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Analysis of the Dynamic Behavior of the Inner Hair Cell Stereocilia by the Finite Element Method∗

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Analysis of the Dynamic Behavior of the Inner Hair Cell Stereocilia by the Finite Element Method∗ 828 Analysis of the Dynamic Behavior of the Inner Hair Cell Stereocilia by the Finite Element Method∗ Toshihiro MATSUI∗∗, Chihiro NAKAJIMA∗∗, Yuichi YAMAMOTO∗∗, Masayoshi ANDOH∗∗, Koji IIDA∗∗, Michio MURAKOSHI∗∗, Shun KUMANO∗∗ and Hiroshi WADA∗∗ The motion of the inner hair cell (IHC) stereocilia, which results in tension in the tip links connected to mechanically gated ion channels, mediates the auditory transduction proc- ess. However, it is difficult to directly observe the motion of the stereocilia because of their minute dimensions and complex structure. In this study, to investigate such motion, a fi- nite element method model of the tall, middle and short IHC stereocilia, including the tip and lateral links extending between the stereocilia, was constructed. By applying an analyt- ically estimated fluid force caused by a stimulus of 60 dB SPL at 500 Hz to the model, the dynamic behavior of the stereocilia was analyzed. Numerical results showed that the stere- ocilia moved in phase and that the maximum tensions of 2.5 fN and 2.1 fN occurred in the tip link connecting the tall and middle stereocilia and in the tip link connecting the middle and short stereocilia, respectively. By contrast, under the condition in which the lateral links were removed, maximum tension in the former increased to 11.6 fN, while that in the latter only increased to 2.3 fN. It was therefore suggested that the lateral links protect the MET channels located at taller stereocilia against large stimuli and subject the channels located in the same IHC to forces of similar size. Key Words: Acoustic, Biomechanics, Finite Element Method, Sound, Cochlea, Hair Cell, Stereocilia Figure 3 shows a cross section of the OC. The OC 1. Introduction contains an inner hair cell (IHC) which operates as a sen- The mammalian auditory system is divided into the sory receptor of the auditory system. When the OC under- external, middle and inner ears. Figure 1 shows a goes vibration, shearing motion of the tectorial membrane schematic of the auditory system of the guinea pig and its relative position within the head. The external ear receives sound waves and transmits them to the middle ear through the external auditory meatus. In the middle ear, the sound waves vibrate the tympanic membrane, and the resulting vibrations are then mechanically transmitted via the os- sicles to the cochlea located in the inner ear. The mam- malian cochlea consists of a snail-shell-shaped duct filled with lymph fluid as shown in Fig. 2. Sound waves, which reach the cochlea, are transmitted through the cochlear fluid and vibrate the basilar membrane (BM). The organ of Corti (OC), which sits on the BM, transforms the vi- bration of the BM to electrical signals. As a result, we perceive sound. ∗ Received 11th July, 2005 (No. 05-4091) Fig. 1 Schematic of the auditory system of the guinea pig. The ∗∗ Department of Bioengineering and Robotics, Tohoku Uni- external auditory meatus conducts sound waves to the versity, 6–6–01 Aoba-yama, Sendai 980–8579, Japan. tympanic membrane. The vibration of this membrane is E-mail: [email protected] mechanically transmitted via the ossicles to the cochlea. Series C, Vol. 49, No. 3, 2006 JSME International Journal 829 Fig. 2 Schematic of the cross section of the guinea pig cochlea. The cochlea consists of a fluid-filled duct that is coiled like a snail shell. The organ of Corti (OC) sits on the basilar membrane (BM). Fig. 4 Schematic of the IHC. The stereocilia are located on the apical surface of the IHC, i.e., the reticular lamina (RL), and are arranged in a staircase pattern. There are two types of links, i.e., tip links, which extend between the tip of the shorter stereocilium and the lateral wall of the next taller one, and lateral links, which extend between the shafts of the stereocilia. (a) (b) Fig. 3 Schematic of the cross section of the OC. (a) Resting state of the OC. The OC contains a sensory cell, i.e., the inner hair cell (IHC). (b) Bending state of the OC. When the BM vibrates vertically, the stereocilia are deflected by the shearing motion of the tectorial membrane (TM) against the reticular lamina (RL) through endolymph. Fig. 5 Internal structure of the stereocilium. The stereocilium consists of cross-linked actin filaments, the so-called intracellular actin cytoskeleton, and the rootlet. The (TM) against the reticular lamina (RL) occurs(1). A bun- density of the actin filaments which compose the rootlet dle of stereocilia located on the apical surface of the IHC is larger than that of the intracellular actin cytoskeleton. is deflected by Couette flow of the endolymph caused by The rootlet is the