intracochlear pressure and derived quantities from …chasst/cochlea/bingabr/yoonpur...intracochlear...

Intracochlear pressure and derived quantities from athree-dimensional model

Yong-Jin Yoon,a� Sunil Puria, and Charles R. SteeleDepartment of Mechanical Engineering and Department of Otolaryngology-Head and Neck Surgery,Stanford University, Stanford, California 94305-4035

�Received 10 October 2006; revised 9 May 2007; accepted 10 May 2007�

Intracochlear pressure is calculated from a physiologically based, three-dimensional gerbil cochleamodel. Olson �J. Acoust. Soc. Am. 103, 3445–3463 �1998�; 110, 349–367 �2001�� measured gerbilintracochlear pressure and provided approximations for the following derived quantities: �1� basilarmembrane velocity, �2� pressure across the organ of Corti, and �3� partition impedance. Theobjective of this work is to compare the calculations and measurements for the pressure at pointsand the derived quantities. The model includes the three-dimensional viscous fluid and the pectinatezone of the elastic orthotropic basilar membrane with dimensional and material property variationalong its length. The arrangement of outer hair cell forces within the organ of Corti cytoarchitectureis 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 seriesexpansions is used to provide an efficient procedure that requires about 1 s to compute the responsefor a given frequency. Results show reasonably good agreement for the direct pressure and thederived quantities. This confirms the importance of the three-dimensional motion of the fluid for anaccurate cochlear model. © 2007 Acoustical Society of America. �DOI: 10.1121/1.2747162�

PACS number�s�: 43.64.Kc, 43.64.Bt �BLM� Pages: 952–966

I. INTRODUCTION

The cochlea is a snail-shaped, fluid-filled duct which isdivided along its longitudinal direction by the compliantbasilar membrane �BM�, on which is located the organ ofCorti �OC�. The fluid and compliant structures within thecochlea are set in motion in response to sound input at thestapes, and the detection of this motion by inner hair cellsinitiates hearing through afferent auditory nerve firing trans-mitted to the auditory brainstem. The pressure differenceacross the OC is one of the driving forces of the motion ofOC, and this motion has been the subject of intracochlearexperimentation and cochlear models for analysis.

This study is motivated by the measurements of intra-cochlear pressure �Olson, 1998, 2001�. Pressure near thestapes at scala vestibule �SV�, which is the “input” pressureto the inner ear, and pressure at the scala tympani �ST�through the round window �RW� opening was measuredfrom the gerbil cochlea in vivo. Intracochlear pressure at anumber of positions spaced by tens of micrometers was mea-sured to obtain the localized pressure and the pressure gra-dients which indicate fluid motion in the base of the gerbilcochlea.

In this study, the mechanical behavior of the cochlea issimulated with a physiologically-based, three-dimensional�3D� cochlear model. Results are compared with the experi-mental data for the best frequency �best frequency �BF��-to-place map, BM velocity, intracochlear pressure, and quanti-

a�
Electronic mail: [email protected]
952 J. Acoust. Soc. Am. 122 �2�, August 2007 0001-4966/2007/1

ties derived from the pressure, using the formulas of: �1� BMvelocity, �2� pressure difference across OC, and �3� OC im-pedance.

Numerous cochlear models have been used to explainthe biomechanical behavior of the cochlea. Models extendthe passive cochlear model with the inclusion of the motionof the OC, particularly the active behavior of the outer haircells �OHCs�. The simplified one-dimensional model withnegative damping by de Boer �1983� was extended to includenonlinearity in the activity using a quasilinear method �Kanisand de Boer, 1996, 1997�. Higher dimensional active modelshave also been developed. Two-dimensional finite differencemodels were constructed by using a feedback law �Neely,1985, 1993�. Numerically intense 3D finite-element modelshad been developed with the inclusion of varying details andcomplexities of the OC �Kolston and Ashmore, 1996;Böhnke and Arnold, 1998�. However, the fluid was modeledas inviscid, which does not require as fine a mesh. Finally,models including the activity in the OC as a feed-forwardmechanism which took into account the longitudinal tilt ofthe OHCs had been developed. Two-dimensional �Geislerand Sang, 1995� and 3D models with the active feed-forwardmechanism has been developed �Steele et al., 1993; Steeleand Lim, 1999; Lim and Steele, 2002�.

The present study uses the physiologically -based, linear3D feed-forward model. The model uses a combination ofthe asymptotic phase integral method that is commonlyknown as the Wentzel–Kramers–Brillouin �WKB� methodand the fourth-order Runge-Kutta �RK4� numerical forward
integration. This hybrid approach provides significantly
© 2007 Acoustical Society of America22�2�/952/15/$23.00

faster computations than the finite difference or finite ele-ment methods and more accuracy than the WKB alone �Limand Steele, 2002�.

The present model is as simple as possible to capture theessential features in the cochlea. Included in the model arethe variation of the dimensions and material properties alongthe cochlear duct, and 3D viscous fluid effects. On the organof Corti, only one degree of freedom, the flexing of the pec-tinate zone of the orthotropic BM, is considered. The spiralcoiling of the cochlea is also neglected, as it has been shown,in general, to have no significant effects on the model re-sponse �Loh, 1983; Steele and Zais, 1985�. The results fromthis active model successfully demonstrate various aspects ofa live cochlea, as observed by in vivo measurements.

II. MATHEMATICAL METHODS

The passive and active cochlear models are presented.First, the passive model with macroscopic features of thecochlea without OHCs motility is described. Next, the feed-forward active mechanism of the OHCs from the motion ofOC is formulated. Quantities of interest, including the BF-to-place map, BM velocity, and intracochlear pressure, werecomputed with the hybrid method which combines the WKBand RK4 methods. Intracochlear pressure combines the pres-sure from the compressive fast wave and the slow travelingwave. Finally, the derived quantities from the intracochlearpressure, �1� BM velocity, �2� pressure difference across OC,and �3� OC impedance, are compared with Olson’s measure-ments.

A. Passive model

The physical cochlea consists of a rigid bony housingcontaining two coiled, fluid-filled ducts, separated by a par-tition that is composed of rigid and compliant regions. Thegeometric properties of the ducts and the mechanical prop-erties of the partition vary along the length of the cochlea.The entire system is stimulated when the stapes displaces thecochlear fluid adjacent to the oval window �OW�, which liesat the base of the top fluid duct called scala vestibuli �SV�.
The model is based on these physiological features of the
J. Acoust. Soc. Am., Vol. 122, No. 2, August 2007

cochlea. A schematic drawing with the side, cross section,and top view is shown in Fig. 1. For simplicity, the duct hasbeen uncoiled and all boundaries made vertical or horizontal.

