To cite this document: Foucaud, Simon and Michon, Guilhem and Morlier, Joseph and Gourinat, Yves Harmonic response of the organ of corti: results for wave dispersion. (2010) In: The 11th International Mechanics of Hearing Workshop, 16–22 Jul 2011, Williamstown, USA.
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Harmonic Response of the Organ of Corti:
Results for Wave Dispersion
Simon Foucaud, Guilhem Michon, Joseph Morlier, and Yves Gourinat
Université de Toulouse, ICA, ISAE, Toulouse, France
Abstract. Inner ear is a remarkable multiphysical system and its modelling is a great challenge.
The approach used in this paper aims to reproduce physic with a realistic description of the radial cross section of the cochlea. A 2D-section of the organ of Corti is fully described. Wavenumbers and corresponding modes of propagation are calculated taking into account passive structural responses. The study is extended to six cross sections of the organ of Corti and a large frequency bandwidth from 100 Hz to 3 kHz. Dispersion curves reveal the influence of fluid structure interactions with a dispersive behavior at high frequencies. Longitudinal mechanical coupling provides new interacting modes of propagation.
Keywords: cochlea, organ of Corti, dispersion curves, acoustic waves, finite element analysis PACS: 43.64.Bt, 43.64.Kc
INTRODUCTION
The inner ear plays a major role in perception and orientation. Consequently its global modelling is a great challenge. The previous studies focused on the modelling of the vestibular system [8] and the present paper deals with the project for the modelling of the cochlea. The mechanism of sound transformation in the cochlea is complex. A global approach taking into account every physical phenomenon seems still too complex. Comprehensive mathematical model gave good results for passive and active cochlea [6]. Some authors gave more physical sense to the models using physical parameters instead of mathematical functions [1, 7, 11]. This paper follows this way involving only physical equations in a finite elements analysis. A focus on the deformation of the radial structure during the propagation of acoustic waves in cochlear ducts is proposed. Modelling is inspired from the finite elements analysis of Cai and Chadwick [1]. The influence of longitudinal position is studied with the modelling of six varying sections from the base to the apex of a gerbil’s cochlea. The frequency analysis is extended to a frequency bandwidth from 100 Hz to 3 kHz. Longitudinal mechanical coupling is investigated in this frame.
MODEL
Geometry of the radial cross section
A simplified representation of the organ of Corti and cochlear ducts is described as a radial cross section of the cochlea. The proposed model comprises basilar membrane (BM), tectorial membrane (TM), pillar cells (PC), Deiter’s cells (DC), tissues of the organ of Corti (OC), fluids ducts (FD), reticula lamina (RL) and outer hair cells (OHC)
FIGURE 1. (A) Zoomed view of a cross section of the organ of Corti in the middle turn. (B) Parameters. (a) width of the basilar membrane (pectinate zone); (b) length of the tectorial membrane; (c) height of the inner pillar cell; (d) height of the outer pillar cell; (e) height of the inner sulcus; (g) thickness of the tectorial membrane; (f) free part of the basilar membrane; (r) radius of the cochlear ducts.
(see Fig. 1 A). Inner hair cells (IHC) do not participate to structural response and are here omitted. A section can be defined with only few parameters (see Fig. 1B). The method
FIGURE 2. Cross sections (1, 2, 3, 4, 5, 6) of the organ of Corti.
developed by Edge et al. [3] provided morphological parameters from observations of three sections of the mongolian gerbil cochlea (one basal, one in the midlle turn and the last one at the apex). Six representations of cross sections are reconstructed from these datas with a CAD-software and then imported to finite elements software COMSOL Multiphysics®. . From the base to the apex, the width of the free part (f)(see Fig. 1B) of the BM increases while the radius (r) of FD reduces (see Fig. 2). Due to these morphological variations along the cochlea the last cross section will be more sensistive to low frequencies while the first cross section to higher frequencies.
Finite Elements Analysis
Fluid and solid domains
COMSOL Multiphysics® software uses Finite Elements Analysis (FEA) for solving partial differential equations from multiple physics. Following the small displacements assumption, solids are ruled by the classical linear elastic law for isotropic solids in a plane strain approach. No structural damping is considered for solids. Fluid follows linearized Navier-Stokes equation with incompressibility and no viscosity assumptions.
Materials properties are differents for each solid domain but are considered constant along the six cross sections. Young’s moduli are adjusted from literature [1] (see table 1).
TABLE 1. Young’s moduli for solids from [1].
Description Symbol Value
BM, TM, OC, RL, PC, DC, FD density ρs 1000 kg/m−3 BM Young’s modulus EBM 2· 108Pa TM Young’s modulus ET M 1· 103Pa OC Young’s modulus EOC 1· 104Pa RL Young’s modulus ERL 3· 104Pa PC Young’s modulus EPC 4· 104Pa DC Young’s modulus EDC 1· 104Pa
Boundaries
Boundary conditions rule the interaction between the domains. On fluid/solid bound-aries the normal acceleration of the solid is applied to the fluid and reciprocally the fluid pressure is imposed to the solid. For this preliminary study no loss is considered on boundaries. From lubrication theory, as suggested by Cai [1], the normal displacement of TM is imposed equal to that of RL.
