puri a steele article

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HANDBOOK OF THE SENSES Audition Volume

Chapter 7: Mechano-Acoustical Transformations

Sunil Puria1,2 and Charles R. Steele1

1Stanford University, Mechanical Engineering Department, Mechanics and

Computation Division, 496 Lomita Mall, Durand Building, Room 206, Stanford, CA94305

2Department of Otolaryngology-Head and Neck Surgery, 801 Welch Road,Stanford, CA 94304


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1. KEYWORDS2. BIOMECHANICS, OTOBIOMECHANICS, COCHLEA,

COLLAGEN FIBERS, EXTERNAL EAR, IMPEDANCE, INNER

HAIR CELLS, LEVERS, MATHEMATICAL MODELS,

MICROCT, MIDDLE EAR, ORGAN OF CORTI, OUTER HAIRCELLS, PRESSURE, TYMPANIC MEMBRANE, VIBRATION


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I. OutlineIn this chapter we examine the underlying biophysics of acoustical and mechanical

transformations of sound by the sub components of the ear. The sub components include

the pinna, the ear canal, the middle ear, the cochlear fluid hydrodynamics and the organof Corti. Physiological measurements and the deduced general biophysics that can beapplied to the input and output transformations by the different sub components of the earare presented.

II. AbstractThe mammalian auditory periphery is a complex system, many components of which

are biomechanical. This complexity increases sensitivity, dynamic range, frequency

range, frequency resolution, and sound localization ability. These must be achieved

within the constraints of available biomaterials, biophysics and anatomical space in theorganism. In this chapter, the focus is on the basic mechanical principles discovered forthe various steps in the process of transforming the input acoustic sound pressure into the

correct stimulation of mechano-receptor cells. The interplay between theory andmeasurements is emphasized.

III. Main Body

A. Introduction

The auditory periphery of mammals is one of the most remarkable examples of a biomechanical system. It is highly evolved with tremendous mechanical complexity.

What is the reason for such complexity? Why can’t hair cells tuned to various frequencies just sit on the outside and detect motion due to sound? Clearly, the complexity serves the

animal by providing greater functionality. This can be appreciated by looking at simplerauditory systems.

One of the simplest hearing organs is that of the fly ( Drosophila melanogaster ), which

has a tiny feather-like arista that produces a twisting force directly exerted by sound. Thissound receiver mechanically oscillates to activate the Johnston’s organ auditory receptors

with a moderately damped resonance at about 430 Hz (Gopfert and Robert 2001). The

level required to elicit a response, due to wing-generated auditory cues involved incourtship, are in the 70 to 100 dB SPL range (Eberl et al. 1997). An example of a simpleranatomy with more complex function than that of the fly is the Müller’s organ of the

locust. This invertebrate is capable of discriminating sounds at broadly tuned frequenciesof approximately 3.5-5, 8, 12 and 19 kHz corresponding to the four mechanotransduction

receptors attached to the tympanic membrane (Michelsen 1966). The best threshold forthe receptor cells is about 40 dB SPL. The resonances of the tympanic membrane and

attached organs provide the greater number of frequency channels than the fly (Windmill


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et al. 2005). Amphibians evolved to have a basilar papilla with a few hundred receptorcells in a fluid medium. Other amphibians, birds and mammals have many thousands of

hair cells. Other such examples, where structure that is more complex leads to greaterhearing capability, are found in some of the other chapters in this volume.

The peripheral part of the auditory system comprising of the external ear, middle ear

and the inner ear systematically transform and transduce environmental sounds to neuralimpulses in the auditory nerve. The precise biophysical mechanisms relating the input

variables to the output variables of some of the subsystems are still being debated.However, there is general agreement that these transformations lead to improvedfunctionality. Five of the most important functional considerations are described below.

1. Sensitivity. The human ear is most sensitive to a range of sounds from theloudest at about 120 dB SPL to the softest at about –3 dB SPL. At its mostsensitive frequency near 4 kHz, the displacement at the tympanic membrane is

less than 1/10th

the diameter of a hydrogen atom. At this threshold, the amountof work that is done at the eardrum

i is 3 x 10

-18 J. In comparison, the amount

of work done for the perception of light at the retinaii is 4x10-18 J, which isclose to the calculated value at the threshold of hearing. This suggests that at

its limits, the two sensory modalities have comparable thresholds.2. Dynamic range. The dynamic range of psychophysical hearing in humans is

about 120 dB SPL corresponding to environmental sounds and vocalizedsounds. However, the neurons of the auditory nerve have a dynamic range that

is typically less than 60 dB SPL. The organ of Corti mechanics must facilitatethis dynamic range mismatch problem.

3. Frequency range. Hearing has about 8.5-octave frequency range in human andin some other mammals this range can be as wide as 11.5-octaves (ferrets). To

handle this processing mechanically, the sensory receptors should have

physical variations on a similar scale. However, the large range is achievedover an extraordinarily small space in comparison to the wavelengths ofsound.

4. Frequency resolution. One of the most important functions of the cochlea isthe tonotopic organization, which maps different input frequencies to its

characteristic place in the cochlea. Like a Fourier frequency analyzer, eachcharacteristic place has narrow frequency resolution, which provides greater

sensitivity to narrow-band signals by reducing bandwidth and thus input noiseat the individual mechano-receptor hair cells and thus the auditory nerve.

5. Sound localization. The physical characteristics of the pinna and head diffractsound in a spatially dependent manner. The diffraction pattern provides

important cues that allow the more central mechanisms to localize, segregateand stream different sources of sound.

In this chapter, we follow the chain of acousto-mechanical transformations of sound

from the pinna to modulation of tension in inner hair cell tip links which is the finalmechanical output variable of the cochlea from our vantage point. The tension in the tip

links opens ion channels in the stereocilia, which then starts a chain of biochemicalevents that leads to the firing of the auditory neurons. We designate the output of a given


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sub system a proximate variable. The chain of proximate variables leads to the ultimateoutput variable, the tip-link tension in hair cells. Input variables are sound pressure level,

morphometry of anatomical structures, and mechanical properties of those structures.Biomechanical processes combined with the input variables lead to the proximatevariables, which are physiologically measurable.

B. Theories of sound transmission in the ear

Starting with Helmholtz, mathematical models have played a key role in improving ourunderstanding of the underlying biomechanical processes of the auditory periphery. In

comparison to using natural languages to describe the observed phenomena,mathematical formulations have advantages and disadvantages. The advantages include a

methodology for the possibility to describe compactly a correspondence to reality. Thedisadvantages are that the description may be incomplete or its validity difficult to test. A

mathematical model is also often a statement of a scientific theory that captures theessence of the current state of the empirical observations. The power of a specific model

is its ability to evolve as more facts become available and to be able to predict facts notyet observed. Thus the interplay between theory and experiments allows one to testdifferent hypotheses and generate new hypotheses.

In this chapter we provide a foundation for physiological measurements in the form of

mathematical models. Below we present some common principles, found all through theauditory periphery, to transform the input variables to the ultimate output variable of hair

cell tip link tension. Several general concepts are presented. These include how levers areformed, how Newton’s laws apply not only to celestial mechanics as originally

formulated but also in otobiomechanics, how sound transmission through differentmaterials is described by transmission line formulations, and how modes of vibrations

arise in structures. A combination of these principles is used to understand how the earimproves sensitivity, frequency range, frequency resolution, dynamic range, and soundlocalization within the constraints of biological materials and anatomic space.

1. Mechanical and acoustic levers

One of the simplest transformations of energy is achieved with a simple mechanical

lever. There are numerous places in the auditory periphery where levers produce forceand velocity transformations through anatomical changes in lengths and areas. These

transformations take place in the context of improving sound transmission at theinterfaces of different anatomical structures where there is a change in the impedance. An

example of a change in impedance is the low impedance of air to the high impedance ofthe fluid filled cochlea. Examples of the lever action at work include an increase in sound

pressure due to a decrease in area of the concha of the pinna to the ear canal, increase in pressure from the decrease in surface area from the tympanic membrane to the stapes

footplate, increase in force due to the lever ratio in the ossicular chain, increase in volumevelocity from the stapes footplate to the basilar membrane due to a decrease in surface

area, and transformation of the basilar membrane displacement to hair cell stereocillia tiplink tension due to relative shearing motion between the reticular lamina and the tectorialmembrane.


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2. Newton’s second law of motion

A key principle in describing dynamic transformation of forces in mechanical systems

to accelerations is the well-celebrated Newton’s second law of motion elegantly written

as F = ma , (Eq. 1)

which states that a force F acted upon a mass, results in acceleration a. Newton’ssecond law, transformed to the frequency domain, is:

F (! ) = M + R

j! +

K

j! ( )2

"

#

$$

%

&

''a(! ) . (Eq. 2)

Here the sinusoidal force F (w) , with radian frequency ! , acts upon an object

described by the variables in the square bracket. This object has now been generalized to

include mass M, resistance R, and stiffness K . An alternative form of Eq. (2) in terms of particle velocity v(! ) is:

F (! ) = j! " M + R + K

j!

