An Analytical Mechanical Model of Corti in the Cochlea

*e organ of Corti (OC) in the cochlea is a significant structure for feeling sound. *e components of OC and the interaction of the part with the surroundings contribute to the fact that the passive tuning of the cochlear macrostructure is unclear. Based on the interaction between the basilar membrane (BM), tectorial membrane (TM), reticular lamina (RL), and various parts of OC, a mechanical model of the cochlea is established to study the motion patterns of each part under the action of a certain pressure. *e variational principle is applied to the calculation of the analytical expression of the displacement of the BM.*e results of the analytical solution differ little from the experimental value, and the variation trend is consistent, which presents the correctness of the model. *e parameter sensitivity analysis is carried out for obtaining the interaction principle and the primary and secondary roles of each component in the process of the sense of sound. *e results show that the absence of the TM and the decrease in the stiffness of the outer hair cells (OHCs) and OHC bundles will shift vibratory response patterns to lower frequencies, in which the lack of TM will result in the greatest reduction of CF. *e absence of RL exerts a negative influence on the CF as well as the amplitude of BM and thereby loss of hearing. *erefore, both TM and RL are essential structures during the process of the sense of sound. At the same time, the resonance frequency at the base of the BM is concentrated on the high-frequency segment, while the apex of the BM is mainly in the low frequency. Different points of BM correspond to different CF, which demonstrates the frequency selectivity of the BM.


Introduction
As a part of the section of the cochlea, OC plays a significant role in the process of the sense of sound. e incoming vibration from the middle ear will cause sound pressure waves in the inner ear, thereby moving the OC relative to surrounding structures. e relative motion will cause the impulse of the auditory nerve, which is the key to producing hearing in the brain. With the tuning of OC, the amplitude of the BM can increase vibratory magnitudes by 1000 times of the original value [1]. How does this remarkable acoustic behavior come about? e reasons for this phenomenon are thought to be dominated by the active effect of OHCs [2][3][4]. However, the contributions from the various components of OC itself could not be excluded [5]. So far, scientists have not been able to accurately interpret the sound-sensing mechanism of the cochlea. e effect of various components of OC on the passive tuning of the cochlea is also unclear.
In the meantime, noninvasive measurement of the motion inside OC is very inconvenient. erefore, cochlear mechanics has entered the stage of exploring the soundsensing mechanism of the cochlea and has become a research hotspot in this century. However, the physical properties of each part of OC are unknown. Consequently, in the early stage, many experts and scholars focused on the structural parameters and material properties of the OC. Fernandez [6] measured the length, width, and thickness of the BM of the guinea pig cochlea with a calibration eyepiece. Edge et al. [7] observed the structure of living cochlea from a radial perspective for the first time and obtained the length and thickness of OHCs at different points of BM. Glueckert et al. [8] measured the length and thickness of OHC bundles using high-power scanning electron microscopy. Nilsen and Russell [9] obtained the relative position relationship of various parts of OC in guinea pigs through histological sections. Sugawara et al. [10] used atomic force microscopy to measure the local mechanical properties of guinea pig OHCs. Zagadou et al. [11] obtained the elastic modulus of OHC bundles and BM by the method of point-stiffness measurements. Cai et al. [12] used the formula to convert the measured stiffness of OHCs into the elastic modulus of OHCs. With further study, Cooper [13] and Nilse and Russell [14] found that the vibration of the center of the BM was larger than the vibration at both ends. TM and RL also have the largest amplitude in the middle [15,16]. At the same time, some scholars believe that the modes of the cochlear partition are not single during the vibration process [17]. ese experimental data and analytical studies provide a theoretical basis for the development of the cochlear mechanics model.
For the study of the model, many scholars have focused on the interaction between the lymph fluid and the BM at the macrostructural level of the cochlea. In the early stage, researchers generally spread the spiral cochlea into a straight cochlea for analysis. Allen [18] established a 2D cochlea mechanical model and greatly simplified the model, and the cochlear interface was assumed to be rectangular. After that, Mammano and Nobili [19] built a 2D model considering the variation of cochlea cross section. Steele and Taber [20] established a 3D linear cochlear mechanical model, which included fluid action, and analyzed the mechanical behavior of the cochlea using the WKB method. However, with the development of research, it is found that the spiral shape of the cochlea may not only improve the space utilization of the cochlea [21,22]. Recent experiments [23,24] have demonstrated that the spiral shape of the cochlea can enhance the hearing ability of low-frequency signals in the cochlea. erefore, some scholars gradually began to establish spiral cochlea for analysis. Lieberstein [25,26] established a cochlea with a spiral shape BM but did not consider the fluidsolid coupling between the BM and lymph. Ma and Yao [27] built a 3D spiral cochlea to study the flow field distribution of lymph fluid in the cochlea and the vibration of BM. At the same time, many scholars used numerical methods to study the mechanical behavior of the cochlea. Gan et al. [28] first built a complete cochlea model containing a linear cochlea, which was applied to study the mechanical behaviors of BM. Later, Wang et al. [29] established a cochlea model containing a linear cochlea and combined it with the active feedforward mechanism to analyze the nonlinear vibration performance of the active cochlea. To make the finite element model closer to reality, Givelberg and Bunn [30] established a spiral finite element model of the cochlea and used the model to reproduce some known features of the cochlea mechanics. After that, in the study of implantable hearing devices, Gan established a spiral cochlear model [31], which can more intuitively analyze the implanted hearing equipment and the operation process. Yao et al. [32][33][34][35] built a 3D model of the cochlea with a spiral shape to describe the fluid-solid coupling motion of the lymphatic fluid and BM.
However, these models assumed that the motion of each part in OC is consistent with the movement of the BM and ignored the relative motion of the RL, the TM, and so forth during the vibration process. Moreover, the stiffness of the RL is far greater than that of the BM. e effect of the RL is essential for the interaction of the structures of OC. erefore, in this paper, a cochlear mechanical model containing all parts of the OC was established according to the relative motion relations among the BM, OHCs, RL, and TM.
rough parameter sensitivity analysis, we mainly explored the role of RL in the cochlear sensory mechanism and the effect of the RL on the passive tuning of OC. At the same time, the role of OHC bundles, OHCs, and TM in the passive mechanism of the cochlea was also studied.