Longitudinal propagation
The COMSOL’s preset equations are modified to fit with the following eigenvalue problem. The searched eigenvalue is here the longitudinal wavenumber kz :
∇ · (−∇p)− ω cf 2 − k2 z p= 0, (1) −ρsω2u− ∇ ·C· 1 2 (∇u)T+ (∇u)= p bnd, (2)
where u is the displacement field in solids, p is the pressure field, pbnd is the pressure on the fluid/solid boundaries andω is the chosen frequency. The solid density (ρs), the compliance matrix (C), the sound celerity in the fluid (cf) are calculated from materials properties. These equations are deduced from the following assumption : the propagating waves are slowly varying in the distance of one wavelength. The propagating wave is therefore represented by the following expression :
F = A · Ftei[ωt− z
0kz(s)ds], (3)
where A is the amplitude of the wave, Ft the transverse modal shape and z the lon-gitudinal position. The wavenumber kz and the corresponding mode are calculated for each cross section. The enveloppe function A(z) can be found using the WKB (Wentzel, Kramers, Brillouin) method (see [1, 9–11]). This procedure is not detailed here.
Longitudinal mechanical coupling
A mechanical longitudinal coupling is simulated in the BM. The plane strain assump-tion is modified with an out-of-plane strain artificially added in the BM solid. The WKB assumption provides the expression for the derivatives along the out-of-plane axis. For example, εzz= −∂ 2v ∂z2(y − y0) = k 2 zv(y − y0) . (4)
RESULTS
Modes of propagation
For each imposed pulsation, wavenumbers and corresponding pressure and displace-ment modes are obtained. The real wavenumbers are related to propagating modes and only these ones are selected. The first flexural mode of the basilar membrane is chosen among the propagating ones.
0 1000 2000 3000 0 5000 10000 15000 Frequency [Hz] Wave number [m −1 ]
Dispersion curves for 6 cochlear cross sections
Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 0 500 1000 1500 2000 2500 3000 3500 4000 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Frequency [Hz] Wave number [m −1 ]
Dispersion curves for a cross section including longitudinal mechanical coupling
Re(kz), Dz=0,1D Im(kz), Dz=0,1D 2nd coupled flexural mode
veering 1st coupled flexural mode
Mode conversion at CF
1st mode of propagation imaginary part associated
with mode conversion 2nd mode of propagation
FIGURE 3. Dispersion curves plotted for the six cross sections without mechanical longitudinal coupling (leftside) and for the section 4 with an orthotropic (ratio of 0.1 between longitudinal and transverse stiffness) mechanical longitudinal coupling for BM (rightside).
Wavenumber for the first transversal mode of the basilar membrane is plotted for the six cross sections and for a frequency bandwidth of [0.1-3 kHz] (see Fig. 3). For low frequency, the relation between frequency and wavenumber is linear and shows a non-dispersive behavior. For high frequency, the behavior is clearly dispersive reveal-ing the interaction between the fluid and the structure of the organ of Corti. The en-tire real wavenumbertending to infinity is not realistic but is ocherent with the no loss assumption. Dispersion curves including mechanical longitudinal coupling show inter-esting features. Additional propagatives modes appear and the previous ones (without coupling) interacts with them. A non-zero imaginary part after the characteristic fre-quency shows an attenuation of the wave with the no loss assumption. This results from the mode conversion between different propagative modes.
Pressure and displacement
The deformed shape of the third cross section for a 500 Hz excitation shows a constant pressure in the two ducts (see Fig. 4A). For higher frequencies a pressure gradient
FIGURE 4. Mode of propagation of the third cross section for 0.5 kHz (A) and 2 kHz (B). Solids displacements are plotted wireframe while fluid pressure is plotted with plain colors. A strong gradient of pressure is present for higher frequencies (B) compared to the constant repartition of pressure for lower frequencies (A). Displacements of the solids are scaled with the same factor (20) for both pictures (A and B).
appears close to the basilar membrane (see Fig. 4B). In both cases, displacements of the elements of the organ of Corti are similar. The basilar membrane deflection’s shape is closed to the first modal shape of a clamped-clamped beam. The tectorial membrane moves in a shearing displacement. The mode shape are quite similar in the both cases with or without mechanical coupling.
DISCUSSION
In this study, the purpose was to prove that the use of a multiphysical FEA software could improve the modelling of wave propagation in the cochlea. This approach permits to reduce computational times (1 minute for a frequency solving) compared to full 3D computation. The results presented are very promising. At lower frequencies, the linear relation between frequency and wavenumber is coherent with Cai’s results [1]. Results are here extended to a larger frequency bandwidth, revealing a different behavior at higher frequencies. The tonotopy is here evident showing the role of fluid structure interactions for the propagation of the cochlear acoustic waves. Interesting features appeared with mechanical longitudinal coupling but still have to be investigated.
However some simplifying assumptions have been applied in this paper :
1. No loss is considered. That is the reason why the solver give purely propagating modes. Shearing stresses on fluid structure boundaries (particularly in the RL-TM gap) should increase the imaginary part of the wavenumber [1] and the attenuation of propagating waves [2].