#

$%

&

'( "v(! ) , (Eq. 3)

where the term in the square bracket is the mechanical impedance Z m

. Sound

pressure P (! ) , measured with a microphone, is defined as the force per unit area A.

Thus, Eq. (3) can be rewritten for acoustics as

P (!

) =

j!

" M a +

Ra +

K a

j!

#

$%

&

'( "V (!

) (Eq. 4)

The term in the square bracket is now the acoustic impedance Z a, which for

uniform properties is the mechanical impedance Z

m divided by A

2, and V (! ) = v(! ) A is

the volume velocity.

One thing to keep in mind is that impedance concepts are limited to linear steady state

analyses. Despite this limitation, Eqs. (3) and (4) play a prominent role in helping usunderstand transformations of forces and pressures to velocities and volume velocities

throughout the ear. It is clear from these equations that the velocity of any structure is proportional to the applied force but inversely proportional to the impedance due to its

mass (M), damping (R) and stiffness (K). At resonance the velocity reaches a maximum because the impeding effect of mass is cancelled by the impeding effect of stiffness. One

of the challenges in efficient sound transmission to the hair cell detectors is in minimizingthe impeding effect of fluid damping and stiffness and mass of structures.

3. Transmission lines

Many problems in sound and vibration are described by the wave equation that resultsfrom Newton’s laws of motion. The one-dimensional (1-D) version of the wave equation


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was formulated by d’Alembert in 1747 for the vibrating string. It did not take Euler verylong (1759) to formulate the first derivation of the wave equation for sound transmission

in one dimension and later in three dimensions (3-D). The wave equation has stood thetest of time as is evident by its use in disciplines that include electromagnetic theory,

transverse waves in stretched membranes, blood vessels, and electromagnetic

transmission lines. Because it was used so extensively in telephone communication and power line transmission problems, the 1-D wave equation is also known as thetransmission line equation. In these equations the properties of the transmission system

are assumed to be constant along the direction of propagation. A special form of the waveequation exists when a property along the propagation direction varies exponentially. As

reviewed by Eisner (1967), these equations were originally formulated by Lord Rayleighand are now known as Webster’s horn equation.

Subsequent sections will show that the transmission line formulation can be used to

describe ear canal acoustics, the coupling between the canal and tympanic membrane,wave propagation in the cochlea, and transverse motion on the basilar membrane. The

series of transmission lines that are sequentially coupled may improve frequency

bandwidth while maintaining sensitivity of the proximate variables.

4. Modes of vibration

Anatomical structures and membranes have various modes of vibration with peakresponses at modal frequencies due to resonance. These modes of vibrations are not very

different from modes of vibrations in the strings of violins, guitars and pianos where theends of the strings are fixed are both ends. The resonant frequency is directly

proportional to the string tension and density but inversely proportional to its length.More complicated modes of vibrations are found in membranes and plates. In the ear,

examples where resonances are a characteristic feature include the pinna and ear canal,tympanic membrane, ossicles, the basilar membrane, organ of Corti, and hair cell

stereocilia. Despite the presence of structural resonances in many of the proximatevariables, the overall sensitivity of hearing varies smoothly with frequency and does notexhibit sudden changes

iii. Understanding this dichotomy has been challenging.

5. The input and output variables

Which input variable, at the ear-canal entrance determines sensitivity? Which outputvariable characterizes changes in tension of the inner-hair-cell stereocilia? Possible

candidates for the input variable are pressure measured with a microphone, volume-velocity (acceleration and displacement), power, or transmittance and reflectance. Since

these variables are interrelated, it is difficult to truly separate the effects of one variablefrom another. However, the use of pressure has some advantages.

Dallos (1973) showed that there is good agreement between hearing sensitivitymeasured behaviorally and the eardrum-to-cochlear pressure transfer function, also calledthe middle ear pressure gain resulting from ossicular coupling. It appears that the

combined properties of the middle ear and its cochlear load are the dominantdeterminants of the animal’s measured behavioral sensitivity. This has been directly

measured in cat (Nedzelnitsky 1980), guinea pig (Dancer and Franke 1979; 1980;Magnan et al. 1997), gerbil (Olson 1998) and human (Puria et al. 1997; Aibara et al.

2001; Puria 2003). In agreement with Dallos (1973), Puria et al (1997) show that there is


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good correlation between the human middle ear pressure gain and behavioral threshold.This suggests that an important proximate variable, at least at the base of cochlea, is fluid pressure in the vestibule

iv.

In the organ of Corti it is well accepted that tension in the tip links is the ultimatemechanical variable for the mechano-electric transduction (Corey and Hudspeth 1983;

Howard and Hudspeth 1988). This tension opens ion channels and initiates the flow ofions through the stereocilia bundle resulting in depolarization of the hair cell body whichresults in firing of the auditory nerve.

In the sections that follow we generally follow the path taken by sound from theexternal ear, through the middle ear, into the fluid filled cochlea. We then analyze the

mechanisms that cause the basilar membrane and the organ of Corti to vibrate which thenresults in tension modulations of the hair cell stereocilia tip-links.

C. External ear

The external ear consists of the highly visible cartilaginous pinna flange, the cavumconcha and the ear canal buried in the skull. It is generally accepted that sound sourcelocalization in a free field consists of two processes. The sound source azimuth is

determined using interaural time or interaural intensity, whichever is the dominant, whilesound source elevation is based on spectral cues from the pinna.

There is significant variability in both size and shape of the external ear amongst

mammals and the resulting pressure transformation from the free field to the tympanicmembrane. Examples of anatomical variations include cone shaped pinna in cats to

almost flat pinna in ferrets, numerous invaginations and protuberances of the pinna flangeand concha, and changes in ear canal cross-sectional area often accompanied by bends in

the canal. The ear canal and concha boost the sound field in the middle frequency range.A key role of the pinna is to diffract the sound in a spatially dependent manner and thus

augment the sound field spectral cues. The torso also adds to elevation cues particularlyat low elevations and low frequencies in the form of a shadowing effect (Algazi et al. 2002).

A common measure of the effect of external ear function is the free field to tympanicmembrane pressure ratio Ptm/Pff . When measured as a function of spatial angle, the

magnitude of the ratio is often called the head related transfer function (HRTF). Notsurprisingly, the effect of the anatomical structures on the HRTF is likely unique to eachanimal and varies significantly in individuals for a given species.

The transformation of the free field sound pressure to that measured at the tympanic

membrane is determined by diffraction, scattering, and resonances due to the asymmetricstructures along the way. The frequency region where different structures becomeimportant occurs when the wavelength of sound becomes smaller than the physicaldimensions of a feature of the external ear.

1. Concha and ear-canal resonance

Dimensionally, the largest feature of the human ear with some acoustic consequence is

the ear canal, which is approximately 25 mm in length, and 7 mm in diameter with a


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corresponding quarter-wavelength resonance near 2.5 kHz with an approximate pressuregain of about 10 dB (Békésy 1960; Shaw and Teranishi 1968; Shaw 1974). Significant

developmental changes in the ear canal dimensions and wall properties take place evenup to the age of 24 months (Keefe et al. 1993).

The next larger feature is the concha with a height of 19 mm, a width of 16 mm and a

depth of about 10 mm. There is significant individual variation in these dimensions withvery little correlation between them or with other pinna dimensions (Algazi et al. 2001).

The depth mode resonance in the 4-5 kHz range, results in a pressure gain of about 10dB. Both the canal and concha-depth resonances are complementary effects and are

approximately independent of angle of the free-field sound and produce a pressure gainthat starts at about 1.5 kHz reaching a maximum gain of up to 20 dB near 3-4 kHz and

then decreasing again. At frequencies above 5 kHz, the width and depth modes of theconcha becomes important and excitation of these modes is dependent on the angle ofincident sound (Shaw and Teranishi 1968; Teranishi and Shaw 1968).

2. Spatial diffraction by the Pinna

To a first order approximation, the pinna flange and the surface of the headmechanically behave as rigid bodies to acoustic waves. In humans and in some animalslike ferrets the pinna is immobile while in other animals like mice and cats the pinna are

mobile and able to move due to muscular control independent of the skull. Many of themobile pinnae have a horn like structure, which improves their sound collecting ability.

The larger cone may allow an effective interaural time delay that is greater than is possible for the head alone while the mobility allows for the possibility to modulate theinteraural time difference (Shaw and Teranishi 1968).

In humans the pinna is relatively large (64 mm x 29 mm) but it does not seem to bestrongly correlated with a resonant mode (Algazi et al. 2001). One role for the larger pinna is to increase directivity and thus reduce background noise. There are several

unique geometric features of the pinna that contribute to resonance modes at frequenciesabove 6-7 kHz. These modes are dependent on the angle of the incident sound and areclearly important for determining the HRTFs measured in individual subjects.

