University of Cincinnati

UNIVERSITY OF CINCINNATI

Date: 23-Feb-2010

I, Won Joon Song , hereby submit this original work as part of the requirements for the degree of: Doctor of Philosophy in Mechanical Engineering It is entitled: Study on Human Auditory System Models and Risk Assessment of Noise

Induced Hearing Loss

Student Signature: Won Joon Song

This work and its defense approved by: Committee Chair: J. Kim, PhD J. Kim, PhD

William Murphy, PhD William Murphy, PhD

Mark Schulz, PhD Mark Schulz, PhD

Teik Lim, PhD Teik Lim, PhD

3/3/2010 412 Study on Human Auditory System Models and Risk Assessment of Noise Induced Hearing Loss

A dissertation submitted to the

Division of Research and Advanced Studies of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

DOCTORATE OF PHILOSOPHY

in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering and Applied Science

2010

Won Joon Song

B.S. Mechanical Engineering Hanyang University, Seoul, Korea, 1995

M.S. Mechanical Engineering Hanyang University, Seoul, Korea, 1997

Committee Chair: Dr. Jay H. Kim

Abstract

Simulation-based study of human auditory response characteristics and development of a prototype for advanced noise guideline are two major focuses of this dissertation research.

This research was conducted as a part of the long-term effort to develop an improved noise guideline for better protection of the workers exposed to various noise environments.

The human auditory responses were studied with simulation models. A human full-ear model derived from an existing model, Auditory Hazard Assessment Algorithm for Human

(AHAAH), was utilized as a baseline for the study. Frequency- and time-domain responses of well-known human middle ear network models were cross-compared to estimate expected accuracy of the models and understand their proper use. Responses of the stapes to impulsive noises were investigated by using the middle ear models to understand the effects of the temporal characteristics of impulsive noises on the responses. Available measured transfer functions between the free-field pressure and the stapes response for human and chinchilla were also used to study the auditory response characteristics. The measured transfer functions were refined and reconditioned to make them have equivalent formats. Using the reconstructed transfer functions, time-domain stapes responses of human and chinchilla to impulsive and complex type noises were calculated and compared.

Applicability of the noise metrics defined in terms of the stapes response to assess the risk of the noise induced hearing loss was studied.

A prototype of an improved noise guideline was developed from existing chinchilla noise iv

exposure data. Applying a new signal processing technique to the time histories of the exposed noises and studying the relationship between the noise metric and the permanent threshold shift (PTS), the dose-response relationship (DRR) was established in a compatible way with the definition used in current human noise guidelines. From the DDR, noise induced hearing loss (NIHL) threshold is estimated as a function of frequency. An advanced noise guideline that enables quantitative, frequency by frequency assessment of risk of the noise was developed by utilizing the identified NIHL threshold. The guideline was developed so that it can be easily transformed to a human noise guideline. Therefore, the guideline serves as a prototype of a future human noise guideline.

Acknowledgements

Theory , practice , knowledge and experience, inscribed at the entrance of Swift Hall and Old

Chemistry, are four elements that should be equipped to be a real engineer.

I must acknowledge the dedicated efforts of Prof. Jay Kim in guiding and assisting my research. With his valuable advices and innovative suggestions, I could proceed and finalize this work. I would like to express my appreciation to Prof. Teik Lim, Prof. Mark Schulz and

Dr. William Murphy for accepting to be in my dissertation committee and providing helpful review comments. My special thanks go to Drs. Price and Kalb in the U.S. Army Research

Laboratory (USARL) and Drs. Hamernik and Qui in SUNY Plattsburgh for supplying useful data. The financial support by the National Institute for Occupational Safety and Health

(NIOSH), Grant number R21 OH008510, is to be highly appreciated.

I am deeply grateful to Shrikant Pattnaik, Steve Goley and Ed Zechmann for the friendship they showed to me. I really appreciate to my Korean friends in UC and international friends in Baldwin 445 for sharing great times with me. I wish all of them to make great strides in their future careers. I also would like to offer my special gratitude to Prof. J. K. Lim, a role model to me, in Hanyang University.

