Finite Element Dynamics of Human Ear System Comprising Middle Ear and Cochlea in Inner Ear

A human middle ear consists of an eardrum and three ossicles which are linked by each other, and connect with the eardrum and an inner ear. The inner ear consists of a cochlea and a vestibular system. An abnormality of the human middle ear such as ossicular dislocation may cause conductive hearing loss. The conductive hearing loss is generally treated by surgery using artificial ossicles. The treatments of conductive hearing loss require a better understanding of characteristics and dynamic behaviors of the human middle ear when the sounds transmit from outer inner to inner ear. The purpose of this research is to simulate the dynamic behaviors of a human ear system comprising the middle ear and the cochlea in the inner ear using the finite element method (FEM). Firstly, the eigen-value analysis was performed to obtain the natural frequencies and vibration modes of the total ear system. Secondly, the frequency response analysis was carried out. Thirdly, the time history response analyses were performed using human voices as the external forces. In the time history response analyses, the sounds created as input sound pressures were used. Human voices, for example vowels “I”, “u” and “e” as input sound pressures were created by using the sound pressures downloaded from the opening samples of human voices as wav files in a website. Then it was clarified that the high frequency components of sounds are reduced by the middle ear system.

hearing loss is the most common case due to problems in the middle ear or the outer ear at times. A human middle ear which includes an eardrum and three ossicles (malleus, incus and stapes) is a mechanical system to transmit sounds from outer ear to inner ear. The conductive hearing loss occurs when sounds hardly transfer through the ear canal to the eardrum and three ossicles. The otosclerosis in which the ossicles in middle ear become stiff or the dislocation of three ossicles causes severe conductive hearing loss. Conductive hearing loss can often be treated with a surgery using artificical ossicles.
Several researchers on middle ear system use finite element analysis in their study.
Finite Element Analysis (FEA) is a type of numerical analyses that uses finite element method to simulate/analyze the behaviors of mechanical structures and components under a variety conditions. The FEA is a powerful tool to analyze the vibration of a middle ear because the middle ear has a complicated shape. The major benefits of finite element analysis in this study are to calculate natural frequencies, vibration modes, frequency response and time history response analyses of a human middle ear which cannot be obtained by using conventional methods. The past researchers on middle ear system are introduced as follows. Wada et al. measured Young's modulus, thickness and damping ratio of the human eardrum by using measuring aparatus developed by themselves [1]. As for dynamics analysis on the middle ear system, a cat eardrum was firstly investigated by a finite element analysis on the curved conical eardrum by W. R.
I. Funnell and C. A. Laszlo [2]. D. D. Greef et al. performed dynamics analysis on a new anatomically-accurate model composed of the tympanic membrane and malleus using the finite element method [3]. Computational modeling methodology of multi-body system was examined in order to simulate and study the middle ear mechanical response to acoustic stimuli by F. Bohnke et al. [4]. Y. Liu et al. carried out a three-dimensional finite element analysis of human ear in order to analyze lesion of ossicular chain [5]. E. Skrodzka and J. Modlawska performed modal analysis of the human tympanic membrane of middle ear using the finite element method [6]. C. F. Lee et al.
proposed a practical approach that uses high-resolution computed tomography (HRCT) to derive models of the middle ear for finite element analysis [7]. R. Z. Gan et al. performed a three-dimensional finite element analysis of the human ear that included the external ear canal, eardrum, ossicular bones, middle ear suspensory ligaments/muscles, and middle ear cavity [8]. A practical and systematic method for reconstructing accurate computer and physical models of entire human middle ear were proposed by Q.
Sun et al. [9]. T. Koike et al. performed the finite element analysis of the human middle ear and compared calculated results with measurement data [10]. The displacements of the tympanic membrane in a human ear were measured by using a Laser Doppler Vibration (LDV) and compared with those of a finite element analysis of the middle ear by T. S. Ahn et al. [11]. S. Ihrle et al. developed a nonlinear numerical model of the human middle ear based on the elastic multibody system (EMBS) method [12]. Middle ear responses to both air conduction and bone conduction excitations were measured at the umbo and lateral process of the malleus by Kenji Homma et al. [13].
Then the authors reported on a three dimensional model of human middle ear system, eigen-value analysis, frequency response analysis and time history response analysis. In the time history response analysis of the previous research, formant frequencies were used as the frequencies of the external forces. Here, the frequencies, F 1 = 750 [Hz], F 1 = 1250 [Hz] and F 1 = 2500 [Hz] are the first Formant frequencies of "o", "a" and "i" in human simple vowels respectively [14].
In the present paper, time history response analyses were carried out using human voices created as external forces. The Hyper Works (finite element analysis code) was used to carry out dynamics analyses of the total ear system. Firstly, the eigen-value analysis was carried out to obtain the natural frequencies and the vibration modes.
Then, the frequency response analysis was performed to show the response of total ear system in the frequency range 100 to 10,000 [Hz]. In time history response analysis, firstly, the sound pressures were downloaded from a website opening samples of human voices as wav files for the input sound pressure. Then, the sounds as input data were created by using downloaded sound pressures. The sound pressure on three vowels "i", "u" and "e" as wav files were used. Finally, it was obtained that the high frequency components of sounds were reduce by the middle ear system. Figure 1 shows the finite element model of total ear system containing a middle ear, a cochlea in an inner ear, four ligaments, a tendon and a tensor tympanic membrane. The middle ear consists of an eardrum and three ossicles, namely malleus, incus and stapes.