The brain continually calibrates and interprets the HRTFs to infer the location of soundindicating that there is plasticity in the perception of the spectral cues (Hofman et al.

1998). This was demonstrated by modifying the pinna of adult human subjects with a prosthesis so as to disrupt the spectral cues resulting in poor spatial localization in the

vertical plane. However, after a relearning period of about 30-45 days the subjects wereable to localize accurately again. Furthermore, the subjects did just as well after removal

of the prosthesis suggesting that the new cues did not interfere with the perception of previous cues.

3. Tympanic-membrane and ear-canal interface

The delicate tympanic membrane is located at the end of the long ear canal deep inside

the skull likely for protection from mechanical damage. At frequencies aboveapproximately 1 kHz the membrane response is very complex, while the cochlea provides

a mainly resistive load (Onchi 1961; Møller 1963; Zwislocki 1963; Khanna and Tonndorf1969; Lynch et al. 1994; Puria and Allen 1998). This resistive load is the primary


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damping factor of the external ear resonances.

D. Middle ear

The ear canal is filled with air that is continuous with the free field. On the other hand

the cochlea is filled with cerebro-spinal and other salty fluids. The mechanical propertiesof these media are shown in Table 1. What matters for effective wave propagation is the

specific impedance, which is the product of density and wave-speed of the medium. Eventhough the fluid of the cochlea has mechanical properties close to saline, the flexibility of

the cochlear partition greatly slows the wave speed, which causes a lower specificimpedance

v and an air-to-cochlea impedance ratio of about 1/200. Such a large

impedance mismatch would cause most of the sound energy entering the ear canal toreflect and not enter the cochlea.

Table 1 – Acoustical and mechanical properties of air, saline and the input widow to thecochlea.

medium density

! (kg/m3)

speed of soundc (m/s)

specificimpedance

z = ! c (Pa-s/m)

impedanceratio

" =z /z cochlea

Air 1.29 350 448 1/212=0.0047

Saline 1000 1500 1.5x106 15.7

Cochlear input 1000 95 (approx) 9.5x104 1

The above shows that the slower speed of sound in the cochlea fluid reduces the air to

fluid impedance mismatch by a factor of 15.7 (24 dB). A simple model in Figure 1illustrates this concept. The model consists of two semi-infinite tubes of cross-sectional

areas A1 and A2, with the ratio # = A1/ A2, filled with fluids with the densities ! 1 and ! 2

and speeds of sound c1 and c2. The acoustic impedances are z1 = ! 1c1 and z2 = ! 2c2 , with

the ratio " = z 1/ z 2. The piston has one face in tube 1, and the other face in tube 2.

Figure 1: Greatly simplifiedmodel for the middle earconsisting of a pistonconnecting two acoustic tubes.Tube 1 represents the earcanal, with an incident waveand a wave reflected from thepiston. Tube 2 represents thefluid filled inner ear with atransmitted wave.

The hypothetical piston is free from constraint and is massless, so the force on the twosides of the piston must be equal. An incoming acoustic wave in tube 1 (the ear canal)


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impinges upon the piston, causing the generation of a transmitted wave in tube 2 (thecochlea), as well as a reverse reflected wave in tube 1. The standard 1-D transmission

line analysis for acoustic waves yields the ratios of the amplitudes of transmitted andincident pressure and energy:

p2

p1in

=

2!

1+!" E

2

E 1in

=

4!"

1+!" ( )2 (Eq. 5)

The ratios for the areas of the tympanic membrane and the stapes footplate typical for

human and cat give the results in Table 2. For conduction in air, the large ratio greatlyimproves the energy flowing into the cochlea. Since this is far from 100%, it is not

impedance “matching”, but rather impedance mismatch alleviation. Perfect impedance

matching #" = 1 would provide for humans only a 15 dB improvement in the transmitted

pressure at the considerable cost of a 10 times larger tympanic membrane. It must benoted that larger areas enhance the signal-to-noise ratio at the hair cell level (Nummela1995).

So the large tympanic membrane is advantageous to human and cat for hearing in air. Itis interesting to consider a change to hearing under water. For this, the air in tube 1 is

replaced by water, which yields the results in the bottom section of Table 2. The acoustic pressure transmitted to the cochlea is greatly reduced to a value insensitive to the area

ratio. The difference in pressure in air and water of 49 dB is close to the behavioralthreshold difference measured in divers (Brandt and Hollien 1967; DPA 2005). This

supports the simple relation in Eq. 5 as a fundamental consideration for the design of themiddle ear.

Table 2 – Effect of middle ear area ratio and specific impedance ratio in transmitting

sound pressure and energy into the cochlea, according to the basic model in Figure 1.

Replacing the air in the ear canal (tube 1) with saline simulates underwater hearing, whichhas a great reduction in the transmitted pressure.

Tube 1 (EC) " = z 1/z 2 # = A1/A2 p 2/p 1 (lin) p 2/p 1 (dB) E 2/E 1 (lin)

Air 0.0047 1 2 6 1.8%

20 (human) 36 31 25

40 (cat) 67 36 53

212 212 46 100

Water 15.7 1 0.12 -18 22

20 (human) 0.13 -18 1.7

40 (cat) 0.13 -18 0.6

In Table 3 the amplitude of the incident sound wave at threshold is given for hearing inair and water (Fay 1988). The pinnipeds (marine mammals including sea lions, walruses,and true seals) spend time in both air and water and have hearing sensitivity worse than

humans by a factor 10 (20 dB) in air and better by a factor of 5 (14 dB) in water.However, the cetaceans (whales and dolphins) have better hearing sensitivity in water

than humans by factor of 54 (36 dB). It is interesting that the intensity of the sound atthreshold is about the same for human in air and pinniped in water, and for human in

water and pinniped in air. Obviously, the middle ear of the pinniped is designed for the


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water environment. Quite a different middle ear design provides the extraordinarysensitivity under water of cetaceans (Hemila et al. 1999).

Table 3 – Some approximate thresholds of hearing in air and water.

Air Water

Pressure(µPa)

Intensity(Watts/m2)

Pressure(µPa)

Intensity(Watts/m2)

Human 20 8.9 x10–13 5400 2 x 10–11

Pinnipeds 200 8.9 x 10–11 1000 6.7 x 10–13

Cetaceans - - 100 6.7 x 10–15

As the simple estimate indicates, without an effective middle ear, the sensitivity of thecochlea would be compromised and so would the overall bandwidth as is evident by

pathological conditions of the ear repaired by otologists. As discussed in a subsequentsection, another important role of the middle ear is in exerting some degree of dynamic

range control at high input levels via the three sets of muscles.The simple model of Figure 1 is useful to certain degree but has significant limitations.

In order to build an acoustic lever with an area change from the ear canal to the cochlea

requires using biological materials consisting of bone and soft tissues. A rigid piston witha large area requires a large mass, which limits its ability to transduce sound particularly

at the higher frequencies. A membrane is lighter but has a significant number of resonantmodes particularly at frequencies above 2-3 kHz. In a very thorough study, Nummela

(1995) show that malleus and incus masses scale with eardrum area, which further limitshigh frequency hearing. These factors must be considered when formulatingmathematical models of the middle ear.

More sophisticated models describing sound transmission in the middle ear have been

around for some time. Early studies allocated various acoustic influences to the differentmiddle ear structures interconnected in 5-6 functional blocks. The blocks were thenassigned more detailed elements, which consist of masses, springs, and dashpots. Some

of the earliest models by Onchi (1949; 1961), Zwislocki (1961), and Møller (1961) usedynamic analogies and represent the middle ear in the form of electrical circuit models.

These phenomenological models have evolved and continue to be useful forunderstanding surgical interventions of the middle ear (Rosowski and Merchant 1995;

Merchant et al. 1997; Rosowski et al. 2004). Nevertheless, they have limitations in thatthere is not a tight relationship between the underlying anatomical structure and function.

To overcome these limitations requires models that explicitly incorporate morphometryof the middle ear into the formulation.

1. Tympanic membrane shape and internal structure

There remain many unanswered questions regarding the biomechanics of the tympanicmembrane. For example, why does the tympanic membrane have a conical shape? Why

do the tympanic membrane sublayers have a highly organized collagen fiber structure?What is the advantage of its angular placement in the ear canal? Why is there

symmetrical malleus attachment to the eardrum in some animals while in others there isasymmetry? The functional significance of many of these gross anatomical features of the


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tympanic membrane is just beginning to be understood and current status is discussed below.

