1 Introduction

The organ of Corti (OC) is mainly composed of hair cells and supporting cells. Moreover, the hair cells are divided into the inner hair cells (IHCs) and outer hair cells (OHCs). The vibration of the basilar membrane (BM) leads to the shear movement between the tectorial membrane (TM) and the reticular lamina (RL) in the OC, which makes the stereocilia bending. Moreover, the deflection of the stereocilia transforms the mechanical movement into neural pulses that are signaled to the brain in the end.

In the early stage, the research on the microscopic mechanical properties of the OC is mainly focused on the parameter information of the structure of the OC. In vitro preparations, experimental data on the stiffness parameters of sensory and supporting cells were obtained ^[1-4]. The mechanical properties of the functionally important TM had been studied in elegant experiments in the living gerbil by Zwislocki and Cefaratti^[5]. Through the experiment on the gerbil cochlea, the geometric data of different parts of Corti were achieved by Edge et al.^[6]. And with the deeper experiment, the internal structure properties of Corti had been understood. Based on the experiment on the excised gerbil cochlea, Naidu and Mountain^[7] found that the cells of the OC increased the overall coupling exhibited by the BM, and at a given location, the longitudinal coupling increased the effective stiffness of the OC near the characteristic frequency. Brundin et al.^[8] found tuned motility of the cell body of OHCs through experiment, which amplified the frequency selectivity in the OC, and the results had been reported on 《nature》. These experimental data lay the foundation for the numerical and theoretical modeling in the future.

With the further study, more and more attention has been paid to the study of the active characteristics of the cochlea. Many are extensions of passive cochlear models with the inclusion of the micromechanics of the OC, in particular, the active behavior of OHCs. In the early stage, a negative damping in one-dimensional and lumped parameter models had been used by de Boer^[9] and Diependaal and Viergever^[10]. Then, Kanis and de Boer ^[11-12] extended this model to include nonlinearity in the activity and obtained both frequency and time domain solutions using a quasi-linear method. Other researchers used a `second filter' to describe the micro-mechanics of the OC ^[13-14]. Lim and Steele^[15] established a three-dimensional nonlinear active cochlear model by using the Wentzel Kramers Brillouin (WKB) method, and the nonlinear characteristics of the BM were obtained. Yoon et al.^[16] also established a linear active three-dimensional model by using the WKB method, and found that the active process was represented by adding the motility of OHCs to the passive model with the feed-forward approximation of the OC. However, in these studies, the movement of the BM, the OHC, and the RL in the OC is not involved. Therefore, a more effective approach, the variational principle, is used to analyze in this paper.

In this paper, the motion of the BM, OHCs, and RL in the OC is studied from the perspective of mechanics. According to the vibration characteristics of the OC, a mechanical OC model is established. Using the variational principle, the displacement analytical expression is achieved under a certain pressure. Combined with the damage made by noise in clinic, the effect on the motion of BM is studied, which is caused by the hardening of OHCs and outer stereocilia.

2 Mechanical model of OC 2.1 Characteristic frequency (CF) of mammals

As first shown by von Bekesy, the systematic mapping of CF upon the longitudinal position on the BM is a general and fundamental principle of the mechanical processing of acoustic signals in the cochleae of mammals. The tonotopic cochlear map has been worked out in some species, such as cat and mongolian gerbil, by measuring the sites of innervation of auditory nerve fibers of known CF ^[17-18]. In these cochleae, as well as in the cochleae of several other species that are known with less precision, the tonotopic map is as follows:

(1)

where f is the CF, and the unit is kHz. x, the distance from the apex, is expressed as a proportion of the BM length, from 0 to 1. The constant α is 2.1 in many cochleae differing widely in length (11.1 mm to 60 mm) including those of gerbil, chinchilla, guinea pig, cat, macaque monkey, humans, cow, and elephant ^[19-20]. The constant k also varies between 0.8 and 1, typically 0.85, in many species. The constant A determines the range of CF which is different in different mammals that is 0.456 in cat, 0.164 in chinchilla, 0.35 in guinea pig, 0.36 in the macaque monkey, and 0.4 in gerbil.

2.2 Mechanical model

According to the vibration characteristics of the OC, the model is assumed as follows:

(ⅰ) The surrounding for the BM is assumed as two spring supports, and the overall translational motion is expressed as u₁ =u₁ (x₀), where x₀ is the distance from the base of BM, and u₁ is the displacement of BM.

