# Simulation-Based Study on Round Window Atresia by Using a Straight Cochlea Model with Compressible Perilymph

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Acoustics of Cochlea Based on Even and Odd-Mode Analysis

#### 2.1. Fluid Equations for Compressible Media

**u**stands for the fluid velocity,

**I**for the identity tensor, p for the fluid pressure, ρ for the fluid density, μ for the viscosity, and t for time. ∇ is a differential operator defined by

**M**is the mass matrix,

**C**is the damping matrix,

**K**is the stiffness matrix, and

**f**(t) is the external force applied to the material.

#### 2.2. Modeling of the Cochlea

_{c}= 35 mm [26]. At the cochlea’s base, the total width was W

_{bc}= 1.2 mm [27] and the total height was H

_{bc}= 2.5 mm, while, at the cochlea’s apex, the total width was W

_{ac}= 0.7 mm and the total height was H

_{ac}= 1.5 mm. The scala media was sandwiched symmetrically by the scala vestibuli and scala tympani, and the basilar membrane was embedded in the scala media, whose dimensions were defined by a whole length of L

_{m}= 34 mm, a width of W

_{bm}= 100 µm, and a thickness of H

_{bm}= 30 µm at the base, and a width of W

_{am}= 500 µm and a thickness of H

_{am}= 10 µm at the apex [28,29]. The Young’s modulus, Poisson ratio, and mass density of the basilar membrane were E = 1 MPa, v = 0.49, and ρ = 1200 kg/m

^{3}, respectively. These values were determined to satisfy Greenwood’s tonotopy equation [3]. Additionally, a helicotrema with a diameter of D

_{h}= 0.65 mm was set at 34.675 mm from the cochlea’s base so as to connect the scala vestibuli and scala tympani [28]. Then, the scala vestibuli, scala tympani, and helicotrema were filled with compressible perilymph with a viscosity of µ = 0.7027 mPa·s, a density of ρ = 994.6 kg/m

^{3}, a sound velocity of c = 1520 m/s, and a temperature of T = 36 degrees Celsius to allow the sound waves to propagate as a compression wave. On the other hand, to represent the air-filled environment of the middle ear cavity, an air space with a length of L

_{a}= 3.0 mm, a width of W

_{a}= 1.2 mm, and a height of H

_{a}= 1.235 mm was connected to the base end of the scala tympani. An elliptical round window with a width of W

_{r}= 1.12 mm, a height of H

_{r}= 0.69 mm, a thickness of T

_{r}= 70 µm, a Young’s modulus of E = 1 MPa, a Poisson’s ratio of v = 0.49, and a mass density of ρ = 1200 kg/m

^{3}was modeled at the boundary between the air space and the scala tympani [27,30,31,32]. In addition, an ideal hard boundary condition, that is, the condition that the normal component of the particle velocity is always zero on the wall’s surface, was applied to the walls in the model, except for the round window and basilar membrane, so that the sound wave reflected perfectly without losses on their surfaces. Detailed settings of the finite element method and PC specification for simulations are summarized in Table 1.

#### 2.3. Sound Waves in Cochlea and Generation of Traveling Waves

^{3}) and the perilymph-filled inside region (an acoustic impedance of 1.5 MPa·s/m

^{3}) across the round window. In such a case, an acoustic wave traveling in the scala tympani heading to the round window was reflected mostly with a free-end reflection condition due to the large impedance mismatch between the air and the perilymph. As a result, the velocity of the medium became the maximum, and the pressure level became zero at the border.

_{SV}(x, t) in the scala vestibuli and P

_{ST}(x, t) in the scala tympani, where x and t denote the position in the cochlea and the time variation, respectively. Then, we calculated:

_{SV}(x, t) and P

_{ST}(x, t) were expressed as follows.

_{SV}(x, 38.84 ms) and P

_{ST}(x, 38.84 ms) are drawn in the upper region of Figure 2b in blue and red, respectively. In addition, the displacement of the basilar membrane was also calculated and presented in light gray in the same figure together with the P

_{EVEN}(x, 38.94 ms) in black. In the same way, the pressure levels of the P

_{SV}(x, 38.94 ms), P

_{ST}(x, 38.94 ms), and P

_{EVEN}(x, 38.94 ms), and the displacement of the basilar membrane, are presented in the lower region of the figure. Based on these results, the combination of the P

_{EVEN}(x, 38.84 ms) and P

_{ODD}(x, 38.84 ms), and that of the P

_{EVEN}(x, 38.94 ms) and P

_{ODD}(x, 38.94 ms), were calculated as expressed in Figure 2c. This result indicates that even and odd symmetric sound wave modes, P

_{EVEN}(x, t) and P

_{ODD}(x, t), respectively, exist in the cochlea, and they configure the sound waves in the scala vestibuli P

_{SV}(x, t) and scala tympani P

_{ST}(x, t).