This model consists of a tapered chamber with rigidwalls filled with viscous fluid. The chamber is divided by acochlear partition into two equal ducts representing scalavestibule and scala tympani �ST�. The two fluid ducts arejoined at the apical end of the fluid chamber via a hole rep-resenting the helicotrema. The cochlear partition represents acollapsed scala media �SM� with its structural propertiesdominated by the pectinate zone of the BM. The pectinatezone of the BM is considered to be an orthotropic plate, inwhich the Young’s modulus in the transverse direction ismuch greater than that in the longitudinal direction. By in-cluding the variations of BM width, thickness, and fiber den-sity, the stiffness of the partition varies in the longitudinaldirection.

Two types of waves are set up in such a model, thesymmetric and the anti-symmetric pressure waves �Petersonand Bogert, 1950�. The symmetric pressure wave is a fastcompression wave with equal pressure on both sides of thepartition. The anti-symmetric pressure wave is a slow travel-ing wave that has pressure of opposite sign acting on the topand bottom of the partition. Consequently, the antisymmetricpressure wave causes a significant displacement of the parti-tion while the symmetric pressure wave does not result inmotion of the partition. In the present model, the antisym-metric pressure slow wave is taken into account for the BMmotion and the symmetric pressure fast wave is added to theslow traveling wave pressure for the intracochlear pressurecalculation.

Due to symmetry present in the model, only one fluidduct needs to be considered in the simulation. Also, takingadvantage of its slender nature, the cochlear duct is dividedalong its length into discrete cross-sectional slices. For eachcross section the 3D fluid displacement and pressure fieldsare computed using a Fourier series expansion. For eachcross section, the explicit expressions for the fluid displace-ments and fluid pressure at the partition are obtained, andthese are matched with the plate’s displacement and pressure

FIG. 1. Schematic drawing of the passive cochlearmodel geometric layout. Distances are parametrized bythe Cartesian coordinates �x ,y ,z�, which represent thedistance from the stapes, the distance across the scalawidth, and the height above the partition, respectively.�a� Side, �b� cross section �A-A�, and �c� top views ofthe model.

to give an eikonal equation. Solving the eikonal equation

Yoon et al.: Intracochlear pressure and derived quantities 953

yields the complex wave numbers for each cross section inthe cochlear duct. Using the continuity condition for the fluidacross the cross-sectional slices, a transport equation is ob-tained from which the amplitudes of the waves are obtained.The detailed derivations are given in the following.

The fluid displacement vector field u in the ducts is de-composed into divergence of a scalar field � �irrotationalcomponent� and the curl of a vector field � �rotational com-ponent�

u = �� + � Ã � . �1�

The displacement field from � satisfies the rigid wall bound-ary conditions at y=0, y=L2, and z=L3 where the normalfluid displacements are zero. A functional form of � thatsatisfies the above-presented boundary conditions is

��x,y,z,t� = e−i�t�j

��x�Tj�x�cos� j�y

L2

�cosh�� j�x��L3 − z�� , �2�

where � is the frequency, with a Fourier cosine series expan-sion used in the y direction. ��x� is the amplitude functionalong the x direction, while the Tj�x� coefficients allow thefluid to match the arbitrary displacement on the BM. Thecoefficients � j�x�, related to the wave-number n by the con-tinuity equation for the incompressible fluid ��=0�, re-duces to

� j�x� =n2 + � j�

L22

, �3�

where the wave-number n is defined by

n2 = −�xx

�. �4�

Due to the low viscosity of the fluid, the boundary layersare localized such that the boundary layers at the rigid wallshave no significant effects on the partition motion. Hence,only the boundary layer at the partition is considered. This isdescribed by the vector field � �with x, y, and z components,�1, �2, and �3� that assumes the form

��x,y,z,t� = ��1

�2

�3� = e−i�t�j

ej�x�z�� j1 sin� j�y

L2

� j2 cos� j�y

L2

0� .

�5�

Note that vector field � describes the rotational compo-nent of the fluid displacement field due to viscosity. Here, aharmonic excitation with frequency � is applied at thestapes. The coefficients � j

1 and � j2 are related to the ampli-

tude ��x� and Tj�x� through no-slip boundary conditions onthe cochlear partition where the tangent displacements arealso zero.

The Navier-Stokes equation for no body force, incom-
pressibility, and small displacement is
954 J. Acoust. Soc. Am., Vol. 122, No. 2, August 2007

f�� + � � �� = − �p + � � � �� . �6�

Matching the terms on each side of the equation gives thefollowing:

f�� = − p , �7a�

f�� = ��. �7b�

Equation �7a� relates the pressure acting on the fluid to thescalar potential � and the vorticity Eq. �7b� gives the condi-tion on the vector field, �.

The vorticity equation �Eq. �7.2�� can be expressed as

j�x� =� j2 − i�

f

�, �8�

where � is the dynamic viscosity of the fluid.For the BM partition, the plate bending equation is

phw� +�2

�x2�D11�2w

�x2 + 2�2

�x�y�D12

�2w

�x�y +

�2

�y2�D22�2w

�y2 = pp, �9�

where pp is the pressure acting on the pectinate zone, p isthe density of the plate, and D11, D12, and D22 are the bend-ing stiffness components which take into account the fiberdensity �f� and sandwiched construction of the BM,

Dij =fEij

1 − �2 I , �10�

where Eij is the Young’s modulus, � is Poisson’s ratio, andthe area moment of inertia I for symmetric layers is I=2 h/2−g

h/2 2d , where h is the thickness of the membrane, g isthe layer thickness, and is the distance through the thick-ness. Unlike most mammals, the gerbil BM is not symmetric,but has radial fibers concentrated in a curved tympanic bandand a flat band on the OC side in the pectinate zone. Sch-weitzer et al. �1996� find that the thickness of the tympanicband correlates with the postnatal maturity of hearing. Sincethe fiber density for gerbil is not known, the details of theBM mechanics into an effective volume fraction f , consistentwith the values from Cabezudo �1978� for cat, are lumpedwith I=h3 /12.

From the plate bending equation �Eq. �9��, the displace-ment profile of the partition in harmonic motion is

wp�x,�,t� = e−i�tW�x�sin��

b , �11�

where W�x� is the amplitude function. The fluid and partitiondisplacements are matched at their interface �with W�x�=��x�� and the coefficients Tj�x� are determined from thisassumed shape function of the displacement.

