# An Analytical Mechanical Model of Corti in the Cochlea

Authors: Jiangtao Su; Wenjuan Yao; Zhengshan Zhao
Journal: Shock and Vibration
 (2020)

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## Abstract

The organ of Corti (OC) in the cochlea is a significant structure for feeling sound. The 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. The variational principle is applied to the calculation of the analytical expression of the displacement of the BM. The results of the analytical solution differ little from the experimental value, and the variation trend is consistent, which presents the correctness of the model. The 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. The 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. The absence of RL exerts a negative influence on the CF as well as the amplitude of BM and thereby loss of hearing. Therefore, 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.


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## Body

## 1. Introduction

As a part of the section of the cochlea, OC plays a significant role in the process of the sense of sound. The incoming vibration from the middle ear will cause sound pressure waves in the inner ear, thereby moving the OC relative to surrounding structures. The 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?The reasons for this phenomenon are thought to be dominated by the active effect of OHCs [2–4]. 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. The 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. Therefore, cochlear mechanics has entered the stage of exploring the sound-sensing 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]. These 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. Therefore, 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 fluid-solid 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 feed-forward 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. [32–35] 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. The effect of the RL is essential for the interaction of the structures of OC. Therefore, 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. Through 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.


## 2. Model

### 2.1. Establishment of a Mechanical Model

The mechanical model of OC was established based on the medical anatomical diagram of the guinea pig cochlea (Figure1) and the geometric relationship of each part in OC. The mechanical model of OC includes RL, TM, OHCs, and OHC bundles. Both ends of the BM are fixed constraints. The Deiters cells are considered as a rigid connection due to their high stiffness. The arch of OC composed of outer pillar cell and inner pillar cell is assumed to be a rigid structure. There are three OHC bundles on the RL. The bottom of the OHC bundles connects with the top of the OHC. The top of the OHC bundles is embedded in the TM to form a fixed constraint. The OHC bundles can deform with vibration. The part of TM near the cochlear pivot is a fixed connection. The model does not include the motion of TM because of its high quality and rigidity. The mechanical model of OC is shown in Figure 2.Figure 1
Cross-sectional anatomy of guinea pig [36]. IP, inner pillar cell; OP, outer pillar cell; DC, Deiters cell; BM, basilar membrane; OHCs, outer hair cells; RL, reticular lamina; TM, tectorial membrane.


### 2.2. 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.Figure 2
Micromechanical model of OC structure.Lpc is the distance between the left end and the right end of the arch structure, L0 is the distance between the bottom of the middle row of OHCs and the left end of the BM, ϕ is the acute angle between OHCs and z-axis, α is the acute angle between OHCs bundles and z-axis, α and ϕ have the same value, and β1=β2=β is the inside angle of the arch structure.The mode of BM vibrations is assumed to be sinusoidal vibration. When the calculation point isx and the time is t, the displacement of the BM of Cochlea section is given by(1)ubmx,y,t=Asinπy/b,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, andA represents the vibration amplitude of the BM.The arch structure is assumed to be rigid. When the BM is displaced, the arch structure will rotate in response. The displacement of the apex of the arch structure is given by(2)uoc=AsinπLpc/b2cosβ,In the formula,Lpc 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.The compression deformation of the OHCs depends on the relative displacement of the top and the bottom of the OHCs. The motion of the top of the OHCs in the middle row is given by(3)uohc−middlea=AsinπLpc/b2cosβ−cosβ−ϕ+cosβ−α,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.The displacement of the bottom of the middle row of the OHCs is(4)uohc−middleb=Acosϕ,The shear displacement of the OHC bundles in the middle row is(5)uhb−middle=AsinπLpc/b2cosβsinβ−ϕ.There was no significant difference in the displacement of three rows of OHCs. Their values are considered to be the same.


