1 Introduction

Sound can be transmitted into external auditory canal by the sound waves affecting the vibrations of the eardrum and ossicular chain and spreading to the basilar membrane (BM) through the cochlea lymph fluctuations. The vibration of a basilar membrane drives the shear movement between the tectorial membrane (TM) and the reticular lamina (RL) in the organ of Corti (OC) ^[1], affecting the Couette flow in the endolymph ^[2] and making the inner stereocilia bending.

A stereocilium consists of cross-linked actin filaments, intracellular actin cytoskeleton, and rootlet ^[3-8]. Rootlet is the core of a stereocilium, and is thought to be responsible for its stiffness. When the bundle is deflected in the excitatory direction, e.g., toward the tallest stereocilium, the tip link will be under tension, and the mechanoelectrical transduction (MET) channel located at the end of the tip link will be thought to be open ^[9], which transforms the mechanical movement into the neural pulses signaled to the brain in the end.

Zetes and Steel ^[10] established a multiple degree of freedom linear motion equation based on the fluid-solid interaction of stereocilia, and found stereocilia with different frequencies. Duncan and Grant ^[11] established a finite element (FE) model of a single stereocilium, and described the relationship between the stiffness and the elastic modulus of the stereocilium. Eiichi Ishiyama and Koichi Ishiyama ^[12] established a three-dimensional model to study the effects of the bending between the stereocilia on the stiffness of the stereocilia. Matsui et al. ^[2] proposed a three-dimensional model of stereocilia to study the dynamic characteristics of stereocilia, and found that the lateral links protected the MET channels. However, these studies only consider the effects of the fluid made by the shear motion of the tectorial membrane and a reticular lamina on the stereocilia, and the force generated by the active or passive hair cells and the longitudinal fluid pressure of the cochlear duct are seldom considered.

Therefore, in this paper, a three-dimensional model of stereocilia based on the experimental data of guinea pig is established. The effects made by the force generated by the active or passive hair cells and the longitudinal fluid pressure of the cochlear duct on the x-displacements of the inner stereocilia (the displacements of the inner stereocilia along the cross section of the basilar membrane) are studied.

2 Model 2.1 Geometry

To study the displacements of the stereocilia along the cross section of the basilar membrane, based on the experimental data of guinea pig ^[12-17], a three-dimensional simplified stereocilium model is established by PATRAN (see Fig. 1). The inner stereocilia at the basal turn of the guinea pig cochlea is located at 17 mm from the base. The tip link is constructed between the center of the shorter stereocilium and the lateral wall of the next taller one at an angle of 45 degrees with the length 283 nm. The length of the lateral link is 50 nm, and is set as the spring element. The length between the stereocilia is about 200 nm ^[17]. The other geometric data of the stereocilia are described in Table 1. The tip link length is 283 nm, and the lateral link length is 50 nm. The boundary condition at the bottom of the model is fixed. The fluid force f expressed in Subsection 2.3 is applied on one side of the stereocilia.

The FE mesh model is created in the software PATRAN. The rootlet is meshed by 360 eight-noded hexahedra with 693 nodes. The stereocilia are meshed by 1 890 eight-noded hexahedra with 2 910 nodes. The tip link and lateral link are spring elements (see Fig. 2).

2.2 Material properties

The mechanical properties in the stereocilium model include Young's moduli and densities of the rootlet and intracellular actin cytoskeleton. According to the relevant data made by Matsui et al. ^[2], Young's moduli of the rootlet and intracellular actin cytoskeleton are 1.3× 10⁹ Pa and 8.6× 10⁷ Pa, respectively, the densities of the rootlet and intracellular actin cytoskeleton are 1.2× 10³ kg/m³, and Poisson's ratios of the rootlet and intracellular actin cytoskeleton are 0.4. The spring constants of the tip and lateral links are 5.3× 10^-4 N/m and 2.5× 10^-3 N/m, respectively ^[2].