Middle Ear
The eardrum is a concave membrane that separates the outer ear and the middle one.
The ossicles are three linked small bones that connect the eardrum to the cochlea. A sound wave transmits from the ear canal to the eardrum. Then the eardrum vibrates due to the sound wave. After that, the vibration transmits from the eardrum to the three ossicles, and form the three ossicles to the cochlea.

Cochlea in Inner Ear
A cochlea is a part of an inner ear. The cochlea has a snail-like shape and is filled with fluid. There are tiny hair cells as a sensory receptor inside the cochlea. The stapes, namely a part of three osiccles is in contact with the surface of cochlea [15]. In this research, the cochlea was modelized with the translational springs as shown in Figure   1.

Ligaments
An anterior malleal ligament and a lateral malleal one in four ligaments are muscles supporting the malleus in the middle ear as shown in Figure 1. Then a superior malleal ligament and a posterior incudal one in the other two ligaments are muscles supporting the incus in the middle ear as shown in Figure 1. The four ligaments were modelized

Tendon
A posterior stapedial tendon is a muscle located at the head of the stapes as shown in

Tensor Tympanic Membrane
A tensor tympanic membrane is a muscle attaching to the malleus as shown in Figure   1. The tensor tympanic membrane can regulate the motion of the malleus. If loud sounds are heard, the tensor tympanic membrane reduce the vibration by pulling the malleus away from the eardrum. The tensor tympanic membrane was modelized with the translational spring in this research. The end of the tensor tympanic membrane connects to the wall of middle ear cavity. Then the end of translational spring was clamped.

Total Ear System
A total ear system is composed of the middle ear, the cochlea in inner ear, the four ligaments, the tendon and the tensor tympanic membrane as shown in Figure 1. The middle ear consists of an eardrum and three ossicles, namely malleus, incus and stapes.
The shapes and dimensions of the eardrum were decided by considering the other researchers [1] [2]. Then, the shapes and dimensions of the three ossicles were decided by considering the references [3] [4]. The CAD software (Solidworks 2015) was used to create the three dimensional model of the middle ear.
In the finite element model of total ear system, the eardrum and the three ossicles were meshed by using six-node triangular elements and ten-node tetrahedron elements, respectively. Table 1 shows the material properties of middle ear used in the present calculations.