Integrating the pressure across the width and summingup the Fourier harmonics gives the force per unit length inthe time domain for the partition and fluid. For the pectinatezone �PZ� of the partition �Lim and Steele, 2002�,

FPZ�x,t� = � e−i�tFPZ�x,� j� , �12�

where FPZ�x ,� j�= �Kp�n ,x ,� j�−� j2Mp�x��W�x ,� j� with the

plate stiffness given by

Yoon et al.: Intracochlear pressure and derived quantities

Kp�n,x,�� =�

2b�− p�2h + D11n

4 + 2D12n2��

b2

+ D22��

b4 �13�

and mass given by

Mp =�

2bph . �14�

For the fluid

FBMf �x,t� = � e−i�tFf�x,�� , �15�

where

Ff�x,� j� = f� j2bHfW�x,� j� = � j

2MfW�x,� j� �16�

with Mf and Hf being the effective mass and thickness of thefluid layer over the width of the plate �Lim and Steele, 2002�.

Taking into account the asymmetric fluid pressure fromthe two fluid ducts �SV and ST�, and matching the coeffi-cients of stresses in Eqs. �12� and �15� give

FPZ�x,t� = 2FBMf �x,t� . �17�

Equation �17� represents the eikonal equation for the passivemodel, and it provides a physically consistent and systematicreduction of the 3D model to a one-dimensional formulationin spatial coordinates. The stiffness and mass quantities arereminiscent of those used in lumped parameter models, butthese are derived from the physics, and there are no freeparameters except for the exact value of volume fraction fand the effective thickness h that are selected for thefrequency-to-place map.

B. Feed-forward active model

The active elements in the cochlea are presumed as theOHCs which act like piezoelectric actuators that push on theBM partition to improve the cochlea’s sensitivity and fre-quency selectivity. In this model, the force applied by theOHCs on the BM partition is assumed to be proportional tothe total force acting on the BM. Equation �17� states thetotal force acting on the �PZ� results from the fluid forcedifference across the two scalae. To include forces resultingfrom the OHCs’ motility, an effective OHC force on the BM,FBM

C , is added to Eq. �17�:

FPZ = 2FBMf + FBM

C . �18�

The OHC force acting at x+� is proportional to the BMdisplacement sensed at x by the effect of the OHC longitu-dinal tilt shown in Fig. 2:

FBMC �x + �� = ��x�FPZ�x� , �19�

where � is the feed-forward gain factor and � is the longi-tudinal distance between the apex and base of the OHC. Thisdepends on the length of the OHC �lOHC� and angle to the
longitudinal direction ��,

� = lOHC cos � . �20�

Combining Eqs. �18� and �19� provides a relation that entersthe eikonal equation for the feed-forward active model �Limand Steele, 2002�.

C. BM displacement and intracochlear pressure

1. BM displacement

Each individual equation from Eq. �17� or Eq. �18� givesan eikonal equation from which the complex wave numberscan be obtained by Newton-Raphson iterative scheme. In thepresent formulation, positive and negative real parts corre-spond to forward and backward propagating waves, respec-tively. The amplitude of the propagating wave can be ob-tained from the transport equations which are obtained byconsidering the volume integral over a thin slice of the duct’scross section with differential volume �V=L2L3�x, �V��dV=0 and the transport equation is then reduced to anODE in x in the form of the well-known reduced wave equa-tion

G,xx + n�x,��2G = 0, �21�

in which n�x� is the local wave numbers, determined by solv-ing the 3D fluid equations for each cross section. The depen-dent variable G�x� provides the potential ��x� for the fluid,

��x� =G�x�n

T0 sinh�nL3�, �22�

where T0�x� is the Fourier coefficient for zeroth componentof scalar potential for fluid displacement and L3 is the heightof the fluid chamber. The function G�x� is obtained using acombination of the WKB asymptotic method in the shortwavelength region �n is large� and the RK4 forward integra-tion in the long wavelength region �n is small�. The boundaryconditions of matching the volume displacement at thestapes and zero pressure at the helicotrema are satisfied.

2. Intracochlear pressure

The pressure field in the real cochlea is a summation oftwo components. The first component is the traveling pres-sure wave where the fluid displacement is antisymmetric

FIG. 2. Schematic of the longitudinal view of organ of Corti, showing thelongitudinal tilt of the outer hair cells. The longitudinal distance between thebase and apex of the outer hair cells is defined as �. The force on the BM tothe neighboring OHCs is FBM

C .

about the partition. The second component is a compressive


wave where the fluid displacement is symmetric. The twopressure wave components are needed to satisfy simulta-neous boundary conditions located at the OW and RW.

a. Pressure from the slow traveling wave For a har-monic excitation with a frequency �, the fluid pressurethroughout the cochlear duct associated with the slow trav-eling wave follows from the functional form of the scalarpotential for the fluid displacement that satisfies boundaryconditions and the pressure acting on the fluid �Eqs. �2� and�7a� respectively�:

pt�x,y,z� = f�2�

j

��x�Tj�x�cosh�� j�x��L3 − z��

�cos� j�y

L2 . �23�

b. Pressure from the compressive fast wave The equi-librium and continuity equations are one dimensional for thefast wave acoustics subject to time-harmonic displacementsas follows:

dpc

dx= f�

2u , �24�

dq

dx= −

A

fc2 pc, �25�

where pc is the fluid pressure associated with the compres-sive fast wave, u is the fluid displacement in the x direction,q is the fluid flux, and A is the cross-sectional area of theducts. Combining Eqs. �24� and �25� yields the first-ordersystem of differential equations:

�pc

q�

,x= � 0

f�2

A

−A

fc2 0 ��pc

q� . �26�

c. Intracochlear pressure from the combined waves.The total intracochlear pressure �ptotal� is obtained by the

summation of the pressure field from the slow traveling wave�pt� and the pressure field from the compressive wave �pc�.The boundary conditions of zero total pressure at the RW andthe prescribed total pressure �pstapes� at the stapes yield

ptotal = atpt + acpc, �27�

where two unknown coefficients at and ac are determined bythese two boundary conditions,

at = pstapes� pc�xrw�pt�xow�pc�xrw� − pt�xrw�pc�xow� �28�

and

ac = pstapes� pt�xrw�pt�xrw�pc�xow� − pt�xow�pc�xrw� �29�

with xrw and xow representing the RW and OW coordinates,respectively.

III. RESULTS

The cochlear model is used to calculate the response ofa gerbil cochlea. The material property values in Table I were
taken from a number of sources �Smith, 1968; Lim, 1980;

Miller, 1985; Steele et al., 1995; Karavitaki, 2002� and thedimensions in Table II were from the anatomical measure-ments for gerbil cochlea �Sokolich et al., 1976; Greenwood,1990; Dannhof et al., 1991; Cohen et al., 1992; Edge et al.,1998; Thorne et al., 1999�.

The model is meshed into 1200 sections along the12 mm length of the gerbil cochlea. Forty terms are used inthe Fourier expansion across the width of the scala. Calcula-tion with 80 terms for the Fourier expansion shows no dif-ference from 40 terms. Running on an Intel Pentium IX�3.40 GHz� processor, the average time taken for a singleharmonic excitation calculation is about 1 s. This methodprovides a fast and efficient solution compared to a full-scalefinite element model. Note that the computation time indi-cated by Parthasarathi et al. �2000� is measured in hours ofcomputing time for the linear solution for a single frequency.