2.3 Fluid force exerted on stereocilia caused by endolymph flow

When the BM vibrates vertically, the shearing motion of the TM against the RL occurs. Assume that the Couette flow exists between the TM and the RL caused by the shearing motion. Then, the fluid velocity for the Couette flow V is

(1)

where z is the axis perpendicular to the RL, H is the distance between the TM and the RL, and V_TM is the velocity of the TM relative to the RL. According to the relevant literature ^[18], H=7.3 μm. Based on a previously reported OC model ^[19], V_TM is determined to be 1.3× 10^-5 m/s by the stimuli of 60 dB at 500 Hz. Based on the study of Tomotika and Aoi ^[20], the fluid force F exerted on the cylindrical component is

(2)

where h is the height of the cylindrical component, which is 200 nm, and μ is the viscosity coefficient of the endolymph, which is assumed to be that of water, i.e., 0.659× 10^-3 Pa·s. The variable S is

(3)

where γ=0.57721 is the Euler number, and Re is the Reynolds number defined by

(4)

In the above equation, d is the diameter of the cylindrical component and varies with z from the base diameter of the stereocilium to that of its shaft, ranging from 0.12 μm to 0.3 μm (see Table 1). ρ is the density of the endolymph, which is set to be that of water, i.e., 1× 10³ kg/m³. In order to analyze the effect of the fluid pressure on the structure qualitatively, the force F resulted from z is mapped into 0~1. Moreover,

The force is applied to the stereocilium model to analyze the obtained results (see Fig. 3).

2.4 Motion equation of stereocilia

The motion equation of the stereocilia is represented by the following matrix differential equation:

(5)

where M, C, and K are the mass, damping, and stiffness matrices, respectively. u is the displacement vector, F is the force, and t is the time.

2.5 Mechanical stereocilium model

Castigliano's theorem on deflections is a method based on the concept of complementary energy, and is expressed in terms of the total strain energy. The method is assumed to be used in single stereocilia. The total strain energy is defined as the integral of the strain energy density over the structure volume (see Fig. 4).

The strain energy U in the pure bending case is

(6)

The internal bending moment M at any height y is

(7)

Castigliano's theorem from the material mechanics and the relevant literature ^[21] gives the deflection x at the top of the beam (y=L) as follows:

(8)

Therefore, the deflection can be considered to be the deflection sum due to bending and shear. The deflection due to bending is

(9)

where L₁ is the height of the tapered base region, L is the height of the total stereocilia, and I₁ and I₂ represent the inertia area moments for the tapered and uniform regions, respectively. The inertia area moments are

(10)

(11)

3 Results 3.1 Verification of FE model and mechanical model

x-displacement of each inner stereocilium is achieved. The x-displacement of the tall stereocilium is set as the standard, the ratio of the x-displacement difference between each inner stereocilium and the tall stereocilium to the x-displacement of the tall stereocilium is plotted in Fig. 5. From the figure, we can see that the ratio obtained from the FE model is similar to that obtained by Matsui et al. ^[2], which verifies the correctness of the mode.

Excitation is applied to one side of the inner stereocilia, and the tip links and lateral tips are neglected. Then, the top x-displacement of each inner stereocilium can be achieved from Eq. (9). The x-displacement of the tall stereocilium is set as the standard, the ratio of the x-displacement difference between each inner stereocilium and the tall stereocilium to the x-displacement of the tall stereocilium is plotted in Fig. 6. As shown in the figure, the theoretical results are close to that derived from the FE model with a max difference at the middle stereocilium, which confirms the validity of the formula.

3.2 Effects of vertical excitation (z-direction) on x-displacements of stereocilia

When the vibrations of the Corti organ are affected by the vibration of the basilar membrane, the pressure will be affected not only by the Couette flow but also by the vertical force (the z-direction that perpendicular to the RL) made by the active or passive motion in the hair cells. To study the effect of the vertical excitation (z-direction) on the x-displacements of the stereocilia, the vertical force is set to be 1 time, 5 times, 10 times, 100 times, 500 times, and 1 000 times of the lateral force F_max (x-direction), respectively.

Figure 7 shows the effects of the vertical pressure (z-direction) made by the hair cells and the pressure made by the Couette flow on the x-displacements of the stereocilia. The vertical axis represents the ratio of the x-displacement under two kinds of pressure to that under the pressure affected by the Couette flow. It is seen that the difference in the inner stereocilia is small. To better understand the small difference in the inner stereocilia, we set the horizontal coordinate as the logarithmic coordinate. It can be seen from the figure that, when the inner stereocilia are subjected to the pressure from the hair cells, the x-displacement of the middle stereocilium increases, while the x-displacements of the tallest and shortest stereocilia decrease. However, the trend is similar. When the pressure increases, the effect on the shortest stereocilium is the greatest. Moreover, the changing rate of the shortest stereocilium is the largest, the changing rate of the middle stereocilium is the smallest, and the changing rate of the the tallest stereocilium is between those of the shortest stereocilium and the middle stereocilium.

