*2.2. Material Properties*

The middle layer of the eardrum pars tensa and the BM were assumed to be orthotropic. Other components of the FE model were assumed to be isotropic. Poisson's ratios were assumed to be 0.3 for all components in the middle ear. The material properties of each component of the middle ear in the FE model were mainly referred to Gentil et al. et al. [20] and Zhang et al. [22], as listed in Table 1.


**Table 1.** Material properties of the middle ear components.

The components of the middle ear and the cochlea were modelled as elastic properties, except for the eardrum, eardrum annulus ligament, incudostapedial joint, incudomallear joint, stapedial annulus ligament, and RW membrane, which were modelled as linear viscoelastic materials. The Rayleigh damping was specified for the elastic components. The Rayleigh damping parameters were taken as α = 0 s<sup>−</sup>1, β = 0.0001 s [23]. The relaxation modulus of the linear viscoelastic materials was expressed as Equation (1):

$$E(t) = E\_0(1 + \varepsilon\_1 \exp(-\frac{t}{\tau\_1}))\tag{1}$$

where *E*0, *e*1, and τ1 were viscoelastic parameters with constant values for each type of soft tissue, and *t* is the time. *E*0 is the elastic modulus listed in Table 1. *e*1, and τ1 are listed in Table 2. The viscoelastic parameters were referenced to Zhang et al.'s report [24]. These parameters were obtained by dynamic material tests on these components and the cross-calibration method.

The BM is assumed to be the anisotropic membrane with a density of 1200 kg/m3. The stiffness of BM was decreased from the base to the apex. The BM's longitudinal modulus was assumed as 600 MPa at the base, linearly decreased to 10 MPa at the apex along the BM length. Similarly, the transverse modulus and the vertical modulus of the BM decrease linearly from 6 MPa, and 12 MPa at the base to 0.1 MPa, and 0.2 MPa at the apex, respectively. The density of the RW membrane was set to 1200 kg/m<sup>3</sup> with an elastic modulus of 2.32 MPa [25]. The bulk modulus of the cochlear fluid and the air in the external ear canal and the middle ear cavity were set as 2250 MPa and 0.142 MPa, respectively. The viscosity of the cochlear fluid is 0.001 Ns/m<sup>2</sup> [22].


**Table 2.** Parameters of linear viscoelastic materials.

#### *2.3. Piezoelectric Transducer Simulation*

Since the purpose of this paper is to study the stimulating site's influence rather than the piezoelectric transducer's structural design, we simplified the piezoelectric transducer as an ideal displacement-driven transducer. This idealized representation of the piezoelectric transducer is possible as small displacements and forces are required for hearing compensation in IMEHDs [18]. Based on this simplification, a displacement excitation with the magnitude of 0.1 μm was applied at the commonly used stimulating sites, i.e., the eardrum's umbo, the incus long process, the incus body, the stapes, and the RW membrane, respectively. The magnitude of the applied displacement excitation was ascertained as it can produce a sound pressure level equivalent to 100 dB, which is a design criterion for an IMEHD transducer [4]. The stimulating sites were plotted in Figure 3. When stimulating the eardrum's umbo, the incus long process, the incus body, the stapes, and the direction of the applied displacement excitation was along the longitudinal axis of the stapes, which is efficient for IMEHD stimulation [18]. For stimulating the round window membrane, the excitation's direction was normal to the surface of the round window membrane. Under these forces' stimulation, harmonic analysis was conducted over the frequency range of 0.25–6 kHz using the finite element software package ABAQUS (Dassault Systèmes, Johnston, RI, USA).

The surgical procedure, e.g., the transmastoidal approach for the piezoelectric transducer's implantation, will possibly change the direction of the excitation. To study the stimulating site's sensitivity to the direction changes of their excitations, the excitations were also applied in different directions at each stimulating site with the same magnitude of 0.1 μm. For stimulating the ossicular chain (umbo, incus long process, incus body, stapes), the reference direction was along the stapes' longitudinal axis. The other directions are defined by rotating the direction relative to the reference direction in the plane based on the longitudinal axis and the long axis of the stapes' footplate. The rotation is 20◦, 45◦, and 60◦ off the reference direction to crus posterior (20◦, 45◦, and 60◦ to CP). For stimulating the RW membrane, the reference direction is the normal direction of the RW membrane. The other directions are rotated 20◦, 45◦, and 60◦ off the reference direction.

**Figure 3.** The simulation of the piezoelectric transducer's simulation. (**a**) Stimulating sites on the finite element model; (**b**) the anatomy of the three ossicles (malleus, incus, and stapes).