### 2.3. Analytical Solution of BM

The work done by external forces is mainly absorbed by the movement of BM, OHCs, and OHC bundles. When the calculation point isx and the time is t, the energy expression of OC components can be obtained according to the content of material mechanics.The BM is regarded as an isotropic plate [20], and the strain potential energy of the BM is given by(6)Ebm=∫0b12π∫02πD2∂2ubm∂x22+∂2ubm∂y22+2v∂2ubm∂x2∂2ubm∂y2+21−v∂2ubm∂x∂y2dθdy,where(7)D=E1hbmc2,c2=hbm2121−v2,The formula is used to analyze the section of the cochlea, and the integration formula of the strain energy of the BM is(8)Ebm=π42b3E1hbm3241−v2A2hohc,where E1 is the elastic modulus of the BM, ubm is the displacement of the BM, hbm is the thickness of the BM, hohc is the thickness of the OHCs, and v is Poisson’s ratio of the BM.The axial deformation of OHC depends on the relative displacement of the top and bottom of OHC. The strain energy generated by OHCs is(9)Eohc=3×12Fohcuohc,Fohc=uohcE2AohcLohc,and the integration formula of the strain energy of the OHCs is(10)Eohc=32×E2Aohcuohc2Lohc,uohc=AsinπLpc/b2cosβcosβ−α−cosβ−ϕ−Acosϕ,Eohc=32×E2AohcsinπLpc/b2cosβcosβ−α−cosβ−ϕ−cosϕ2A2Lohc,where E2 is the elastic modulus of OHCs, Aohc is the cross-sectional area of OHCs, Lohc is the length of OHCs, and uohc is the axial deformation of OHCs.The strain energy equations generated by OHC bundles are(11)Este=3×6E3Isteuste2/Lste3,Iste=Lstehste3/12,uste=uhb−middle=AsinπLpc/b2cosβsinβ−ϕ,and the integration formula of the strain energy of the OHC bundles is(12)Este=6E3hste3sin2πLpc/bcos2βsin2β−ϕA2Lste2,where E3 is the elastic modulus of the OHC bundles, hste is the thickness of the OHC bundles, Lste is the length of the OHC bundles, and uste is the shear displacement perpendicular to the axis of OHC bundles.The total kinetic energy generated by BM is given by(13)Tbm=12mbmvbm2=∫0b12ρbmhbmbwAsinπyb2dyhohc=b24ρbmhbmwA2hohc,where mbm is the mass of the BM, vbm is the velocity of the BM, ρbm is the density of the BM, and w is the angular frequency.The kinetic energy generated by OHCs is given by(14)Tohc=12mohcvohc2=3×12×ρohcLohchohcwu¯ohc2hohc,u¯ohc=12Acosϕ+AsinπLpc/b2cosβ−cosβ−ϕ+cosβ−α,and the integration formula of the kinetic energy of the OHCs is(15)Tohc=38ρohcLohchohc2cosϕ+2sinπLpc/bcosβ−cosβ−ϕ+cosβ−α2wA2,where ρohc is the density of OHCs, mohc is the mass of OHCs, u¯ohc is the average displacement of OHCs, and vohc is the speed of OHCs.The kinetic energy generated by OHC bundles is given by(16)Tste=12mstevste2=3×12ρsteLstehstewu¯ste2hste,u¯ste=12AsinπLpc/b2cosβsinβ−ϕ,and the integration formula of the kinetic energy of the OHC bundles is(17)Tste=32ρsteLstehste2sinπLpc/bcosβsinβ−ϕ2wA2,where mste is the mass of the OHC bundles, vste is the velocity of the OHC bundles, ρste is the density of the OHC bundles, and u¯ste is the average displacement of the OHC bundles.The energy equationWof external force P is given by(18)W=∫0bPsinwtAsinπybdyhohc=2bπPAsinwthohc,the structural strain energyV is given by(19)V=Ebm+Eohc+Este−W,and the total potential energy of OC is given by(20)U=V−T=Ebm+Eohc+Este−W−Tbm−Tohc−Tste,By substituting the expressions of each term into equation (20), the total potential energy can be obtained from the following equation:(21)U=32×E2AohcsinπLpc/b2cosβcosβ−α−cosβ−ϕ−cosϕ2A2Lohc+6E3hste3sin2πLpc/bcos2βsin2β−ϕA2Lste2+π42b3E1hbm3241−v2A2hohc−2bπPAsinwthohc−b24ρbmhbmw2hohcA2−32ρsteLstehste2sinπLpc/bcosβsinβ−ϕ2w2A2−−38ρohcLohchohc2cosϕ+2sinπLpc/bcosβ−cosβ−ϕ+cosβ−α2w2A2.The first-order variation ofA is(22)δU=0,When the calculation point isx and the time is t, the expression of BM amplitude is as follows:(23)A=2b/πPsinwthohcπ4E1hbm3hohc/24b31−v2+3E2AohcsinπLpc/b2cosβcosβ−α−cosβ−φ−cosφ2/Lohc+12E3hste3sin2πLpc/bcos2βsin2β−φ/Lste2−3ρsteLstehste2sinπLpc/bcosβsinβ−φ2w2−b2/2ρbmhbmw2hohc−3/4ρohcLohchohc2cosφ+2sinπLpc/bcosβ−cosβ−φ+cosβ−α2w2.IfA=A0sinwt, the expression of A0 is(24)A0=2b/πPhohcπ4E1hbm3hohc/24b31−v2+3E2AohcsinπLpc/b2cosβcosβ−α−cosβ−φ−cosφ2/Lohc+12E3hste3sin2πLpc/bcos2βsin2β−φ/Lste2−3ρsteLstehste2sinπLpc/bcosβsinβ−φ2w2−b2/2ρbmhbmw2hohc−3/4ρohcLohchohc2cosφ+2sinπLpc/bcosβ−cosβ−φ+cosβ−α2w2.Therefore,A0 is the maximum amplitude of the calculation point in the process of vibration.Sincew is the angular frequency, from w=2πf, we can get the maximum amplitude A0 expressed by frequency f.(25)A0=2b/πPhohcπ4E1hbm3hohc/24b31−v2+3E2AohcsinπLpc/b2cosβcosβ−α−cosβ−φ−cosφ2/Lohc+12E3hste3sin2πLpc/bcos2βsin2β−φ/Lste2−12π2ρsteLstehste2sinπLpc/bcosβsinβ−φ2f2−2b2π2ρbmhbmf2hohc−3π2ρohcLohchohc2cosφ+2sinπLpc/bcosβ−cosβ−φ+cosβ−α2f2.In the formula,P is the external force acting on the BM, which is replaced by the uniform distributed force. The pressure difference between the upper and lower parts of the BM decides the size of uniform pressure [37]. Moreover, the coefficient in equation (25) depends on the selected calculation point except for f.Therefore, when the calculation point on the BM is determined, equation (25) is a function that changes with frequency f. There is a frequency f that makes A0 get the maximum value, which is the CF of the solution point.


