HANDBOOK OF THE SENSES Audition Volume Chapter 7: Mechano-Acoustical Transformations

Transcription

1 HANDBOOK OF THE SENSES Audition Volume Chapter 7: Mechano-Acoustical Transformations Sunil Puria 1,2 and Charles R. Steele 1 1 Stanford University, Mechanical Engineering Department, Mechanics and Computation Division, 496 Lomita Mall, Durand Building, Room 206, Stanford, CA Department of Otolaryngology-Head and Neck Surgery, 801 Welch Road, Stanford, CA

2 1. KEYWORDS 2. BIOMECHANICS, OTOBIOMECHANICS, COCHLEA, COLLAGEN FIBERS, EXTERNAL EAR, IMPEDANCE, INNER HAIR CELLS, LEVERS, MATHEMATICAL MODELS, MICROCT, MIDDLE EAR, ORGAN OF CORTI, OUTER HAIR CELLS, PRESSURE, TYMPANIC MEMBRANE, VIBRATION 2

3 I. Outline In 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 organ of Corti. Physiological measurements and the deduced general biophysics that can be applied to the input and output transformations by the different sub components of the ear are presented. II. Abstract The 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 the organism. In this chapter, the focus is on the basic mechanical principles discovered for the various steps in the process of transforming the input acoustic sound pressure into the correct stimulation of mechano-receptor cells. The interplay between theory and measurements 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 simpler auditory 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. This sound 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 in courtship, are in the 70 to 100 db SPL range (Eberl et al. 1997). An example of a simpler anatomy 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 frequencies of 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 for the 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 3

4 et al. 2005). Amphibians evolved to have a basilar papilla with a few hundred receptor cells 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 greater hearing 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 neural impulses 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 improved functionality. Five of the most important functional considerations are described below. 1. Sensitivity. The human ear is most sensitive to a range of sounds from the loudest at about 120 db SPL to the softest at about 3 db SPL. At its most sensitive frequency near 4 khz, the displacement at the tympanic membrane is less than 1/10 th the diameter of a hydrogen atom. At this threshold, the amount of work that is done at the eardrum i is 3 x J. In comparison, the amount of work done for the perception of light at the retina ii is 4x10-18 J, which is close 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 vocalized sounds. 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 facilitate this dynamic range mismatch problem. 3. Frequency range. Hearing has about 8.5-octave frequency range in human and in 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 achieved over an extraordinarily small space in comparison to the wavelengths of sound. 4. Frequency resolution. One of the most important functions of the cochlea is the tonotopic organization, which maps different input frequencies to its characteristic place in the cochlea. Like a Fourier frequency analyzer, each characteristic place has narrow frequency resolution, which provides greater sensitivity to narrow-band signals by reducing bandwidth and thus input noise at the individual mechano-receptor hair cells and thus the auditory nerve. 5. Sound localization. The physical characteristics of the pinna and head diffract sound in a spatially dependent manner. The diffraction pattern provides important cues that allow the more central mechanisms to localize, segregate and 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 final mechanical 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 biochemical events that leads to the firing of the auditory neurons. We designate the output of a given 4

5 sub system a proximate variable. The chain of proximate variables leads to the ultimate output 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 proximate variables, which are physiologically measurable. B. Theories of sound transmission in the ear Starting with Helmholtz, mathematical models have played a key role in improving our understanding 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. The disadvantages 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 the essence 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 not yet observed. Thus the interplay between theory and experiments allows one to test different 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 the auditory 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 are formed, how Newton s laws apply not only to celestial mechanics as originally formulated but also in otobiomechanics, how sound transmission through different materials 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 ear improves sensitivity, frequency range, frequency resolution, dynamic range, and sound localization 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 force and velocity transformations through anatomical changes in lengths and areas. These transformations take place in the context of improving sound transmission at the interfaces 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 of the 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 volume velocity 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 tip link tension due to relative shearing motion between the reticular lamina and the tectorial membrane. 5

6 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 s second 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 + R a + K & a % ( "V (!) (Eq. 4) $ j! ' 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 us understand 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 minimizing the 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 results from Newton s laws of motion. The one-dimensional (1-D) version of the wave equation 6

