic material behaviour as well Lateral malleolar 3 4 2.5 × 10 [12] 6.7 × 10 [12] 0.3 [13] as a mean thickness for the tympanic membrane model can ligament fit experimental data for the frequency range studied. Anterior malleolar 3 6 2.5 × 10 [12] 2.1 × 10 [12] 0.3 [13] According to the results shown, it could be stated than ligament 32 MPa seems to be a good value for the TM EM; neverthe- Superior malleolar 3 4 less, important variations of this parameter lead to result that 2.5 × 10 [12] 4.9 × 10 [12] 0.3 [13] ligament could be argued as valid too, depending of the experimental Posterior incudal 3 6 data used to compare. 2.5 × 10 [12] 6.5 × 10 [14] 0.3 [13] ligament 3 5 Stapedial tendon 2.5 × 10 [12] 5.2 × 10 [12] 0.3 [13] 3. Results Stapedial annular 3 5 2.5 × 10 [12] 2 × 10 [15] 0.3 [13] ligament Different modal analyses were performed. As a reference, a Incudomalleolar 3 10 modal analysis of the whole system (TOS) was made. Some 3.2 × 10 [13] 1.41 × 10 [13] 0.3 [13] joint selected mode shapes are drawn in Figure 5. Different types Incudostapedial of shapes are present. Some clearly reflect a TM vibration 3 5 1.2 × 10 [13] 6 × 10 [36] 0.3 [13] joint pattern (modes 8, 12, 13, 14, 25, and 29). Mode 12 is the clas- sic piston-like motion. Mode 8 presents a similar membrane motion but with a lower stapes displacement. Modes 3 and 11 show the movement of the joints and ossicles while the mem- brane and the stapes present a very low motion. Mode 27 4 Applied Bionics and Biomechanics 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 100 1000 10,000 Frequency (Hz) Exp. Hato et al. 2003 (intact cochlea) FE model (properties in Table 1) (a) 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 1.0E − 11 100 1000 10,000 Frequency (Hz) Exp. Hato et al. 2003 (intact cochlea) FE model (properties in Table 1) (b) Figure 2: Amplitude of displacement versus frequency of umbo (a) and stapes footplate (b), for a range from 100 to 10,000 Hz at 80 dB SPL sound pressure. shows a tilting movement of the stapes with hardly no mem- published results (see [38]). The modal analysis was per- brane displacement. Modes 14 and 25 correspond to the formed with this model. As the ossicular chain has been transition to complex pattern (as described in Rosowski removed, only the TM modes are present on the solution. et al. [4]), and mode 29 would be the starting point of the Figure 6 presents some selected mode shapes calculated ordered pattern. In this model, high-frequency modes are less with the reference properties (Table 1). It can be seen how accurate than lower due to the limited number of element modes 1, 2, and 3 (Figure 6) are equivalent to modes 12, 8, used. Nevertheless, they are a good reference as it was proved and 13, respectively (Figure 5) and can be considered simple patterns. Mode 14 (Figure 5) is equivalent to mode 4 with TM models. A list of the 30 first modal frequencies is in Table 2, where (Figure 6) and mode 5 (not drawn), being the first complex the type of dominant pattern has been marked (due to TM; patterns. Modes 11 and 14 (Figure 5) are representative of joint, J; or ligament and tendon motion, LT). This classifica- the starting point of the transition zone to ordered pattern. tion is made based on the part of the system where higher Finally, mode 50 corresponds clearly to the ordered pattern, motion is present. Different mechanisms will contribute to where a huge number of vibration modes can be observed the sound transmission in this frequency range. The sound with similar shapes. The small element size used in this pressure stimuli at the membrane will make dominant those model to mesh the TM captures these modes very accurately. modes involving the TM motion. The presence of the other At this frequency range, the absence of the ossicular chain modes will be lower or negligible. has a reduced influence [18, 19], so for the purpose of the The TM simplified model was developed in order to have present study, this simplified model is acceptable. As a rule the opportunity to make many calculations with a reasonably of thumb for the interpretation of the following figures, mode computational cost and enough accuracy at higher frequen- 5 may represent the transition to complex pattern and mode cies. It also provided the chance to compare with previously 15 the transition to ordered pattern. Amplitude of umbo displacement (m) Amplitude of stapes footplate displacement (m) Applied Bionics and Biomechanics 5 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 100 1000 10,000 Frequency (Hz) E = 3.2 MPa E = 160 MPa E = 16 MPa E = 320 MPa E = 32 MPa Figure 3: Influence of the TM elastic modulus of the pars tensa at umbo, for a range from 100 to 10,000 Hz at 80 dB sound pressure. SPL 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 100 1000 10,000 Frequency (Hz) th = 0.0125 mm th = 0.1 mm th = 0.025 mm th = 0.15 mm th = 0.05 mm Figure 4: Influence of the thickness of tympanic membrane at umbo, for a range from 100 to 10,000 Hz at 80 dB sound pressure. SPL In order to clarify the influence of the TM elastic modu- we can see how the first frequencies are related to the stiffness lus on the dynamic response, modes 1, 2, and 3 have been of the membrane. plotted in Figure 7. Results provided by Volandri et al. [38] Since each model provides different results, it is diffi- have also been included. cult to argue which of them is the best. Excepting at very The same references have been used (see [38] to consult low elastic modulus, which can be discarded as unreal, the the original references). The main idea we can extract from other results could be considered acceptable as they are these figures is something somehow expected. Considering ranging on the different proposed elastic modulus. This is some exceptions due to particular modelling circumstances the same conclusion obtained from the sensitivity analysis (as the use of different TM thickness or boundary condition), on previous section. Amplitude of umbo displacement (m) Amplitude of umbo displacement (m) 6 Applied Bionics and Biomechanics Mode 3:330.8 Hz Mode 8:704.6 Hz Mode 11: 880.5 Hz Mode 12: 970.7 Hz Mode 13: 1073.3 Hz Mode 14: 1552.7 Hz Mode 25: 2343.3 Hz Mode 27: 2571.7 Hz Mode 29: 2683.1 Hz Figure 5: Selected mode shapes. Middle ear FE model including the tympanic-ossicular system (TOS). TM elastic modulus E = 32 MPa; thickness th = 50 μm. Nevertheless, if we observe the behaviour of higher black triangle), the first thing we can see is how the whole sys- modes, we can make some distinction between these results tem results (TOS) are coincident with the simplified model. comparing with the experimental observation. In Figure 8, However, if we focus on the higher modes, now we the values of the first 60 TM vibration modes are plotted can detect significant differences with the experimental and calculated with different elastic moduli (Figure 8(a)) results observed. Considering mode 50 (Figure 8) as a ref- and different TM thickness (Figure 8(b)). If we observe the erence, this type of pattern has been detected in human at curve corresponding to the reference values (E = 32 MPa, frequencies above 8 kHz (see Figure 4 in [4]); in this case, Applied