2. Longitudinal structural coupling is taken into account with an out-of-plane strain corresponding to the deflection of the BM. A strain corresponding to the shearing motion could be added for the others part of the OC.
3. Quantitative results are shifted from the cochlear place-frequency map described by Müller [5]. Materials properties should be more precisely defined. Including a variation of the Young’s modulus along the cochlea could help to reproduce the stiffness gradient measured by Emadi et al. [4] and a better correlation between experimental measurements and model.
CONCLUSIONS
This paper shows the feasibility of a comprehensive model of the cochlea with a standard FEM software. The implementation of an eigenvalue problem for solving propagating modes in a 2D-section is very promising. First results are coherent with literature. Non-dispersive behavior is shown at lower frequencies while a dispersive behavior appears with fluid/solid interactions at higher frequencies. The transition happens at different frequencies for each cross section reproducing the basis for tonotopy. An effect of the longitudinal mechanical coupling appears through the mode conversion. This phenomenon reveals an interesting attenuation of travelling waves without any introduced damping. The passive model presented here shows interesting features but the integration of active elements is the main purpose of this modelling. The next step is the definition of an active model for OHC solids for observing the influence of electromotility on wave propagation.
REFERENCES
[1] Cai H, Chadwick R (2003) Radial structure of traveling waves in the inner ear. SIAM J Appl Math 63:1105–1120
[2] De Boer E, Viergever M (1984) Wave propagation and dispersion in the cochlea. Hear Res 13:101– 112
[3] Edge R, Evans B, Pearce M, Richter C, Hu X, Dallos P (1998) Morphology of the unfixed cochlea. Hear Res 124:1–16
[4] Emadi G, Richter C, Dallos P (2004) Stiffness of the gerbil basilar membrane: radial and longitudi-nal variations. J Neurophysiol 91:474
[5] Muller M (1996) The cochlear place-frequency map of the adult and developing Mongolian gerbil. Hear Res 94:148–156
[6] Nobili R, Mammano F, Ashmore J (1998) How well do we understand the cochlea? Trends Neurosci 21:159–167
[7] Ramamoorthy S, Deo NV, Grosh K (2007) A mechano-electro-acoustical model for the cochlea:
Response to acoustic stimuli. J Acoust Soc Am 121:2758–2773
[8] Selva P, Morlier J, Gourinat Y (2009) Development of a dynamic virtual reality model of the inner ear sensory system as a learning and demonstrating tool. Modelling and Simulation in Engineering 2009:1–10
[9] Steele C, Taber L (1979) Comparison of WKB calculations and experimental results for three-dimensional cochlear models. J Acoust Soc Am 65:1007–1018
[10] Whitham G (1974) Linear and Nonlinear Waves. Wiley New York
[11] Yoon Y, Puria S, Steele C (2009) A cochlear model using the time-averaged lagrangian and the push-pull mechanism in the organ of Corti. J Mech Mater Struct 4:977
COMMENTS AND DISCUSSION
Steve Elliot: This paper presents an interesting study of using finite element models of sections at
different positions along the cochlea to deduce the longitudinal wavenumber. Such a model predicts many wave numbers however, which is perhaps not very clear. The wave selected involves the first flexural mode of the basilar membrane, but there may be some numerical problems in tracking the corresponding pressure distribution along the cochlea that gives rise to the irregularities noted in the dispersion curves. Are only six sections of uniform properties used with the wave numbers interpolated between them?
Presumably the organ of Corti dynamics are assumed to be passive? With no losses, the wavenumber is entirely real below the characteristic frequency but then tends to infinity. If a passive cochlea is modeled, damping could just be added to the solid domain to obtain more realistic real and imaginary components of the wavenumber.
Reply: Thank you for this comment.
Th dispersion curves plotted here were selected from a large number of solutions. The tracking of solutions along the frequency domain starting from the previous solution is not available using the FEM software. This solving issue could be solved for future works. This also explains the choice of a lossless model since propagative solutions can be segregated easily from evanescent ones. Once the tracking problem is solved, complex solutions could be found with damped model and viscid fluid.
Irregularities are due to crossing modes which give rise to veering. These modes are not represented here. For a longitudinal mechanical coupling in the basilar membrane some other propagative modes appear. It is interesting to observe that even with a lossless model the imaginary part of the wavenumber increases from the veering point between two modes. This feature was shown on the poster.
For now the organ of Corti dynamics are passive. One of the points of this model is that the modeling of active behavior is possible. After the validation of the passive damped behavior with comparison with experimental data, this will be the next step.
Only six cross sections are solved here. For the global WKB solutions, many more sections are necessary to obtain a good prediction of the wavenumber variations along the longitudinal axis. Envelope functions will be calculated from these variations. For now only six cross sections are tested but even one is sufficient to see some interesting features. The changes in the section produce only a shift in frequency. The curves are qualitatively the same.
![TABLE 1. Young’s moduli for solids from [1].](https://thumb-eu.123doks.com/thumbv2/123doknet/3658862.108092/4.918.247.679.226.394/table-young-s-moduli-for-solids-from.webp)