Helmholtz

(1868) discussed the need for impedance matching of the air in the

environment and the fluid of the inner ear and suggested that the tympanic membrane behaved as a piston. This assumption is widely used in lumped parameter (circuit) models

of the middle ear, which build upon the free piston model (Eq. 5) by adding springs andthe resonances of the malleus-incus complex and of the middle ear cavity. However,

instead of piston behavior, surface displacement measurements revealed multiple modesof vibration for frequencies above a few kHz (Tonndorf and Khanna 1972). Since the to-

and-fro motion of a resonance mode would reduce the effective area for the sound pressure, the presence of these modes has been difficult to explain. Pioneering work by

Rabbitt and Holmes (1986) formulated a continuum analytic model with asymptoticapproximations for the cat tympanic membrane. They included the membrane geometry

and anisotropic ultrastructure in combination with curvilinear membrane equations, butdid not analyze the effects of the eardrum angle and the conical shape of the eardrum, nor

have Eiber and Freitag (2002). Current finite-element models represent the eardrum as an

isotropic membrane (Wada et al. 1992; Koike et al. 2002; Gan et al. 2004) and thus donot explain the need for the detailed fiber structure (Lim 1995).

Two breakthroughs have increased our understanding of tympanic membrane mechanics. First, was the observation that there is significant acoustic delay in eardrum

transduction (Olson 1998; Puria and Allen 1998). Second, the multiple modes ofvibration seen on the surface of the eardrum are not transmitted to the cochlea. Rather,

the pressure inside the cochlea as a function of frequency remains relatively smooth, evenwhen measured at a high frequency resolution (Magnan et al. 1997; Puria et al. 1997;

Olson 1998; Aibara et al. 2001; Puria 2003). Clearly these observations are tied to thecomplicated motions of the eardrum observed by Khanna and Tonndorf (1972) but needexplanation.

2. Tympanic Membrane Biomechanics

To understand the functional consequences of the tympanic membrane structure on its

sound transducing capabilities, a biocomputation model has been formulated which leadsto some insights on the posed questions (Fay 2001; Fay et al. 2006). The model

incorporates measurements of the geometry of the ear canal (Stinson and Khanna 1994),the 3-D cone shape of the eardrum (Decraemer et al. 1991), and details of the eardrumfiber structure (Lim 1995).


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Figure 2: Human eardrum photograph with its biomechanical model representation composed of

adjacent wedges. The zoomed box shows the four layer composite of each wedge. The inner radial and

circumferential collagen fiber layers, unique to mammals, provide the scaffolding for the tympanic

membrane. Dimension and material property differences of the wedges lead to mistuned resonances at

high frequencies. The thickness of the eardrum layers increases from the umbo to the tympanic annulus.

The discrete model for the human eardrum is shown in Figure 2, in which a series of

adjacent wedges approximate the eardrum. Near the center, the eardrum is attached to themalleus, while the outer edge is attached to the bony annulus (not shown). The 1-D

acoustic horn equation is used for a small cross-section of the ear canal. The change inarea from the adjacent section, the curvature of the centerline, and the flexibility of the

portion of the eardrum that intersects with that section of the ear canal are taken intoaccount. Each strip of the eardrum has a curvature near the outer edge (locally a toroidal

surface) and is straight in the central portion (locally conical). Because the main conical portion has few circumferential fibers, the approximation is that the radial strips areweakly coupled in the circumferential direction.

The tympanic membrane is represented as a four-layer composite (Figure 2). The input


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parameters for the formulation are the thickness of each layer as a function of positionand the Young’s modulus of elasticity (a measure of resistance to deformation) for each

layer. The outer most epithelial layer and the inner most submucosal layers are relativelyflexible. Because the sub-epidermal layer and the sub-mucosal layers consist of

connective tissue and are also relatively flexible, they are part of the epidermal and

mucosal layers respectively (Figure 2). The inner two layers have collagen fibers that areradially oriented in one layer and circumferentially oriented in the layer directly below.These two layers, unique to mammals, provide the majority of the scaffolding for the

eardrum and thus those layers mostly determine the compliance of the membrane. Themass on the other hand comes from overall thickness of the membrane. Quantitative

measurements for cat were used for the overall thickness (Kuypers et al. 2005). Fromthese measurements and from sparse measurements of collagen sublayers, the thickness

of each sub layer was estimated for human (Figure 2) and cat (Fay et al. 2006; Fay et al. 2005).

Direct measurements of the static elasticity of portions of the eardrum (Békésy 1960;

Decraemer et al. 1980) indicate an effective modulus of elasticity of around 0.03 GPa.

This was re-examined using three very different methods to determine the eardrummodulus of elasticity (Fay et al. 2005). First, constitutive modeling incorporating theYoung’s modulus of collagen and experimentally observed fiber densities in cat and

human were used. Second, the experimental tension and bending measurements (Békésy1960; Decraemer et al. 1980) were reinterpreted using composite laminate theory. And

third, dynamic measurements of the cat surface displacement patterns were combinedwith a composite shell model. All three methods lead to similar modulus of elasticity

value of 0.1-0.4 GPa for near the center of the eardrum. The corresponding values nearthe outer edge are approximately ! these values due to the liner taper in the elastic

modulus. In previous models the eardrum is treated as a single layer having a uniformelastic modulus resulting in a low value of elastic modulus (Funnell et al. 1987;

Prendergast et al. 1999; Koike et al. 2002; Gan et al. 2004). In the four-layer model, thecollagen fiber sub layer is much thinner than the overall thickness and hence theestimated elastic modulus is higher.

The modulus of elasticity was combined with the sub layer thickness to formulate acomplete model of the cat tympanic membrane. The calculation for the dynamic response

of each strip was performed with an algorithm for elastic shells (Steele and Shad 1995),which has no restriction on wavelength along the strip.


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Figure 3: Effect of modification of the eardrum depth. (a) In the center is the anatomically normal eardrum. The z -

coordinate of all the points is divided by a factor of 10 to obtain the shallow eardrum on the left, and multiplied by a

factor of 2 to obtain the steep eardrum on the right. (b) Effect of eardrum depth on the middle ear pressure transfer

function, which is the ratio in dB of the pressure delivered to the vestibule inside the cochlea ( pv) divided by the input

pressure in the ear canal ( pec). The deep eardrum calculation is nearly the same as the normal, but the shalloweardrum has more than a 20 dB loss at higher frequencies. For the normal and deep eardrums, the phase delay is

steeper than it is for the shallow drum, indicating more acoustic delay. (Reproduced from Fay et al., 2006 with

permission).

The full 1-D interaction of the air in the ear canal and the eardrum is included. Behind

the eardrum are the middle ear cavities and the middle-ear bones connected to thecochlea, for which lumped-element approximations were used. Verification involved

mesh refinement studies, comparison with exact solutions for limiting cases, anatomicalvalues of geometry, best estimate for elasticity, and comparison with physiologicalmeasurements to 20 kHz, all for the cat middle ear.

Different depths of the eardrum play an important role as shown in Figure 3. With ashallow eardrum (no cone shape) there is a loss of more than 25 dB for frequencies above

about 4 kHz (Figure 3b, top panel). A deep eardrum shows a response similar to that seenin anatomic specimens, with little loss for low frequencies. Above 4 kHz, the phase for

the normal and deep eardrum continues to decrease while for the shallow drum the phasetends to go in the opposite direction and increases. This suggests that there is more phase

delay for the deep and normal shape than for shallow eardrums. In comparison to the


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normal eardrum the deep drum requires more real estate in the skull, which competes forspace with other organs.

The effect of the two collagen-fiber sub-layers was also analyzed. This was done by

examining the effects of isotropic eardrums that had the same stiffness in the radial andcircumferential directions and orthotropic eardrums where there were radial fibers but no

circumferential fibers (Fay 2001; Fay et al. 2006). Results indicate that there is anadvantage of the orthotropic microstructure with a dominance of radial fibers in the

central region. In the normal drum when both are present, the radial fibers on the inner portion of the tympanic membrane result in an effectively orthotropic membrane while

the outer circumferential fibers provide a low-impedance beam-like support. Theorthotropic central portion allows maximal sound transmission at both low and highfrequencies.

The model calculations indicate that sound transmission from the ear canal to thecochlea varies smoothly despite the fact that there are a significant number of resonances

at different points on the eardrum. This suggests a design where drum sections aredeliberately mistuned. Because these resonant points are added together at the malleus,

no single mode ever dominates. Thus the ensemble of eardrum modes produces arelatively large and yet fairly smooth response at the malleus at the higher frequencies.

Understanding of eardrum biomechanics is of critical importance to the development

and improvement of “myringoplasty” which is a surgical procedure for repairingdamaged eardrums. The underlying disease process is often chronic inflammatory disease

of the middle ear and mastoid, referred to as chronic otitis media (COM), which leads toa partial or total loss of the tympanic membrane or ossicles. Clinically, isotropic materials

like temporalis fascia are used for myringoplasties. To improve hearing results at thehigher frequencies, orthotropic material with collagen scaffolding preferentially oriented

in the radial direction would be a better choice for improved high frequency hearingoutcomes. Improving post-operative high frequency results may be important for the

perception of sound localization cues present at high frequencies. Currently the standard practice is to measure clinically to 6 kHz. The above results suggest that clinical

measurements at frequencies above 6 kHz might better show the effects of differentmaterials.