Bionics and Biomechanics 7 Table 2: Modal frequencies for the tympanic-ossicular system (TOS) FE model. Mode Frequency Main Mode Frequency Main Mode Frequency Main number (Hz) pattern number (Hz) pattern number (Hz) pattern 1 266.9 LT 11 880.55 J 21 1865 LT 2 267.03 LT 12 970.76 TM 22 1866 LT 3 330.82 J, LT 13 1073.3 TM 23 1894 LT 4 562.27 LT 14 1552.7 TM 24 2032 TM 5 580.91 LT 15 1725.5 LT 25 2343 TM 6 612.84 LT 16 1750.8 LT 26 2527 TM 7 613.67 LT 17 1751.2 LT 27 2572 J 8 704.62 TM 18 1782.7 LT 28 2683 TM, J 9 853.5 LT 19 1809.4 TM 29 2683 TM, J 10 854.34 LT 20 1831 LT 30 2765 TM M Mod ode 1: 870.2 H e 1: 870.2 Hz z M Mo ode 2: 1263.5 H de 2: 1263.5 Hz z M Mod ode 3: 1555.3 H e 3: 1555.3 Hz z M Mod ode 4: 1743.3 H e 4: 1743.3 Hz z M Mo ode 7: 2457.4 H de 7: 2457.4 Hz z Mo Mod de 9: 2772.7 H e 9: 2772.7 Hz z Mode 11: 3008 Hz Mode 14: 3355 Hz Mode 50: 6336.3 Hz Figure 6: Selected mode shapes. Tympanic membrane FE model. TM elastic modulus E = 32 MPa; thickness th = 50 μm. the frequency 6.3 kHz could be considered very low and (Figure 8(a)). This is coherent with the values proposed by wrong. Increasing the value of the elastic modulus (keeping [27] for the elastic modulus. constant the other parameter of the model), we can reach Regarding the influence of the TM thickness the experimentally observed frequencies. Result obtained in (Figure 8(b)), we can see how it hardly affects the lower fre- the range 100 to 320 MPa would fit this requirement quencies but it increases the higher modes. Its effect is not 8 Applied Bionics and Biomechanics 3000 3000 2500 2500 2000 2000 1500 1500 1000 1000 500 500 0 0 0 50 100 150 200 250 300 350 31 33 35 37 39 41 TM elastic modulus (MPa) TM elastic modulus (MPa) Liu09 Mik04 Koi05 Liu09 Mik04 Koi05 Sun02 Gan06 Lee10 Sun02 Gan06 Lee10 Fer03pre Fer03post Pre08 Pre08 E = 32 MPa Fer03pre TOS; E = 32 MPa E = 3.2 MPa E = 32 MPa TOS; E = 32 MPa E = 100 MPa E = 160 MPa E = 320 MPa (a) (b) Figure 7: (a) 1st, 2nd, and 3rd tympanic membrane modal frequencies in terms of the TM elastic modulus and (b) scale enlarged. 16,000 12,000 14,000 10,000 12,000 10,000 8000 6000 0 0 0 102030405060 0 10 20 30 40 50 60 Mode number Mode number E = 3.2 MPa E = 32 MPa th = 0.025 mm th = 0.1 mm E = 100 MPa E = 160 MPa th = 0.05 mm TOS; th = 0.05 mm E = 320 MPa TOS; E = 32 MPa th = 0.074 mm (a) (b) Figure 8: Tympanic membrane modal frequencies in terms of the (a) TM elastic modulus and the (b) TM thickness. as significant as the elastic modulus, and the range of uncer- and accurate estimation of a complex prestrain pattern is dif- tainty is smaller. ficult to do without speculations, so this approach is consid- In any case, we can conclude that multiple combinations ered a good solution to evaluate its qualitative influence. of both values could lead to similar results as many authors Figure 9 shows the result obtained using two different have pointed out. This is the main critique to FE models. elastic moduli. It can be seen how in both cases the effect is Finally, this TM model has been used to evaluate the similar to the increase of the elastic modulus. Therefore, influence of a potential prestrain in the membrane. It is again, it is possible to find different combinations of elastic another aspect of the problem that has not been considered modulus and prestrain that provide similar responses. This previously in numerical simulations but presents a significant has been plotted in Figure 10, where it can be observed how influence on the results. In this case, it has been considered in the cases E = 32 MPa, pst = 1%; E = 100 MPa, pst = 0.3%; and the simplest possible way. A homogeneous and uniform iso- E = 160 MPa and E = 320 MPa without prestrain provide tropic strain value has been applied to the membrane, rang- results in a narrow band which could be accepted as realistic. ing from 0.1% to 1% (ɛ = ɛ = 0.001 to 0.01 in the plane It is interesting to observe how the effect of prestrain 11 22 of the TM and ɛ = − 2 ɛ ). These values are very below and elastic modulus increase is different at higher frequen- 33 11 those assumed by Aernouts and Dirckx [30] and cause a cies and at lower (Figure 10(b)). At higher frequencies, the static imperceptible displacement of the umbo very below slope of the curve is lower when the increase of the stiff- 10 μm. The presence of prestrain must be related to the posi- ness is due to the prestrain instead of the EM. This differ- tion and orientation of the fiber on the membrane. A detailed ent behaviour opens a possible future working line where Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) Applied Bionics and Biomechanics 9 14,000 16,000 14,000 12,000 12,000 10,000 10,000 0 0 0 102030405060 0 10 20 30 40 50 60 Mode number Mode number E = 32 MPa; pst = 0 % E = 32 MPa; pst = 0.1 % E = 32 MPa; pst = 0 % E = 100 MPa; pst = 0.3 % E = 100 MPa; pst = 0 % TOS; E = 32 MPa E = 32 MPa; pst = 0.3 % E = 32 MPa; pst = 0.5 % E = 32 MPa; pst = 1 % TOS; E = 32 MPa E = 100 MPa; pst = 0.1 % (a) (b) Figure 9: Tympanic membrane modal frequencies with different prestrain (pst) level: (a) TM elastic modulus 32 MPa and (b) TM elastic modulus 100 MPa. 16,000 10,000 14,000 12,000 10,000 0 0 0 102030405060 0 5 10 15 20 Mode number Mode number E = 32 MPa E = 160 MPa E = 32 MPa E = 160 MPa E = 32 MPa; pst = 1 % E = 320 MPa E = 32 MPa; pst = 1 % E = 320 MPa E = 100 MPa; pst = 0.3 % TOS; E = 32 MPa E = 100 MPa; pst = 0.3 % TOS; E = 32 MPa (a) (b) Figure 10: Tympanic membrane modal frequencies with different TM elastic modulus and prestrain (pst) level combination. joint numerical and experimental works could be done in Nevertheless, this lack of accuracy on the estimation of order to fit more accurately the vibration patterns in order the mechanical parameter has been widely criticized from to estimate these parameters. the side of the experimental researcher and is the main rea- son to the limited confidence of the results based on numer- ical models of the hearing system. 4. Discussion In this section, a brief discussion on the consequences of this source of uncertainty on the model and the potential In previous section, a great deal of numerical results has been research lines to solve it is made. shown. From them, the main general conclusion is the According to the results shown previously and regarding dependency of the results with the values of the mechanical numerical modelling, the use of bending stiffness parameters properties used on the model. This dependency cannot be estimation (thickness and elastic modulus) may be accept- distinguished due to the general good agreement between able in order to represent the TM behaviour in terms of numerical and experimental results for a wide range of umbo or stapes displacement, especially at lower frequencies. values. This comparison is normally made in terms of umbo It also seems that rather than estimating separately prop- and stapes movement whose experimental measurement is erties very accurately, it would be valid fitting a combination of both properties to describe the bending stiffness of the TM based on the very common and extended LDV technique. No significant differences are observed at lower frequencies, over the whole range of