# Numerical Study of Tympanosclerosis Including Its Effect on Human Hearing
## 2 Methods
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ear Components** & **Density \(\rho\) (kg/m\({}^{3}\))** & **Young’s Modulus E (Pa)** \\ \hline \multirow{3}{*}{Eardrum} & _Pars tensa_ & \multirow{3}{*}{\(1.2\times 10^{3}\)} & \(3.2\times 10^{7}\) (radia) \\ \cline{2-3}  & _Pars flaccida_ & & \(2.0\times 10^{7}\) (circumferential) \\ \cline{2-3}  & & & \(1.0\times 10^{7}\) \\ \hline \multirow{3}{*}{Malleus} & \multicolumn{3}{c}{\(2.55\times 10^{3}\) (Head)} \\ \cline{2-3}  & & \(4.53\times 10^{3}\) (Neck) \\ \cline{2-3}  & & \(3.70\times 10^{3}\) (Handle) \\ \hline \multirow{3}{*}{Incus} & \multicolumn{3}{c}{\(2.36\times 10^{3}\) (Body)} \\ \cline{2-3}  & & \(2.26\times 10^{3}\) (Short process) \\ \cline{2-3}  & & \(5.08\times 10^{3}\) (Long process) \\ \hline \multicolumn{3}{c}{Stapes} & \(2.2\times 10^{3}\) \\ \hline \multicolumn{3}{c}{Incudomalleolar joint} & \(3.2\times 10^{3}\) \\ \hline \multicolumn{3}{c}{Incudostapedial joint} & \(1.2\times 10^{3}\) & \(6.0\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the eardrum, ossicles and joints [20,21]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Ear Components** & **Density \(\rho\) (kg/m\({}^{3}\))** & **Young’s Modulus E (Pa)** & **Bulk Modulus B (Pa)** \\ \hline \multirow{3}{*}{Superior malleolar ligament} & \multirow{3}{*}{\(2.5\times 10^{3}\)} & \(4.9\times 10^{4}\) \\ \cline{2-3}  & & \(6.7\times 10^{4}\) \\ \cline{2-3}  & & \(2.1\times 10^{6}\) \\ \cline{2-3}  & Superior incudal ligament & \(4.9\times 10^{4}\) \\ \cline{2-3}  & Posterior incudal ligament & \(6.5\times 10^{5}\) \\ \cline{2-3}  & Tensor typnani tendon & \(2.6\times 10^{6}\) \\ \hline \multicolumn{3}{c}{Stapedius tendon} & \(5.2\times 10^{5}\) \\ \hline \multicolumn{3}{c}{Stapedius annular ligament} & \(2.0\times 10^{4}\) \\ \hline \multicolumn{3}{c}{Tympan annulus} & \(1.2\times 10^{3}\) & \(6.0\times 10^{4}\) \\ \hline \multicolumn{3}{c}{Tympan cavity air} & \(1.164\) \\ \hline \multicolumn{3}{c}{Tympan cavity fluid} & \multirow{3}{*}{\(1000\)} & - \\ \cline{2-3}  & & \(2.2\times 10^{9}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Material properties of the ligaments, muscles and acoustic medium [20,21]. the fluid in the cochlea. The constitutive behavior of the fluid was defined mathematically in the work of [PERSON] et al. [17]. According to Equation (1), sound pressure levels (SPLs) of 80, 90 and 100 dB SPL can be converted into pressure (\(\rho\)), respectively, as 0.200, 0.632 and 2.000 Pa. These acoustic pressures were applied in the external auditory canal (Figure 2). \[\mathrm{dB\,SPL}=20\times\mathrm{log}_{10}(\rho/\rho_{0}) \tag{1}\]  
In Equation (1), \(\rho_{0}=20\) uPa is the reference sound pressure, which corresponds to the audibility threshold. In order to transmit the sound pressures from the acoustic elements into the structural elements, the Abaqus(r) Standard TIE command was used [18]. According to the work of [PERSON] et al. [13], the mechanical properties of the intermediate layer of the eardrum were modified, using the classical rule of mixtures for composites containing particles, assuming that the plaques were made of HA particles imbedded in a matrix of connective tissue. The HA plaque properties were obtained using the rule of mixtures for composite materials, with the fibers uniformly distributed [13,14]. The plaques were considered isotropic, and the particles of HA were assumed as having a uniform distribution throughout the plaque. The density considered for the HA was 3160 kg/m\({}^{3}\) (Table 3) [24]. In a steady-state dynamic analysis, the equations of the steady harmonic motion of the system were solved directly. Such an analysis provides the steady-state amplitude and phase of the umbo and stapes footplate response due to the fact of  
Figure 2: Surface area of the external auditory canal, where the pressure was applied.