Since the modulus of elasticity and the biocomputation approach using asymptotic

methods is already developed for the cat, the challenge will be to estimate eardrummorphometry for other species such as human (Figure 2 shows an approximate guess). Of

particular interest is determining how the shape and thickness of the tympanic membranevaries from subject to subject. Such quantification will allow for the possibility of using

the eardrum biocomputation on individual subjects. Non-destructive high-resolutionimaging methods are needed to obtain morphometry on individual subjects. A promising

new imaging technology is described in the next section.

3. Middle ear imaging

To obtain morphometry of the ear, histological methods have been the primary

technique. However, this age old technique is destructive and certainly not appropriatefor in-vivo imaging of individual subjects. One of the most recent advances for obtaining

anatomical information is micro computed tomography (microCT). This has been used to


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obtain volume reconstructions of the temporal bone of living subjects at a resolution ofless than 125 "m (Dalchow et al. 2006). In-vitro resolution can be increased by an orderof magnitude (Decraemer et al. 2003).

Figure 4: MicroCT image of anintact cadaver temporal bone. Thisis image #769, of 1897 imagesspanning a length of 28.455 mm.The image illustrates that most ofthe middle ear structures can bevisualized from an intact temporalbone ear scan. The resolution forboth in-plane and out-of-plane

(slice thickness) is 15 m. The

tympanic membrane althoughvisible is faint, suggesting thatthe basic geometry and anapproximate thickness can be

obtained. The 30.72 mm scandiameter outline is clearly seen.

Figure 4 shows an image slice from an intact human cadaver temporal bone ear. The

image resolution in the x, y, and z planes is 15 µm (iso-volume). Most of the middle ear

structures, including the tympanic membrane cone shape and thickness, ossicles, and

suspensory soft tissue, can be visualized because there is good density contrast betweenthese structures and air in the ear canal and middle ear cavity. Because they provide the

best resolution, histological methods remain the standard. However, µCT imaging offers

some distinct advantages (Decraemer et al. 2003). These include: (1) elimination of

stretching distortions commonly found in histological preparations, (2) use of a non-destructive method, (3) shorter preparation time (hours rather than 12-16 months), and (4)

results already in digital format. This imaging technology is rapidly evolving and it islikely that similar resolutions will be possible for in-vivo imaging in the near future.

4. Malleus-incus complex

The middle ear of most non-mammalian terrestrial animals consists of the tympanicmembrane and a columella, while mammals have a tympanic membrane and a malleus-

incus complex. Amongst vertebrates a great majority of mammals are sensitive toultrasonic sounds (above 20 kHz), while non-mammals are not

vi. This suggests that the

mammalian hearing organ evolved to be a superior organ for high-frequency responsecompared to that of non-mammals and that the incorporation of the malleus-incus

complex may have something to do with this capability (Fleischer 1978; 1982). However,the biomechanics of this sub system of the middle ear are not well understood.

Since the time of Helmholtz (1868) the handle of the malleus and the long process of

the incus were described as the two arms of a lever with a fixed axis. Ossicle suspensionalso further supported the notion that the malleus and the incus rotate about a fixed axis


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while driving the stapes in a piston like manner. However, detailed measurements of theossicles have changed this view (Decraemer et al. 1991; Decraemer and Khanna 1995).

The malleus motion changes with frequency and all 3-D components of translation androtation are present at biologically relevant stimulation levels. These measurementssuggest that a full 3-D model of ossicle motion is required.

Between the malleus and incus is a saddle-shaped joint formed from an indentation inthe head of the malleus into which the surface of the body of the incus fits (Figure 5). The

incus also has a depression into which a part of the malleus head fits, forming a cog-likemechanism as described by Helmholtz (1868). The significance of such a mechanism is

thought to be a locking of the joint causing one part to move with the other duringrotation in one direction but leaving the parts free to rotate in the orthogonal direction

(Wever and Lawrence 1954). However, measurements (e.g., Helmholtz 1868; Békésy1960) suggested that the incus and malleus are fused together indicating that there is no

slippage at the incudo-malleolar joint (IMJ). Making measurements in the cat ear, Guinanand Peake (1967) showed clear evidence of slippage at the IMJ above about 8 kHz. Using

time-averaged holography measurements Gundersen and Høgmoen (1976) concluded that

the “malleus and incus rotate like one stiff body” for frequencies below about 2 kHz. Dueto these measurements, mathematical models of the human middle ear generally treat thetwo ossicles as fused and do not include slippage (Goode et al. 1994; Koike et al. 2002).

More recent measurements suggest slippage between the incudo-malleolar joint and lackof slippage in previous measurements was possibly due to methodological reasons

including a possible lack of a cochlear load and insensitive measurement techniques(Willi et al. 2002). In some animals, like guinea pig and chinchilla, the IMJ is fused and

thus there is no slippage (Puria et al. 2006). On the other hand, there is no controversyregarding slippage at the joint between the incus and the stapes, and most mathematicalmodels currently include it (e.g., Goode et al. 1994).

Natural mode shape calculations indicate that the ossicles can be treated as rigid bodies

only for frequencies below about 3.5 kHz (e.g., Beer et al. 1999). Consequently, theossicles have been modeled as finite elements, which require much more computationtime. An alternative approach is to model the ossicles as elastic bodies incorporating justthe first two or three modes in each body (Sim et al. 2003).

Not unlike the biological ligaments found in other parts of the body, the suspensoryligaments and tendons of the middle ear are a composite, consisting of collagen and

elastin embedded in an amorphous intercellular material often called ground substance ormatrix which is composed of proteoglycans, plasma constituents, metabolites, water and

ions. Almost two-thirds of the weight of ligaments is water, while about three-quarter ofthe remaining weight can be attributed to the fibrillar protein collagen (reviewed by

Weiss and Gardiner 2001). Like the eardrum, the primary component resisting tensile

stress in ligaments and tendon is collagen. The primary role of the ground substance is inmaintenance of the collagen scaffolding. As such, the biomechanical behavior of aligament is determined by its geometry, shape of the articulating joint surfaces,

orientation and type of insertions to bone, in-situ pretension, and material properties.What role do the suspensory ligaments play in the complicated 3-D vibrations of themiddle ear bones? This question has yet to addressed with any degree of satisfaction.

In the cat study discussed above, a simple ball and stick model for the malleus-incuscomplex was used (Fay et al. 2006). This was a gross simplification but allowed


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concentration on the tympanic membrane biomechanics. A goal of several laboratories isto combine anatomical data with human cadaver temporal bone malleus-incus complex 3-

D motions into a computational model for individual ears, which should increaseunderstanding of the functional consequences of the anatomy of the ossicles andsuspensory ligaments and tendons.

a) b)

Figure 5 : Volume reconstruction of the malleus and incus from uCT slices. (a) The incus is made transparent to allow

better visualization of the incudo-malleolar joint. (b) The incudo-malleolar joint saddle shape and thickness map (0 is

dark green while about 300 !m is red).

The biomechanical characterization of the malleus-incus complex requiresmorphological and dynamical measurements from individual ears. The center of mass,

moments of inertia, anatomical location and orientation of the ligaments and tensor-

tympani tendon, are obtained from 3-D volume reconstructions (Figure 6) based on µCT

images of the isolated preparation.

Figure 6: Three-dimensional

volume reconstruction of the

malleus, incus, suspensory

ligaments and the tensor tympani

tendon. The soft tissue is

represented as tapered cylinders

or as a polyhedron. The origin is

at the umbo. All dimensions are

in mm.

The morphometry is used to construct a computational biomechanical model for themalleus-incus complex that includes ligament and tendon attachments to the bony walls

and muscle, and slippage at the incudo-malleolar joint. Bending of the malleus and incus


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handles is also allowed. The viscoelastic parameters of each ligament, tensor tympanitendon, and the incudo-malleolar joint cannot be determined from the morphometry andthus 3-D motion measurements are required.

As discussed in previous sections, the biomechanics of the tympanic membrane can befairly complicated. This implies that the input to the malleus-incus complex is also

relatively complicated and thus it is difficult to deduce the dynamics of ossicles and softtissue attachments with the sound driven eardrum. To better understand ossicle dynamics

an isolated malleus-incus complex preparation was developed where the tympanicmembrane and the stapes were dissected. Without an eardrum or a cochlea, the middle

ear bones have to be directly driven. A tiny magnet and a coil around the tympanicannulus were used to drive the malleus-incus complex (Sim et al. 2003). The magnet on

the tip of the malleus is oriented to drive it in the forward direction. The preparation is placed on a set of goniometers and malleus-incus motion measurements made at several

points at several different angles. The resulting three-dimensional x, y and z vectorcomponents of velocity at each point is used within the biomechanical model to obtain

the soft tissue viscoelastic parameters. The 3D volume reconstruction of the magnet and

coil combined with electro-magnetic theory allows accurate calculation of the 3D forcesand moments exerted by the magnet to the malleus. The combined, imaging, physiologyand biomechanics approach should help us better understand the structure and functionalrelationships at audio frequencies in normal and pathological ears.