frequency of interest. and differences at higher frequencies can be attributable to This picture increases its complexity if we include an several modelling issues. additional potential effect of stiffness on the membrane: Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) 10 Applied Bionics and Biomechanics 1.0E − 07 1.0E − 07 1.0E − 08 1.0E − 08 1.0E − 09 1.0E − 09 1.0E − 10 1.0E − 10 1.0E − 11 100 1000 10,000 100 1000 10,000 Frequency (Hz) Frequency (Hz) Exp. Hato et al. 2003 E = 320 MPa Exp. Hato et al. 2003 E = 320 MPa E = 32 MPa E = 32 MPa; pst = 1 % E = 32 MPa E = 32 MPa; pst = 1 % E = 160 MPa E = 100 MPa; pst = 0.3 % E = 160 MPa E = 100 MPa; pst = 0.3 % (a) (b) Figure 11: Influence of the prestrain of the TM (a) at umbo and (b) stapes footplate, for a range from 100 to 10,000 Hz at 80 dB SPL sound pressure. prestrain. The presence of prestrain alters significantly the displacement curves as shown in Figure 2 [37] are not fre- whole response of the system. This presence is difficult to dis- quency response function of the mechanical system, because tinguish from the estimation of the other mechanical proper- input function on the mechanical system is not measured. ties. Figure 11 shows the result of a harmonic analysis made They are normalized by the sound pressure at the source, with two combinations of parameter E = 32 MPa, pst = 1% but certain acoustic coupling effects are not included. and E = 100 MPa, pst = 0.3%, showing a similar behaviour Some recent numerical models include this acoustic for the case with higher TM elastic modulus (E = 160 MPa effect ([15, 17–19]), but even in this case, the condition in which the experiment was made may show significant differ- and E = 320 MPa without prestrain). Even when the reference model (with the properties shown in Table 1) ences, making the comparison limited. Different conditions also fits properly experimental data (Figure 2), we should on the experiment as the position of the source of sound go back to the observation of higher-order modes [31, 32] or the presence of cavities open or closed lead to dif- (Figures 8–10, see Discussion on previous section) to ferent responses of the same mechanical system. conclude that these new combinations of parameters (including prestrain) are closer to the dynamic behaviour of the membrane (as described by the experiment of [4]). 5. Conclusions Experimental work should be done to evaluate and esti- mate potential prestrain in the TM. Its probable influence A contribution to evaluate the quality of the parameter esti- on the dynamic response of the TM and the ME has been mation of the mechanical properties of the TM on the hear- stated in qualitative terms. It also has been shown as a possi- ing system has been done in this paper. This is a key aspect ble source of discrepancy between statically and dynamically when developing numerical models. estimation of the elastic modulus. Supported on experimental observation provided with Elucidating the real presence of this effect is a key aspect holography techniques, especially at high-frequency pattern, to be studied in future as it can help to understand the conse- TM stiffness properties suggested on bibliography have been quence of some surgical intervention (as tympanoplasty evaluated by means of FE numerical models. [39]). Different combinations of prestrain in the two TM One of the conclusion of this work is that different fiber layers make difficult to speculate about this phenome- parameter combinations (TM thickness and EM) may lead non. Detailed numerical models, including laminate theory, to similar result that could be considered correct when com- should be developed to account for this effect. paring with experimental data. This makes necessary addi- Numerical work supported on holography or in LDV tional external means of validation to clarify which of these measurement along a line should be an option. The high parameters are acceptable. number of combined vibration modes in the range of fre- It also has been shown that prestrain presence causes an quency of interest is a difficulty, but techniques used in increase on the TM stiffness that makes difficult to distin- dynamic system identification could be adapted for this pur- guish its effect from the effect of the elastic modulus of the pose. It is the case of modal experimental techniques used for material. This could be the reason for the lack of agreement model update. of some TM elastic modulus estimation. The different pat- In this sense, proper numerical-experimental techniques tern observed due to the prestrain effect would allow distin- should be developed being a key aspect its capacity to account guishing it from the EM. Adapted modal experimental for the acoustomechanic coupling [31]. Experimental techniques can be applied to determine this prestrain. Amplitude of umbo displacement (m) Amplitude of stapes footplate displacement (m) Applied Bionics and Biomechanics 11 Prestrain effect must be studied and evaluated because The Journal of the Acoustical Society of America, vol. 92, no. 6, pp. 3157–3168, 1992. even a small value alters significantly the response of the sys- tem. Additionally, orthotropic behaviour must be included [12] T. Koike, H. Wada, and T. Kobayashi, “Modeling of the human middle ear using the finite-element method,” The Jour- and probably related to this prestrain. nal of the Acoustical Society of America, vol. 111, no. 3, Numerical and experimental joint works should be done pp. 1306–1317, 2002. to identify the mechanical properties. In this sense, numerical [13] Q. Sun, R. Z. Gan, K.-H. Chang, and K. J. Dormer, “Computer- models must include the acoustic part of the system in order integrated finite element modeling of human middle ear,” Bio- to account for the acoustomechanic coupling effects and to mechanics and Modeling in Mechanobiology, vol. 1, no. 2, allow numerical-experimental fitting. pp. 109–122, 2002. [14] R. Z. Gan, B. Feng, and Q. Sun, “Three-dimensional finite Data Availability element modeling of human ear for sound transmission,” Annals of Biomedical Engineering, vol. 32, no. 6, pp. 847– The data used to support thefindings of this study are available 859, 2004. from the corresponding author upon request. [15] R. Z. Gan, Q. Sun, B. Feng, and M. W. Wood, “Acoustic– structural coupled finite element analysis for sound trans- Conflicts of Interest mission in human ear—pressure distributions,” Medical Engineering & Physics, vol. 28, no. 5, pp. 395–404, 2006. The authors declare no conflict of interest. [16] L. 