## 2.1. Establishment of a Mechanical Model

The mechanical model of OC was established based on the medical anatomical diagram of the guinea pig cochlea (Figure1) and the geometric relationship of each part in OC. The mechanical model of OC includes RL, TM, OHCs, and OHC bundles. Both ends of the BM are fixed constraints. The Deiters cells are considered as a rigid connection due to their high stiffness. The arch of OC composed of outer pillar cell and inner pillar cell is assumed to be a rigid structure. There are three OHC bundles on the RL. The bottom of the OHC bundles connects with the top of the OHC. The top of the OHC bundles is embedded in the TM to form a fixed constraint. The OHC bundles can deform with vibration. The part of TM near the cochlear pivot is a fixed connection. The model does not include the motion of TM because of its high quality and rigidity. The mechanical model of OC is shown in Figure 2.Figure 1
Cross-sectional anatomy of guinea pig [36]. IP, inner pillar cell; OP, outer pillar cell; DC, Deiters cell; BM, basilar membrane; OHCs, outer hair cells; RL, reticular lamina; TM, tectorial membrane.


## 2.2. 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.Figure 2
Micromechanical model of OC structure.Lpc is the distance between the left end and the right end of the arch structure, L0 is the distance between the bottom of the middle row of OHCs and the left end of the BM, ϕ is the acute angle between OHCs and z-axis, α is the acute angle between OHCs bundles and z-axis, α and ϕ have the same value, and β1=β2=β is the inside angle of the arch structure.The mode of BM vibrations is assumed to be sinusoidal vibration. When the calculation point isx and the time is t, the displacement of the BM of Cochlea section is given by(1)ubmx,y,t=Asinπy/b,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, andA represents the vibration amplitude of the BM.The arch structure is assumed to be rigid. When the BM is displaced, the arch structure will rotate in response. The displacement of the apex of the arch structure is given by(2)uoc=AsinπLpc/b2cosβ,In the formula,Lpc 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.The compression deformation of the OHCs depends on the relative displacement of the top and the bottom of the OHCs. The motion of the top of the OHCs in the middle row is given by(3)uohc−middlea=AsinπLpc/b2cosβ−cosβ−ϕ+cosβ−α,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.The displacement of the bottom of the middle row of the OHCs is(4)uohc−middleb=Acosϕ,The shear displacement of the OHC bundles in the middle row is(5)uhb−middle=AsinπLpc/b2cosβsinβ−ϕ.There was no significant difference in the displacement of three rows of OHCs. Their values are considered to be the same.


