^{1,a)}and Jont B. Allen

^{2}

### Abstract

In order to better understand signal propagation in the ear, a time-domain model of the tympanic membrane (TM) and of the ossicular chain (OC) is derived for the cat. Ossicles are represented by a two-port network and the TM is discretized into a series of transmission lines, each one characterized by its own delay and reflection coefficient. Volume velocity samples are distributed along the ear canal, the eardrum, and the middle ear, and are updated periodically to simulate wave propagation. The interest of the study resides in its time-domain implementation—while most previous related works remain in the frequency domain—which provides not only a direct observation of the propagating wave at each location, but also insight about how the wave behaves at the ear canal/TM interface. The model is designed to match a typical impedance behavior and is compared to previously published measurements of the middle ear (the canal, the TM, the ossicles and the annular ligament). The model matches the experimental data up to .

I. INTRODUCTION

II. BASIC ASSUMPTIONS

III. METHODS

A. Tympanic membrane

1. Anatomical description

2. Distributed model

3. Reflection coefficient function

4. Interface with the ear canal

5. Power conservation

B. Ossicular chain

1. Anatomical description

2. Lumped-parameter OC model

IV. RESULTS

A. Impedance-related measurements

1. Case I: Blocked tympanic membrane

2. Case II: Pathological ears

3. Case III: Intact ear

B. Ossicles displacement ratios

V. DISCUSSION

A. Attic ligaments

B. Tympanic membrane cutoff

VI. SUMMARY

### Key Topics

- Auditory system models
- 38.0
- Middle ear mechanics
- 26.0
- Reflection coefficient
- 12.0
- Acoustic waves
- 9.0
- Acoustic modeling
- 5.0

## Figures

Circuit representation for an element of transmission line. The series impedance is noted and the shunt admittance is noted , in terms of their per unit length distributions (i.e., an impedance will be of the form ).

Circuit representation for an element of transmission line. The series impedance is noted and the shunt admittance is noted , in terms of their per unit length distributions (i.e., an impedance will be of the form ).

Discretized tympanic membrane model for . (a) Position of the TM in the ear canal. (b) The decomposition in annuli, respecting the circular geometry of the membrane. (c) The one-dimensional model that we actually use, derived from the previous one by applying mass conservation, indicated by the different widths of the stripes. The number of samples on TM is , i.e., nine in this example.

Discretized tympanic membrane model for . (a) Position of the TM in the ear canal. (b) The decomposition in annuli, respecting the circular geometry of the membrane. (c) The one-dimensional model that we actually use, derived from the previous one by applying mass conservation, indicated by the different widths of the stripes. The number of samples on TM is , i.e., nine in this example.

Tympanic membrane reflection coefficient for a discretized membrane with 71 annuli, i.e., 141 samples. Note the asymmetry in , and its maximum, at the TM center. At the canal termination, the canal impedance is actually greater than the TM impedance, which results in negative reflection coefficients.

Tympanic membrane reflection coefficient for a discretized membrane with 71 annuli, i.e., 141 samples. Note the asymmetry in , and its maximum, at the TM center. At the canal termination, the canal impedance is actually greater than the TM impedance, which results in negative reflection coefficients.

Interface between the ear canal and the tympanic membrane. Forward and backward propagation path are represented by solid and hollow arrow heads, respectively. Rectangles represent transmission line delays with length proportional to delay. In the first section from the left, at the canal termination, the last forward sample is multiplexed into the interface transmission lines and scaled according to the ratio of the stripe area over the total TM surface area . The second section from the left represents transmission lines showing the wave in air under the TM; due to the inclination of the drum, they bring different delays. At the interface between the canal (air) and the membrane (the middle or third section), the wave is split into a transmitted and a reflected part, computed from the knowledge of the reflection coefficient on the TM (see Fig. 3). In the fourth section the transmitted part propagates on the TM toward the umbo (right-most series of transmission lines) where finally, in the fifth section, all contributions are added and then feed the OC. The reflected part at the canal/TM interface propagates in air back to the canal input. In the backward propagation, the wave coming from the OC is only input into the TM superior region, where it is in contact with the manubrium, and is multiplexed and scaled according to the ratio of the stripes areas over the superior region area, . It propagates to the canal input using the same path than the forward-going wave. All contributions are then added before being input into the canal transmission line. Note that our actual implementation takes into account the future possibility of adding an input from the middle ear cavity space into the inferior region, but these (velocity) inputs are presently zeroed (far lower-right corner).

Interface between the ear canal and the tympanic membrane. Forward and backward propagation path are represented by solid and hollow arrow heads, respectively. Rectangles represent transmission line delays with length proportional to delay. In the first section from the left, at the canal termination, the last forward sample is multiplexed into the interface transmission lines and scaled according to the ratio of the stripe area over the total TM surface area . The second section from the left represents transmission lines showing the wave in air under the TM; due to the inclination of the drum, they bring different delays. At the interface between the canal (air) and the membrane (the middle or third section), the wave is split into a transmitted and a reflected part, computed from the knowledge of the reflection coefficient on the TM (see Fig. 3). In the fourth section the transmitted part propagates on the TM toward the umbo (right-most series of transmission lines) where finally, in the fifth section, all contributions are added and then feed the OC. The reflected part at the canal/TM interface propagates in air back to the canal input. In the backward propagation, the wave coming from the OC is only input into the TM superior region, where it is in contact with the manubrium, and is multiplexed and scaled according to the ratio of the stripes areas over the superior region area, . It propagates to the canal input using the same path than the forward-going wave. All contributions are then added before being input into the canal transmission line. Note that our actual implementation takes into account the future possibility of adding an input from the middle ear cavity space into the inferior region, but these (velocity) inputs are presently zeroed (far lower-right corner).