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Khanna, “On the tympanic membrane by nanoindentation,” Journal of the damped frequency response of a finite-element model of the Mechanical Behavior of Biomedical Materials, vol. 2, no. 1, cat eardrum,” The Journal of the Acoustical Society of America, pp. 82–92, 2009. vol. 81, no. 6, pp. 1851–1859, 1987. [23] J. A. N. Buytaert, J. E. F. Aernouts, and J. J. J. Dirckx, “Inden- [8] W. R. J. Funnell, S. M. Khanna, and W. F. Decraemer, “On the tation measurements on the eardrum with automated projec- degree of rigidity of the manubrium in a finite-element model tion moiré profilometry,” Optics and Lasers in Engineering, of the cat eardrum,” The Journal of the Acoustical Society of vol. 47, no. 3-4, pp. 301–309, 2009. America, vol. 91, no. 4, pp. 2082–2090, 1992. [24] J. Aernouts, I. Couckuyt, K. Crombecq, and J. J. J. Dirckx, [9] K. R. Williams and T. H. J. Lesser, “A finite element analysis of “Elastic characterization of membranes with a complex shape the natural frequencies of vibration of the human tympanic using point indentation measurements and inverse model- membrane. Part I,” British Journal of Audiology, vol. 24, ling,” International Journal of Engineering Science, vol. 48, no. 5, pp. 319–327, 1990. no. 6, pp. 599–611, 2010. [10] H. Wada and T. Kobayashi, “Dynamical behavior of middle [25] S. M. Hesabgar, H. Marshall, S. K. Agrawal, A. Samani, and ear: theoretical study corresponding to measurement results obtained by a newly developed measuring apparatus,” The H. M. Ladak, “Measuring the quasi-static Young’s modulus of the eardrum using an indentation technique,” Hearing Journal of the Acoustical Society of America, vol. 87, no. 1, pp. 237–245, 1990. Research, vol. 263, no. 1-2, pp. 168–176, 2010. [11] H. Wada, T. Metoki, and T. Kobayashi, “Analysis of dynamic [26] J. Aernouts, J. R. M. Aerts, and J. J. J. Dirckx, “Mechanical behavior of human middle ear using a finite element method,” properties of human tympanic membrane in the quasi-static 12 Applied Bionics and Biomechanics regime from in situ point indentation measurements,” Hearing Research, vol. 290, no. 1-2, pp. 45–54, 2012. [27] J. Fay, S. Puria, W. F. Decraemer, and C. Steele, “Three approaches for estimating the elastic modulus of the tympanic membrane,” Journal of Biomechanics, vol. 38, no. 9, pp. 1807– 1815, 2005. [28] F. Gentil, M. Parente, P. Martins et al., “The influence of muscles activation on the dynamical behaviour of the tympano-ossicular system of the middle ear,” Computer Methods in Biomechanics and Biomedical Engineering, vol. 16, no. 4, pp. 392–402, 2013. [29] M. M. Henson, V. J. Madden, H. Rask-Andersen, and O. W. Henson Jr, “Smooth muscle in the annulus fibrosus of the tym- panic membrane in bats, rodents, insectivores, and humans,” Hearing Research, vol. 200, no. 1-2, pp. 29–37, 2005. [30] J. Aernouts and J. 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Analysis of the Mechanical Properties of the Human Tympanic Membrane and Its Influence on the Dynamic Behaviour of the Human Hearing System

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Copyright © 2018 L. Caminos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Applied Bionics and Biomechanics Volume 2018, Article ID 1736957, 12 pages https://doi.org/10.1155/2018/1736957 Research Article Analysis of the Mechanical Properties of the Human Tympanic Membrane and Its Influence on the Dynamic Behaviour of the Human Hearing System 1 2 2 2 L. Caminos, J. Garcia-Manrique, A. Lima-Rodriguez, and A. Gonzalez-Herrera Departamento de Ingeniería Mecánica, Universidad Nacional Experimental del Táchira, San Cristobal, Venezuela Departamento de Ingeniería Civil, de Materiales y Fabricación, Universidad de Málaga, Malaga, Spain Correspondence should be addressed to A. Gonzalez-Herrera; agh@uma.es Received 18 January 2018; Revised 25 April 2018; Accepted 29 April 2018; Published 9 May 2018 Academic Editor: Thibault Lemaire Copyright © 2018 L. Caminos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The difficulty to estimate the mechanical properties of the tympanic membrane (TM) is a limitation to understand the sound transmission mechanism. In this paper, based on finite element calculations, the sensitivity of the human hearing system to these properties is evaluated. The parameters that define the bending stiffness properties of the membrane have been studied, specifically two key parameters: Young’s modulus of the tympanic membrane and the thickness of the eardrum. Additionally, it has been completed with the evaluation of the presence of an initial prestrain inside the TM. Modal analysis is used to study the qualitative characteristics of the TM comparing with vibration patterns obtained by holography. Higher-order modes are shown as a tool to identify these properties. The results show that different combinations of elastic properties and prestrain provide similar responses. The presence of prestrain at the membrane adds more uncertainty, and it is pointed out as a source for the lack of agreement of some previous TM elastic modulus estimations. 1. Introduction numerical simulation and limit the extension of the conclu- sions. This is particularly significant in the case of the TM. The function of the tympanic membrane (TM) in the sound A great deal of work has been devoted to estimate transmission process is easy to understand intuitively. The these properties, especially to the determination of the piston-like motion, which transfers air sound pressure wave TM elastic modulus (EM). Most of the results range from into the cochlea, has been clearly identified long time ago 20 to 40 MPa or are close to these values. They are based on tension tests on small samples [1, 2, 20, 21] and are [1, 2]. Nevertheless, at higher frequencies, the TM motion is not so simple and new characteristic patterns appear. subject to important uncertainties. Different indentation There have been many experimental and numerical techniques have been developed providing result in the studies to evaluate the behaviour of the TM. These pat- same range [22–25]. Nevertheless, recently, values on the terns have been widely studied experimentally since the range of 3 MPa have been reported [26], showing the dif- development of techniques as holography [3–5]. With this ficulty to obtain accurately this parameter separately from technique, the complex TM vibration patterns at high fre- the estimation of the thickness. quencies have been revealed. Fay et al. [27] suggested that these values could be under- The alternative is numerical simulation. Since the early estimated; they used composite laminate theory and evalu- works led by Funnell and Laszlo [6] and Funnell et al. [7, 8], ated their result comparing with the dynamic response of finite element (FE) models have been used to study the behav- the TM. They calculated the effect of the orientation of the iour of the system [9–19]. Nevertheless, uncertainties regard- fiber on the EM obtained on tension or bending tests when ing the accuracy of the material properties hinder the an isotropic assumption was made. They showed that the 2 Applied Bionics and Biomechanics elastic material behaviour and small displacement condition EM could be in a broad range depending of the angle of ori- entation of the fibers of the sample tested. They confirmed are assumed. Both models have been simplified and limited their finding by means of correlating experimental dynamic to the components necessary for the purpose of this study. wavelength pattern. These patterns were obtained from live The anatomic measures and functional properties were anesthetized cat but conclusions can be transferred to the based on published data. The geometrical model is divided human case. Fitting a mathematical model and by means of into three parts: TM, ossicular