## 2.3. Analytical Solution of BM

The work done by external forces is mainly absorbed by the movement of BM, OHCs, and OHC bundles. When the calculation point isx and the time is t, the energy expression of OC components can be obtained according to the content of material mechanics.The BM is regarded as an isotropic plate [20], and the strain potential energy of the BM is given by(6)Ebm=∫0b12π∫02πD2∂2ubm∂x22+∂2ubm∂y22+2v∂2ubm∂x2∂2ubm∂y2+21−v∂2ubm∂x∂y2dθdy,where(7)D=E1hbmc2,c2=hbm2121−v2,The formula is used to analyze the section of the cochlea, and the integration formula of the strain energy of the BM is(8)Ebm=π42b3E1hbm3241−v2A2hohc,where E1 is the elastic modulus of the BM, ubm is the displacement of the BM, hbm is the thickness of the BM, hohc is the thickness of the OHCs, and v is Poisson’s ratio of the BM.The axial deformation of OHC depends on the relative displacement of the top and bottom of OHC. The strain energy generated by OHCs is(9)Eohc=3×12Fohcuohc,Fohc=uohcE2AohcLohc,and the integration formula of the strain energy of the OHCs is(10)Eohc=32×E2Aohcuohc2Lohc,uohc=AsinπLpc/b2cosβcosβ−α−cosβ−ϕ−Acosϕ,Eohc=32×E2AohcsinπLpc/b2cosβcosβ−α−cosβ−ϕ−cosϕ2A2Lohc,where E2 is the elastic modulus of OHCs, Aohc is the cross-sectional area of OHCs, Lohc is the length of OHCs, and uohc is the axial deformation of OHCs.The strain energy equations generated by OHC bundles are(11)Este=3×6E3Isteuste2/Lste3,Iste=Lstehste3/12,uste=uhb−middle=AsinπLpc/b2cosβsinβ−ϕ,and the integration formula of the strain energy of the OHC bundles is(12)Este=6E3hste3sin2πLpc/bcos2βsin2β−ϕA2Lste2,where E3 is the elastic modulus of the OHC bundles, hste is the thickness of the OHC bundles, Lste is the length of the OHC bundles, and uste is the shear displacement perpendicular to the axis of OHC bundles.The total kinetic energy generated by BM is given by(13)Tbm=12mbmvbm2=∫0b12ρbmhbmbwAsinπyb2dyhohc=b24ρbmhbmwA2hohc,where mbm is the mass of the BM, vbm is the velocity of the BM, ρbm is the density of the BM, and w is the angular frequency.The kinetic energy generated by OHCs is given by(14)Tohc=12mohcvohc2=3×12×ρohcLohchohcwu¯ohc2hohc,u¯ohc=12Acosϕ+AsinπLpc/b2cosβ−cosβ−ϕ+cosβ−α,and the integration formula of the kinetic energy of the OHCs is(15)Tohc=38ρohcLohchohc2cosϕ+2sinπLpc/bcosβ−cosβ−ϕ+cosβ−α2wA2,where ρohc is the density of OHCs, mohc is the mass of OHCs, u¯ohc is the average displacement of OHCs, and vohc is the speed of OHCs.The kinetic energy generated by OHC bundles is given by(16)Tste=12mstevste2=3×12ρsteLstehstewu¯ste2hste,u¯ste=12AsinπLpc/b2cosβsinβ−ϕ,and