The results include BF-to-place map, BM frequency re-sponse, intracochlear pressure, and derived quantities fromthe intracochlear pressure defined by Olson �1998�; �1� BMvelocity, �2� pressure difference across OC, and �3� OC im-pedance. The modeling results are compared with in vivomeasurements �Olson 1998, 2001�.

A. BF-to-place map

The calculation for BF versus location along the gerbilcochlear �BF range: 0.3–50 kHz� is shown in Fig. 3 with thegerbil BF-to-place map �Sokolich et al., 1976; Greenwood,

TABLE I. Material properties for the gerbil cochlear model.

Basilar membrane p=1.0�103 kg/m3

E11=1.0�10−4 GPaE22=1.0 GPaE12=0.0 GPa�=0.5

Scala fluid f =1.0�103 kg/m3

�=0.7�10−3 Pa sOuter hair cell �=60°, 80°

�=0.15, 0.28

TABLE II. Anatomical dimensions as a function of longitudinal position �x�for the gerbil cochlear model.

x �mm� b �mm�a h �mm�b fc L2, L3 �mm�d lOHC ��m�e

0 0.0210 0.030 1.000 25.01.5 0.0175 0.7072.9 0.162 0.3873.5 0.01315.0 0.3165.9 0.00887.2 0.190 0.0073 0.2828.49.0 0.0055 0.316

10.2 0.205 0.004412.0 0.0031 0.007 0.245 65.0

ab: Width of plate.bh: Effective thickness of plate.cf: Effective fiber density of plate.dL2, L3: Width and height of fluid chamber.e
lOHC: Length of outer hair cell length.

1990� that was constructed from cochlear-microphonic re-cording. The BF-to-place map from the passive model andmeasurement are in excellent agreement �Fig. 3�. Thedashed-dot line represents the BF-to-place map for the feed-forward active model with 0.15 gain factor. Near the stapes�0–4 mm from the stapes�, the active model shows approxi-mately 1/2 octave higher BF, whereas there is no BF shiftnear the helicotrema region. Due to the lower wave numberfor low frequency, the feed-forward gain from the activemodel is less in the apical region.

B. Frequency response of BM velocity

The gerbil cochlear BM velocity magnitude and phasefor 4.2 mm from the base �BF=9.5 kHz� relative to thestapes displacement are computed over a range of excitation

FIG. 3. 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 cochlearBF-to-place map of gerbil �Sokolich et al., 1976; Greenwood, 1990� over0.3–50 kHz range spanning a length of 12 mm.

FIG. 4. Basilar membrane �BM� velocity relative to the stapes Vbm/Vst magnbase �BF=9.5 kHz�. For the active model, �=0.15, �=60° �dashed line�Experimental data �expt.� for 30 and 100 dB SPL corresponding to the acti
2001�. Dashed-dot line in �b�: Phase from the model at the 2.8 mm from the stap

frequencies up to 20 kHz �Figs. 4�a� and 4�b��. Results fromthe model are compared with the gerbil measurements �Renand Nuttall, 2001�. The passive model shows quantitativelyvery good agreement with motion measured at a high stimu-lus level �100 dB SPL at the ear canal� in magnitude �Fig.4�a��.

Karavitaki �2002� evaluated the angle of tilt of gerbilOHC �� to be approximately 84°, which is close to beingperpendicular to the basilar membrane �Fig. 2�. The gainfrom OHCs is calculated for two cases; a nominal mamma-lian value of �=60° and �=80°. The active model showsfairly good agreement with data at low stimulus level �30 dBSPL at the ear canal� with 30 dB gain for either �=60° withfeed-forward gain factor �=0.15 �dashed line in Fig. 4�a�� or�=80° with forward gain factor �=0.28 �dotted line in Fig.4�a��. Thus only a slightly higher gain, still in the physiologi-cally reasonable range, is needed even when the OHC isnearly perpendicular to the BM.

In the relative BM velocity magnitude plot �Fig. 4�a��,BF place shifts from 9.5 kHz �passive model� to 15 kHz �ac-tive model�, which is 3 /5 octave higher. In the animal mea-surement BF is also near 9.5 kHz for the high level passivecase. For the low level active case, BF place shifts to about13 kHz, which is only about 2 /5 octave higher. So the modelappears to overestimate the BF for the active case.

In the model, the phase is normalized to the volume flowrate at x=0, as the stapes is assumed to be a piston at the endof the fluid chamber. As shown in Fig. 4�b�, the phase of theresponse obtained from the model shows 2.5 cycles largerroll-off with frequency than the experimental measurements.In the region of the low frequency input, below 4 kHz, theBM velocity phases both from the model and measurementare similar. However, after 4 kHz, the phase of BM velocityfrom the model shows a larger roll-off than the phase fromthe data, which corresponds to a higher wave number in themodel above 4 kHz. It is well known that the phase excur-sion to the best place is about 1.5–2 cycles, over a wide

�a� and corresponding phase �b� for the gerbil cochlea at 4.2 mm from the�=0.28, �=80° �dotted line� were used while for the passive case �=0.d passive case, respectively, are included for comparison �Ren and Nuttall,

itudeand

ve an
es.

range of the cochlea �Ren and Nuttall, 2001; Overstreet etal., 2002�. Too much phase excursion in the present modelmay come from the unique shape of the basilar membrane inthe gerbil cochlea that is not taken into account. Anotherpossible problem is that the actual position of the stapes inthe cochlea extends over a small portion of the basal end ofthe scala vestibuli which may result in this discrepancy in thephase. The phase calculated at 2.8 mm from the stapes �BFof 15 kHz for the passive case, dashed-dot line in Fig. 4�b��is very close to the measurements.

C. Intracochlear pressure

The frequency response of the intracochlear pressurefrom the gerbil model is presented. First, the intracochlearpressure only from the slow wave is calculated for four dif-ferent locations away from the BM. Next, intracochlear pres-sure from the combined slow and fast waves is calculated attwo different locations which are 1.4 and 2.6 mm from thestapes. These intracochlear pressure simulations for the ger-bil model very close to the stapes �1.4 mm from the stapes�show good agreement both in magnitude and phase with thein vivo measurements �Olson, 1998�, whereas simulation re-sults at 2.6 mm from the stapes shows one cycle more phaseexcursion at BF location than in vivo measurements �Olson,2001�.