chain, and the system of liga- a parameter estimation procedure, they suggested that the ments, tendons, and joints (Figure 1(a)). Details of the geom- EM should be in the range 0.1 to 0.3 GPa. etry and property estimation are described in different A potential reason for this high stiffness value could be references [16, 33]. the presence of an unaccounted prestrain. It is well accepted The FE model was developed using ANSYS. TM was mod- the presence of active muscles on the middle ear system [28], elled with shell elements assuming uniform thickness (50 μm) but apart from hypothesized active tension effects inside the in order to simplify the analysis. The tympanic annulus is TM due to smooth muscles [29], only passive prestrain has modelled as a band (0.2 mm wide and 0.2 mm thickness) with been named by several authors [1, 2], but no quantitative the external border fixed. Solid elements are used for the ossic- estimation has been clearly made. Recently, Aernouts and ular chain, posterior incudal ligament, and incudomalleolar Dirckx [30] assumed an in situ strain for the elastic charac- and incudostapedial joints. Ligaments and tendons support- terization of the gerbil pars flaccida, but no other numerical ing malleus are modelled with beam elements. The stapedial model has included this effect. annular ligament is modelled with shell elements surrounding In the present paper, the parameters that define the bend- the stapedial footplate (0.05 mm wide and 0.05 mm thickness). ing stiffness properties of the membrane have been studied. Figure 1(a) shows the middle ear FE model, and the mechani- Mainly, the influence of two parameters has been studied: cal properties are summarized in Table 1. the Young’s modulus of the tympanic membrane and the The effect of the impedance of the fluid of the cochlea was thickness of the eardrum. Additionally, it has been completed modelled using a mechanical equivalent load consisting of a with the evaluation of the presence of an initial prestrain block of mass and dashpots according to [14]. This simplified inside the TM. model provides a good accuracy with a low computational Firstly, a harmonic analysis is made to evaluate the range cost. When modal analysis is applied to the TOS model, this of influence of these properties. Secondly, a modal analysis is equivalent load is removed to eliminate its own vibration used to check how the properties affect the dynamic modes, which are outside the scope of this study. response. We will demonstrate that variations in membrane Regarding the TM simplified model (Figure 1(b)), most parameters have small effects on the lower natural frequen- of the description previously stated is valid. The area corre- cies; the effects of thickness, EM, and prestrain on the higher sponding to the connection with the manubrium has also natural frequencies are larger and separable. been meshed with shell elements, but the mechanical proper- Rosowski et al. [4] used holography to observe the TM ties have been adapted to represent the inertial and stiffness vibration pattern in different species including human. They effect of the ossicular chain. The umbo was represented by established a qualitative description based on the pattern an area 0.6 mm wide with an EM equal to 14 GPa and the observed and related with the frequency range. They consid- density 1900 kg/m . In this way, the ossicular chain complex ered a simple pattern when only one maximum displacement vibration patterns—out of the scope of this part of the was observed (below 2 kHz in human), complex pattern study—are excluded from the model. Another important dif- when more than one maximum (from 2 to 8 kHz), and ference is that the membrane has been meshed with a higher ordered pattern when a high number of maximum appear number of elements as this model is intended to capture (above 8 kHz). This description will be used to evaluate the accurately high-frequency vibration pattern. response of the TM at different ranges of frequencies. Direct comparison of numerical and experimental results 2.2. Middle Ear Harmonic Response. As a starting point of is not easy to do [31, 32] as the acoustomechanic coupling this study, a sensitivity analysis has been done in order to must be solved; nevertheless, comparing numerical modes evaluate the influence of two key mechanical properties that and experimental vibration pattern leads to identify which strongly influence the response of the system: TM elastic of these material properties combinations are more suitable modulus and TM thickness. and suggest potential future working lines. First, the behaviour of the middle ear model will be shown. A harmonic analysis was conducted from 100 to 10,000 Hz. A uniform harmonic 80 dB stimulus pressure was applied to SPL 2. Materials and Methods the lateral side of the eardrum. The amplitude of umbo and 2.1. FEA Modelling Methodology. A brief summary of the stapedial footplate displacements versus frequency is shown main modelling methodology is presented in this section in Figure 2 compared with experimental measurements [37]. (see detail in [16]). It has been developed from precedent These experimental curves are the average data of 10 human works [6–15]. temporal bones with intact cochlea, using LDV with 80 dB sound pressure applied at the TM in similar condition to the Two models will be described. The first one includes all the elements of the middle ear, referred as tympanic-ossicular sys- present study. Rayleigh damping β = 0.0001 s is assumed. tem (TOS) model. The second one only includes the TM and With Table 1 parameters, the numerical result approximately the approximate effect of the manubrium. In both cases, linear predicts the experimental displacement. Applied Bionics and Biomechanics 3 Superior malleolar ligament Posterior incudal Malleus Incudomalleolar ligament joint Stapes Lateral malleolar Incus footplate ligament Stapedial annular ligament Eardrum (pars flaccida) Stapes Anterior malleolar ligament Stapedial tendon Tensor tympani Incudostapedial joint tendon Tympanic annulus Manubrium Eardrum Umbo (pars tensa) (a) (b) Figure 1: Finite element models: (a) tympanic-ossicular system (TOS) model and (b) tympanic membrane simplified model. Figures 3 and 4 show the influence of the EM and the thickness, respectively. The thick line without marks corre- Table 1: Mechanical properties used in middle ear components for sponds to the results obtained with the reference model finite element model. (Figure 2, Table 1). Values assigned to EM range from 3.2 MPa to 320 MPa, covering the extreme values obtained Density Young’s Poisson’s Component 3 2 from literature. (kg/m ) modulus (N/m ) ratio The TM EM presents a qualitatively similar influence in Eardrum both cases (Figure 3). Below 1000 Hz, the displacements 3 7 Pars tensa 1.2 × 10 [9] 3.2 × 10 [10] 0.3 [13] increased as the stiffness decreases. The peak value is at a 3 7 Pars flaccida 1.2 × 10 [9] 1 × 10 [13] 0.3 [13] lower frequency with a lesser stiffness. The displacement at 3 10 Malleus 1.9 × 10 [34] 1.41 × 10 [35] 0.3 [13] high frequency (above 2000 Hz) presents an opposite rela- 3 10 Incus 1.9 × 10 [34] 1.41 × 10 [35] 0.3 [13] tion; it increases slightly as the stiffness increases. 