the integration formula of the kinetic energy of the OHC bundles is(17)Tste=32ρsteLstehste2sinπLpc/bcosβsinβ−ϕ2wA2,where mste is the mass of the OHC bundles, vste is the velocity of the OHC bundles, ρste is the density of the OHC bundles, and u¯ste is the average displacement of the OHC bundles.The energy equationWof external force P is given by(18)W=∫0bPsinwtAsinπybdyhohc=2bπPAsinwthohc,the structural strain energyV is given by(19)V=Ebm+Eohc+Este−W,and the total potential energy of OC is given by(20)U=V−T=Ebm+Eohc+Este−W−Tbm−Tohc−Tste,By substituting the expressions of each term into equation (20), the total potential energy can be obtained from the following equation:(21)U=32×E2AohcsinπLpc/b2cosβcosβ−α−cosβ−ϕ−cosϕ2A2Lohc+6E3hste3sin2πLpc/bcos2βsin2β−ϕA2Lste2+π42b3E1hbm3241−v2A2hohc−2bπPAsinwthohc−b24ρbmhbmw2hohcA2−32ρsteLstehste2sinπLpc/bcosβsinβ−ϕ2w2A2−−38ρohcLohchohc2cosϕ+2sinπLpc/bcosβ−cosβ−ϕ+cosβ−α2w2A2.The first-order variation ofA is(22)δU=0,When the calculation point isx and the time is t, the expression of BM amplitude is as follows:(23)A=2b/πPsinwthohcπ4E1hbm3hohc/24b31−v2+3E2AohcsinπLpc/b2cosβcosβ−α−cosβ−φ−cosφ2/Lohc+12E3hste3sin2πLpc/bcos2βsin2β−φ/Lste2−3ρsteLstehste2sinπLpc/bcosβsinβ−φ2w2−b2/2ρbmhbmw2hohc−3/4ρohcLohchohc2cosφ+2sinπLpc/bcosβ−cosβ−φ+cosβ−α2w2.IfA=A0sinwt, the expression of A0 is(24)A0=2b/πPhohcπ4E1hbm3hohc/24b31−v2+3E2AohcsinπLpc/b2cosβcosβ−α−cosβ−φ−cosφ2/Lohc+12E3hste3sin2πLpc/bcos2βsin2β−φ/Lste2−3ρsteLstehste2sinπLpc/bcosβsinβ−φ2w2−b2/2ρbmhbmw2hohc−3/4ρohcLohchohc2cosφ+2sinπLpc/bcosβ−cosβ−φ+cosβ−α2w2.Therefore,A0 is the maximum amplitude of the calculation point in the process of vibration.Sincew is the angular frequency, from w=2πf, we can get the maximum amplitude A0 expressed by frequency f.(25)A0=2b/πPhohcπ4E1hbm3hohc/24b31−v2+3E2AohcsinπLpc/b2cosβcosβ−α−cosβ−φ−cosφ2/Lohc+12E3hste3sin2πLpc/bcos2βsin2β−φ/Lste2−12π2ρsteLstehste2sinπLpc/bcosβsinβ−φ2f2−2b2π2ρbmhbmf2hohc−3π2ρohcLohchohc2cosφ+2sinπLpc/bcosβ−cosβ−φ+cosβ−α2f2.In the formula,P is the external force acting on the BM, which is replaced by the uniform distributed force. The pressure difference between the upper and lower parts of the BM decides the size of uniform pressure [37]. Moreover, the coefficient in equation (25) depends on the selected calculation point except for f.Therefore, when the calculation point on the BM is determined, equation (25) is a function that changes with frequency f. There is a frequency f that makes A0 get the maximum value, which is the CF of the solution point.