#### *2.4. Equivalent Sound Pressure Level*

The sound transmission property via normal air conduction is different from that by a piezoelectric transducer's stimulation. Considering the basilar membrane inside the cochlea is responsible for sensing the input vibration energy, we used its response to assess the transducer's hearing compensation performance.

The vibration transmitted into the cochlea propagates in the form of a traveling wave from the base to the apex along the basilar membrane. For excitations of different frequencies, the maximum amplitude position of the traveling wave formed on the basilar membrane is different, with high frequencies maximally activating basal regions of the BM and low frequencies maximally activating apical areas of the BM. For a specific frequency excitation, the position of the basilar membrane that is most responsive in the longitudinal direction is referred to as the characteristic place of this frequency. The frequency is called the characteristic frequency of that position on the basilar membrane. The cochlea senses a pure tone sound of a specific frequency through its corresponding characteristic place along the basilar membrane. Therefore, in order to make the sound-perceived effect of the transducer's stimulation of a specific frequency equivalent to that excited by normal sound stimulation (sound pressure applied at the eardrum), the displacements of the BM's characteristic place of the frequency under the two excitations should be equal.

Based on above principle, in the normal sense of sound, when a sound with the frequency of ω and amplitude of *P*E is applied at the eardrum, its stimulated BM displacement at the characteristic place *x*CF is *<sup>d</sup>*acBM(<sup>ω</sup>, *x* CF:

$$d\_{\rm BM}^{\rm ac}(\omega, \chi\_{\rm CF}) = T F\_{\rm d}^{\rm ac}(\omega) \cdot P\_{\rm E} \tag{2}$$

where *TF*acd (ω) is the transfer function of the normal human ear sensation from the pressure applied at the eardrum to the displacement of the basilar membrane. The human ear functions as a linear system under the normal acoustic sound pressure excitation [26]. Therefore, based on the model-calculated basilar membrane's displacement under 100 dB SPL sound stimulation applied at the eardrum, we can obtain the transfer function:

$$TF\_{\rm d}^{\rm sc}(\omega) = \frac{d\_{\rm BM}^{\rm sc\_{100}}(\omega, \chi\_{\rm CF})}{2 \times 10^{-5} \times 10^{\frac{100}{20}}}.\tag{3}$$

Under the excitation of the ideal piezoelectric transducer, its stimulated BM displacement at the characteristic place *d*piezo BM (<sup>ω</sup>, *<sup>x</sup>*CF) can be calculated by the FE model. Since the basilar membrane vibration is responsible for transmitting the input energy to hair cells, the transducer-stimulated effect is equivalent to that excited by a normal acoustic stimulation *P*E applied at the eardrum, which produces the same displacement amplitude *d* ac BM(<sup>ω</sup>, *<sup>x</sup>*CF) at the characteristic place of the basilar membrane:

$$d\_{\rm BM}^{\rm piezo}(\boldsymbol{\omega}, \mathbf{x\_{CF}}) = \widehat{d}\_{\rm BM}^{\rm 3C}(\boldsymbol{\omega}, \mathbf{x\_{CF}}) = T F\_{\rm d}^{\rm ac}(\boldsymbol{\omega}) \cdot \widehat{P}\_{\rm E} = \frac{d\_{\rm BM}^{\rm AC\_{100}}(\boldsymbol{\omega}, \mathbf{x\_{CF}})}{2 \times 10^{-5} \times 10^{\frac{10}{20}}} \cdot \widehat{P}\_{\rm E}. \tag{4}$$

Based on Equation (4), the transducer's corresponding equivalent sound pressure *P*E applied at the eardrum can be derived as

$$\widehat{P}\_{\rm E} = \frac{d\_{\rm BM}^{\rm pico}(\omega\_{\rm }\mathbf{x}\_{\rm CF})}{d\_{\rm BM}^{\rm ac. 100}(\omega\_{\rm }\mathbf{x}\_{\rm CF})} \times 2 \times 10^{-5} \times 10^{\frac{100}{20}}.\tag{5}$$

Thus, the performance of the transducer's excitation can be evaluated by *<sup>L</sup>*EQ, which is the equivalent sound pressure level (ESPL) of the piezoelectric transducer:

$$L\_{\rm EQ} = 20 \log \frac{\widehat{P}\_{\rm E}}{2 \times 10^{-5}} = 100 + 20 \log (\frac{d\_{\rm BM}^{\rm pole}(\omega, \mathbf{x\_{CF}})}{d\_{\rm BM}^{\rm ak\_{-}(100)}(\omega, \mathbf{x\_{CF}})}). \tag{6}$$