1. Slow wave intracochlear pressure

Intracochlear pressure due to the motion of BM from theslow traveling wave is obtained from Eq. �23�. Fig. 5 givesthe pressure distribution in the SV in the section at 1.4 mmfrom the stapes �BF=26 kHz�. The pressure decreases expo-nentially with distance from the BM. It is also clear that thepressure depends strongly on the transverse direction to thefluid motion, and is fully three dimensional. This confirms

FIG. 5. Radial distribution of intracochlear pressure from the slow wave atdifferent distances from the partition �1.4 mm from the stapes, BF=26 kHz�. The location of the BM is indicated by the thickened line �BMwidth=0.151 mm�. The pressure drops exponentially with the distance fromthe BM in either perpendicular or radial direction.

the results first given by Steele and Taber �1979, Fig. 10�.


2. Combined slow and fast wave intracochlearpressure

a. Combined intraochlear pressure at 1.4 mm from thestapes (passive case). The intracochlear pressure in the STfrom the passive model which includes contributions fromthe traveling wave solution and the compressive wave solu-tion is shown in Figs. 6�a� and 6�b�. The intracochlear pres-sure magnitude and phase at 1.4 mm from the stapes �BF=26 kHz� are calculated for two locations �at 3 and 23 �maway from the BM, Figs. 6�a� and 6�b�� and compared withexperimental measurements �Olson, 1998�.

Intracochlear pressure from the 3D cochlear modelshows good agreement with measurements �Olson, 1998�both in magnitude and phase. Around the BF region, theintracochlear pressure magnitude at 3 �m away from theBM is 5 dB larger than magnitude at 23 �m away from theBM both in the model and measurement. Several distinctpeaks and valleys after the peak region �30–40 kHz� areevident in Fig. 6�a�. These peaks and valleys are from con-structive and destructive interference between the slow andfast wave pressure. However, these are not clearly seen in thedata �Olson, 1998�. This difference between data and modelresults may come from �i� the intracochlear pressure calcu-lated at one point compared to the experiment of pressureaveraged from the region of the transducer, or �ii� the inad-equate frequency sampling in the measurement since morepeaks and valleys are found in the most recent intracochlearpressure measurements �Dong and Olson, 2007�.

Intracochlear pressure from the model also shows atransition from the slow wave to the compressive fast waveafter the BF region which is observed in the measurements.In Fig. 6�a�, intracochlear pressure in the model shows thatthe traveling wave is dominant from the low frequency to theBF region. However, after this BF region, the traveling waveon the BM disappears and the fast acoustic wave becomesthe dominant wave as is evident by the approximately con-stant pressure.

In Fig. 6�b�, the pressure phase remains near zerountil the frequency exceeds 25 kHz, when the phase changesrapidly. The decrease in phase is characteristic of the travel-ing wave component of the pressure field, whereas the phaseplateau is characteristic of the fast compressive wave. Intra-cochlear pressure phase from the model at 3 �m away fromthe BM shows a larger phase accumulation than the measure-ment because of more oscillation in the model after the peakregion �Fig. 6�a�, solid line, 30–40 kHz�.

b. Combined intraochlear pressure at 2.6 mm from thestapes (active case). Intracochlear pressures for the passiveand active case at 2.6 mm from the stapes �BF=15 kHz� and22 �m away from the BM are shown in Figs. 6�c� and 6�d�.Model results are compared with Olson’s �2001� experimen-tal data of expt. 9-8-98-I-usual. Active model results are cal-culated at a low stimulus level �50 dB SPL at the ear canal�;�=60° �OHC angle�, �=0.15 which was used in the Sec.III B. In the magnitude �Fig. 6�c��, the model results andmeasurement data show good agreement in �i� nonlinearitywith 10 dB gain from the OHCs and �ii� 2/3 octave BF shiftin the active case, whereas model shows more peaks andvalley than data. In this case, simulation shows 20 dB lessgain than BM velocity. This smaller gain in the intracochlearpressure than BM velocity is indication that the organ of
Corti is supplying additional force to the BM. Also, this im-

plies that the gain from OHCs can be measured more clearlyfrom the BM velocity than intracochlear pressure measure-ment.

From Fig. 6�d�, the model shows approximately onecycle more phase excursion at the BF position than the mea-surement. The phase difference at the BF for the three dif-ferent locations are: 0 cycle, 1 cycles, and 2.5 cycles at1.4 mm �Olson, 1998�, 2.6 mm �Olson, 2001�, and 4.2 mm�Ren and Nuttall, 2001� from the stapes, respectively. Themodel shows more phase excursion than measurement withincreasing distance from the stapes. These results indicatethat the phase difference between model and measurementaccumulates with increasing distance from the stapes. Thesephase excursion issues in the model may be resolved by amore advanced model.

In the following derived quantities analysis, intra-cochlear pressure both at 1.4 and 2.6 mm from the stapes areused. Especially, intracochlear pressure for the low stimuluslevel �40 dB at the ear canal� at 1.4 mm from the stapes�Olson, 1998� can be considered as nearly passive since thecondition of the cochlea was not optimal. However, there isstill a small gain in the low stimulus level in the measure-

FIG. 6. Combined slow and fast wave intracochlear pressure in the scalasponding phase at 1.4 mm from the stapes �BF=26 kHz� and 3 and 23 �m�1998, Fig. 10� expt. 2-26. �c� Intracochlear pressure magnitude and �d� corrthe BM �80 dB SPL: passive and 50 dB SPL: active case�. Data are from O

ment. Thus, a small gain factor ��=0.05� is used for this


case. Since the intracochlear pressure simulation at 1.4 mmfrom the stapes shows the best agreement both in the phaseand magnitude with data, the simulation results at 1.4 mmfrom the stapes are used for the analysis of exact theoreticalOC impedance.

D. Derived quantities from the intracochlear pressure

Olson �1998� developed formulas in terms of the mea-sured intracochlear pressure as approximations for the BMvelocity, the pressure difference across the OC, and the OCimpedance �ZOC�. Since the results from the pressure differ-ences show interesting behavior, the calculation using Ol-son’s formulas and Olson’s results are compared directly.Finally, the exact theoretical OC impedance from the modelfor the passive and active cases is calculated and comparedto the result of the difference formula for the estimated the-oretical OC impedance.

In this section, quantities derived from the intracochlearpressure as defined by Olson �1998� are calculated and com-pared with measurement �Olson, 1998, 2001�. These are �1�

ani �ST� of the gerbil. �a� Intracochlear pressure magnitude and �b� corre-y from the BM �100 dB SPL at the stapes: passive�. Data are from Olson,ding phase at 2.6 mm from the stapes �BF=15 kHz� and 22 �m away from�2001, Fig. 7� expt. 9-8-98-I-usual.

tympawa

esponlson

BM velocity, �2� pressure difference across OC, and �3� OC


impedance. The derivation was presented in a previous studyand derived results in this section are based on that study�Olson, 1998�.