(ⅱ) The effect of fluid on the BM is the uniform pressure, and according to the relative literature^[21], it can be replaced by the concentrated force p.

(ⅲ) The vibration is undamped, steady and forced.

(ⅳ) Because the movement of the BM mainly causes the movement of the hair cell and the shear movement between TM and RL, the work done by the external forces is assumed to mainly split into the movement of the BM, the movement of the hair cell, and the shear movement of the stereocilia, and all the properties is homogeneous.

(ⅴ) According to the assumption (ⅳ), it can be assumed that u₂ =u₁ cosθ, and u₃ =u₁ sin θ, where θ is the included angle between the RL and the BM, u₂ is the longitudinal elongation of OHCs, and u₃ is the shear displacement of outer stereocilia.

Because the experiment can only provide data at several positions of the BM, according to the variation of the material properties of BM^[22], other assumptions are shown as follows:

(ⅵ) The elastic modulus of OHCs and outer stereocilia changes exponentially, and the cross section of OHCs and outer stereocilia is square.

(ⅶ) Three OHCs are the same, and their lengths are equal to those of the outer supporting cells. The lengths of the three outer stereocilia are the same, too.

According to the seven points above, the mechanical OC model can be established, as shown in Fig. 1.

2.3 Formulation

The energy expression of the spring, OHCs, and outer stereocilia in the OC is expressed according to the expression of the strain energy and kinetic energy of mechanics of materials. The corresponding expressions are as follows:

(ⅰ) The total elastic potential energy equation on the boundary condition of BM, E_BM, is

(2)

(ⅱ) The total strain energy equations of OHCs, E_OHC, are

(3)

(4)

(ⅲ) The total strain energy equations of outer stereocilia, E_S, are

(5)

(6)

(ⅳ) The kinetic energy of BM, T_BM, is

(7)

(ⅴ) The total kinetic energy of OHCs, T_OHC, is

(8)

(ⅵ) The total kinetic energy of outer stereocilia is

(9)

(ⅶ) The work W done by an external force is

(10)

(ⅷ) The structural strain energy V is

(11)

(ⅸ) The structure total potential energy is

(12)

Then, u₁ takes the first order variational,

(13)

The solution is as follows:

(14)

where u₁ is the displacement of BM, u₂ is the longitudinal elongation of OHCs, and u₃ is the shear displacement of outer stereocilia. E₁, E₂, and E₃ are the elastic moduli of BM, OHCs, and outer stereocilia, respectively. L_BM, L_OHC, and L_S are the lengths of BM, OHCs, and outer stereocilia, respectively. h₁, h₂, and h₃ are the thicknesses of BM, OHCs, and outer stereocilia, respectively. A_OHC is the area of cross-section of OHCs, and k is the coefficient of elasticity. Since the dimension of elasticity coefficient is the same as the stiffness, and the elastic coefficient is changed, the stiffness of BM is used to represent the elastic coefficient k of BM which can change with the size of BM. Therefore, k=E₁ I₁ =E₁ L_BM h₁ ³/12.

3 Verification of model

According to the length of BM of gerbils in Ref. [23], the length of the BM in this paper is 11 mm. The lengths of BM, outer pillar cell, and outer stereocilia are achieved according to Ref.[6], which is shown in Table 1. Other relevant parameters ^[24-26] are shown in Table 2.

According to the assumption that the elastic moduli of OHCs and outer stereocilia change exponentially, the expression can be shown as

(15)

where E_n, (n=1, 2, and 3) represent the elastic moduli of BM, OHCs, and outer stereocilia, respectively. E₀ and D are constant which can be solved according to the boundary conditions, x₀ is the distance from the base of the BM, and the unit is mm.