## 3. Verification of Model

### 3.1. 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 frequency-position relationship. The tonotopic map satisfies the following formula:(26)CF=M10αx0−N,where CF is characteristic frequency. The 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. The parameter of the expression is mainly from [39]. The value of α is 2.1 and it has small change when choosing different lengths of BM. The value of the parameter N is generally 0.85. M is a constant that varies with the scope of CF. The value of gerbil is 0.4, the guinea pig is 0.35, and the cat is 0.456. x0 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.


### 3.2. Model Parameters

The model was validated with guinea pig cochlea data.


#### 3.2.1. Geometric Parameters

The geometric parameters of each part of the OC are listed in Table1. The relevant parameters are from [6–10].Table 1
Geometric data of each part of OC.
ParameterParameter descriptionParameter valuebμmThe width of BM168 (base) ∼ 200 (apex)LpcThe distance between the left end and the right end of the arch structureb/3L0The distance between the bottom of the middle row of OHCs and the left end of the BMb/2αThe acute angle between OHCs bundles andz-axis25° (base) ∼ 45° (apex)β1,β2The acute angle of the arch structureβ1=β2=β=60°ϕThe acute angle between OHCs andz-axis25° (base) ∼ 45° (apex)LohcμmThe lengths of OHCs70 (base) ∼ 110 (apex)LsteμmThe lengths of OHC bundles5hbmμmThe thickness of BM7 (base) ∼ 1 (apex)hohcμmThe thickness of OHCs4.5hsteμmThe thickness of OHC bundles0.2According to the presumption that the structural parametersb, Lohc, hbm, α, and ϕ change linearly along the length of BM, the general form is as follows:(27)Y=cx+d,where Y is b, Lohc, hbm, α, and ϕ in Table 1 and x is the distance from the selection point to the bottom of the BM. c and d are the coefficients decided by the geometric parameters of the bottom and the top of the cochlea. The corresponding expressions are given by the following:The width of BM:(28)Lbm=6.8571x+62.8573,the thickness of BM:(29)hbm=−0.3429x+7.8572,the length of OHCs:(30)Lohc=2.2857x+64.2858,and the acute angle:(31)α=ϕ=1.1429x+22.142.


#### 3.2.2. Mechanical Properties of Materials

The material properties of the BM, OHCs, and OHC bundles in OC are shown in Table2. The relevant parameters are from [10–12, 28].Table 2
Material properties of the microstructure of OC.
ParameterParameter descriptionParameter valueE1paThe elastic modulus of BM2×108 (base) ∼ 1×107 (apex)E2paThe elastic modulus of OHCs4.5×103 (base) ∼ 1×103 (apex)E3paThe elastic modulus of OHC bundles1×107 (base) ∼ 1×106 (apex)ρbmkg/m3The density of BM1200ρohckg/m3The density of OHCs1050ρstekg/m3The density of OHC bundles1200vPoisson’s ratio of BM0.3The parametersE1, E2, and E3 are the elastic moduli of BM, OHCs, and OHC bundles, respectively, which are assumed to change exponentially along the length of BM. The general form can be written as(32)E=MeNx,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.The data of the base and apex of BM, OHCs, and OHC bundles are substituted into the formula as boundary conditions. The elastic modulus function of each part of OC is given by(33)E1=3.0663×108e−0.1712x,E2=5.5775×103e−0.086x,E3=1.3889×107e−0.1316x.


### 3.3. Validation of the Model

The 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(34)CF=0.35102.1x0−0.85After determining the calculation point on the BM, the experimental value ofCF can be obtained by equation (34), which is compared with the analytical value achieved from equation (25), as shown in Figure 3.Figure 3
Frequency-position relationship of BM.Figure3 illustrates that the calculated analytical value is slightly larger than the experimental value. The slope of the frequency-place map of analytical value is similar to the measured experimental data. The 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.Figure4 shows the frequency response curve of the calculation point. The 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.Figure 4
Frequency response graph of corresponding points.


## 3.1. 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 frequency-position relationship. The tonotopic map satisfies the following formula:(26)CF=M10αx0−N,where CF is characteristic frequency. The 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. The parameter of the expression is mainly from [39]. The value of α is 2.1 and it has small change when choosing different lengths of BM. The value of the parameter N is generally 0.85. M is a constant that varies with the scope of CF. The value of gerbil is 0.4, the guinea pig is 0.35, and the cat is 0.456. x0 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.


## 3.2. Model Parameters

The model was validated with guinea pig cochlea data.