Derived quantities are estimated from a pair of adjacentST intracochlear pressures, which are 15 �m �Pa� and20 �m �Pb� from the BM �Fig. 7�. The difference betweenthese two adjacent intracochlear pressures is used to estimatethe vertical �z� component of the intracochlear pressure gra-dient. First, the BM velocity �vBM� is found from this intra-cochlear pressure gradient. Based on the estimation thatclose to the BM the fluid moves with BM, vBM is consideredto be this fluid velocity. From the dimensional analysis, theNavier–Stokes equation in an incompressible fluid is simpli-fied as

�P = − f�v�t

. �30�

For the z component,

�P

�z�

�Pa − Pb��z

= − f�vz

�t. �31�

By assuming vBM=vz for a harmonic response,

vBM = −�Pa − Pb�

f��zi �32�

was given by Olson for the calculation of vBM from the in-tracochlear pressure measurements.

Second, the pressure difference across OC ��POC� isobtained by assuming zero pressure at the round window�Olson, 1998�,

�POC = PSV − 2Pa �33�

where Psv is the intracochlear pressure near the stapes at500 �m from the BM in the scala vestibule. Finally, the OCimpedance �ZOC� is defined as

ZOC =�POC

vBM. �34�

Derived quantities from the gerbil cochlear model arecompared with results of expt. 2-26 �Olson, 1998� for thepassive case and expt. 9-8-98-I-usual �Olson, 2001� for the

FIG. 7. Schematic 3D drawing of the cochlear model. Intracochlear pres-sures Pa and Pb are measured and calculated at the indicated positions a andb, 15 and 20 �m away from the BM in the ST respectively, and in SV �PSV�.These are used to obtain an approximation �Olson, 1998, 2001� for the BMvelocity, pressure difference and impedence, and organ of Corti impedancereferred to as derived quantities. Cross indicates distance from the stapes.

more active case.


1. Derived BM velocity

Numerous measurements of basal BM velocity, vBM,have been conducted, and the frequency domain responsehas been studied under various physiological conditions�Khanna and Loenard, 1986; Cooper and Rhode, 1992a,1992b; Xue et al., 1995; Nuttall and Dolan, 1996; Ruggeroet al., 1996, 1997; Overstreet et al., 2002�. Derived vBM fromanimal intracochlear pressure measurements �2–26� from Ol-son �1998� represented a similar response to the vBM mea-sured by direct measurement methods by Xue et al. �1995�.

Figure 8�a� displays the magnitude of the vBM from themodel and Olson’s 1998 expt. 2-26 measurement. Figure8�b� shows the vBM phase relative to SV pressure. Stimuluslevels were 80 and 40 dB SPL in the ear canal for the mea-surements, which corresponds to 110 dB and 70 dB SPL atthe stapes for the model with the consideration of 30 dB gainfrom the middle ear ossicles �Olson, 1998�. For a moderateOHCs’ motility force on the BM, the active model with 0.05gain factor is used in the calculation of the low level stimulus�40 dB SPL at the ear canal�. The derived vBM from themodel at 1.4 mm from the stapes shows excellent agreementwith the experimental results both in the magnitude �Fig.8�a�� and phase �Fig. 8�b��: �i� their peaked shape, �ii� maxi-mum peak region, �iii� the absolute value of maxima, �iv�dropping off rapidly after the peak in the vBM magnitude, and�v� decreasing phase with increasing frequency at a steadilyincreasing rate above about 8 kHz.

Figures 8�c� and 8�d� show magnitude and phase of thevBM from the model and Olson’s 2001 expt. 9-8-98-I-usualmeasurement. For the passive case, derived BM velocity at2.6 mm from the stapes shows good agreement with mea-surement up to 25 kHz. The measurement shows plateau af-ter 25 kHz, which is not observed in the model both in thepassive and active cases. However, this is expected becausethe fast wave should be canceled out in the calculation ofderived BM velocity �Eq. �32��. For the low level stimulus�40 dB SPL at the ear canal�, the active model with �=60°�OHC angle� and �=0.15 is used. Derived BM velocity forthe active case shows good agreement with measurement upto BF region. After the BF region, the derived BM velocitymagnitude from the active model shows less decrease thanthe data. The active case simulation also shows no plateauafter 25 kHz for the same reason mentioned earlier. Phase ofderived BM velocity at 2.6 mm from the stapes shows ap-proximately one cycle more phase excursion than data. Thereasons for this are not clear at this point and need to beresolved.

2. Derived pressure across OC

The derived pressure difference across OC ��POC�simulation and experimental results both in the magnitudeand phase are shown in Fig. 9. For the calculation of �POC,Psv−2Pa from the current symmetric cochlear model is used�Eq. �33��. Pa is obtained at the position of 15 �m from theBM in the ST. �POC values from the model are calculatedand compared with measurements by Olson �1998, 2001�.

�POC magnitude at 1.4 mm from the stapes �Fig. 9�a��
shows �i� a mild peak around the BF region, �ii� notches, and

�iii� small value in the 0–15 kHz region due to long wave.Long wave has small BM velocity value in the z direction,which induces the small pressure difference across organ ofCorti. On the other hand, in the BF region which providesbetter measurement condition than other frequency region,simulation results show good agreement with measurement.

The estimated �POC magnitude from the model showsdecrease from the BF region to 40–45 kHz whereas �POC

from the measurements remains fairly flat. Above the 45 kHzregion where the fast wave dominates, simulation results ofthe estimated �POC shows the plateau which comes from thefast wave in the estimation of �POC �Fig. 9�a��. In Fig. 9�b�,the phase is shown relative to the SV pressure. Since the SVpressure is much greater than the ST pressure at low frequen-cies, the phase is close to zero. Between 20 and 30 kHz, thephase accumulates almost 400°. Similar to ST pressure, thederived �POC is composed of the sum of a fast and slow

FIG. 8. Derived BM velocity from the gerbil cochlear model and measuremof the measurement results for 40 and 80 dB SPL at the ear canal and mrespectively �1.4 mm from the stapes, BF=26 kHz�. �b� Phase relative to SV2-26 at the stimulus levels of 40 and 80 dB SPL at the ear canal. �c� Derived�BF=15 kHz� for the passive �85 dB SPL� and active case �45 dB SPL�. D

wave �Eq. �33��. However, for the exact calculation of �POC,


the fast wave component should be canceled out since themagnitudes of the pressure from the fast wave in SV and STat the same x position are equal.