The above discussion concerns the dynamics of the malleus-incus complex. At high positive and negative static pressures such as during sneezing and coughing the

suspensory ligaments may also play a critical role (Huttenbrink 1989). Incorporation ofthis mode of operation requires extension of the linear models to non-linear models.

5. Lenticular process

The inferior end of the long process of the incus terminates in a short perpendicular

bend called the lenticular process consisting of the pedicle and the lenticular platesurrounded by soft tissue. Between the lenticular plate and the stapes head is the incudo-stapedial joint. Motion from the incus is transmitted to the stapes via this process and thusits mechanical description is of significance.

Most previous modeling work has treated the lenticular process to be a rigid bone thattransmits the incus motion directly to the stapes head or with a slippage representing the

incudo-stapedial joint (Beer et al. 1999; Koike et al. 2002). Recent anatomicalmeasurements suggest that the plate-like bony pedicle is perpendicular to the lenticular

plate and is extremely thin and fragile. In cat the dimensions of the pedicle are 240 µm x

160 µm x 55 µm (Funnell et al. 2005). Model calculations of static displacements suggest

that there is significant relative motion between the incus long process and stapes head(Funnell et al. 2005). Funnell and colleagues have hypothesized that one role for the thin

pedicle and lenticular plate arrangement may be to convert the rotational modes ofvibration of the incus into translational motion of the stapes. More work is needed tofurther test this hypothesis.

It has been observed that at high static pressures, there is a large lateral displacement ofthe lenticular process and this serves to protect the cochlea from large motions(Huttenbrink 1988). Clearly, bending of the pedicle may be involved.


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6. Stapes

The interface between the malleus-incus complex and the vestibule of the fluid filledcochlea is the stapes, which is held in place in the oval window (Fenestra vestibule) by

the annular ligament. The mechanics of the stapes is quite independent of the malleus-incus complex and of the cochlear fluid load. For this reason the stapes can be considered

a semi-independent sub system of the mammalian ear (Fleischer 1978). This treatment ofthe stapes is widely accepted (Wada and Kobayashi 1990; Wada et al. 1992; Goode et al. 1994; Puria and Allen 1998; Beer et al. 1999; Koike et al. 2002).

7. Ossicular reconstruction

While we are discussing the ossicles this is good place to discuss ossiculoplasty, which

is the reconstruction of the middle ear bones to improve hearing sensitivity. Two of themost common pathologies are missing (or eroded) incus and ossified stapes. Both result

in significant conductive hearing loss. Since the introduction of these surgical proceduresmore than fifty years ago, ossiculoplasty continues to pose significant challenges tootologists.

The interposition of passive prostheses between the malleus or tympanic membraneand the stapes head or footplate is used to reconstruct the transfer function of the middleear in the missing or eroded incus condition. These are the incus replacement prostheses.

Two types, depending on the circumstance, are the partial ossicular reconstruction prosthesis (PORP) to the stapes head while another is the total ossicular reconstruction

prosthesis (TORP) to the stapes footplate. The PORP is typically used if there is an intactstapes superstructure. However, ear canal pressure to cochlear pressure transfer function

and clinical measurements suggest that even if the stapes superstructure is present thereare acoustico-mechanical advantages to placing the foot of the prostheses on the footplate(Murugasu et al. 2005; Puria et al. 2005).

In a very different disease process called otosclerosis, the stapes becomes fixed to thesurrounding oval window through ossification. The immobile stapes prevents sounds

from entering the cochlea and results in significant hearing loss. The precise cause ofotosclerosis is not well understood. However, it is becoming well established that

otosclerosis is hereditary. Otolaryngologists repair the condition by a procedure calledstapedotomy. A hole is made in the footplate often with a surgical laser (Perkins 1980)

and then covered with soft tissue to prevent the inner ear fluid perilymph from leakingout. Sound transmission is restored with a piston like prosthesis. One end of the

prosthesis is crimped to the long process of the mobile incus while the other end isinserted in the covered artificial hole in the footplate.

8. Middle-ear muscles

The malleus and stapes each have a tendon attached to a tiny muscle, the tensortympani muscle and the stapedius muscle respectively. The muscles contract when

exposed to high level sounds and are part of the middle ear reflex arc involving the spiralganglion neurons, the auditory nerve, cochlear nucleus, the superior olive, the facial

nerve nucleus, the facial nerve and the two middle ear muscles (Margolis 1993). Thisreflex arc can reduce sound transmission through the middle ear at high levels and may

serve to control the dynamic range of the auditory system and to protect the cochlea at


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high sound levels. The reflex is slow and thus does not provide protection to the cochleaagainst sudden impulsive sounds. The time for the stapedius reflex may be on order ofabout 20 ms while the tensor tympani arc is more than ten times slower.

Two additional functions are attributed to the middle ear muscle reflex. Low frequencysounds, particularly when they are high in level, normally tend to mask mid and high

frequency sounds due to their upward excitation patterns on the basilar membrane. Onerole of the middle ear muscles is to reduce the level of low frequency inputs so they do

not mask the higher frequency sounds on the basilar membrane (Pang and Guinan 1997).A second role of the middle ear reflex is in the reduction of the audibility of generated

sounds during speech, mastication, yawning and sneezing (Simmons and Beatty 1962;Margolis and Popelka 1975). Because the reflex arc involves so many mechanisms, itsmeasurement is clinically used to diagnose central and peripheral pathologies.

Recently it has been discovered that there are smooth muscle arrays on the peripheraledge, annulus fibrous, of the tympanic membrane in all four (bats, rodents, insectivores,

and humans) of mammalian species studied (Henson and Henson 2000; Henson et al. 2005). The role of this rim of contractile muscle cells in the par tensa region is not clear,

but two suggested possibilities are to maintain tension of the tympanic membrane and tocontrol blood flow to the membrane (Henson et al. 2005). Measurements indicate that

these smooth muscles can exert control over the input to the cochlea as measured bycochlear microphonics (Yang and Henson 2002).

9. Middle-ear cavity

One role of the middle-ear cavity is to act as a baffle for the tympanic membrane sothat sound does not impinge on both sides of the eardrum. Without this, the sensitivity of

the membrane, and thus hearing sensitivity, would be significantly reducedvii

. However,the presence of the cavity results in an increase in overall impedance, due to volume

compliance, at low frequencies and resonant modes at high frequencies. An increase in

middle ear impedance results in a decrease in hearing sensitivity (Wiener et al. 1966).

In humans the middle ear cavity is relatively large but is irregular in shape. The

mastoid cavity portion has many air cells, or air pockets, that results in an increase insurface area. Each cell is lined by a mucous membrane of thin epithelial cells. It is

thought that the irregular shape minimizes resonant modes and the air cells effectivelydampen remaining resonance (Fleisher, 1978).

10. Middle-ear acoustic load

The primary load to the middle ear is the acoustic input impedance of the cochlea Zc.

As defined by Zwislocki (1975), Zc is the ratio of sound pressure in the scala vestibuli atthe stapes footplate to the volume velocity of the footplate. Based on simplifications to

the equations of motion at the base of the cochlea, Zwislocki (1948; 1965; 1975) predicted that the cochlear input impedance is primarily resistive. Direct measurements in

the cat (Lynch et al. 1982), guinea pig (Dancer and Franke 1980), and human cadaverears (Aibara et al. 2001) suggest that the prediction by Zwislocki was essentially correctfor a broad range of frequencies.

Zwislocki’s calculation had not included effects from the apical region of the cochlea.Calculations of the cochlear input impedance in the constant scalae area standard box


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models of the cochlea, that include the apical region, shows that below approximately 1-2kHz, the cochlear input impedance magnitude decreases and becomes more mass like.

This calculated result diverges from the measured data and from Zwislocki’s prediction(Puria and Allen 1991; Shera and Zweig 1991). The decrease in the acoustic impedance

and mass like response is shown to be due to the use of constant cross sectional area for

the scala vestibule and scala tympani in all standard box models. Using a more realisticscalae area that decreases from base to apex of the cochlea avoids the divergingcatastrophe in the model calculations of cochlear input impedance at low frequencies.

The resistive nature of the cochlear input impedance, which is the primary damping

component of sound transmission in the middle ear, has two consequences. Foremost isthat a large fraction of the acoustic energy that enters the cochlea is absorbed by it rather

than being reflected by it. Second, is that it smoothes out the peaks and valleys resultingfrom any resonances in the middle ear structures.