Displacement Relationships of Each Point of OC.
In the OC structure, displacement of the BM can express that of the OHCs, OHC bundles, RL, and the arch of OC. e mode of BM vibrations is assumed to be sinusoidal vibration. When the calculation point is x and the time is t, the displacement of the BM of Cochlea section is given by In the formula, y represents the distance between the calculated point and the left end of the BM, b is the width of the BM, x is the distance between the selected point and the basal of the BM, and A represents the vibration amplitude of the BM. e arch structure is assumed to be rigid. When the BM is displaced, the arch structure will rotate in response. e displacement of the apex of the arch structure is given by In the formula, L pc is the distance between the left end and the right end of the arch structure and β is the inner angle of the arch structure. e compression deformation of the OHCs depends on the relative displacement of the top and the bottom of the OHCs. e motion of the top of the OHCs in the middle row is given by 2 Shock and Vibration where ϕ is the acute angle between OHCs and the z-axis and α is the acute angle between OHCs bundles and the z-axis. e displacement of the bottom of the middle row of the OHCs is e shear displacement of the OHC bundles in the middle row is ere was no significant difference in the displacement of three rows of OHCs. eir values are considered to be the same.

Analytical Solution of BM.
e work done by external forces is mainly absorbed by the movement of BM, OHCs, and OHC bundles. When the calculation point is x and the time is t, the energy expression of OC components can be obtained according to the content of material mechanics. e BM is regarded as an isotropic plate [20], and the strain potential energy of the BM is given by where e formula is used to analyze the section of the cochlea, and the integration formula of the strain energy of the BM is where E 1 is the elastic modulus of the BM, u bm is the displacement of the BM, h bm is the thickness of the BM, h ohc is the thickness of the OHCs, and v is Poisson's ratio of the BM. e axial deformation of OHC depends on the relative displacement of the top and bottom of OHC. e strain energy generated by OHCs is and the integration formula of the strain energy of the OHCs is OHCs and z-axis, α is the acute angle between OHCs bundles and z-axis, α and ϕ have the same value, and β 1 � β 2 � β is the inside angle of the arch structure.
Shock and Vibration 3 where E 2 is the elastic modulus of OHCs, A ohc is the crosssectional area of OHCs, L ohc is the length of OHCs, and u ohc is the axial deformation of OHCs. e strain energy equations generated by OHC bundles are and the integration formula of the strain energy of the OHC bundles is where E 3 is the elastic modulus of the OHC bundles, h ste is the thickness of the OHC bundles, L ste is the length of the OHC bundles, and u ste is the shear displacement perpendicular to the axis of OHC bundles. e total kinetic energy generated by BM is given by where m bm is the mass of the BM, v bm is the velocity of the BM, ρ bm is the density of the BM, and w is the angular frequency. e kinetic energy generated by OHCs is given by and the integration formula of the kinetic energy of the OHCs is where ρ ohc is the density of OHCs, m ohc is the mass of OHCs, u ohc is the average displacement of OHCs, and v ohc is the speed of OHCs. e kinetic energy generated by OHC bundles is given by and the integration formula of the kinetic energy of the OHC bundles is where m ste is the mass of the OHC bundles, v ste is the velocity of the OHC bundles, ρ ste is the density of the OHC bundles, and u ste is the average displacement of the OHC bundles. e energy equation Wof external force P is given by the structural strain energy V is given by and the total potential energy of OC is given by By substituting the expressions of each term into equation (20), the total potential energy can be obtained from the following equation: e first-order variation of A is When the calculation point is x and the time is t, the expression of BM amplitude is as follows: If A � A 0 sin(wt), the expression of A 0 is erefore, A 0 is the maximum amplitude of the calculation point in the process of vibration.
