1 Introduction

The outer ear collects sound and funnels it via ossicular chain to cochlea. When vibrations are transmitted to cochlear fluids, a pressure wave travels down the scala vestibuli from base to apex and causes the basilar membrane (BM) and the organ of Corti (OC) sitting on the BM to vibrate. Therefore, the vibration of the BM leads to the shear movement between the tectorial membrane (TM) and the reticular lamina (RL) in the Corti, which makes the stereocilia bend. The deflection of the stereocilia transforms the mechanical movement into neural pulses that are transmitted to the brain finally.

There are two main groups of sensory cells, inner cells and outer hair cells (OHCs). There are different pillar cells that are connected to the hair cells, which are associated with the passive dynamics of the cochlea and the support of the active metabolism. Among them, the pillar cells which are connected to the BM and the auditory system, stabilize the cell structure and form the tunnel of Corti (TC).

In the early stage, the microscopic mechanical properties of the OC were studied, and a number of studies focused on the parameters of the internal structure of OC which were obtained by observing the motion of OC under sound pressure. Ruggero and Rich^[1] found that the BM mechanics changed following alterations to the properties of the OHCs through experiments. With further study on the mechanics of the individual cochlear structures, the experimental data on the stiffness parameters of sensory and supporting cells used in vitro preparations were obtained^[2-5]. The mechanical properties of the functionally important TM were studied in elegant experiments in the living gerbil by Zwislocki and Cefaratti^[6]. Furthermore, Cormack et al.^[7] established a two-dimensional two-chamber finite difference model of the cochlea and studied the effect of BM and TM properties and gradients on cochlear response. Moreover, based on the model, the independent coupling of the BM and TM to the surrounding liquid was also investigated^[8].

With the development of the technology, video microscopy and confocal microscopy were used to visualize motion within the cochlear partition^[9-10]. Based on a special feature of the interferometer set-up which was designed and used by Khanna et al.^[11-17], a number of observations on the micro mechanical behavior of the hearing organ in the isolated cochlea were published. However, these experiments were often based on the cochlea that was without perilymph, and the measurement of the perilymph in the OC was little. Therefore, the effect of the fluid on the structure of OC was rarely involved.

With the in-deep research on the OC structure, the amplification of cochlea was gradually correlated with the fluid in the TC. Mountain and Hubbard^[18] found that at the peak of the traveling wave, the OHC contractions would pump fluid into the TC. The finite element model developed by Zagadou and Mountain^[19] }$ supported the hypothesis that the cochlear amplifier functions as a fluid pump. Therefore, whether there is an oscillatory fluid flow in the TC is a significant subject to study. However, in most of the current cochlear models, longitudinal coupling occurred through the cochlear fluids, which are referred to as ``classical models''. Although they have been successful in replicating some experimental data, they still fail to replicate all the unique features of cochlear responses^[20]. On the contrary, the nonclassical models are referred to the models which require a different source of longitudinal coupling to successfully replicate experimental measurements of BM motion. Geisler and Sang^[21] as well as Steele et al.^[22] assumed that longitudinal coupling existed because the apical end of the OHCs was tilted toward the basal end of the cochlea. A very different approach to longitudinal coupling was incorporated in Hubbard's traveling-wave amplifier model^[23], which was reported in $Science$. The model included two modes of wave propagation. The first mode corresponded to the traditional BM displacement wave. The second hypothesized that the oscillatory fluid flow was included in the TC^[24]. Such a flow was first considered in de Boer's^[25] formulation of the sandwich model, but this model did not address the functional role of TC flow in the cochlear amplification. Hubbard et al.^[26] explicitly incorporated the TC in a multicompartmental hydromechanical model of the cochlea, which successfully replicated BM motion. Furthermore, through the experiment on gerbil, Karavitaki and Mountain^[27] found that the velocity and space constants that propagated in the fluid of TC were similar to those on the BM, which means that it played an important role in the amplification of the motion of BM. These nonclassical models described above posed a challenge because so far there was no experimental evidence to confirm the oscillatory fluid flow in the TC. In the meantime, these previous studies always focused on how the fluid flow forms and did not take its effect on the OC structure into account.

In view of this, a three-dimensional OC model based on the experimental data of gerbil is established. Using the fluid-solid coupling method, the effect of the fluid on the structure of the OC is studied.

