# Numerical Study of Tympanosclerosis Including Its Effect on Human Hearing
## 2 Methods  
The geometrical models used in this work were obtained from the project \"The Visible Ear\". The images were acquired, using a cryosectioning procedure, from the temporal bone of an old woman. Finally, hand segmentation of the different tissues was performed [16]. The slice thickness was 25 \(\upmu\)m, and high-resolution images were captured every 50 \(\upmu\)m. A total of 597 images, with 24 bit RGB and a resolution of 50 \(\upmu\)m/pixel, were obtained. Approximately 26 different organs of the middle ear were identified through the manual segmentation, performed by [PERSON], of each image. A 3D finite element model of the ear was built, and it included the external auditory canal, the eardrum, ossicles (malleus, incus and stapes), tympanic cavity, six ligaments (malleus: anterior, lateral and superior; incus: posterior and superior; stapes: stapedius annular ligament), two muscles (stapes: stapedius; malleus: tensor typnani), incudomale-leolar and incudostapedial joints and the cochlear fluid [17]. The linear elastic properties were incorporated in all parts of the 3D finite element model. The geometry was discretized into a finite element mesh, and numerical simulations were conducted using the Abaqus(r) Standard [18]. The finite element mesh was built using linear tetrahedral elements (C3D4) for the ossicles, linear tetrahedral acoustic elements (AC3D4) for the acoustic components and linear truss elements (T3D2) for the tendons and ligaments. The connection of the Eustachian tube with the nasopharynx was not modeled, since it was assumed closed. The eardrum was considered as having homogeneous properties, with isotropic behavior in the \"_pars flacida_\" and orthotropic behavior in \"_pars tensa_\" [17; 19; 20; 21; 22]. In the \"_pars tensa_\", the radial and circumferential Young's modulus was 32 and 20 MPa, respectively. For the \"_pars flacida_\", a Young's modulus of 10 MPa was considered (Table 1). A homogeneous and isotropic behavior was assumed for the ossicles, with a Young's modulus of 14.1 GPa. The adopted values for the density are in accordance with the literature and are shown in Table 1[20; 21]. A Poisson's ratio of 0.3 was considered for the entire model [20], except for the ear cartilage, where a value of 0.4 was assumed [23]. Figure 1: Visualization of an eardrum perforation occupying its inferior quadrants (of 7 mm). The Raleigh damping matrix, \(\mathbf{C}\), can be expressed as a combination of the stiffness and mass matrixes [17] and requires two parameters. In the middle ear components, the Raleigh's proportional damping coefficients considered were \(\alpha=0\) s\({}^{-1}\) and \(\beta=0.0001\) s.  
The ligaments suspending the ossicles and the muscles were assumed to behave in a linear elastic regime, and the mechanical properties (density and Young's modulus) are shown in Table 2. Table 2 also contains the density and bulk modulus for the acoustic medium, including the air in the tympanic cavity and the fluids in the tympanic cavity and cochlea [20,21]. Fixed boundary conditions were applied to the tympanic annulus, surrounding the eardrum, which is attached to the temporal bone. The bone was assumed to have an isotropic behavior, with a Young's modulus of 6.0 E4 Pa. The free extremities of the sapedius annular ligament (around the stages), the three ligaments of the malleus and the two ligaments of the incus were also considered fixed, and the properties are shown in Table 2.  
Acoustic elements, containing only pressure degrees of freedom, were used in order to simulate the existence of air inside the tympanic cavity, in the external auditory canal and