My parents deserve the greatest gratitude from me. From the bottom of my heart, I appreciate the unconditional supports that I owed to my younger sisters and brother-in-law.

vii

Contents

Introduction ...... 1

I. Introduction to the human auditory system: Structure and functions ...... 6

A. External ear ...... 7

B. Middle ear ...... 9

1. Tympanic membrane ...... 9

2. Ossicular chain ...... 11

3. Middle ear cavities ...... 13

4. Eustachian tube ...... 15

C. Inner ear ...... 15

II. Auditory system modeling ...... 19

A. Sound source and the external ear modeling ...... 19

1. From the free-field to the concha entrance ...... 19

2. Concha and the ear canal modeling ...... 36

B. Middle ear modeling ...... 45

1. Classic configuration of the human middle ear ...... 45

2. Tympanic membrane models ...... 50

3. Ossicular-chain and middle-ear transformer ...... 59

4. Nonlinear elements ...... 72 viii

5. Stapes-cochlea (SC) complex network model ...... 87

6. Network analogues for the middle-ear cavities ...... 105

C. Cochlear modeling ...... 113

1. Passive cochlear model by Zwislocki (2002b) ...... 113

2. Cochlear transfer function ...... 116

III. Comparative study of human middle ear models based on the frequency response solutions ...... 124

A. Middle ear network models ...... 124

1. Classic configuration of the middle ear network model ...... 126

2. Selected network models for comparison ...... 128

3. Impedance characteristics of reference network models ...... 130

B. Middle ear transfer functions of network models ...... 131

1. Pressure transfer function ...... 131

2. Middle-ear transfer admittance ...... 136

3. Displacement-pressure transfer function ...... 139

4. Volume velocity transfer function ...... 144

C. Comparison of frequency-domain stapedial responses of middle ear network models

(Song and Kim, 2008) ...... 147

IV. Comparison of human middle ear models based on their temporal responses to

impulsive waveforms ...... 157

A. Methodology ...... 158

B. Comparison of temporal responses of human middle ear models ...... 159

1. Unit impulse responses of human middle ear models ...... 159

2. Parametric study on the responses of network middle ear models to impulsive

inputs ...... 165

C. Effect of impulse parameters on the inner ear responses ...... 178

V. Time-domain stapes responses obtained from reconstructed transfer functions ...... 184

A. Transfer functions from measurements...... 185

1. External ear pressure transfer function...... 186

2. Middle ear volume velocity-pressure transfer function ...... 188

B. Reconstruction of the transfer functions ...... 191

C. Time-domain responses of the stapes ...... 200

D. Noise metric based on the stapedial responses ...... 205

1. Design of stapes response based noise metric ...... 205

2. Comparison of the stapes response metrics of human and chinchilla...... 207

3. Correlation study ...... 214

VI. Development of a prototype of advanced noise guideline by using noise induced hearing loss threshold level of chinchillas extracted from existing exposure test data ...... 218

A. Reprocessing of the exposure data ...... 218

1. Re-analysis of exposure data ...... 218

B. Estimation of NIHL threshold and its applications ...... 220

1. Permissible exposure level (PEL) and NIHL threshold ...... 220

2. Applications of NIHL threshold data ...... 223

References ...... 230

Appendix: Variable Impedance component modeling using Simulink ...... 247

A. Equivalent model of a nonlinear electric component ...... 248

1. Equivalent modeling of a variable resistor ...... 248

2. Equivalent modeling of a variable capacitor ...... 251

B. Validity check of equivalent network models ...... 253

C. Application of the non-linear component models to the AHAAH ...... 259

Introduction

With approximately 22 million workers exposed to hazardous noise environments in the

United States alone (Tak et al. , 2009), noise induced hearing loss (NIHL) is one of the most frequently reported occupational diseases (NIOSH, 2001). Various noise guidelines (OSHA,

1981; ISO, 1990; ANSI, 1996; NIOSH, 1998) have been used to protect workers from NIHL.

All these noise guidelines recommend the permissible exposure level (PEL) according to a highly simplified method based on the equally energy hypothesis (EEH). For example,

NIOSH 98-126 (1998) and ISO-1999 (1990) adopts 85-dBA as the level that allows 8-hour daily exposure and the 3-dB exchange rate that halves the permissible exposure time for every 3-dB increase from the noise level. The approach is very simple to apply; however may not be accurate because it ignores the effect of temporal and spectral characteristics of the noise on the NIHL. It is a near consensus of researchers that the EEH based approach underestimates the risk of a complex noise environment where impulsive noises are embedded in a continuous background noise (Roberto et al. , 1985; Ahroon et al. , 1993;

Hamernik and Qiu, 2001; Harding and Bohne, 2004; Hamernik et al. , 2007). A new methodology to assess the risk of noises of any general type of occupational noises is urgently necessary, which motivated this research.