2 Model 2.1 Geometry

Based on the experimental data of gerbil acquired by Andoh and Wada^[28] and Edge et al.^[29], a three-dimensional OC model is established by COMSOL. Its longitudinal length (i.e., the $z$-direction in Fig. 1) is 48 $µ$m which is less than 1/4 wavelength of the traveling wave, and the corresponding fluid field is added, as shown in Fig. 1^[30]. The OC at the basal turn of the gerbil cochlea is located at 2.5mm--3.0mm from the base. Therefore, in order to know the effect that the fluid flow has on the structure of the TC better, the extension at both ends of the TC fluid is set to be 10 $µ$m (i.e., the $z$-direction in Fig. 1). One side is a fluid inlet, and the other side is a fluid outlet, as shown in Fig. 2. The geometric parameters such as the BM, the TM, the inner pillar cell (IPC), the outer pillar cell (OPC), and the RL of the OC model are shown in Table 1.

The meshes of the three-dimensional OC model are shown in Figs. 3(a) and 3(b). As shown in Fig. 3(a), the element size is selected as normal, and the refinement is carried out at stereocilia because their geometric area is relatively small. The numbers of triangular elements and vertex elements are 25444 and 272, respectively. As shown in Fig. 3(b), the element size is selected as coarse. The numbers of triangular elements and vertex elements are 15686 and 272, respectively. The solver of the simulation is MUMPS in stationary.

2.2 Material properties

Young's moduli of individual portions in the model are based on measurements in vitro and in situ. Young's modulus assigned to the model is $3\times 10^4$N/m$^{2}$^[31] at the TM, $1\times 10^4$N/m$^{2}$^[32] at the OHCs, $1\times 10^7$N/m$^{2}$^[33] at the phalanxes, and $1\times 10^9$N/m$^{2}$^[34] at the PCs, respectively. Young's moduli of the IHCs and Deiters' cells are assumed to be the same as those of the OHCs and the phalanxes, respectively. Because the RL is stiff enough to support adjacent structures, Young's modulus of the RL is assumed to be $1\times 10^9$N/m$^{2}$. Young's modulus for the osseous spiral lamina is assumed to be $2\times 10^{10}$N/m$^{2}$, which is the same as that of the cortical bone. Young's modulus of the stereocilia is assumed to be $1\times 10^7$N/m$^{2}$^[35]. Based on the experimental data of Naidu and Mountain^[36], Young's moduli of BM and Hensen's cells are assumed to be $1\times 10^7$N/m$^{2}$ and $5\times 10^3$N/m$^{2}$, respectively.

The Poisson's ratios of soft cells and the TM, which are composed of an extracellular matrix, are assumed to be 0.49 because these portions are nearly incompressible, whereas those of hard cells and the osseous spiral lamina are assumed to be 0.3 because this value is commonly used in the structural analysis and the relevant literature. The according mechanical properties are listed in Table 2.

2.3 Boundary condition

The left and right boundaries of the three-dimensional OC model (i.e., the $x$-direction in Fig. 3) are regarded as fixed boundaries. The pressure is applied at two ends of the fluid domain of the TC, and the other free fluid domain is regarded as the hard fluid boundary.

2.4 Formulation

In the coupling calculation of fluid and structure, the basic equation of coupling dynamics can be expressed as

(1)

where

(2)

(3)

Here, $ M $ is the mass matrix. $ {M^\mathrm {fs}} $ is the mass matrix on the coupling surface. $ {M^\mathrm P } $ is the fluid mass matrix. $ {C }$ is the damping matrix. $ {C^\mathrm P }$ is the fluid damping matrix. $ {K^\mathrm P } $ is the fluid stiffness matrix. $u $ is the displacement vector. $P $ is the pressure.

3 Results

Based on the three-dimensional OC model, the initial pressure is applied in the fluid of TC. According to the relevant literature^[37], the fluid pressure value at this location is between 1Pa and 7Pa. In this paper, the fluid pressure value is set as the average value which is 3Pa for study, and the distribution of fluid pressure in the TC is obtained, as shown in Fig. 4. When the initial fluid pressure in the TC is 3Pa, the frequency response curve of displacement amplitude of the BM is plotted in Fig. 5. As shown in the figure, under the initial fluid pressure, with the increase of frequency, the displacement amplitude of the BM increases first and then decreases with a maximum at 16kHz. Therefore, the characteristic frequency of the model is about 16kHz. Furthermore, as seen in the figure, with different kinds of meshes, the displacements of BM are close, which confirms the independence of meshes. Moreover, according to the relevant literature^[38-39], the equation which contains the characteristic frequency and the location of the BM is shown as

(4)

where $f_\mathrm c$ is the characteristic frequency whose unit is kHz. $x$ is the ratio of the length between any location on the BM and the distance from the top of the BM and is between 0 and 1.

From the formula, the characteristic frequency at 2.5mm from the base is about 16.43kHz, which agrees well with that in this model. It verifies correctness of the three-dimensional OC model.