### 3.2.1. Geometric Parameters

The geometric parameters of each part of the OC are listed in Table1. The relevant parameters are from [6–10].Table 1
Geometric data of each part of OC.
ParameterParameter descriptionParameter valuebμmThe width of BM168 (base) ∼ 200 (apex)LpcThe distance between the left end and the right end of the arch structureb/3L0The distance between the bottom of the middle row of OHCs and the left end of the BMb/2αThe acute angle between OHCs bundles andz-axis25° (base) ∼ 45° (apex)β1,β2The acute angle of the arch structureβ1=β2=β=60°ϕThe acute angle between OHCs andz-axis25° (base) ∼ 45° (apex)LohcμmThe lengths of OHCs70 (base) ∼ 110 (apex)LsteμmThe lengths of OHC bundles5hbmμmThe thickness of BM7 (base) ∼ 1 (apex)hohcμmThe thickness of OHCs4.5hsteμmThe thickness of OHC bundles0.2According to the presumption that the structural parametersb, Lohc, hbm, α, and ϕ change linearly along the length of BM, the general form is as follows:(27)Y=cx+d,where Y is b, Lohc, hbm, α, and ϕ in Table 1 and x is the distance from the selection point to the bottom of the BM. c and d are the coefficients decided by the geometric parameters of the bottom and the top of the cochlea. The corresponding expressions are given by the following:The width of BM:(28)Lbm=6.8571x+62.8573,the thickness of BM:(29)hbm=−0.3429x+7.8572,the length of OHCs:(30)Lohc=2.2857x+64.2858,and the acute angle:(31)α=ϕ=1.1429x+22.142.


### 3.2.2. Mechanical Properties of Materials

The material properties of the BM, OHCs, and OHC bundles in OC are shown in Table2. The relevant parameters are from [10–12, 28].Table 2
Material properties of the microstructure of OC.
ParameterParameter descriptionParameter valueE1paThe elastic modulus of BM2×108 (base) ∼ 1×107 (apex)E2paThe elastic modulus of OHCs4.5×103 (base) ∼ 1×103 (apex)E3paThe elastic modulus of OHC bundles1×107 (base) ∼ 1×106 (apex)ρbmkg/m3The density of BM1200ρohckg/m3The density of OHCs1050ρstekg/m3The density of OHC bundles1200vPoisson’s ratio of BM0.3The parametersE1, E2, and E3 are the elastic moduli of BM, OHCs, and OHC bundles, respectively, which are assumed to change exponentially along the length of BM. The general form can be written as(32)E=MeNx,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.The data of the base and apex of BM, OHCs, and OHC bundles are substituted into the formula as boundary conditions. The elastic modulus function of each part of OC is given by(33)E1=3.0663×108e−0.1712x,E2=5.5775×103e−0.086x,E3=1.3889×107e−0.1316x.


## 3.2.1. Geometric Parameters

The geometric parameters of each part of the OC are listed in Table1. The relevant parameters are from [6–10].Table 1
Geometric data of each part of OC.
ParameterParameter descriptionParameter valuebμmThe width of BM168 (base) ∼ 200 (apex)LpcThe distance between the left end and the right end of the arch structureb/3L0The distance between the bottom of the middle row of OHCs and the left end of the BMb/2αThe acute angle between OHCs bundles andz-axis25° (base) ∼ 45° (apex)β1,β2The acute angle of the arch structureβ1=β2=β=60°ϕThe acute angle between OHCs andz-axis25° (base) ∼ 45° (apex)LohcμmThe lengths of OHCs70 (base) ∼ 110 (apex)LsteμmThe lengths of OHC bundles5hbmμmThe thickness of BM7 (base) ∼ 1 (apex)hohcμmThe thickness of OHCs4.5hsteμmThe thickness of OHC bundles0.2According to the presumption that the structural parametersb, Lohc, hbm, α, and ϕ change linearly along the length of BM, the general form is as follows:(27)Y=cx+d,where Y is b, Lohc, hbm, α, and ϕ in Table 1 and x is the distance from the selection point to the bottom of the BM. c and d are the coefficients decided by the geometric parameters of the bottom and the top of the cochlea. The corresponding expressions are given by the following:The width of BM:(28)Lbm=6.8571x+62.8573,the thickness of BM:(29)hbm=−0.3429x+7.8572,the length of OHCs:(30)Lohc=2.2857x+64.2858,and the acute angle:(31)α=ϕ=1.1429x+22.142.


## 3.2.2. Mechanical Properties of Materials

The material properties of the BM, OHCs, and OHC bundles in OC are shown in Table2. The relevant parameters are from [10–12, 28].Table 2
Material properties of the microstructure of OC.
ParameterParameter descriptionParameter valueE1paThe elastic modulus of BM2×108 (base) ∼ 1×107 (apex)E2paThe elastic modulus of OHCs4.5×103 (base) ∼ 1×103 (apex)E3paThe elastic modulus of OHC bundles1×107 (base) ∼ 1×106 (apex)ρbmkg/m3The density of BM1200ρohckg/m3The density of OHCs1050ρstekg/m3The density of OHC bundles1200vPoisson’s ratio of BM0.3The parametersE1, E2, and E3 are the elastic moduli of BM, OHCs, and OHC bundles, respectively, which are assumed to change exponentially along the length of BM. The general form can be written as(32)E=MeNx,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.The data of the base and apex of BM, OHCs, and OHC bundles are substituted into the formula as boundary conditions. The elastic modulus function of each part of OC is given by(33)E1=3.0663×108e−0.1712x,E2=5.5775×103e−0.086x,E3=1.3889×107e−0.1316x.