3 10 Stapes 1.9 × 10 [34] 1.41 × 10 [35] 0.3 [13] Regarding the thickness, its response is inversely propor- tional to the increase of stiffness and mass due to the incre- 1.2 × 10 Tympanic annulus 6 × 10 [13] 0.3 [13] ment of thickness (Figure 4). The influence is more (assumed) 3 9 significant at lower frequencies. The peak value close to Manubrium 1.0 × 10 [12] 4.7 × 10 [12] 0.3 [13] 1000 Hz is similar for all curves. Tensor tympanic 3 6 2.5 × 10 [12] 2.6 × 10 [12] 0.3 [13] These results are similar to those obtained by Gan et al. tendon [14] suggesting that an isotropic material behaviour as well Lateral malleolar 3 4 2.5 × 10 [12] 6.7 × 10 [12] 0.3 [13] as a mean thickness for the tympanic membrane model can ligament fit experimental data for the frequency range studied. Anterior malleolar 3 6 2.5 × 10 [12] 2.1 × 10 [12] 0.3 [13] According to the results shown, it could be stated than ligament 32 MPa seems to be a good value for the TM EM; neverthe- Superior malleolar 3 4 less, important variations of this parameter lead to result that 2.5 × 10 [12] 4.9 × 10 [12] 0.3 [13] ligament could be argued as valid too, depending of the experimental Posterior incudal 3 6 data used to compare. 2.5 × 10 [12] 6.5 × 10 [14] 0.3 [13] ligament 3 5 Stapedial tendon 2.5 × 10 [12] 5.2 × 10 [12] 0.3 [13] 3. Results Stapedial annular 3 5 2.5 × 10 [12] 2 × 10 [15] 0.3 [13] ligament Different modal analyses were performed. As a reference, a Incudomalleolar 3 10 modal analysis of the whole system (TOS) was made. Some 3.2 × 10 [13] 1.41 × 10 [13] 0.3 [13] joint selected mode shapes are drawn in Figure 5. Different types Incudostapedial of shapes are present. Some clearly reflect a TM vibration 3 5 1.2 × 10 [13] 6 × 10 [36] 0.3 [13] joint pattern (modes 8, 12, 13, 14, 25, and 29). Mode 12 is the clas- sic piston-like motion. Mode 8 presents a similar membrane motion but with a lower stapes displacement. Modes 3 and 11 show the movement of the joints and ossicles while the mem- brane and the stapes present a very low motion. Mode 27 4 Applied Bionics and Biomechanics 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 100 1000 10,000 Frequency (Hz) Exp. Hato et al. 2003 (intact cochlea) FE model (properties in Table 1) (a) 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 1.0E − 11 100 1000 10,000 Frequency (Hz) Exp. Hato et al. 2003 (intact cochlea) FE model (properties in Table 1) (b) Figure 2: Amplitude of displacement versus frequency of umbo (a) and stapes footplate (b), for a range from 100 to 10,000 Hz at 80 dB SPL sound pressure. shows a tilting movement of the stapes with hardly no mem- published results (see [38]). The modal analysis was per- brane displacement. Modes 14 and 25 correspond to the formed with this model. As the ossicular chain has been transition to complex pattern (as described in Rosowski removed, only the TM modes are present on the solution. et al. [4]), and mode 29 would be the starting point of the Figure 6 presents some selected mode shapes calculated ordered pattern. In this model, high-frequency modes are less with the reference properties (Table 1). It can be seen how accurate than lower due to the limited number of element modes 1, 2, and 3 (Figure 6) are equivalent to modes 12, 8, used. Nevertheless, they are a good reference as it was proved and 13, respectively (Figure 5) and can be considered simple patterns. Mode 14 (Figure 5) is equivalent to mode 4 with TM models. A list of the 30 first modal frequencies is in Table 2, where (Figure 6) and mode 5 (not drawn), being the first complex the type of dominant pattern has been marked (due to TM; patterns. Modes 11 and 14 (Figure 5) are representative of joint, J; or ligament and tendon motion, LT). This classifica- the starting point of the transition zone to ordered pattern. tion is made based on the part of the system where higher Finally, mode 50 corresponds clearly to the ordered pattern, motion is present. Different mechanisms will contribute to where a huge number of vibration modes can be observed the sound transmission in this frequency range. The sound with similar shapes. The small element size used in this pressure stimuli at the membrane will make dominant those model to mesh the TM captures these modes very accurately. modes involving the TM motion. The presence of the other At this frequency range, the absence of the ossicular chain modes will be lower or negligible. has a reduced influence [18, 19], so for the purpose of the The TM simplified model was developed in order to have present study, this simplified model is acceptable. As a rule the opportunity to make many calculations with a reasonably of thumb for the interpretation of the following figures, mode computational cost and enough accuracy at higher frequen- 5 may represent the transition to complex pattern and mode cies. It also provided the chance to compare with previously 15 the transition to ordered pattern. Amplitude of umbo displacement (m) Amplitude of stapes footplate displacement (m) Applied Bionics and Biomechanics 5 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 100 1000 10,000 Frequency (Hz) E = 3.2 MPa E = 160 MPa E = 16 MPa E = 320 MPa E = 32 MPa Figure 3: Influence of the TM elastic modulus of the pars tensa at umbo, for a range from 100 to 10,000 Hz at 80 dB sound pressure. SPL 1.0E − 07 1.0E − 08 1.0E − 09 1.0E − 10 100 1000 10,000 Frequency (Hz) th = 0.0125 mm th = 0.1 mm th = 0.025 mm th = 0.15 mm th = 0.05 mm Figure 4: Influence of the thickness of tympanic membrane at umbo, for a range from 100 to 10,000 Hz at 80 dB sound pressure. SPL In order to clarify the influence of the TM elastic modu- we can see how the first frequencies are related to the stiffness lus on the dynamic response, modes 1, 2, and 3 have been of the membrane. plotted in Figure 7. Results provided by Volandri et al. [38] Since each model provides different results, it is diffi- have also been included. cult to argue which of them is the best. Excepting at very The same references have been used (see [38] to consult low elastic modulus, which can be discarded as unreal, the the original references). The main idea we can extract from other results could be considered acceptable as they are these figures is something somehow expected. Considering ranging on the different proposed elastic modulus. This is some exceptions due to particular modelling circumstances the same conclusion obtained from the sensitivity analysis (as the use of different TM thickness or boundary condition), on previous section. Amplitude of umbo displacement (m) Amplitude of umbo displacement (m) 6 Applied Bionics and Biomechanics Mode 3:330.8 Hz Mode 8:704.6 Hz Mode 11: 880.5 Hz Mode 12: 970.7 Hz Mode 13: 1073.3 Hz Mode 14: 1552.7 Hz Mode 25: 2343.3 Hz Mode 27: 2571.7 Hz Mode 29: 2683.1 Hz Figure 5: Selected mode shapes. Middle ear FE model including the tympanic-ossicular system (TOS). TM elastic modulus E = 32 MPa; thickness th = 50 μm. Nevertheless, if we observe the behaviour of higher black triangle), the first thing we can see is how the whole sys- modes, we can make some distinction between these results tem results (TOS) are coincident with the