At the same time, the elastic moduli of the apical and basal outer stereocilia and OHCs are set to the corresponding boundary conditions, and the corresponding expressions are as follows:

(16)

(17)

(18)

Figure 2 shows the absolute value of the displacement at different locations on the BM in response to the frequency. As shown in Fig. 2, as the frequency increases, the displacement amplitude of the BM fluctuates with a maximal value at the CF. Furthermore, at different positions of the BM, the corresponding CF is different, which is 500~Hz at basal (2.5 mm), 3~kHz at middle (5 mm), and 10~kHz at apical (10 mm) of the BM. Furthermore, it can be seen from the figure that, at the low frequency, with the distance away from the basal of the BM, the displacement amplitude increases, while at the high frequency, the situation is opposite, which is in accordance with Ref. [27]. Because the subject chosen in this paper is gerbil, Eq. (1) can be changed into

(19)

According to Eq.(19), the CF of basal, middle, apical of the BM is calculated, which is compared with the results achieved from Eq.(14), as shown in Fig. 3. As it can be seen from Fig. 3, the CF of different locations on the BM is different, which reflects the frequency selective characteristics of the BM. At the same time, it can be seen from the figure that the slope of the experimental value obtained from the experiment is similar to the analytical value obtained in this paper, which proves the correctness of the formula.

4 Discussion and results

It is assumed that the length and thickness of BM, the length of OHCs, and the included angle between the RL and the BM are two linear changes. The basal and apical data in Table 1 are the left and right boundary conditions, and the middle part is the dividing point of the two-segment linear variation.

(ⅰ) The length of BM, L_BM, is

(20)

(ⅱ) The thickness of BM, h₁, is

(21)

(ⅲ) The length of OHCs, L_OHC, is

(22)

(ⅳ) The included angle between the RL and the BM is

(23)

Then, by taking Eqs. (20) -(23) into Eq. (14) and according to the boundary condition, the function is obtained between the displacement of BM and the position of BM.

The frequency response curves of displacement on the BM with different x₀ are plotted and shown in Fig. 4. As it can be seen in Fig. 4, the displacement of the BM increases first and then decreases with the increasing frequency. And for different positions on the BM, the CF is different, which is that, at the high frequency, the bottom of the BM resonates, and at the low frequency, the top of the BM resonates. It reveals the frequency selective characteristics of BM well. At the same time, the corresponding frequency map is drawn in Fig. 5. Compared with the theoretical value calculated by Eq. (19), the analytical solution obtained in this paper shows that the results are slightly larger than the theoretical values at the low frequency, however, at the high frequency, they are slightly lower. Moreover, the general trend of the analytical solution is closer to the theoretical value, and the slope is similar.

In clinic, the damage of OC in the cochlea is made by the noise, especially to the hair cells. When threshold shift occurs, the stereocilia will be disordered, and its response to stimulation is poor. In this paper, the decrease of the stimulus response is reflected by increasing the stiffness of the stereocilia. Further, the location that is 10 mm from the top of the BM is analyzed. In the meantime, according to Refs. [28] and [29] about the hardening of ossicular ligaments in clinic, the elastic modulus of the hardening stereocilia is 100 times as large as that of the normal physiological state. As shown in Fig. 6, it can be found that as hardening of the stereocilia, the displacement amplitude of BM decreases, which means that with the decline of the stimulus response of the stereocilia, it will lead to the hearing loss. Furthermore, the peak values are shifted which cause the phenomenon of threshold shift. This phenomenon is consistent with reality. Moreover, when OHCs are hardening, the effect made on the BM is plotted in Fig. 7. The elastic modulus of the hardening OHCs is 100 times as large as that of the normal physiological state. As can be seen in the figure, with hardening of OHCs, the displacement amplitude of BM decreases obviously. However, the overall trend with frequency is similar. At the same time, it can also be found that with the hardening of OHCs, threshold shift phenomenon of the BM also appears.

5 Conclusions

According to the vibration characteristics of the OC, seven hypotheses are made to simplify the structure of the model, and a mechanical OC model is established. Using the variational principle, the displacement analytical expression is solved under the certain pressure. The results are in good agreement with the experimental data, showing the validity of the formula. At the same time, by using this formula, the vibration of the OC under the certain pressure can be obtained quickly, and it can also avoid interference of the environment and the technical level of the test personnel and evaluate performance of the OC objectively.

By assuming that the corresponding structure size of the OC two-linearly changes, the relationship between the displacement amplitude of the BM and the distance from the base of the BM is obtained. The correctness of the formula is verified by comparing with the experimental data. At the same time, combined with the cause of the OC damage in clinic, it is found that the hardening of OHCs and outer stereocilia can lead to loss of hearing and generation of threshold shift. In addition, since this method is simple and easy to use, as long as the input parameters are given, the displacement of the BM can be achieved. Therefore, it provides a theoretical basis for clinical medicine.