_{ODD}(x, t), reduced with the position x, and finally, it vanished. Instead, the displacement of the basilar membrane grew largely in the cochlea. This fact indicates that the odd mode P

_{ODD}(x, t) played an important role in exciting the traveling wave on the basilar membrane. However, more importantly, the even mode P

_{EVEN}(x, t) was not independent of the traveling wave, and it was deeply related to the odd mode P

_{ODD}(x, t). The even sound wave mode excited at the cochlea base (x = 0 mm) traveled along the cochlea, and it was reflected perfectly at the cochlea apex (x = 35 mm) with a fixed-end reflection condition. As a result, the even-mode sound wave generated a standing wave in the cochlea. Therefore, the pressure level of the even mode P

_{EVEN}(x, t) became a function of the cochlea length. Under such behaviors of the even and odd-mode sound waves, P

_{ODD}(0 mm, t) should be equal to P

_{EVEN}(0 mm, t) to meet the condition P

_{ST}(0 mm, t) = 0 at the base of the scala tympani.

#### 2.4. Validity of Even and Odd Mode Approaches in Other Frequencies

_{EVEN}(x, t) and P

_{ODD}(x, t). In this section, we describe the verification of the validity of this approach at frequencies of 2000 Hz, 5000 Hz, and 10,600 Hz.

_{SV}(x, t), P

_{ST}(x, t), P

_{EVEN}(x, t), and P

_{ODD}(x, t), and the displacement of the basilar membrane when the cochlea was excited by a continuous sinusoidal plane wave with a sound pressure level of 1 Pa and a frequency of 2000 Hz. When the velocity of the sound wave traveling in the perilymph was assumed to be 1520 m/s, the wavelength of the sound wave at 2000 Hz was estimated to be 0.76 m. Since the wavelength was long enough at this frequency compared to the total length of the cochlea of 35 mm, the pressure level of the even mode P

_{EVEN}(x, t) became almost constant anywhere in the cochlea, while the waveform of the odd mode P

_{ODD}(x, t) vibrated at the position from 0 mm to 20 mm, and, finally, it vanished beyond 20 mm. At the same time, the displacement of the basilar membrane was greatly generated. Here, the sound pressure levels of the even mode and odd mode at the base of the cochlea could be read as P

_{EVEN}(0 mm, t) = +0.12 Pa and P

_{ODD}(0 mm, t) = +0.14 Pa, respectively. When we applied the even and odd-mode approach to the model, as explained in the previous section, P

_{ST}(0 mm, t) was −0.02 Pa, not 0 Pa. The reason for this is that the traveling wave excited on the basilar membrane did not converge sufficiently at the apex of the cochlea, and we considered that it was reflected slightly at the end of the basilar membrane. The reflected traveling wave was transformed into the sound wave again and traveled back to the cochlea base. However, if we increase the simulation time, P

_{ST}(0 mm, t) will converge to zero.

_{EVEN}(0 mm, t) = +0.2 Pa and P

_{ODD}(0 mm, t) = +0.2 Pa, respectively, and P

_{SV}(0 mm, t) = +0.4 Pa and P

_{ST}(0 mm, t) = 0.0 Pa were obtained. To obtain such an ideal result, it is necessary that the traveling wave excited by the odd mode does not cause reflection at the end of the basilar membrane.