core of the stereocilium and makes it ff such shearing motion(1). Figure 4 shows a schematic of the sti . IHC. About sixty stereocilia, a few micrometers in height, are arranged in a staircase pattern and interconnected by cross-linked actin filaments(3) – (8), the so-called intracellu- two types of links, i.e., tip links, which extend between lar actin cytoskeleton, and the rootlet. The rootlet is the the tip of the shorter stereocilium and the lateral wall of core of the stereocilium and is thought to be responsible the next taller one, and lateral links, which extend between for its stiffness(9). When the bundle is deflected in the ex- the shafts of the stereocilia(2). Figure 5 shows the internal citatory direction, i.e., toward the tallest stereocilium, the structure of the stereocilium. The stereocilium consists of tip link is under tension and the mechanoelectrical trans- JSME International Journal Series C, Vol. 49, No. 3, 2006 830 duction (MET) channel located at the end of the tip link is 2. Modeling of Stereocilia thought to open(10). Due to this, the membrane potential of the IHC is depolarized and action potentials are then 2. 1 Geometry produced in the auditory nerve fibers(11). The geometry of the FEM model of the IHC stere- To clarify the transduction process of the auditory ocilia located 17 mm from the basal end in a guinea pig system, investigation of the motion of the stereocilia, cochlea shown in Fig. 6 (a) was determined from measure- which results in the tension in the tip link, is necessary. ment data reported in the literature(2), (15) – (17), (20), (21).The However, it is difficult to obtain such motion from ex- characteristic frequency (CF, the most effective stimulus periments because the stereocilia are a few micrometers frequency) at this location is approximately 500 Hz(22). in height and they interact with one another due to the The numbers of hexahedral elements assigned to the mod- links(2). Therefore, numerical analyses, which are use- els of tall, middle and short stereocilia were 888, 456 and ful in the investigation of such motion, have been done 336, respectively. As shown in Fig. 6 (b), a cylindrical to date(12) – (14). Zetes and Steele(12) employed the linear, component of the model consists of twelve elements. In multi-degree of freedom equation of motion for the fluid- the components in which the rootlet exists, four inner el- structure interaction of the stereocilia and obtained the fre- ements and eight outer elements represent the rootlet and quency responses of the displacement of the stereocilia the intracellular actin cytoskeleton, respectively. On the and the tension in the tip links. They showed that vis- other hand, in the components in which the rootlet does cous force caused by endolymph limited the motion of the not exist, all twelve elements represent the intracellular stereocilia, resulting in the reduction of the tension in the actin cytoskeleton. The boundary condition at the bottom tip links in a fashion analogous to that of a low pass fil- of the model was fixed. ter. However, the connection of the lateral link between the stereocilia was not considered. In addition, although all stereocilia are subject to the fluid force caused by the flow of endolymph in vivo, force was applied to only the tip of the tallest stereocilium in their analysis. Duncan and Grant(13) presented a finite element method (FEM) model of the IHC stereocilia and analyzed the deflection of the stereocilia by applying force to the tip of the tallest stereocilium. The relationship between the stiffness of the stereocilium and its Young’s modulus was elucidated in their analysis. Cotton and Grant(14) showed the reduction of the stiffness of the stereocilia caused by the buckling of the links by using a three-dimensional FEM model of the vestibular hair cell bundle. However, the dynamic be- havior of the bundle was not examined by Duncan and Grant(13) nor by Cotton and Grant(14). Furthermore, in the above three analyses, as the stereocilium was assumed to be homogeneous, the contribution of Young’s modulus of the rootlet to the stiffness of the stereocilia was not con- sidered. In this study, a three-dimensional FEM model of the IHC stereocilia of the guinea pig including the rootlet and the lateral link was constructed based on measurement (a) (b) data reported in the literature(2) – (6), (15) – (21). The unknown Fig. 6 FEM model of the stereocilia. (a) Side view of the Young’s modulus was estimated by comparing the numer- model. The numbers of hexahedral elements assigned ically obtained stiffness of the stereocilia with that experi- to the models of tall, middle and short stereocilia mentally obtained in previous study(16). Then, by applying are 888, 456 and 336, respectively. The boundary the analytically estimated fluid force exerted on the stere- condition at the bottom of the model is fixed.
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