�POC at 2.6 mm from the stapes is calculated and com-pared with measurements �Olson, 2001� for the high and lowlevel stimulus which correspond to the passive and activecase, respectively. In Fig. 9�c�, both simulation and measure-ment for the passive case �85 dB SPL at the ear canal� show�i� a mild peak near the BF �15 kHz� region, �ii� drop afterthis region about 15 dB, and �iii� plateau after 23 kHz whichis from the fast wave. More peaks and valleys in the simu-lation of �POC than measurements are still observed. Simu-lation of �POC magnitude for the low level stimulus�45 dB SPL at the ear canal� shows �i� a peak at 19 kHz with10 dB gain from the feed-forward mechanism, �ii� less sharptuning than data near the BF region, �iii� plateau after30 kHz which is one octave higher than measurement, and

using the formulas in Olson �1998, Appendix 1, Eq. �A7��. �a� Magnituderesults at 70 and 110 dB at the stapes are plotted re: 0.01 and 1 mm/s,sure at 1.4 mm from the stapes. Data are from Olson, �1998, Fig. 18� expt.velocity magnitude and �d� corresponding phase at 2.6 mm from the stapese from Olson �2001, Figs. 15�a� and 15�b�� expt. 9-8-98-I-usual.

ents,odelpresBM

ata ar

�iv� 2/3 octave BF shift in active case. �POC phases relative


�200

to the SV pressure are calculated and compared with mea-surements for the passive and active case �Fig. 9�d��. Nearthe BF reason, phases show one cycle more excursion thanmeasurements both in the active and passive case.

3. Derived OC impedance

Olson �1998� estimated the specific acoustic impedanceof the OC �ZOC� which is defined as �POC divided by vBM

�Eq. �34��. The same approach in the gerbil model is con-ducted to allow quantitative comparison with Olson’s ZOC

estimation. First, the derived ZOC is studied by comparing thegerbil model to measurements. Next, the exact value of ZOC

is compared with the derived ZOC. Last, the values of ZOC ofthe passive and active models are discussed based on theexact calculations.

The magnitude �Fig. 10�a�� and phase �Fig. 10�b�� of theZOC is shown for the gerbil model and 2–26 animal measure-ments which were obtained near the stapes �Olson, 1998�.The model results show good qualitative and quantitative

FIG. 9. Derived pressure across the OC complex, �POC, from the gerbil cocEq. �A10��. �a� Magnitude. �b� Phase relative to SV pressure at 1.4 mm fromstimulus levels of 40 and 80 dB SPL at the ear canal. �c� Derived �POC magthe passive �85 dB SPL� and active case �45 dB SPL�. Data are from Olson

agreement with measurements: �i� a tuning to frequencies


between 22 and 26 kHz, �ii� a primary minimum�5 Pa/ �mm/s�� at 24 kHz, with secondary minimum close tohalf an octave above at 32 kHz, �iii� constant slope of themagnitude in the low frequency region �−6 dB/oct� whichrepresents stiffness dominated impedance, and �iv� phasefluctuation after the BF.

The estimated real and imaginary part of ZOC for thepassive and active case at 2.6 mm from the stapes are calcu-lated and compared with Olson’s 2001 9-8-98 measurements�Figs. 10�c� and 10�d��. The real part of estimated ZOC fromthe model stays near zero values �±10 Pa/ �mm/s�� and theimaginary part of estimated ZOC stays at a negative valueover the whole frequency range �5–25 kHz�. This shows thatthe estimated ZOC phases for the passive and active modelstay near −90° �stiffness dominated region�. Magnitude ofZOC can be calculated �not shown�. Measured ZOC magni-tudes decrease up to 20 kHz then increase after 20 kHz inboth the active and passive cases. On the other hand, mag-nitude from the passive model decreases by −15 dB/oct up

model and measurements, using the formulas in Olson �1998, Appendix 2,stapes �BF=26 kHz�. Data are from Olson �1998, Fig. 19� expt. 2-26 at thee and �d� corresponding phase at 2.6 mm from the stapes �BF=15 kHz� for1, Figs. 15�a� and 15�c�� expt. 9-8-98-I-usual.

hlearthe

nitud

to 13 kHz and increases after 23 kHz whereas magnitude


from the active model decreases over the whole frequency�5–25 kHz� with different decreasing rate: −15 dB/oct�5–13 kHz� and −3 dB/oct �13–25 kHz�.

The impedance phase of a classical mechanical reso-nance begins at −90° �stiffness dominated region� at lowfrequencies, then increase through 0° at the resonance fre-quency, and ends up at +90° �mass dominated region�. Thephase of gerbil model stays near −90° up to 10 kHz, whichindicates stiffness dominated response whereas measurementimpedance phase shows a more complicated value which isconsidered as a combination of stiffness and damping. Mostnotably, in the phase results at 1.4 mm from the stapes �Fig.10�b��, the derived ZOC phase of the passive model and mea-surement �80 dB SPL� appears beyond the region between+90° and −90°; this indicates that the real part of the imped-ance is negative. Since the passive system always has posi-tive real components of impedance, the ZOC phase for thepassive model and measurement �80 dB SPL� should staybetween +90° and −90°. This discrepancy may come fromthe estimation approach, thus it can be resolved by finding

FIG. 10. Derived impedance of organ of Corti �ZOC� from the gerbil cochle�b� Phase of ZOC for model and measurements at 1.4 mm from the stapes �Band �d� imaginary part of ZOC at 2.6 mm from the stapes �BF=15 kHz� for thFigs. 15�d� and 15�e�� expt. 9-8-98-I-usual.

the exact ZOC from the model.


4. Comparison between estimated and exacttheoretical OC impedance

Near the stapes �1.4 mm from the stapes�, the physi-ologically based three-dimensional gerbil cochlear modelshows the best agreement with measured derived quantities.With this approach, exact theoretical value of those quanti-ties can be studied at this location. In this section, the differ-ence between exact and estimated theoretical OC impedanceis discussed.

The derived �POC includes the compressive fast wavebecause it is estimated by Psv−2Pst which contains the fastwave component. This fast wave in the derived �POC causesdeep notches in the ZOC magnitude �Fig. 10�a�� and ZOC

phase fluctuation �Fig. 10�b��. However, the fast wave com-ponent should be canceled out in the exact �POC which iscalculated from �Psv-Pst�z=0.

In Fig. 11, the exact and estimated theoretical ZOC areshown. Magnitude �Fig. 11�a�� and phase �Fig. 11�b�� of theexact and estimated theoretical ZOC for the passive case are

del and measurements, using the formulas in Olson �1998�. �a� Magnitude.kHz�. Data are from Olson �1998, Fig. 20� expt. 2-26. �c� Real part of ZOC

sive �85 dB SPL� and active case �45 dB SPL�. Data are from Olson �2001,

ar moF=26e pas

compared. The exact theoretical ZOC was obtained from the


�vBM�z=0 and ��POC�z=0. The exact theoretical ZOC shows: �i�smooth decrease with almost constant slope in the wholefrequency region �−6 dB/oct� which represents stiffnessdominated impedance, �ii� lower magnitude than estimatedtheoretical ZOC except for the BF region, and �iii� the ex-pected phase between −45° and −90°. The magnitude of theexact theoretical ZOC decreases smoothly with increasing fre-quency without the fast wave mode �Fig. 11�a��. Less exacttheoretical ZOC magnitude implies that larger vBM and/orsmaller �POC in the real case than derived theoretical quan-tities.