7 was formulated by d Alembert in 1747 for the vibrating string. It did not take Euler very long (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 the test 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 the transmission 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 wave equation exists when a property along the propagation direction varies exponentially. As reviewed by Eisner (1967), these equations were originally formulated by Lord Rayleigh and 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 peak responses 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 the ends 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 proximate variables, the overall sensitivity of hearing varies smoothly with frequency and does not exhibit 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 output variable characterizes changes in tension of the inner-hair-cell stereocilia? Possible candidates for the input variable are pressure measured with a microphone, volumevelocity (acceleration and displacement), power, or transmittance and reflectance. Since these variables are interrelated, it is difficult to truly separate the effects of one variable from another. However, the use of pressure has some advantages. Dallos (1973) showed that there is good agreement between hearing sensitivity measured behaviorally and the eardrum-to-cochlear pressure transfer function, also called the middle ear pressure gain resulting from ossicular coupling. It appears that the combined properties of the middle ear and its cochlear load are the dominant determinants 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 7

8 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 ultimate mechanical variable for the mechano-electric transduction (Corey and Hudspeth 1983; Howard and Hudspeth 1988). This tension opens ion channels and initiates the flow of ions through the stereocilia bundle resulting in depolarization of the hair cell body which results in firing of the auditory nerve. In the sections that follow we generally follow the path taken by sound from the external 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 then results 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 cavum concha and the ear canal buried in the skull. It is generally accepted that sound source localization in a free field consists of two processes. The sound source azimuth is determined using interaural time or interaural intensity, whichever is the dominant, while sound 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 tympanic membrane. Examples of anatomical variations include cone shaped pinna in cats to almost flat pinna in ferrets, numerous invaginations and protuberances of the pinna flange and 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 particularly at 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 tympanic membrane pressure ratio P tm /P ff. When measured as a function of spatial angle, the magnitude of the ratio is often called the head related transfer function (HRTF). Not surprisingly, the effect of the anatomical structures on the HRTF is likely unique to each animal 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 asymmetric structures along the way. The frequency region where different structures become important occurs when the wavelength of sound becomes smaller than the physical dimensions 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 8

9 corresponding quarter-wavelength resonance near 2.5 khz with an approximate pressure gain 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 even up 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 with very 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 10 db. 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 gain that 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 the concha becomes important and excitation of these modes is dependent on the angle of incident 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 head mechanically behave as rigid bodies to acoustic waves. In humans and in some animals like 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 the mobile 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 the interaural time difference (Shaw and Teranishi 1968). In humans the pinna is relatively large (64 mm x 29 mm) but it does not seem to be strongly 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 frequencies above 6-7 khz. These modes are dependent on the angle of the incident sound and are clearly important for determining the HRTFs measured in individual subjects. The brain continually calibrates and interprets the HRTFs to infer the location of sound indicating 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 days the subjects were able 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 above approximately 1 khz the membrane response is very complex, while the cochlea provides a mainly resistive load (Onchi 1961; Møller 1963; Zwislocki 1963; Khanna and Tonndorf 1969; Lynch et al. 1994; Puria and Allen 1998). This resistive load is the primary 9

10 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 properties of 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. Even though 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 specific impedance 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 to reflect and not enter the cochlea. Table 1 Acoustical and mechanical properties of air, saline and the input widow to the cochlea. medium density ρ (kg/m 3 ) speed of sound c (m/s) specific impedance z = ρc (Pa-s/m) impedance ratio β =z/z cochlea Air /212= Saline x Cochlear input (approx) 9.5x 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 1 illustrates this concept. The model consists of two semi-infinite tubes of cross-sectional areas A 1 and A 2, with the ratio α = A 1 /A 2, filled with fluids with the densities ρ 1 and ρ 2 and speeds of sound c 1 and c 2. The acoustic impedances are z 1 =! 1 c 1 and z 2 =! 2 c 2, with the ratio β = z 1 /z 2. The piston has one face in tube 1, and the other face in tube 2. Figure 1: Greatly simplified model for the middle ear consisting of a piston connecting two acoustic tubes. Tube 1 represents the ear canal, with an incident wave and a wave reflected from the piston. Tube 2 represents the fluid filled inner ear with a transmitted wave. The hypothetical piston is free from constraint and is massless, so the force on the two sides of the piston must be equal. An incoming acoustic wave in tube 1 (the ear canal) 10