Since w is the angular frequency, from w � 2πf, we can get the maximum amplitude A 0 expressed by frequency f.
In the formula, P is the external force acting on the BM, which is replaced by the uniform distributed force. e pressure difference between the upper and lower parts of the BM decides the size of uniform pressure [37]. Moreover, the coefficient in equation (25)

CF of Mammals.
By comparison with the experimental data of CF, the accuracy of the analytical solution can be verified. In the early days of cochlear research, Békésy discovered the systematic mapping between the position of the BM and CF. Greenwood [38,39] also measured several species to obtain the formula relationship of frequencyposition relationship. e tonotopic map satisfies the following formula: where CF is characteristic frequency. e unit of CF is kHZ. α and N are the parameters associated with the selected species, and they have the same value for the same species. e parameter of the expression is mainly from [39]. e value of α is 2.1 and it has small change when choosing different lengths of BM. e value of the parameter N is generally 0.85. M is a constant that varies with the scope of CF. e value of gerbil is 0.4, the guinea pig is 0.35, and the cat is 0.456. x 0 is the ratio of the distance between the top of the BM and the selected position to the total length of the BM, ranging between 0 and 1.

Model Parameters.
e model was validated with guinea pig cochlea data.
According to the presumption that the structural parameters b, L ohc , h bm , α, and ϕ change linearly along the length of BM, the general form is as follows: where Y is b, L ohc , h bm , α, and ϕ in Table 1 the thickness of BM: the length of OHCs: and the acute angle:

Mechanical Properties of Materials.
e material properties of the BM, OHCs, and OHC bundles in OC are shown in Table 2. e relevant parameters are from [10][11][12]28].
e parameters E 1 , E 2 , and E 3 are the elastic moduli of BM, OHCs, and OHC bundles, respectively, which are assumed to change exponentially along the length of BM. e general form can be written as where E is the elastic modulus, M and N are the parameters derived from the corresponding data at the top and bottom of the cochlea, and x is the distance from the selection point to the bottom of the BM. e data of the base and apex of BM, OHCs, and OHC bundles are substituted into the formula as boundary conditions. e elastic modulus function of each part of OC is given by

Validation of the Model.
e BM length of the guinea pig in this paper is 25 mm [5]. According to equation (25), the CF of eight different points (2.5 mm, 5.0 mm, 7.5 mm, 10 mm, 12.5 mm, 15 mm, 17.5 mm, and 20 mm) of the BM is calculated.
According to [38,39], the CF of the experimental value is given by After determining the calculation point on the BM, the experimental value of CF can be obtained by equation (34), which is compared with the analytical value achieved from equation (25), as shown in Figure 3. Figure 3 illustrates that the calculated analytical value is slightly larger than the experimental value. e slope of the frequency-place map of analytical value is similar to the measured experimental data. e accuracy of the analytic equation is verified. In addition, CF corresponding to different points on the BM is different. At the base, the CF is larger, about 31 kHz; at the apex, the CF is smaller, about 900 Hz; this phenomenon is consistent with our understanding of the frequency selectivity of the BM. Figure 4 shows the frequency response curve of the calculation point. e displacement of the BM in Figure 4 represents the absolute value. Each response curve has a different peak. Moreover, the resonance frequency at the base of the BM is concentrated in the high-frequency segment, while the apex of the BM is mainly in the low frequency. Different points correspond to different CF, which confirms the frequency selectivity of the BM.