E. Cochlear hydrodynamics

In the preceding section, methods of imaging, physiology, and computational biomechanics were presented in the context of understanding the relationship betweenacousto-mechanical transformations of sound by the middle ear. The end result is that the

proximate output variable of the middle ear, which is the vestibule pressure at the base ofcochlea, smoothly varies with frequency and typically with pressure gain for a wide

bandwidth relevant to the specific species. In the following sections we analyze howsound energy at the base of the cochlea propagates in the cochlea. Much effort has been

devoted to this topic, on which many survey papers have been written, as represented byAllen and Neely (1992), Nobili et al. (1998), and deBoer (1991). DeBoer (2006) provides

a summary of current thought. In addition, other articles in this Handbook addressdifferent aspects of cochlear function. Our focus is on what appear to be key acousto-

mechanical mechanisms that have a basis in the physiology.

1. Vestibule pressure

A simple description of what happens to the pressure transmitted into the cochlea by

the middle ear is shown in Figure 7 for a given frequency. This represents a standardtapered box model for the cochlea with two symmetric fluid ducts divided by a partition.

The stapes provides the input pressure. The wall of the cochlea is bone, which isnormally assumed to be rigid, so for air-conducted sound the stapes and round window

have equal and opposite volume displacement, preserving the volume of fluid in thecochlea. However, a very compliant membrane covers the round window, so the fluid

pressure at this point is nearly zero. Therefore the total pressure is divided into an “even”

and an “odd” solution (Peterson and Bogert 1950), as indicated in Figure 7. The evendistribution must cause a compression of the fluid. This corresponds to a wave thattravels with the speed of sound in the fluid, which is relatively “fast”. The odd solution

produces net pressure acting on the partition that causes an elastic deformation of theflexible portion of the partition, the basilar membrane (BM). This interacts with the fluid

motion, causing a wave that is relatively “slow”. This slow wave is the “traveling wave”first observed in the guinea pig by Békésy (1952). Because the BM is narrow at the

stapes and wide at the apex, there is a gradient in stiffness of the partition, which causes


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the traveling wave to have a long wavelength near the stapes, then build up to amaximum as the wavelength becomes short. In the very short wavelength region, the

viscosity of the fluid causes this wave to die out exponentially. The traveling wave is soslow, relative to the fast wave, that the fast wave can often be approximated asinstantaneous, i.e., for incompressible fluid.

For simplicity, we consider the properties of the partition to be continuous. The actualtissue consists of discrete elements. As shown by Békésy (1960, p. 510) by models with

coupled, discrete elements, behavior similar to that of a continuous system can beobtained. This holds, of course for wavelengths of response that are long in comparison

with the spacing between elements. Many authors use discrete systems directly foradvantage in computation and/or construction.

The description in Figure 7 for the spatial distribution for a fixed frequency also holds

for the waves seen at a fixed point as frequency varies. For frequencies less than the bestfrequency (BF), the slow wave has long wavelength, and for frequencies greater than BF,the slow wave decays to negligible magnitude, leaving only the fast wave.

Figure 7: Simple tapered box model for the pressure in the cochlea. The fluid regions scala vestibuli(SV) and scala tympani (ST) with tapered areas are divided by the partition containing the elasticbasilar membrane (BM). At the apex the partition has an opening, the helicotrema. The input soundpressure acts at the round window (stapes). The response for a single frequency is divided into an

even (symmetric) solution with equal pressure in SV and ST, and an odd (asymmetric) solution withthe pressures in SV and ST of opposite sign. The symmetric solution causes a compression of thefluid, so the wave travels with the speed of sound in saline, which is the “fast wave” in the upperdrawing. In contrast, the asymmetric solution has a net pressure on the partition, which causes adisplacement of the BM that slows the wave considerably. This is the “slow wave” in the lowerdrawing. Because of the taper of the BM the stiffness changes and the slow wave has a wavelengththat is long near the input but becomes short near the region of maximum amplitude. The drivingfrequency is the “best frequency” (BF) for this “place”. In the region of short wavelength, the fluidmotion is 3-D, with a pressure that is maximum on the BM and decays exponentially with both thedistance from the BM and the distance toward the apex. The round window is compliant, so the total


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fluid pressure at that location is nearly zero. Thus at the input end, the pressures from the slow andfast wave must cancel in ST and so are equal in magnitude.

The first direct evidence for this behavior is provided by the measurement of pressure

in the gerbil (Olson 1998) at a distance 1.2 mm from the stapes. Some of theexperimental values are shown in Figure 8, along with calculated values from a 3-D

cochlear model to be discussed later. Near BF the pressure is strongly dependent on thedistance from the BM, with much larger values near the BM (Figure 8a). This shows the

3-D behavior of the fluid in the short wavelength region. For low frequencies, the pressures at the different distances from the BM converge, showing the long wavelength

region. The phase response shows the near cancellation of the waves for low frequencies(Figure 8b). For higher frequencies the slow wave dominates, and the rapid accumulation

in phase is characteristic of a traveling wave. For even higher frequencies, the travelingwave disappears, so all that is left is the fast wave with constant phase, which at different

distances from the BM differ by one cycle, so these are in fact the same. The phase

measurements show that far from the BM (305 µm) the traveling wave quickly

disappears, while the points closer (3 – 228 µm) all have the same phase. In contrast, the

calculation shows differences at these points. This may be due to the large pressure probeinterfering with the fluid motion, which is only simulated in the calculation by taking the

average of the pressure at nine points in the 100 µm diameter of the probe. This is for 80

dB SPL input eardrum pressure. The measured pressure shows a constant value for high

frequency equal to 100 dB SPL. This corresponds to the 80 dB input to the eardrum, witha 26 dB gain through the middle ear, and a 6 dB drop because the fast wave has half theamplitude of the vestibular pressure at the stapes.

Figure 8 (a): Pressure magnitude in thecochlea at the distance 1.2 mm from thestapes in ST measured in the gerbil (Olson

1998) and calculated (Baker 2000). Forfrequencies higher than BF (> 40 kHz), theslow wave is negligible and only the fastwave remains (Figure 7). For lowfrequencies, the fast and slow wavesnearly cancel in ST. Near BF (25 kHz), thefast wave dominates, with 3-D fluidmotion that has much higher pressure

near the BM. The discrete points showthe measurements at the distances fromthe BM of 3 and 305 m, while the

calculated values, shown by thecontinuous curves, include distances inbetween.

Figure 8(b): Pressure phase relative to thesimultaneously measured scala vestibulepressure at the base of the cochlea. Forfrequencies above BF (> 40 kHz), only thefast wave remains. The plateaus differ byone cycle, which shows that the fast waveis uniform with distance from the BM andexactly in phase with the eardrumpressure. For low frequency, the phases atthe different distances from the BM are


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also the same, corresponding to the fastwave and the long wavelength region ofthe slow wave. For frequency approachingBF (25 kHz), the slow wave dominates andshows the rapid decrease in phasesignifying the traveling wave.

2. Partition resonance

At one time or another almost every component of the cochlea has been suggested as akey tuned resonator that will cause a significant local response for a given frequency (the

BF in Figure 7). The basilar membrane (BM) is the thin compliant portion of the partitionthat divides the two fluid ducts in Figure 7. The component for which the tuning can be

best related to the physical dimensions is the pectinate zone of the BM. Mathematicaltreatments of the BM include both bending stiffness and tension in addition to their

interaction with the surrounding fluid. From the mathematical formulation, the frequencyrange and the place to frequency map of the cochlea given the anatomical dimensionswith material properties can be predicted.

a) Plate

A cross section of the basilar membrane is sketched in Figure 9. For many mammals,

the BM pectinate zone consists of a sandwich of collagen fibers in the radial ( y) direction

embedded in amorphous ground substance. For the same amount of material thickness,the sandwich provides increased bending stiffness. For simplicity, the details of thesandwich are omitted, and only the motion in the cross section ( y – z plane) is considered.For such a plate, the equation of motion in response to an applied pressure is:

D !

4w

! y4 " T

!2w

! y2 + #

Pt P

!2w

!t 2 = "2 p

F (Eq. 6)

in which w is the displacement of the plate, T is the tension, ! P is the plate density, t P is

the plate thickness, pF is the pressure in the fluid above the plate, which is doubled in Eq.

6 for fluid above and below the plate, and the bending stiffness is D = f Et P

312 , where E

is the Young’s modulus of the fibers and f is the volume fraction of the fibers. For hingededges at y = 0 and b, the solution is:

w =We j! t

sinny n = ! / b (Eq. 7)

in which W is the amplitude and $ the frequency. For static loading and for zero tension,

the results for the volume stiffness and point load stiffness are:


28/61

K Vol

=

2 pF

bw=

!

8 Dn5 k

PtL =

P

W =

48 Dd

b3

(Eq. 8)

where w is the average displacement, and P is the magnitude of a load on a probe at thecenter with diameter d . With all terms retained, Eq. 6 gives the impedance, the relation

between the pressure and velocityv =

!w:

2 pF

v= !

Dn4+Tn

2! t

P" P#

2( ) j#

(Eq. 9)

Figure 9: Cross section consisting of anelastic plate in vacuum with tension T . Theplate thickness is t and the width betweenthe support points is b . The resonant

frequency is proportional to T / tb2( )( )1/2

.