_{EVEN}(0 mm, t) = 0. According to the previous section, the following conditions were required at the healthy cochlea base:

_{SV}(0 mm, t) = P

_{EVEN}(0 mm, t) + P

_{ODD}(0 mm, t)

_{ST}(0 mm, t) = P

_{EVEN}(0 mm, t) − P

_{ODD}(0 mm, t) = 0

_{ODD}(0 mm, t) should be zero, and, accordingly, the pressure level at the oval window also should be zero; that is, P

_{SV}(0 mm, t) = 0, which means that the oval window works as a reflector with the free-end reflection condition. In other words, the input impedance of the cochlea becomes 0 Pa·s/m

^{3}, and the cochlea does not accept any sound waves coming into the cochlea [25]. Even though our simulation result cannot be compared to the loudness curve reported in [33] directly because our simulation results do not consider the frequency characteristics of the outer and middle ears and the nonlinear response of the cochlea amplifier caused by the protein motor Prestin, the loudness curve in [33] also showed hearing deterioration in that area. The simulated hearing loss around 10,600 Hz seems to be more severe than the measured results in [33]. This is because an ideal hard boundary condition was given to the side walls of the scala vestibule and scala tympani, and the leakage of the sound waves through the temporal bone was ignored in the simulation. We expect that, if the leakage of the sound waves was considered as demonstrated in [34,35], the severe hearing loss in our simulation would be milder and closer to the actual hearing properties of humans.

## 3. Hearing Loss by Round Window Atresia

#### 3.1. Round Window Atresia

#### 3.2. Reflection Properties of the Round Window with Round Window Atresia

_{s}= 35 mm and cross-sections of W

_{bs}= 1.2 mm and H

_{bs}= 1.235 mm at x = 3 mm, and W

_{as}= 0.7 mm and H

_{as}= 0.745 mm at x = 38 mm) and the air-filled region (dimensions of L

_{a}= 3 mm, W

_{a}= 1.2 mm, and H

_{a}= 1.253 mm) faced each other across the round window. Since the actual round window has an elliptical shape, the lengths of the major and minor axes of the round window were set as W

_{r}= 1.12 mm and H

_{r}= 0.69 mm. The membrane of the round window was treated as an elastic material with a thickness of T

_{r}= 70 µm, a Young’s modulus of E, a Poisson ratio of 0.49, and a mass density of 1200 kg/m

^{3}, whose peripheral part was fixed to a partition separating these two regions. An ideal hard boundary condition, that is, the condition that the normal component of the particle velocity is always zero on the wall surface, was applied to the inner walls and the partition. An observation line was set on the x-axis to evaluate the sound pressure level in the air-filled and perilymph-filled regions. Then, a continuous sinusoidal plane wave with a sound pressure level of 1 Pa and a frequency of 5000 Hz was excited from Input a of the perilymph-filled region.

- -
- E = 1 MPa (normal mobility—healthy ear);
- -
- E = 100 MPa (lack of mobility to some extent—mild round window atresia);
- -
- E = 10 GPa (complete lack of mobility—severe round window atresia).

#### 3.3. Demonstration of Round Window Atresia

_{EVEN}(0 mm, t) = 1.1 Pa at the base and P

_{EVEN}(35 mm, t) = 1.5 Pa at the apex, while in the case of the healthy ear, the sound pressure levels of the even mode were P

_{EVEN}(0 mm, t) = 0.20 Pa at the base and P

_{EVEN}(35 mm, t) = 0.26 Pa at the apex, as presented in Figure 3b. This means that the even mode becomes dominant when round window atresia is becoming severe, and the excitation of the odd mode is weakened. As a result, the displacement of the basilar membrane, which is excited by the odd mode, is largely reduced and the hearing loss will become more severe.

#### 3.4. Hearing with Round Window Atresia

## 4. Conclusions

- (1)
- The sound waves traveling in the cochlea were classified into the even and odd symmetric modes and were expressed by the sum of these modes;
- (2)
- The odd mode excited the displacement of the basilar membrane and generated the Békésy’s traveling wave on the membrane;
- (3)
- The even mode generated a standing wave in the cochlea due to a fixed-end reflection at the cochlea apex;
- (4)
- The acoustic properties of the cochlea were determined by the contributions of the even and odd modes.