11 impinges upon the piston, causing the generation of a transmitted wave in tube 2 (the cochlea), 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 and incident pressure and energy: p 2 p 1in = 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 greatly improves 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 αβ = 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 be noted that larger areas enhance the signal-to-noise ratio at the hair cell level (Nummela 1995). So the large tympanic membrane is advantageous to human and cat for hearing in air. It is 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 behavioral threshold 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 the middle ear. Table 2 Effect of middle ear area ratio α and specific impedance ratio β 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, which has a great reduction in the transmitted pressure. Tube 1 (EC) β = z 1 /z 2 α = A 1 /A 2 p 2 /p 1 (lin) p 2 /p 1 (db) E 2 /E 1 (lin) Air % 20 (human) (cat) Water (human) (cat) In Table 3 the amplitude of the incident sound wave at threshold is given for hearing in air 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 at threshold 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 11

12 water environment. Quite a different middle ear design provides the extraordinary sensitivity under water of cetaceans (Hemila et al. 1999). Table 3 Some approximate thresholds of hearing in air and water. Air Intensity (Watts/m 2 ) Water Intensity (Watts/m 2 ) Pressure (µpa) Pressure (µpa) Human x x Pinnipeds x x Cetaceans x As the simple estimate indicates, without an effective middle ear, the sensitivity of the cochlea 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 subsequent section, 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 with a 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 resonant modes 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 limits high frequency hearing. These factors must be considered when formulating mathematical 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 different middle ear structures interconnected in 5-6 functional blocks. The blocks were then assigned more detailed elements, which consist of masses, springs, and dashpots. Some of the earliest models by Onchi (1949; 1961), Zwislocki (1961), and Møller (1961) use dynamic analogies and represent the middle ear in the form of electrical circuit models. These phenomenological models have evolved and continue to be useful for understanding surgical interventions of the middle ear (Rosowski and Merchant 1995; Merchant et al. 1997; Rosowski et al. 2004). Nevertheless, they have limitations in that there is not a tight relationship between the underlying anatomical structure and function. To overcome these limitations requires models that explicitly incorporate morphometry of the middle ear into the formulation. 1. Tympanic membrane shape and internal structure There remain many unanswered questions regarding the biomechanics of the tympanic membrane. 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 is asymmetry? The functional significance of many of these gross anatomical features of the 12

13 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 and the resonances of the malleus-incus complex and of the middle ear cavity. However, instead of piston behavior, surface displacement measurements revealed multiple modes of vibration for frequencies above a few khz (Tonndorf and Khanna 1972). Since the toand-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 asymptotic approximations for the cat tympanic membrane. They included the membrane geometry and anisotropic ultrastructure in combination with curvilinear membrane equations, but did 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 do not 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 of vibration 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, even when 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 the complicated motions of the eardrum observed by Khanna and Tonndorf (1972) but need explanation. 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 leads to 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 eardrum fiber structure (Lim 1995). 13

14 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 the malleus, 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 in area 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 into account. 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 are weakly coupled in the circumferential direction. The tympanic membrane is represented as a four-layer composite (Figure 2). The input 14

15 parameters for the formulation are the thickness of each layer as a function of position and 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 relatively flexible. 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 are radially 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. The mass 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). From these 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 eardrum modulus of elasticity (Fay et al. 2005). First, constitutive modeling incorporating the Young s modulus of collagen and experimentally observed fiber densities in cat and human were used. Second, the experimental tension and bending measurements (Békésy 1960; Decraemer et al. 1980) were reinterpreted using composite laminate theory. And third, dynamic measurements of the cat surface displacement patterns were combined with a composite shell model. All three methods lead to similar modulus of elasticity value of GPa for near the center of the eardrum. The corresponding values near the outer edge are approximately ½ these values due to the liner taper in the elastic modulus. In previous models the eardrum is treated as a single layer having a uniform elastic 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, the collagen fiber sub layer is much thinner than the overall thickness and hence the estimated elastic modulus is higher. The modulus of elasticity was combined with the sub layer thickness to formulate a complete 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. 15

