^{1,a)}, Sunil Puria

^{1}and Charles R. Steele

^{1}

### Abstract

Intracochlear pressure is calculated from a physiologically based, three-dimensional gerbil cochlea model.Olson [J. Acoust. Soc. Am.103, 3445–3463 (1998); 110, 349–367 (2001)]measured gerbil intracochlear pressure and provided approximations for the following *derived quantities*: (1) basilar membrane velocity, (2) pressure across the organ of Corti, and (3) partition impedance. The objective of this work is to compare the calculations and measurements for the pressure at points and the derived quantities. The model includes the three-dimensional viscous fluid and the pectinate zone of the elastic orthotropic basilar membrane with dimensional and material property variation along its length. The arrangement of outer hair cell forces within the organ of Corti cytoarchitecture is incorporated by adding the feed-forward approximation to the passive model as done previously. The intracochlear pressure consists of both the compressive fast wave and the slow traveling wave. A Wentzel–Kramers–Brillowin asymptotic and numerical method combined with Fourier series expansions is used to provide an efficient procedure that requires about to compute the response for a given frequency. Results show reasonably good agreement for the direct pressure and the derived quantities. This confirms the importance of the three-dimensional motion of the fluid for an accurate cochlear model.

This work was funded by HFSP Grant No. RGP0051 and NIDCD of NIH Grant No. DC007910.

I. INTRODUCTION

II. MATHEMATICAL METHODS

A. Passive model

B. Feed-forward active model

C. BM displacement and intracochlear pressure

1. BM displacement

2. Intracochlear pressure

III. RESULTS

A. BF-to-place map

B. Frequency response of BM velocity

C. Intracochlear pressure

1. Slow wave intracochlear pressure

2. Combined slow and fast wave intracochlear pressure

D. Derived quantities from the intracochlear pressure

1. Derived BM velocity

2. Derived pressure across OC

3. Derived OC impedance

4. Comparison between estimated and exact theoretical OC impedance

5. Comparison between exact theoretical passive and active OC impedance

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Fluid equations
- 15.0
- Sound pressure
- 14.0
- Auditory system models
- 11.0
- Anatomy
- 10.0
- Acoustic waves
- 9.0

## Figures

Schematic drawing of the passive cochlear model geometric layout. Distances are parametrized by the Cartesian coordinates , which represent the distance from the stapes, the distance across the scala width, and the height above the partition, respectively. (a) Side, (b) cross section , and (c) top views of the model.

Schematic drawing of the passive cochlear model geometric layout. Distances are parametrized by the Cartesian coordinates , which represent the distance from the stapes, the distance across the scala width, and the height above the partition, respectively. (a) Side, (b) cross section , and (c) top views of the model.

Schematic of the longitudinal view of organ of Corti, showing the longitudinal tilt of the outer hair cells. The longitudinal distance between the base and apex of the outer hair cells is defined as . The force on the BM to the neighboring OHCs is .

Schematic of the longitudinal view of organ of Corti, showing the longitudinal tilt of the outer hair cells. The longitudinal distance between the base and apex of the outer hair cells is defined as . The force on the BM to the neighboring OHCs is .

Best frequency (BF) vs position for the passive cochlear model (solid line) compared to measurements (asterisk), and active cochlear model (dashed-dot line). The present 3D cochlear model represents the cochlear BF-to-place map of gerbil (Sokolich *et al.*, 1976; Greenwood, 1990) over range spanning a length of .

Best frequency (BF) vs position for the passive cochlear model (solid line) compared to measurements (asterisk), and active cochlear model (dashed-dot line). The present 3D cochlear model represents the cochlear BF-to-place map of gerbil (Sokolich *et al.*, 1976; Greenwood, 1990) over range spanning a length of .

Basilar membrane (BM) velocity relative to the stapes magnitude (a) and corresponding phase (b) for the gerbil cochlea at from the base . For the active model, , (dashed line) and , (dotted line) were used while for the passive case . Experimental data (expt.) for 30 and SPL corresponding to the active and passive case, respectively, are included for comparison (Ren and Nuttall, 2001). Dashed-dot line in (b): Phase from the model at the from the stapes.

Basilar membrane (BM) velocity relative to the stapes magnitude (a) and corresponding phase (b) for the gerbil cochlea at from the base . For the active model, , (dashed line) and , (dotted line) were used while for the passive case . Experimental data (expt.) for 30 and SPL corresponding to the active and passive case, respectively, are included for comparison (Ren and Nuttall, 2001). Dashed-dot line in (b): Phase from the model at the from the stapes.

Radial distribution of intracochlear pressure from the slow wave at different distances from the partition ( from the stapes, ). The location of the BM is indicated by the thickened line (BM ). The pressure drops exponentially with the distance from the BM in either perpendicular or radial direction.

Radial distribution of intracochlear pressure from the slow wave at different distances from the partition ( from the stapes, ). The location of the BM is indicated by the thickened line (BM ). The pressure drops exponentially with the distance from the BM in either perpendicular or radial direction.

Combined slow and fast wave intracochlear pressure in the scala tympani (ST) of the gerbil. (a) Intracochlear pressure magnitude and (b) corresponding phase at from the stapes and 3 and away from the BM ( SPL at the stapes: passive). Data are from Olson, (1998, Fig. 10) expt. 2-26. (c) Intracochlear pressure magnitude and (d) corresponding phase at from the stapes and away from the BM ( SPL: passive and SPL: active case). Data are from Olson (2001, Fig. 7) expt. 9-8-98-I-usual.