- (5)
- When the Young’s modulus E of the round window membrane is normal (e.g., a healthy ear with E = 1 MPa), the round window membrane provides a free-end reflection condition against the sound waves traveling in the scala tympani heading to the round window;
- (6)
- When the Young’s modulus E of the round window membrane is higher (e.g., an ear with severe round window atresia E = 10 GPa), the round window membrane provides a fixed-end reflection condition against the sound waves traveling in the scala tympani heading to the round window;
- (7)
- It is reported from the clinical medicine perspective that patients who have round window atresia tend to have their hearing ability degraded by 10 dB to 20 dB below 4000 Hz. Our simulation results show good agreement with the reported symptoms, and this ensures that our approach to round window atresia is correct.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A straight-tapered cochlea model configured by scala vestibuli, scala media, and scala tympani. The scala vestibuli and scala tympani were connected by a helicotrema at the cochlea apex. The scala vestibuli, scala tympani, and helicotrema were filled with compressible perilymph, and a basilar membrane was embedded in the scala media. A continuous sinusoidal plane wave was excited at the oval window. The dimensions were L

_{c}= 35 mm, W

_{bc}= 1.2 mm, H

_{bc}= 2.5 mm, W

_{ac}= 0.7 mm, H

_{ac}= 1.5 mm, L

_{m}= 34 mm, W

_{bm}= 100 µm, H

_{bm}= 30 µm, W

_{am}= 500 µm, H

_{am}= 10 µm, D

_{h}= 0.65 mm, L

_{a}= 3 mm, W

_{a}= 1.2 mm, H

_{a}= 1.235 mm, W

_{r}= 1.12 mm, H

_{r}= 0.69 mm, and T

_{r}= 70 µm.

**Figure 2.**Evaluation of sound pressure levels in the scala vestibuli and scala tympani, and the displacement of the basilar membrane when a continuous sinusoidal plane wave with a pressure level of 1 Pa and a frequency of 5000 Hz was excited at the oval window. The horizontal axis shows the position in the cochlea. The observation times t = 38.84 ms and 38.94 ms were chosen by trial and error so that the waveform swing P

_{EVEN}became the maximum or the minimum after the waveform reached a steady state. (

**a**) Maximum (solid line, t = 38.84 ms) and minimum (dashed line, t = 38.94 ms) sound pressure levels in the scala vestibuli (in blue) and scala tympani (in red). (

**b**) Sound pressure levels in the scala vestibule (in blue) and scala tympani (in red), and the displacement of the basilar membrane (in light gray). The upper and lower graphs present the simulation results observed at t = 38.84 ms and t = 38.94 ms, respectively. (

**c**) Sound pressure levels of the even and odd modes, and the displacement of the basilar membrane (in light gray), observed at t = 38.84 ms (the upper graph) and t = 38.94 ms (the lower graph). The sound pressure levels of the odd modes in the scala vestibuli and scala tympani are shown in blue and in red, respectively.

**Figure 3.**The upper graph shows the sound pressure levels in the scala vestibuli (in blue) and scala tympani (in red). The lower graph presents the sound pressure levels of the even mode (in black) and odd modes in the scala vestibuli (in blue) and scala tympani (in red), and the displacement of the basilar membrane (in light gray) when a continuous sinusoidal plane wave with a pressure level of 1 Pa was excited at the oval window. The horizontal axis shows the position in the cochlea. The observation times t = 30.096 ms, 38.840 ms, and 21.090 ms were chosen by trial and error so that the waveform swing P

_{EVEN}became the maximum after the waveform reached a steady state. (

**a**) f = 2000 Hz, t = 30.096 ms. (

**b**) f = 5000 Hz, t = 38.840 ms. (

**c**) f = 10,600 Hz, t = 21.090 ms.

**Figure 4.**Frequency dependence of the displacement of the basilar membrane when a continuous sinusoidal plane wave of 1 Pa was excited at the oval window.

**Figure 5.**Rectangular acoustic tube model configured by an air-filled region and a perilymph-filled region, both of which were connected by an elliptical-shaped elastic membrane. The dimensions were L

_{s}= 35 mm, W

_{bs}= 1.2 mm, H

_{bs}= 1.235 mm, W

_{as}= 0.7 mm, H

_{as}= 0.745 mm, L

_{a}= 3 mm, W

_{a}= 1.2 mm, H

_{a}= 1.253 mm, W

_{r}= 1.12 mm, H

_{r}= 0.69 mm, and T

_{r}= 70 µm.

**Figure 6.**Sound pressure levels evaluated on sound observation line aa’ in Figure 5. Assuming three stages of round window atresia, the Young’s modulus of the round window membrane was set as E = 1 MPa (a healthy ear, in black), E = 100 MPa (an ear with mild round window atresia, in blue), and E = 10 GPa (an ear with severe round window atresia, in red). The position from 0 mm to 3 mm corresponds to the air-filled middle ear region, and the position from 3 mm to 38 mm corresponds to the perilymph-filled scala tympani region. A sinusoidal plane wave with a sound pressure level of 1 Pa and a frequency of 5000 Hz was set for the excitation.