16 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 (p v ) divided by the input pressure in the ear canal (p ec ). The deep eardrum calculation is nearly the same as the normal, but the shallow eardrum 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 the cochlea, for which lumped-element approximations were used. Verification involved mesh refinement studies, comparison with exact solutions for limiting cases, anatomical values of geometry, best estimate for elasticity, and comparison with physiological measurements to 20 khz, all for the cat middle ear. Different depths of the eardrum play an important role as shown in Figure 3. With a shallow 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 seen in 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 phase tends 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 16

17 normal eardrum the deep drum requires more real estate in the skull, which competes for space 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 and circumferential directions and orthotropic eardrums where there were radial fibers but no circumferential fibers (Fay 2001; Fay et al. 2006). Results indicate that there is an advantage 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. The orthotropic central portion allows maximal sound transmission at both low and high frequencies. The model calculations indicate that sound transmission from the ear canal to the cochlea 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 are deliberately mistuned. Because these resonant points are added together at the malleus, no single mode ever dominates. Thus the ensemble of eardrum modes produces a relatively 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 repairing damaged 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 to a 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 the higher frequencies, orthotropic material with collagen scaffolding preferentially oriented in the radial direction would be a better choice for improved high frequency hearing outcomes. 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 different materials. Since the modulus of elasticity and the biocomputation approach using asymptotic methods is already developed for the cat, the challenge will be to estimate eardrum morphometry 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 membrane varies from subject to subject. Such quantification will allow for the possibility of using the eardrum biocomputation on individual subjects. Non-destructive high-resolution imaging 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 appropriate for 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 17

18 obtain volume reconstructions of the temporal bone of living subjects at a resolution of less than 125 µm (Dalchow et al. 2006). In-vitro resolution can be increased by an order of magnitude (Decraemer et al. 2003). Figure 4: MicroCT image of an intact cadaver temporal bone. This is image #769, of 1897 images spanning a length of mm. The image illustrates that most of the middle ear structures can be visualized from an intact temporal bone ear scan. The resolution for both in-plane and out-of-plane (slice thickness) is 15 µm. The tympanic membrane although visible is faint, suggesting that the basic geometry and an approximate thickness can be obtained. The mm scan diameter 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 between these 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 nondestructive method, (3) shorter preparation time (hours rather than months), and (4) results already in digital format. This imaging technology is rapidly evolving and it is likely 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 tympanic membrane and a columella, while mammals have a tympanic membrane and a malleusincus complex. Amongst vertebrates a great majority of mammals are sensitive to ultrasonic 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 response compared 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 suspension also further supported the notion that the malleus and the incus rotate about a fixed axis 18

19 while driving the stapes in a piston like manner. However, detailed measurements of the ossicles 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 and rotation are present at biologically relevant stimulation levels. These measurements suggest 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 in the 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-like mechanism 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 during rotation in one direction but leaving the parts free to rotate in the orthogonal direction (Wever and Lawrence 1954). However, measurements (e.g., Helmholtz 1868; Békésy 1960) 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, Guinan and 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. Due to these measurements, mathematical models of the human middle ear generally treat the two 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 lack of 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 controversy regarding slippage at the joint between the incus and the stapes, and most mathematical models 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, the ossicles have been modeled as finite elements, which require much more computation time. An alternative approach is to model the ossicles as elastic bodies incorporating just the 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 suspensory ligaments and tendons of the middle ear are a composite, consisting of collagen and elastin embedded in an amorphous intercellular material often called ground substance or matrix 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 of the 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 in maintenance of the collagen scaffolding. As such, the biomechanical behavior of a ligament 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 the middle 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-incus complex was used (Fay et al. 2006). This was a gross simplification but allowed 19

20 concentration on the tympanic membrane biomechanics. A goal of several laboratories is to combine anatomical data with human cadaver temporal bone malleus-incus complex 3- D motions into a computational model for individual ears, which should increase understanding of the functional consequences of the anatomy of the ossicles and suspensory 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 µm is red). The biomechanical characterization of the malleus-incus complex requires morphological and dynamical measurements from individual ears. The center of mass, moments of inertia, anatomical location and orientation of the ligaments and tensortympani 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 the malleus-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 20