simplified model. comparing with the experimental observation. In Figure 8, However, if we focus on the higher modes, now we the values of the first 60 TM vibration modes are plotted can detect significant differences with the experimental and calculated with different elastic moduli (Figure 8(a)) results observed. Considering mode 50 (Figure 8) as a ref- and different TM thickness (Figure 8(b)). If we observe the erence, this type of pattern has been detected in human at curve corresponding to the reference values (E = 32 MPa, frequencies above 8 kHz (see Figure 4 in [4]); in this case, Applied Bionics and Biomechanics 7 Table 2: Modal frequencies for the tympanic-ossicular system (TOS) FE model. Mode Frequency Main Mode Frequency Main Mode Frequency Main number (Hz) pattern number (Hz) pattern number (Hz) pattern 1 266.9 LT 11 880.55 J 21 1865 LT 2 267.03 LT 12 970.76 TM 22 1866 LT 3 330.82 J, LT 13 1073.3 TM 23 1894 LT 4 562.27 LT 14 1552.7 TM 24 2032 TM 5 580.91 LT 15 1725.5 LT 25 2343 TM 6 612.84 LT 16 1750.8 LT 26 2527 TM 7 613.67 LT 17 1751.2 LT 27 2572 J 8 704.62 TM 18 1782.7 LT 28 2683 TM, J 9 853.5 LT 19 1809.4 TM 29 2683 TM, J 10 854.34 LT 20 1831 LT 30 2765 TM M Mod ode 1: 870.2 H e 1: 870.2 Hz z M Mo ode 2: 1263.5 H de 2: 1263.5 Hz z M Mod ode 3: 1555.3 H e 3: 1555.3 Hz z M Mod ode 4: 1743.3 H e 4: 1743.3 Hz z M Mo ode 7: 2457.4 H de 7: 2457.4 Hz z Mo Mod de 9: 2772.7 H e 9: 2772.7 Hz z Mode 11: 3008 Hz Mode 14: 3355 Hz Mode 50: 6336.3 Hz Figure 6: Selected mode shapes. Tympanic membrane FE model. TM elastic modulus E = 32 MPa; thickness th = 50 μm. the frequency 6.3 kHz could be considered very low and (Figure 8(a)). This is coherent with the values proposed by wrong. Increasing the value of the elastic modulus (keeping [27] for the elastic modulus. constant the other parameter of the model), we can reach Regarding the influence of the TM thickness the experimentally observed frequencies. Result obtained in (Figure 8(b)), we can see how it hardly affects the lower fre- the range 100 to 320 MPa would fit this requirement quencies but it increases the higher modes. Its effect is not 8 Applied Bionics and Biomechanics 3000 3000 2500 2500 2000 2000 1500 1500 1000 1000 500 500 0 0 0 50 100 150 200 250 300 350 31 33 35 37 39 41 TM elastic modulus (MPa) TM elastic modulus (MPa) Liu09 Mik04 Koi05 Liu09 Mik04 Koi05 Sun02 Gan06 Lee10 Sun02 Gan06 Lee10 Fer03pre Fer03post Pre08 Pre08 E = 32 MPa Fer03pre TOS; E = 32 MPa E = 3.2 MPa E = 32 MPa TOS; E = 32 MPa E = 100 MPa E = 160 MPa E = 320 MPa (a) (b) Figure 7: (a) 1st, 2nd, and 3rd tympanic membrane modal frequencies in terms of the TM elastic modulus and (b) scale enlarged. 16,000 12,000 14,000 10,000 12,000 10,000 8000 6000 0 0 0 102030405060 0 10 20 30 40 50 60 Mode number Mode number E = 3.2 MPa E = 32 MPa th = 0.025 mm th = 0.1 mm E = 100 MPa E = 160 MPa th = 0.05 mm TOS; th = 0.05 mm E = 320 MPa TOS; E = 32 MPa th = 0.074 mm (a) (b) Figure 8: Tympanic membrane modal frequencies in terms of the (a) TM elastic modulus and the (b) TM thickness. as significant as the elastic modulus, and the range of uncer- and accurate estimation of a complex prestrain pattern is dif- tainty is smaller. ficult to do without speculations, so this approach is consid- In any case, we can conclude that multiple combinations ered a good solution to evaluate its qualitative influence. of both values could lead to similar results as many authors Figure 9 shows the result obtained using two different have pointed out. This is the main critique to FE models. elastic moduli. It can be seen how in both cases the effect is Finally, this TM model has been used to evaluate the similar to the increase of the elastic modulus. Therefore, influence of a potential prestrain in the membrane. It is again, it is possible to find different combinations of elastic another aspect of the problem that has not been considered modulus and prestrain that provide similar responses. This previously in numerical simulations but presents a significant has been plotted in Figure 10, where it can be observed how influence on the results. In this case, it has been considered in the cases E = 32 MPa, pst = 1%; E = 100 MPa, pst = 0.3%; and the simplest possible way. A homogeneous and uniform iso- E = 160 MPa and E = 320 MPa without prestrain provide tropic strain value has been applied to the membrane, rang- results in a narrow band which could be accepted as realistic. ing from 0.1% to 1% (ɛ = ɛ = 0.001 to 0.01 in the plane It is interesting to observe how the effect of prestrain 11 22 of the TM and ɛ = − 2 ɛ ). These values are very below and elastic modulus increase is different at higher frequen- 33 11 those assumed by Aernouts and Dirckx [30] and cause a cies and at lower (Figure 10(b)). At higher frequencies, the static imperceptible displacement of the umbo very below slope of the curve is lower when the increase of the stiff- 10 μm. The presence of prestrain must be related to the posi- ness is due to the prestrain instead of the EM. This differ- tion and orientation of the fiber on the membrane. A detailed ent behaviour opens a possible future working line where Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) Applied Bionics and Biomechanics 9 14,000 16,000 14,000 12,000 12,000 10,000 10,000 0 0 0 102030405060 0 10 20 30 40 50 60 Mode number Mode number E = 32 MPa; pst = 0 % E = 32 MPa; pst = 0.1 % E = 32 MPa; pst = 0 % E = 100 MPa; pst = 0.3 % E = 100 MPa; pst = 0 % TOS; E = 32 MPa E = 32 MPa; pst = 0.3 % E = 32 MPa; pst = 0.5 % E = 32 MPa; pst = 1 % TOS; E = 32 MPa E = 100 MPa; pst = 0.1 % (a) (b) Figure 9: Tympanic membrane modal frequencies with different prestrain (pst) level: (a) TM elastic modulus 32 MPa and (b) TM elastic modulus 100 MPa. 16,000 10,000 14,000 12,000 10,000 0 0 0 102030405060 0 5 10 15 20 Mode number Mode number E = 32 MPa E = 160 MPa E = 32 MPa E = 160 MPa E = 32 MPa; pst = 1 % E = 320 MPa E = 32 MPa; pst = 1 % E = 320 MPa E = 100 MPa; pst = 0.3 % TOS; E = 32 MPa E = 100 MPa; pst = 0.3 % TOS; E = 32 MPa (a) (b) Figure 10: Tympanic membrane modal frequencies with different TM elastic modulus and prestrain (pst) level combination. joint numerical and experimental works could be done in Nevertheless, this lack of accuracy on the estimation of order to fit more accurately the vibration patterns in order the mechanical parameter has been widely criticized from to estimate these parameters. the side of the experimental researcher and is the main rea- son to the limited confidence of the results based on numer- ical models of the hearing system. 