e Effect of RL.
To assess the effect of RL, we remove the RL from the model, which will result in only axial deformations of the OHC bundles and OHCs during vibration. e frequency response curve is plotted in Figure 5. Figure 5 shows that the amplitude of the response curve decreases significantly, which indicates that the lack of RL causes hearing loss. Moreover, CF seriously deviates from the experimental value. e reason for these effects is that the stiffness of RL is much greater than that of BM. e force generated by OHCs can be effectively transferred to the BM. At the same time, a freely moving RL is highly responsive to the movement of OHCs and enhances the sensitivity of OHCs. It is worth noting that the shear action of the TM and the RL will cause the opening of ion channels, which is the main reason for producing hearing. erefore, the RL acts as an effective component in OC to enhance the movement of BM.    Shock and Vibration

e Effect of OHC Bundles Stiffness.
OHC bundles play a significant role in the process of the sound-sensing of the cochlea. A key conclusion is that it has a direct relationship with the generation of the active force [40,41], and the mechanical properties of OHC bundles may also have an impact on the tuning of the cochlea. e stiffness of the OHC bundle is an important property because it affects mechanotransduction and, thus, cochlear amplification [42]. Hence, we explore the implications of OHC bundles by reducing the stiffness to half of the original. e results are presented in Figure 6. Figure 6 shows that the vibration response curve moves towards the low-frequency direction. e reason for this phenomenon is that reducing OHC bundles stiffness will reduce the transverse stiffness of OC. At the same time, the fluctuations of CF value are very little, and the OHC bundles contribute minimally to transverse partition stiffness.

e Effect of OHCs Stiffness.
To explore the role of the stiffness of OHCs, the stiffness of OHCs is reduced by half. Figure 7 shows that the reduction of OHCs stiffness affects amplitude in different frequency segments. e response curve shifts to lower frequencies, which indicates that the stiffness of the OHCs also contributes to the transverse partition stiffness of cochlea. Moreover, the decrease of CF is slightly above that of OHC bundles. It is worth noting that the OHCs connect the RL and BM, and they act as a conductive force during the vibration process.

e Effect of TM.
To assess the impact of TM, establishing a model includes a TM that is exfoliated from the epithelium. erefore, the analytical expression will not consider the potential energy of the OHC bundles. e results are plotted in Figure 8. Figure 8 presents that the decrease of CF is more than that of OHC bundles.
is finding implies that the TM contributes substantially more to the transverse stiffness of the partition than do the OHC bundles. In addition, TM directly affects the shear displacement of OHC bundles because it connects OHC bundles and, thus, cochlear amplification. TM is also an essential part of OC.

Conclusions
Based on the interaction between BM, OHCs, RL, TM, and so on, a mechanical model of OC is established. e accuracy of the model has been verified by comparing the calculated results with the experimental data. On the basis of the foregoing analysis, the conclusions are expressed by the following.
It is noted that CF of the calculation point seriously deviates from the experimental value when not considering the role of the RL. e amplitude of the frequency-amplitude response curve of the BM decreased significantly, which will cause hearing loss. e RL is an indispensable structure in the process of the sound-sensing mechanism of the cochlea. e reduction of the stiffness of OHC bundles and OHCs will shift vibratory response patterns to lower frequencies.
e absence of TM makes CF decrease seriously. ese effects are consistent with a reduction in the transverse stiffness of the partition. Significantly, the shear action of the TM and the RL leads to the opening of ion channels at the bottom of OHC bundles, which strongly affects mechanotransduction. us, TM plays an essential role in the process of the sense of sound. e bottom of the cochlea resonates at high frequencies, and the top resonates at low frequencies. Different points of BM correspond to different CF, which demonstrate the frequency selectivity of the BM. Moreover, the analytical solution of this paper not only reveals the fabric laws of each part of OC but also can be used to carry out parameter sensitivity analysis quickly and effectively. erefore, it can provide theoretical support for relevant experiments and medical research.
Data Availability e data used in this manuscript are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.