Material Properties of Middle Ear
The values in Table 1 were decided by considering the references [5] [10]. Table 2 shows the translational spring constants of ligaments, a tendon, a tensor tym-  panic membrane and a cochlea. As for the ligaments, the tendon and the tensor tympanic membrane, each of them was considered as three translational springs in x, y and z directions. The indexes, x, y and z of each spring constant denote the local coordinates of itself. The x-direction of each local coordinate frame was defined in the normal direction to the surface of an ossicle. The values of translational spring constants, K i (i = x, y, z) for the ligaments, the tendon, the tensor tympanic membrane and the cochlea were decided by trial and error so that the stapes can perform a piston motion in the x-direction of local coordinate frame. The stapes contacting with the cochlea moves like a piston in the x-direction.

Spring Constants and Boundary Conditions
As for the boundary conditions of the eardrum, a local coordinate frame was defined at each node of boundary of the eardrum. In each the local coordinate frame, three translational motions in the x-, yand z-directions and two rotational ones around the x-, and z-axes were clamped. Then, the torsional springs of K θy were applied to the rotational motions around the y-axes, on the nodes of boundary of the eardrum. The torsional springs of K θy were used to adjust the stiffness of boundary of the eardrum in order to make the boundary conditions of the eardrum similar to those of a human eardrum. As for the torsional springs on boundary of the eardrum, the same values, K θy = 3.0 × 10 −5 [Nmm/rad] as the authors' previous report [15] were used.

Calculation Method
The Optistruct of Hypermesh was used to carry out eigen-value analysis of total earsystem. The eigen-value analysis of the total ear system was carried out to obtain the natural frequencies and the vibration modes in the frequency range from 100 [Hz] to 10,000 [Hz].  Figure 3 shows the structural dampings used for the total ear system using the struc-

Time History Response Analysis of Total Ear System
The FEA by the Hypermesh was used to carry out time history response analysis of total ear system. In this analysis, the human voices created as input sound pressure were used as the external forces. that they may not exceed the limitation of input data in Hypermesh. As for the second type of sounds, they were created as data in 2[s] so that the authors can evaluate them, by the same method as the first type of them. The waveforms of created sound pres-sures on vowel "e" are a little more complex than vowels "i" and "u". created in the same way as input data of Hypermesh of wav files were played in order to evaluate them before using them as input data of Hypermesh. Then, the original sounds and the created sounds were compared for each vowel. The results of evalutions showed that the created sounds on vowels "i", "u" and "e" were identical to its original sounds.

Creating Input Sound Pressures
Therefore, the created sounds as input data for the three vowels were feasible to input them into the Hypermesh in order to carry out the time history response analysis of the total ear system.

Calculation Method
The Optistruct in HyperWorks was used to carry out time history responses analysis of total ear system. The sound pressure in 0.022 [s] on vowel "i", "u" or "e" created for input data was loaded to the eardrum of total ear system as the external force.   Figure 8, it is understood that high frequency components of the calculated output data reduce due to an effect of the structural damping.

Results
The sounds in 2 [s] created by using the calculated output data of csv files were con- results of evaluation, it was obtained that the sounds of calculated output data could be heard more clearly than those of the created input ones because the high frequency components of calculated output data reduced due to the effect of the structural damping.

Conclusions
In the present research, the simple model proposed by the authors was evaluated by dynamic analyses such as eigen-value, frequency response and time history response analyses. As for the time history response analysis, the current research was performed using the human voices as the external forces. The summary of the results is shown below.
1) The eigen-value analysis had been carried out to obtain natural frequencies and vibration modes of the total ear system using the torsional springs on the boundary of the eardrum and the translational springs for ligaments, a tensor tympanic membrane, a tendon and a cochlea.
2) It was determined that the structural damping coefficient, Gis 0.4 for less than 1500 [Hz] and 2.5 for more than 1500 [Hz] by the frequency response analysis of the total ear system.
3) It was obtained that the sounds of calculated output data could be heard more clearly than those of the created input ones because the high frequency components of calculated output data reduced due to the effect of the structural damping.
In order to improve the simple model, the variation of external forces in the time history response analysis of the total ear system such as created sound pressures on consonants will be considered in the future.