The exact theoretical ZOC phase decrease from −50° atthe low frequency region to −85° at the high frequency re-gion. This corresponds to OC impedance that becomes morestiffness dominated near the BF. Also, the exact theoreticalZOC phase remains between −90° and 90°, which representspassive mechanics �Fig. 11�b��.

FIG. 11. Exact and derived theoretical impedance of organ of Corti �ZOC� fromodel is presented. �a� Magnitude. �b� Phase.

FIG. 12. Exact theoretical impedance of organ of Corti �ZOC� from the gerb
0.15 feed-forward gain factor is used in the active model. �a� Magnitude. �b� Pha

5. Comparison between exact theoretical passive andactive OC impedance

In Fig. 12, the magnitude �Fig. 12�a�� and the phase�Fig. 12�b�� of the exact theoretical ZOC are shown for thepassive and active cases. The active model ��=60°, �=0.15� has a lower ZOC magnitude than the passive case inFig. 12�a�. This difference is due to the lower slow wavepressure gain than BM velocity gain, as was discussed inSec. III C 2. An octave or more below BF, in the tail-frequency region, the active model shows more compliancethan the passive model with a magnitude difference of 2 dB.Similar differences were observed due to medial efferent ac-tivation on auditory-nerve responses in tail-frequency regionby Stankovic and Guinan �1999, 2000�. In Fig. 12�b�, thephase of the exact theoretical ZOC from the active model isbelow −90° in the BF region where that of the passive model

e gerbil cochlear model �1.4 mm from the stapes, BF=26 kHz�. The passive

hlear passive and active model �1.4 mm from the stapes, BF=26 kHz�. The

m th

il coc
se.

is above −90°. Below −90° phase represents negative realcomponent of the OC impedance which is from the forceacting on the BM due to OHCs motility.

IV. DISCUSSION

The measurements of gerbil intracochlear pressure �Ol-son, 1998, 2001� offer an excellent opportunity to test modelcalculations. Presently, the macromechanical cochlear modelfor the chinchilla anatomy �Yoon et al., 2006� is extended tothe gerbil anatomy. The BM properties are physical, withorthotropic elastic properties and no fictitious mass or damp-ing.

The comparison of results from the model and experi-ment is promising, but not fully satisfactory. Using the singleset of anatomically based parameters, the model predicts sev-eral significant features of the cochlea. The BF-to-place mapin the passive model and frequency responses of BM veloc-ity and intracochlear pressure were in close agreement withthose observed in animal measurement. The feed-forwardlinear active model, the most speculative feature of theframework presented, showed excellent agreement with ex-perimental data in the BM relative velocity and intracochlearpressure magnitude. However, the calculated phases for theBM velocity and intracochlear pressure show a larger excur-sion at the BF by 2.5 cycles and 1 cycle at a fixed point; x=4.2 mm and x=2.6 mm, respectively. In contrast, the cal-culated amplitude and phase show excellent agreement forthe fixed point �x=1.4 mm�. This phase excursion issue inthe current model should be improved. Preliminary resultsfrom the most recent model which has a modified BM plateand push-pull mechanism for the organ of Corti may providebetter results in the large phase excursion phenomenon.

By virtue of the 3D cochlear model, intracochlear pres-sure in the ST was obtained by adding the fast wave to thetraveling pressure slow wave. From the intracochlear pres-sure simulation, derived quantities: �1� BM velocity, �2� pres-sure difference across OC, and �3� OC impedance in thebase, were calculated by following Olson’s estimation�1998�. These quantities were compared with animal mea-surements and showed excellent agreement. From the vali-dated gerbil cochlear model, the exact theoretical OC imped-ance was obtained and compared with the estimatedtheoretical OC impedance. By comparing exact and esti-mated theoretical OC impedances, a fast wave component inthe estimated theoretical OC impedance is found and itcauses phase fluctuation out of the reasonable range �nega-tive real part of impedance in the passive response� andnotches in the estimated theoretical OC impedance. Finally,the exact theoretical OC impedances for the passive and ac-tive model were compared in the magnitude and phase. Theexact theoretical OC impedance from the active modelshows negative real components which represents active pro-cess from the OHCs’ motility.

An important consideration of the feed-forward activemechanism is that BM impedance for the active case is 2 dBless than for the passive case below BF in the tail region.Near BF, the change in impedance is due to an apical shift in
resonance for the low-level active case, but the magnitude

and phase change is also small �Fig. 12�. This indicates thatthe zero crossings of the time domain response for the highlevel passive case and the low level active case will be nearlyinvariant. This suggests that force generation by OHCs in thefeed-forward formalism satisfies the near-invariance of finetime structure of the organ of Corti response predicted byShera �2001�. Further calculations of the model in the timedomain will provide a more definitive test.

V. CONCLUSION

In the current work, the pressure in the cochlear fluidcomputed from the model is found to agree with the intraco-chlear pressure measurements �Olson, 1998, 2001�. Thisgives support to our proposition that the present model isclose to the actual behavior of the gerbil cochlea, and that theremaining discrepancies can be resolved. Extension of thecochlear model can be achieved by including more detailedstructures of the OC to the current model �Steele and Puria,2005�.

ACKNOWLEDGMENTS

This work was funded by HFSP Grant No. RGP0051and NIDCD of NIH Grant No. DC007910.

NOMENCLATURE

BM � basilar membraneOC � organ of Corti

OHCs � outer hair cellsBF, EC � characteristic frequency, ear canal

FPZ, FBMf � forces acting on the pectinate zone and fluid

FBMC � force exerted by OHC� � feed-forward gain factor

lOHC � length of OHC� � angle of tilt of OHCn � wave number� � scalar potential for fluid displacement� � coefficient of �Tj � Fourier coefficient for jth component of �� j � decay coefficient for jth component of �

L2 ,L3 � width and height of fluid chamberpt , pc � fluid pressure associated with the slow trav-

eling and compressive fast waveu � fluid displacement in the x directionq � fluid fluxA � cross-sectional area of ducts f � fluid density

t � timex ,y ,z � Cartesian coordinates

� � angular velocityZoc � mechanical impedance of the OC

�POC � pressure difference across the OC�BM � z component of BM velocityPsv � pressure near the stapes in the scala vestibuliPst � pressure near the BM in the scala tympani

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