Combined slow and fast wave intracochlear pressure in the scala tympani (ST) of the gerbil. (a) Intracochlear pressure magnitude and (b) corresponding phase at from the stapes and 3 and away from the BM ( SPL at the stapes: passive). Data are from Olson, (1998, Fig. 10) expt. 2-26. (c) Intracochlear pressure magnitude and (d) corresponding phase at from the stapes and away from the BM ( SPL: passive and SPL: active case). Data are from Olson (2001, Fig. 7) expt. 9-8-98-I-usual.

Schematic 3D drawing of the cochlear model. Intracochlear pressures and are measured and calculated at the indicated positions and , 15 and away from the BM in the ST respectively, and in SV . These are used to obtain an approximation (Olson, 1998, 2001) for the BM velocity, pressure difference and impedence, and organ of Corti impedance referred to as derived quantities. Cross indicates distance from the stapes.

Schematic 3D drawing of the cochlear model. Intracochlear pressures and are measured and calculated at the indicated positions and , 15 and away from the BM in the ST respectively, and in SV . These are used to obtain an approximation (Olson, 1998, 2001) for the BM velocity, pressure difference and impedence, and organ of Corti impedance referred to as derived quantities. Cross indicates distance from the stapes.

Derived BM velocity from the gerbil cochlear model and measurements, using the formulas in Olson [1998, Appendix 1, Eq. (A7)]. (a) Magnitude of the measurement results for 40 and SPL at the ear canal and model results at 70 and at the stapes are plotted re: 0.01 and , respectively ( from the stapes, ). (b) Phase relative to SV pressure at from the stapes. Data are from Olson, (1998, Fig. 18) expt. 2-26 at the stimulus levels of 40 and SPL at the ear canal. (c) Derived BM velocity magnitude and (d) corresponding phase at from the stapes for the passive ( SPL) and active case ( SPL). Data are from Olson (2001, Figs. 15(a) and 15(b)] expt. 9-8-98-I-usual.

Derived BM velocity from the gerbil cochlear model and measurements, using the formulas in Olson [1998, Appendix 1, Eq. (A7)]. (a) Magnitude of the measurement results for 40 and SPL at the ear canal and model results at 70 and at the stapes are plotted re: 0.01 and , respectively ( from the stapes, ). (b) Phase relative to SV pressure at from the stapes. Data are from Olson, (1998, Fig. 18) expt. 2-26 at the stimulus levels of 40 and SPL at the ear canal. (c) Derived BM velocity magnitude and (d) corresponding phase at from the stapes for the passive ( SPL) and active case ( SPL). Data are from Olson (2001, Figs. 15(a) and 15(b)] expt. 9-8-98-I-usual.

Derived pressure across the OC complex, , from the gerbil cochlear model and measurements, using the formulas in Olson [1998, Appendix 2, Eq. (A10)]. (a) Magnitude. (b) Phase relative to SV pressure at from the stapes . Data are from Olson (1998, Fig. 19) expt. 2-26 at the stimulus levels of 40 and SPL at the ear canal. (c) Derived magnitude and (d) corresponding phase at from the stapes for the passive ( SPL) and active case ( SPL). Data are from Olson [2001, Figs. 15(a) and 15(c)] expt. 9-8-98-I-usual.

Derived pressure across the OC complex, , from the gerbil cochlear model and measurements, using the formulas in Olson [1998, Appendix 2, Eq. (A10)]. (a) Magnitude. (b) Phase relative to SV pressure at from the stapes . Data are from Olson (1998, Fig. 19) expt. 2-26 at the stimulus levels of 40 and SPL at the ear canal. (c) Derived magnitude and (d) corresponding phase at from the stapes for the passive ( SPL) and active case ( SPL). Data are from Olson [2001, Figs. 15(a) and 15(c)] expt. 9-8-98-I-usual.

Derived impedance of organ of Corti from the gerbil cochlear model and measurements, using the formulas in Olson (1998). (a) Magnitude. (b) Phase of for model and measurements at from the stapes . Data are from Olson (1998, Fig. 20) expt. 2-26. (c) Real part of and (d) imaginary part of at from the stapes for the passive ( SPL) and active case ( SPL). Data are from Olson [2001, Figs. 15(d) and 15(e)] expt. 9-8-98-I-usual.

Derived impedance of organ of Corti from the gerbil cochlear model and measurements, using the formulas in Olson (1998). (a) Magnitude. (b) Phase of for model and measurements at from the stapes . Data are from Olson (1998, Fig. 20) expt. 2-26. (c) Real part of and (d) imaginary part of at from the stapes for the passive ( SPL) and active case ( SPL). Data are from Olson [2001, Figs. 15(d) and 15(e)] expt. 9-8-98-I-usual.

Exact and derived theoretical impedance of organ of Corti from the gerbil cochlear model ( from the stapes, ). The passive model is presented. (a) Magnitude. (b) Phase.

Exact and derived theoretical impedance of organ of Corti from the gerbil cochlear model ( from the stapes, ). The passive model is presented. (a) Magnitude. (b) Phase.

Exact theoretical impedance of organ of Corti from the gerbil cochlear passive and active model ( from the stapes, ). The 0.15 feed-forward gain factor is used in the active model. (a) Magnitude. (b) Phase.

Exact theoretical impedance of organ of Corti from the gerbil cochlear passive and active model ( from the stapes, ). The 0.15 feed-forward gain factor is used in the active model. (a) Magnitude. (b) Phase.

## Tables

Material properties for the gerbil cochlear model.

Material properties for the gerbil cochlear model.

Anatomical dimensions as a function of longitudinal position for the gerbil cochlear model.

Anatomical dimensions as a function of longitudinal position for the gerbil cochlear model.

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