Figure 8 shows the effects of the vertical tension (the negative direction of the z-direction) made by the hair cells and the pressure made by the Couette flow on the x-displacements of the stereocilia. It can be seen from the figure that, when the inner stereocilia are subjected to the tension from the hair cells, the x-displacement of the middle stereocilium decreases, while the x-displacements of the tallest and shortest stereocilia increase. When the tension increases, the changing rate of the shortest stereocilium is the largest. Besides, the whole tension trend is similar to that of the pressure, but with opposite values.

Figures 9-11 show the effects of each stereocilium on the x-displacement between the tension and the pressure. From the figures, we can see that the changing trend of the middle stereocilium is opposite to that of the tallest and the shortest stereocilia. In general, the effects on the x-displacements of the inner stereocilia under tension are greater than those under pressure, but the differences are small.

3.3 Effects of flow pressure (y-direction) on x-displacements of stereocilia

Since the inner stereocilia are within the cochlear duct, the longitudinal motion of the cochlear perilymph may also have an effect on the x-displacements of the inner stereocilia. Therefore, the y-force caused by the flow pressure along the cochlear duct is set to be 0.1 time, 0.5 time, 1 time, and 10 times of the lateral force F_max, respectively.

Figure 12 shows the effects of the inner stereocilia on the x-displacements under the flow pressure (the y-direction). As can be seen from the figure, when the y-force increases, the x-displacements of the inner stereocilia increase, but the overall trend is similar. Moreover, the effects of the y-force on the middle stereocilium is the greatest. This may be because that the tip links and the lateral links between the inner stereocilia do not work.

4 Discussion

Figures 8 and 9 show the effects when the stereocilia are affected not only by the shear motion of the Couette flow pressure but also by the active or passive motion of the hair cells on the x-displacements. From the figures, we can see that, the effects of the inner stereocilia on the x-displacements under the pressure and the tension are opposite, but the overall trend is similar. When the stress ratio is 1 000, the difference between the maximum inner stereocilium and the minimum inner stereocilium is 0.009 545 706 under the pressure and 0.009 545 916 under the tension, which are small. The y-force caused by the hair cells has a great effect on the shortest stereocilium. Moreover, the movement of the middle stereocilium is always opposite to that of the tallest and shortest stereocilia, which may be because of the existence of the tip links and the lateral links.

Figures 9-11 show that, for each stereocilium, the effects of the pressure and tension are similar, and only the directions are the opposite. In general, the effects on the x-displacements under tension are slightly larger than those under pressure, but the differences are small.

From Fig. 7 and Fig. 12, we can see that, the z-direction force will make the movement directions of the inner stereocilia different, while the y-direction force will make the movement directions of the inner stereocilia consistent. This may be because that, when the z-direction force is used, the tip links and the lateral links between the inner stereocilia do not work.

The effects on the x-displacements of the inner stereocilia will affect the shear movement between the TM and the RL, which will have an effect on the sensing of the cochlea in the end.

Table 2 lists the results of 10 times of F_max obtained by the y-force and the z-force. From Table 2, we can see that, the effects of the y-force on the inner stereocilia are larger than those of the z-force. Moreover, the difference increases with the increase in the pressure. It can also be seen from the table that when it is subjected to different force directions, the middle stereocilium is affected the most, while the shortest stereocilium is affected the least.

5 Conclusions

In this paper, a three-dimensional FE model of inner stereocilia is established by PATRAN and calculated by NASTRAN. The results achieved from the FE model are compared with the relevant data in Ref. [2], and the good agreement verifies the correctness of the model. Based on Castigliano's theorem, the displacements of single stereocilia (the x-direction) are achieved. The results are in good agreement with the data obtained from the FE model, which confirms the validity of the formula. Based on the model, the effects of the hair cells and fluid on the x-displacements of the stereocilia are studied. The conclusions are as follows:

(ⅰ) The pressure (tension) of the z-direction applied to the stereocilium has an effect on the x-displacements of the inner stereocilia. When the pressure increases, the x-displacement of the middle stereocilium increases, while the x-displacements of the tallest and shortest stereocilia decrease. It is found that the changing ratio of the shortest stereocilium is the largest, the changing ratio of the tallest stereocilium is between those of the shortest stereocilium and the middle stereocilium, and the changing ratio of the middle stereocilium is the smallest. Moreover, the effect on the x-displacement under tension is slightly larger than that under pressure, but the difference is small.

(ⅱ) The y-direction force applied to the stereocilia will increase the x-displacements of the inner stereocilia, and the middle stereocilium increases the most.

(ⅲ) Inner stereocilia are more susceptible to the y-direction force than to the forces in other directions. When the forces come from different directions, the effect on the middle stereocilium is the largest.