## 3.3. Validation of the Model

The 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(34)CF=0.35102.1x0−0.85After determining the calculation point on the BM, the experimental value ofCF can be obtained by equation (34), which is compared with the analytical value achieved from equation (25), as shown in Figure 3.Figure 3
Frequency-position relationship of BM.Figure3 illustrates that the calculated analytical value is slightly larger than the experimental value. The slope of the frequency-place map of analytical value is similar to the measured experimental data. The 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.Figure4 shows the frequency response curve of the calculation point. The 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.Figure 4
Frequency response graph of corresponding points.


## 4. Parametric Study

### 4.1. The 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. The frequency response curve is plotted in Figure5. 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.Figure 5
The effect of the lack of RL.The reason for these effects is that the stiffness of RL is much greater than that of BM. The 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. Therefore, the RL acts as an effective component in OC to enhance the movement of BM.


### 4.2. The 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. The 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. The results are presented in Figure 6.Figure 6
The elastic modulus of OHC bundles reducing by half.Figure6 shows that the vibration response curve moves towards the low-frequency direction. The 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.


### 4.3. The Effect of OHCs Stiffness

To explore the role of the stiffness of OHCs, the stiffness of OHCs is reduced by half.Figure7 shows that the reduction of OHCs stiffness affects amplitude in different frequency segments. The 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.Figure 7
The elastic modulus of OHCs reducing by half.


### 4.4. The Effect of TM

To assess the impact of TM, establishing a model includes a TM that is exfoliated from the epithelium. Therefore, the analytical expression will not consider the potential energy of the OHC bundles. The results are plotted in Figure8.Figure 8
The effect of the lack of TM.Figure8 presents that the decrease of CF is more than that of OHC bundles. This 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.


## 4.1. The 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. The frequency response curve is plotted in Figure5. 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.Figure 5
The effect of the lack of RL.The reason for these effects is that the stiffness of RL is much greater than that of BM. The 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. Therefore, the RL acts as an effective component in OC to enhance the movement of BM.


## 4.2. The 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. The 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. The results are presented in Figure 6.Figure 6
The elastic modulus of OHC bundles reducing by half.Figure6 shows that the vibration response curve moves towards the low-frequency direction. The 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.


## 4.3. The Effect of OHCs Stiffness

To explore the role of the stiffness of OHCs, the stiffness of OHCs is reduced by half.Figure7 shows that the reduction of OHCs stiffness affects amplitude in different frequency segments. The 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.Figure 7
The elastic modulus of OHCs reducing by half.


## 4.4. The Effect of TM

To assess the impact of TM, establishing a model includes a TM that is exfoliated from the epithelium. Therefore, the analytical expression will not consider the potential energy of the OHC bundles. The results are plotted in Figure8.Figure 8
The effect of the lack of TM.Figure8 presents that the decrease of CF is more than that of OHC bundles. This 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.


## 5. Conclusions

Based on the interaction between BM, OHCs, RL, TM, and so on, a mechanical model of OC is established. The 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 thatCF of the calculation point seriously deviates from the experimental value when not considering the role of the RL. The amplitude of the frequency-amplitude response curve of the BM decreased significantly, which will cause hearing loss. The RL is an indispensable structure in the process of the sound-sensing mechanism of the cochlea.The reduction of the stiffness of OHC bundles and OHCs will shift vibratory response patterns to lower frequencies. The absence of TM makesCF decrease seriously. These 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. Thus, TM plays an essential role in the process of the sense of sound.The bottom of the cochlea resonates at high frequencies, and the top resonates at low frequencies. Different points of BM correspond to differentCF, 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. Therefore, it can provide theoretical support for relevant experiments and medical research.



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*Source: 8856159-2020-12-09.xml*