4. Discussion In this section, a brief discussion on the consequences of this source of uncertainty on the model and the potential In previous section, a great deal of numerical results has been research lines to solve it is made. shown. From them, the main general conclusion is the According to the results shown previously and regarding dependency of the results with the values of the mechanical numerical modelling, the use of bending stiffness parameters properties used on the model. This dependency cannot be estimation (thickness and elastic modulus) may be accept- distinguished due to the general good agreement between able in order to represent the TM behaviour in terms of numerical and experimental results for a wide range of umbo or stapes displacement, especially at lower frequencies. values. This comparison is normally made in terms of umbo It also seems that rather than estimating separately prop- and stapes movement whose experimental measurement is erties very accurately, it would be valid fitting a combination of both properties to describe the bending stiffness of the TM based on the very common and extended LDV technique. No significant differences are observed at lower frequencies, over the whole range of frequency of interest. and differences at higher frequencies can be attributable to This picture increases its complexity if we include an several modelling issues. additional potential effect of stiffness on the membrane: Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) 10 Applied Bionics and Biomechanics 1.0E − 07 1.0E − 07 1.0E − 08 1.0E − 08 1.0E − 09 1.0E − 09 1.0E − 10 1.0E − 10 1.0E − 11 100 1000 10,000 100 1000 10,000 Frequency (Hz) Frequency (Hz) Exp. Hato et al. 2003 E = 320 MPa Exp. Hato et al. 2003 E = 320 MPa E = 32 MPa E = 32 MPa; pst = 1 % E = 32 MPa E = 32 MPa; pst = 1 % E = 160 MPa E = 100 MPa; pst = 0.3 % E = 160 MPa E = 100 MPa; pst = 0.3 % (a) (b) Figure 11: Influence of the prestrain of the TM (a) at umbo and (b) stapes footplate, for a range from 100 to 10,000 Hz at 80 dB SPL sound pressure. prestrain. The presence of prestrain alters significantly the displacement curves as shown in Figure 2 [37] are not fre- whole response of the system. This presence is difficult to dis- quency response function of the mechanical system, because tinguish from the estimation of the other mechanical proper- input function on the mechanical system is not measured. ties. Figure 11 shows the result of a harmonic analysis made They are normalized by the sound pressure at the source, with two combinations of parameter E = 32 MPa, pst = 1% but certain acoustic coupling effects are not included. and E = 100 MPa, pst = 0.3%, showing a similar behaviour Some recent numerical models include this acoustic for the case with higher TM elastic modulus (E = 160 MPa effect ([15, 17–19]), but even in this case, the condition in which the experiment was made may show significant differ- and E = 320 MPa without prestrain). Even when the reference model (with the properties shown in Table 1) ences, making the comparison limited. Different conditions also fits properly experimental data (Figure 2), we should on the experiment as the position of the source of sound go back to the observation of higher-order modes [31, 32] or the presence of cavities open or closed lead to dif- (Figures 8–10, see Discussion on previous section) to ferent responses of the same mechanical system. conclude that these new combinations of parameters (including prestrain) are closer to the dynamic behaviour of the membrane (as described by the experiment of [4]). 5. Conclusions Experimental work should be done to evaluate and esti- mate potential prestrain in the TM. Its probable influence A contribution to evaluate the quality of the parameter esti- on the dynamic response of the TM and the ME has been mation of the mechanical properties of the TM on the hear- stated in qualitative terms. It also has been shown as a possi- ing system has been done in this paper. This is a key aspect ble source of discrepancy between statically and dynamically when developing numerical models. estimation of the elastic modulus. Supported on experimental observation provided with Elucidating the real presence of this effect is a key aspect holography techniques, especially at high-frequency pattern, to be studied in future as it can help to understand the conse- TM stiffness properties suggested on bibliography have been quence of some surgical intervention (as tympanoplasty evaluated by means of FE numerical models. [39]). Different combinations of prestrain in the two TM One of the conclusion of this work is that different fiber layers make difficult to speculate about this phenome- parameter combinations (TM thickness and EM) may lead non. Detailed numerical models, including laminate theory, to similar result that could be considered correct when com- should be developed to account for this effect. paring with experimental data. This makes necessary addi- Numerical work supported on holography or in LDV tional external means of validation to clarify which of these measurement along a line should be an option. The high parameters are acceptable. number of combined vibration modes in the range of fre- It also has been shown that prestrain presence causes an quency of interest is a difficulty, but techniques used in increase on the TM stiffness that makes difficult to distin- dynamic system identification could be adapted for this pur- guish its effect from the effect of the elastic modulus of the pose. It is the case of modal experimental techniques used for material. This could be the reason for the lack of agreement model update. of some TM elastic modulus estimation. The different pat- In this sense, proper numerical-experimental techniques tern observed due to the prestrain effect would allow distin- should be developed being a key aspect its capacity to account guishing it from the EM. Adapted modal experimental for the acoustomechanic coupling [31]. Experimental techniques can be applied to determine this prestrain. Amplitude of umbo displacement (m) Amplitude of stapes footplate displacement (m) Applied Bionics and Biomechanics 11 Prestrain effect must be studied and evaluated because The Journal of the Acoustical Society of America, vol. 92, no. 6, pp. 3157–3168, 1992. even a small value alters significantly the response of the sys- tem. Additionally, orthotropic behaviour must be included [12] T. Koike, H. Wada, and T. Kobayashi, “Modeling of the human middle ear using the finite-element method,” The Jour- and probably related to this prestrain. nal of the Acoustical Society of America, vol. 111, no. 3, Numerical and experimental joint works should be done pp. 1306–1317, 2002. to identify the mechanical properties. In this sense, numerical [13] Q. Sun, R. Z. Gan, K.-H. Chang, and K. J. Dormer, “Computer- models must include the acoustic part of the system in order integrated finite element modeling of human middle ear,” Bio- to account for the acoustomechanic coupling effects and to mechanics and Modeling in Mechanobiology, vol. 1, no. 2, allow numerical-experimental fitting. pp. 109–122, 2002. [14] R. Z. Gan, B. Feng, and Q. Sun, “Three-dimensional finite Data Availability element modeling of human ear for sound transmission,” Annals of Biomedical Engineering, vol. 32, no. 6, pp. 847– The data used to support thefindings of this study are available 859, 2004. from the corresponding author upon request. [15] R. Z. Gan, Q. Sun, B. Feng, and M. W. Wood, “Acoustic– structural coupled finite element analysis for sound trans- Conflicts of Interest mission in human ear—pressure distributions,” Medical Engineering & Physics, vol. 28, no. 5, pp. 395–404, 2006. The authors declare no conflict of interest. [16] L. 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