Figure 6 shows that, when the initial fluid pressure in the TC increases, the displacement amplitude of the BM varies with the frequency. Different values of the initial fluid pressure are analyzed, which are 0.5 times, 1 time, and 2 times of 3Pa, respectively. As shown in the figure, the displacement amplitude of the BM increases first and then decreases with a peak at 16kHz. At the same time, with the increase of the initial fluid pressure, the displacement amplitude of the BM increases, but the increase rate decreases. When the pressure gradient is absent in the TC and the initial fluid pressure is 3Pa, the change of the displacement amplitude versus frequency at different points in inlet and outlet of the BM is plotted in Fig. 7. As shown in Fig. 7, the trend and the values of the displacements of the two curves are nearly the same.

When the fluid pressure in the TC is 3Pa, the movements of the OHCs and outer stereocilia which are the closest to the TC are plotted in Figs. 8 and 9. As shown in Fig. 8, as the frequency increases, the displacement amplitude of the OHCs that is the closest to the TC increases first and then decreases with a maximum at 16kHz. It means that the movement of the fluid in the TC can amplify the movement of the OHC, especially in the vicinity of the characteristic frequency. Figure 9 shows that, when the initial fluid pressure in the TC is 3Pa, the shear displacement of innermost stereocilia that is the closest to the TC changes little in the low frequency and changes sharply in the vicinity of characteristic frequency with a maximum at 16kHz. This is because that at the characteristic frequency, resonance occurs which causes the shear movement between the TM and RL to increase.

In order to study the effect of pressure gradient on the OC structure, the initial condition is as follows: the inlet fluid pressure is set as 3Pa and the outlet fluid pressure is set as 0Pa, and the corresponding distribution of fluid pressure is shown in Fig. 10. Figure 11 shows the frequency response curves of displacement at different points in inlet and outlet of the BM when there is a pressure gradient in the TC. As seen from the figure, no matter whether the fluid pressure value is large or small, the corresponding trend of BM displacement with the frequency is similar. However, the BM displacement of the small fluid pressure is relatively small. At the same time, it can be found that the change of fluid pressure does not affect the value of its characteristic frequency, and its peak value is about 16kHz.

4 Discussion

The TC is acted as one of the main connections between the BM and the RL in the OC. The movement of the fluid in the TC affects not only the BM but also outer stereocilia and OHCs.

When the fluid pressure in the TC is 3Pa, its effects on the BM and OHCs are plotted in Fig. 5, Fig. 8, and Fig. 9. As can be seen in Fig. 5, the displacement peak value of BM is achieved at 16kHz, which reveals that the fluid in the TC has an effect on the movement of the BM. In the meantime, it proves the BM displacement amplification caused by the movement of fluid in the TC and agrees well with the experimental data^[24]. Figure 8 shows that the displacement of OHCs increases in the vicinity of the characteristic frequency due to the movement of the fluid, which means that the movement of OHCs is amplified. Therefore, the movement of fluid in the TC increases the movement of OHCs and causes the amplification in cochlea. Therefore, in the future study, we shall not only consider the movement of the hair cell due to the movement of the BM but also consider the direct effect of fluid movement in the TC on the hair cell. Figure 9 shows that, when the fluid pressure in the TC is a certain value, the effect of the fluid movement on the shear movement between the TM and RL is small, but the effect is large in the vicinity of the characteristic frequency.

By comparing Fig. 7 with Fig. 9, it can be seen that, when there is a pressure gradient in the fluid, it affects the displacement amplitude of the BM. However, because the longitudinal length of BM in the three-dimensional OC model is very small, the values of thickness and elastic moduli of BM are the same, and the change of fluid pressure does not affect the characteristic frequency of the BM.

5 Conclusions

In this paper, a three-dimensional OC model is established by COMSOL. The results in this paper are in good agreement with the experimental data, confirming validity of the finite element model. In the meantime, based on the model, the effect of the movement of fluid in the TC on the structure of OC is discussed. The conclusions are as follows:

(ⅰ) When the pressure gradient is absent in the TC, with the increase of the initial fluid pressure, the displacement of the BM increases. However, when the initial fluid pressure increases to a certain value, the increase rate decreases. In the meantime, the movement of the fluid has an amplification effect on the BM.

(ⅱ) Under the initial fluid pressure in the TC, the movement of OHCs and outer stereocilia reaches the peak in the vicinity of the characteristic frequency. It can be seen that the movement of fluid in the TC can affect the movement of the hair cell directly and reflect the amplification of the cochlea laterally.

(ⅲ) The pressure gradient in the TC affects the displacement of the BM, but it does not affect the characteristic frequency of the position.