The dashed line shows the deformed

position of the plate.

Figure 10: Cross section consisting of anelastic plate in infinite fluid. When thedensity of plate and fluid are the same, theplate density is negligible, and the resonantfrequency is proportional to

f t 3 b5( )1/2

where f is the volume fraction of

BM fibers.

b) Fluid

The BM is mainly covered with soft cells that are in contact with extra-cellular fluid.

For the effect of the mass on the pressure distribution, there is little difference betweenfluid, soft cells or a gel. As demonstrated by Békésy (1960, p. 445) in an experimental

model there is no change in localization when the fluid is replaced by gel. Consequently,for modeling, the soft cells are often replaced by fluid. Both have similar acoustical

properties. The motion of an inviscid, incompressible fluid (or gel) is governed by thewell-known Laplace’s equation:

!2"

!2 y +

!2"

!2 z = 0

(Eq. 10)

in which ! is the displacement potential. The z -displacement and pressure are:

w =

!"

! z p

F = !"

F

#2$

#t 2

(Eq. 11)

The solution giving the compatible displacement with the plate (Eq. 7) is:


29/61

! = "W

ne j# t "nz

sinny (Eq. 12)

which gives the ratio of pressure to velocity:

pF

v= !

" F#

in=

j" F# t effective t effective

=

1

n= b / !

(Eq. 13)

Thus the inviscid, incompressible fluid has only the effect of a mass attached to the

plate, with the effective thickness t effective

, about a third of the plate width b. Soft cells or a

gel, representing the organ of Corti, also satisfy the same equation, so Eq. 13 holds forthe fluid consisting of a thin or thick layer of soft cells and fluid. The difference betweenthe fluid with and without cells attached to the plate lies in the viscous correction, not theeffective mass.

c) Plate and fluid

Equating the impedance of the plate (Eq. 9) to the impedance of the fluid (Eq. 13)

gives the resonant frequency:

! 2=

Dn4+ Tn

2

" Pt P + 2"

Fn

(Eq. 14)

Helmholtz (1868) proposed that the transverse fibers of the BM behaved as strings

under tension in air, as in a harp. This corresponds to setting the bending stiffness D andthe fluid inertia !

Fto zero in Eq. 14. A sketch of this is in Figure 9 and the result for the

first resonant frequency is:

frequencyTension

=

1

2

T

! Ptb2

"

# $

%

& '

1/ 2

(Eq. 15)

However, there is not much evidence for high tension in the BM and the density of thesurrounding fluid is nearly the same as the BM. Consequently, the plate density term in

Eq. 14 is negligible. Setting the tension T and the plate density ! P

to zero in Eq. 14 gives

the resonant frequency:

frequencyBending

=

1

2!

Dn5

2" F

#

$ %&

' (

1/ 2

=

1

2!

E ! 5

24" F

#

$ %&

' (

1/ 2

ft 3

b5#

$ %&

' (

1/ 2

(Eq. 16)

With the BM dimensions and density of fibers (Cabezudo 1978) Eq. 16 provides areasonable frequency range for several mammals (Steele and Zais 1983). The formula

also works for recent measurements of the gerbil (Emadi et al. 2004). The widefrequency bandwidth capability for the bending (Eq. 16) compared to the tension (Eq. 15)

is shown in Table 4. With a nominal Young’s modulus of elasticity for collagen, thefrequencies computed from Eq. 16 are close to the range for the guinea pig. In contrast, a

variation of tension by 104 would be required for the tension model to work, which is not


30/61

justified by the modest change in the dimensions of the BM support. Shown in Table 4 isthe width g of the spiral ligament. The tension is assumed to be proportional to g , and the

tension frequency is set equal to the bending frequency at the apex for comparison. Theconclusion is that it is bending stiffness that determines the frequency localization on theBM.

Table 4. Frequency range capability of BM pectinate zone (for guinea pig) for bendingstiffness and tension stiffness.

Base Apex Ratio

Elastic modulus E 1 GPa 1 GPa 1

BM fiber vol fract f 0.08 0.01 7

BM width b 80 µm 180 µm 0.44

BM thickness t 7 µm 1 µm 7

Spiral ligamentwidth g

200 µm 40 µm 5

Frequency tension 247 Hz 130 Hz 1.9

Frequency bending 52,000 Hz 130 Hz 400

3. Vestibular fluid pressure to BM displacement

The equations for the slow wave, (Figure 7) in the model with symmetric SV and STwithout Reissner’s membrane (e.g., Peterson and Bogert 1950; Zwislocki 1953) can bewritten as:

!

! x Ap( ) = "#

F

!2

!t 2 Q (Eq. 17)

!

! x Q = "# A

=

"

1

K Vol

2 pBM (Eq. 18)

The distance along the BM from the stapes is x, the average pressure in SV is p , the

pressure acting on the BM is pBM

, the volume displacement of fluid in SV is Q, the area

of SV is A(x), and the area displacement of the partition is ! A = bw . Equation 17 follows

directly Newton’s second law of motion (Eq. 1) while Eq. 18 is a statement about

conservation of mass in a segment of scalae. The equations above are valid for 1-D, 2-Dand 3-D description for the fluid flow by incorporating the dimensionality into the

pressure pBM

acting on the BM.

a) One-dimensional (1-D) approximation

For long wavelengths, the pressure is nearly constant on the cross section, so the

approximation is pBM

= p , and the system is a standard transmission line wave equation.

For a single frequency, with all variables changing with e j ! t

, this reduces to:

d 2

dx2 Ap( )+ m 2 x,! ( ) Ap( ) = 0 (Eq. 19)


31/61

where the coefficient is:

m2 x,! ( ) =

2"! 2

AK Vol

x( ) (Eq. 20)

which can be identified as the local wave number. The wavelength is proportional to the

reciprocal of the wave number. For the cochlea, K Vol decreases with distance from thestapes, so m increases with distance, meaning that the wavelength becomes short.Because of the simple interpretation, the 1-D approximation continues to be the mostwidely used for cochlear modeling. Zwislocki (1953) and then later Dallos (1973)

provide the history of the development. To improve the results, damping and mass areadded to the stiffness, as in Eq. 2. However, these values have no physical basis, so this

must be considered as a phenomenological model. Another approach is to ignore the

physical basis completely and determine the complex wave number m x ,! ( ) from

measurements of BM displacement phase and amplitude, and the neural response, so that

Eq. 19 will produce the measured response. This is often used in studies of the function

of higher neural centers, for which a simple but reasonable input from the cochlea,described by Eq. 19, is desired. Flanagan and Bird (1962) developed this approach using

Békésy’s post mortem measurements of the GP cochlear response. Of interest is their use

of the Hilbert transform to show that the measurements are consistent for a causal and

stable system. Zweig (1991) further developed the approach for the in vivo

measurements of Rhode (1971) in squirrel monkey and obtained self-consistent effective

BM impedance. Although the physical behavior is 3-D, the traveling wave can indeed be

described by an equation of the form Eq. 19. So the choice is to fit the wave number from

the measurements of the response or compute it from the actual stiffness and mass

properties of the BM with the 3-D fluid motion. Since measurements are difficult and

restricted, there is interest in improving and verifying the capability for the direct

calculation. De Boer (2006) provides a recent perspective on the issues.

b) Two-dimensional (2-D) approximation

To place the cochlear model on a physical basis, more details of the fluid motion must

be considered (Ranke 1950; Berkley and Lesser 1973; Neely 1981). When thewavelength is short for large m, the 1-D approximation is not valid because the pressure

on the BM is much larger than the average pressure, as indicated in the measurements(Figure 8). The 2-D approximation (from Laplace’s equation in the x – z plane) is:

pBM

p=

mH

Tanh mH

( )

!

1 for mH 1 short wavelength

!"

#

(Eq. 21)

in which H is the height of SV. With this, Eq. 19 is the same, but the wave number is:

m2=

2!" 2

AK Vol x( )

mH

Tanh mH ( ) (Eq. 22)

For the square SV, A = H 2, and this can be rewritten as:


32/61

mH Tanh mH ( ) =2!"

2

K Vol

x( ), (Eq. 23)

which has the approximation:

mH !

2! F" 2

K Vol x( )# $ % & ' (

1/2

for mH < 1 long wavelength

2! F" 2

K Vol x( ) for mH > 1 short wavelength

)

*++

,

++

(Eq. 24)

Thus in the long wavelength region, the wave number is proportional to the frequency,which is the characteristic of non-dispersive waves. For the short wavelength region, the

wave number is quadratic with the frequency, which indicates dispersion. At the point of“partition resonance” defined by Eq. 16, this has the value:

mH ! 8!

(partition "resonance") (Eq. 25)

A dimensionless form of Eq. 23 is:

!m =8

!

!" 2

Tanh !m (Eq. 26)

in which

!m = mH , !! =! /! ref

, ! ref

=

8

"

K vol

2# F

$

% &'

( )

1/2

, ! x = xH .

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