Ossicular chain circuit representation. Each element is modeled by a two port with four filters given by the matrix of Eq. (17). The multipliers are frequency-dependent, and thus must be implemented as convolutions in the time domain. This is done using a bilinear transformation of the reflectance function in the frequency domain. this operation converts the Laplace-domain formula into its digital domain equivalent. It replaces the Laplace variables by , where is the sampling rate. represents the filter computing the forward output from the backward input, the backward output from the forward input, etc. The stapes/cochlea association is modeled by only one filter (Lynch *et al.*, 1982). The OC lever ratio is represented by the transformer between the malleus and the incus, its ratio is denoted . Malleus impedance is matched with the TM central impedance, and incus impedance is equal to .

Ossicular chain circuit representation. Each element is modeled by a two port with four filters given by the matrix of Eq. (17). The multipliers are frequency-dependent, and thus must be implemented as convolutions in the time domain. This is done using a bilinear transformation of the reflectance function in the frequency domain. this operation converts the Laplace-domain formula into its digital domain equivalent. It replaces the Laplace variables by , where is the sampling rate. represents the filter computing the forward output from the backward input, the backward output from the forward input, etc. The stapes/cochlea association is modeled by only one filter (Lynch *et al.*, 1982). The OC lever ratio is represented by the transformer between the malleus and the incus, its ratio is denoted . Malleus impedance is matched with the TM central impedance, and incus impedance is equal to .

Tympanic membrane for the *blocked-TM* (clamped-umbo) condition: time-domain reflectance and TM output, reflectance and impedance magnitudes. (a) Reflectance (time signal), (b) TM output, (c) reflectance magnitude, (d) impedance magnitude.

Tympanic membrane for the *blocked-TM* (clamped-umbo) condition: time-domain reflectance and TM output, reflectance and impedance magnitudes. (a) Reflectance (time signal), (b) TM output, (c) reflectance magnitude, (d) impedance magnitude.

In this figure we compare the experimental data for the disarticulated stapes (DS) [dashed] with two model simulations, the disarticulated stapes (DS) [solid] and the drained cochlea (DC) [dotted]. In the left column are four input impedance measures: (a) the impedance magnitude, (b) the reflectance magnitude, (c) the impedance imaginary part, and (d) the impedance phase. In the right column are: (e) the power transmittance, (f) the reflectance phase, (g) the impedance real part, and (h) the reflectance group delay. As discussed by Allen (1986), removing the stapes reduces the stiffness below by about a factor of 3, as shown in (a) for the model calculations, but otherwise has only a small effect, especially above .

In this figure we compare the experimental data for the disarticulated stapes (DS) [dashed] with two model simulations, the disarticulated stapes (DS) [solid] and the drained cochlea (DC) [dotted]. In the left column are four input impedance measures: (a) the impedance magnitude, (b) the reflectance magnitude, (c) the impedance imaginary part, and (d) the impedance phase. In the right column are: (e) the power transmittance, (f) the reflectance phase, (g) the impedance real part, and (h) the reflectance group delay. As discussed by Allen (1986), removing the stapes reduces the stiffness below by about a factor of 3, as shown in (a) for the model calculations, but otherwise has only a small effect, especially above .

In this figure we compare the experimental data for the intact ear [dashed] with the model simulation [solid]. In the left column are four input impedance measures: (a) the magnitude impedance, (b) reflectance magnitude, (c) the impedance imaginary part, and (d) the impedance phase. In the right column are: (e) the power transmittance, (f) the reflectance phase, (g) the impedance real part, and and (h) the reflectance group delay. The intact ear brings a major change with respect to the pathological ears: due to the increased load impedance, standing waves are damped out and reflectance is much lower, overall.

In this figure we compare the experimental data for the intact ear [dashed] with the model simulation [solid]. In the left column are four input impedance measures: (a) the magnitude impedance, (b) reflectance magnitude, (c) the impedance imaginary part, and (d) the impedance phase. In the right column are: (e) the power transmittance, (f) the reflectance phase, (g) the impedance real part, and and (h) the reflectance group delay. The intact ear brings a major change with respect to the pathological ears: due to the increased load impedance, standing waves are damped out and reflectance is much lower, overall.

Ossicles displacement ratios: magnitude and phase. The model incus to malleus displacement ratio is the solid line, the model stapes to incus displacement ratio is the dashed line, symbols are cats 58, 65, 68, and 69 from Guinan and Peake (1967). (a) Ratio magnitude, (b) ratio phase.

Ossicles displacement ratios: magnitude and phase. The model incus to malleus displacement ratio is the solid line, the model stapes to incus displacement ratio is the dashed line, symbols are cats 58, 65, 68, and 69 from Guinan and Peake (1967). (a) Ratio magnitude, (b) ratio phase.

## Tables

Model parameter values .

Model parameter values .

Model parameter values .

Model parameter values .

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