**Figure 7.**Displacements of the round window for three stages of round window atresia. The upper graph shows the maximum displacement of the elliptical round window membrane evaluated on its long axis. Position 0 mm corresponds to the center of the round window membrane. The lower ones also show the displacement of the round window membrane expressed by colors. A sinusoidal plane wave with a sound pressure level of 1 Pa and a frequency of 5000 Hz was set for the excitation.

**Figure 8.**Analysis of an ear with severe round window atresia (E = 10 GPa). The upper graph shows the sound pressure levels in the scala vestibuli (in blue) and scala tympani (in red). The middle graph shows the sound pressure level of the even mode (in black). The lower graph presents the sound pressure levels of the odd modes in the scala vestibuli (in blue) and scala tympani (in red), and the displacement of the basilar membrane (in light gray) when a sinusoidal plane wave with a pressure level of 1 Pa and a frequency of 5000 Hz was excited at the oval window. The horizontal axis shows the position in the cochlea.

**Figure 9.**Frequency response of the maximum displacement of the basilar membrane when a sinusoidal plane wave with a pressure level of 1 Pa was excited at the oval window. Displacements of the basilar membrane for a healthy ear (E = 1 MPa, in black), an ear with mild round window atresia (E = 100 MPa, in blue), and an ear with severe round window atresia (E = 10 GPa, in red) are presented, where E stands for the Young’s modulus of the round window membrane.

**Figure 10.**Estimation of the hearing level of the whole auditory system, including the properties of the outer, middle, and inner ears, when the ear is healthy. The upper graph shows the sound pressure level at the oval window when a sinusoidal plane wave with a pressure level of 1 Pa was excited at the pinna [42,43]. The middle graph shows the displacement of the basilar membrane when the pressure levels estimated above were used for sound wave excitation at the oval window. The lower graph presents the hearing level of the whole human auditory system. The result is expressed in “dB”, normalized at 1000 Hz, and redrawn so as to be compared to the general audiogram.

**Figure 11.**Hearing level of the whole auditory system estimated for a healthy ear (E = 1 MPa, in black), an ear with mild round window atresia (E = 100 MPa, in blue), and an ear with severe round window atresia (E = 10 GPa, in red), where E stands for the Young’s modulus of the round window membrane.

FEM Mesh Settings (typical cochlea model) | |
---|---|

Maximum mesh size | 1000 μm |

Minimum mesh size | 10 μm |

Mesh generation | automatic |

Number of mesh elements | 1,033,262 |

FEM Settings for Time Domain Analysis | |

Time step | 0.01 ms |

Time range | 0 ms – 40 ms |

Computation time (depend on convergence) | 48 h – 72 h |

FEM Material Settings | |

Compressible perilymph | |

Viscosity | 0.7027 mPa·s |

Density | 994.6 kg/m^{3} |

Sound velocity | 1520 m/s |

Basilar membrane | |

Density | 1200 kg/m^{3} |

Young’s modulus | 1 MPa |

Poisson’s ratio | 0.49 |

Round window membrane | |

Density | 1200 kg/m^{3} |

Young’s modulus | 1 MPa |

Poisson’s ratio | 0.49 |

PC specification | |

CPU | Corei9–7980XE |

Clock | 2.6 GHz |

Memory | 128 GB |

OS | Win 10 Pro 64bit |

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© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hong, W.; Horii, Y.
Simulation-Based Study on Round Window Atresia by Using a Straight Cochlea Model with Compressible Perilymph. *Acoustics* **2022**, *4*, 345-361.
https://doi.org/10.3390/acoustics4020021

**AMA Style**

Hong W, Horii Y.
Simulation-Based Study on Round Window Atresia by Using a Straight Cochlea Model with Compressible Perilymph. *Acoustics*. 2022; 4(2):345-361.
https://doi.org/10.3390/acoustics4020021

**Chicago/Turabian Style**

Hong, Wenjia, and Yasushi Horii.
2022. "Simulation-Based Study on Round Window Atresia by Using a Straight Cochlea Model with Compressible Perilymph" *Acoustics* 4, no. 2: 345-361.
https://doi.org/10.3390/acoustics4020021