To develop a better noise guideline, an accurate noise dose-response relationship (DRR) is required. The task involves a wide range of scientific approaches including audiology, biology, engineering signal processing and statistics. Almost all past works in this area have 1

been human population studies (Taylor et al. , 1965; Burns et al. , 1970; Passcheir-Vermeer,

1983; Taylor et al. , 1984; Johansson and Arlinger, 2001; Johansson and Arlinger, 2002) or animal noise exposure studies (Nilsson et al. , 1983; Davis et al. , 1989; Dunn et al. , 1991;

Hamernik and Qiu, 2000; Hamernik et al. , 2002). Demographic studies were used to find

DRR by statistical methods. Numerous animal noise exposure tests were conducted to investigate effects of temporal characteristics of the noise (Hamernik et al. , 1974; Hunt et al. ,

1976; Blakeslee et al. , 1977; Roberto et al. , 1983; Henderson et al. , 1985; Roberto et al. ,

1985) and effects of chemicals on noise (Humes, 1984; Jock et al. , 1996; Boettcher et al. ,

1998).

Animal noise exposure studies provide valuable information on DRR because the results were obtained by well-controlled tests. Chinchillas have been used most commonly for exposure tests due to audiometric characteristics similar to those in humans (Miller, 1970;

Vrettakos et al. , 1988; Heffner and Heffner, 1991). Through research studies spanning two decades, Hamernik and his collaborators have obtained a comprehensive set of noise exposure data of chinchillas (Blakeslee et al. , 1977; Hamernik and Hsueh, 1991; Hamernik and Qiu, 2001). The auditory data were measured in a systematic way and the pressure wave forms were digitally recorded. This allowed the reanalysis of the data using newly developed signal analysis techniques to obtain additional information or insights in this study.

Various techniques and related theories of human ear simulation have been extensively 2

studied and applied to investigate the human auditory response characteristics. A simulation model is a convenient tool to estimate the response of the auditory system to any type of noise. If a very accurate simulation technique exists, a noise guideline may be developed solely based on the simulation method. However, quantitatively accurate simulation of the ear system, one of the most delicate organs, is believed to be beyond the current state of the art because much of the operating principle and damage mechanism of the hearing organ are not well understood.

Developing an accurate noise guideline is a challenge that requires a long-term effort. As a part of the long-term effort to develop an improved noise guideline, this dissertation focused on investigating the human auditory responses obtained from simulations and developing a chinchilla version of the noise guideline as a prototype for the future human noise guideline.

This dissertation consists of 6 chapters. Chapters I and II are devoted to explaining human auditory system and discussing typical methods to model the system. Anatomical and functional description of the human auditory system was summarized in chapter I. The modeling techniques of the human ear were discussed in chapter II. Network models for the source field and each functional part of the external and middle ear were represented in detail. Typical procedures to model the passive cochlea and deduce the cochlear transfer function were described at the end of the chapter. Simulation models of the human ear were developed using MATLAB (MATLAB®, 2007b) and Simulink (Simulink ®, 2007b) as 3

programming tools .

Investigation of human ear responses based on the simulation technique is described in chapters III through V. In chapter III, frequency responses of seven popular human middle ear network models were compared. Middle ear transfer functions of the models were cross-compared. Variations of the middle ear transmissibility when a part of the given model is replaced by that used in the other models were utilized to roughly estimate accuracy of the models. In chapter IV, the stapes responses simulated by the seven models were compared to study characteristics of human middle ear time-domain responses to impulsive stimulus. In chapter V, available transfer functions measured for human (Shaw,

1974; Mehrgardt and Mellert, 1977; Kringlebotn and Gundersen, 1985) and chinchilla

(Bismarck, 1967; Bismarck and Pfeiffer, 1967; Ruggero et al. , 1990; Murphy and Davis,

1998) were reconditioned and expanded to make them comparable to one another. Using the reconstructed transfer functions, time-domain stapedial responses of the human and chinchilla were calculated for selected noises and compared. Possibility of using the stapes responses to develop a noise metric was explored.

In chapter VI, identification of the NIHL threshold levels of chinchillas and development of a prototype of an advanced noise guideline were discussed. Existing chinchilla noise exposure data provided by collaborators in SUNY Plattsburgh (Hamernik et al. , 1984;

Hamernik et al. , 1987; Hamernik et al. , 1989; Hamernik et al. , 1994; Hamernik and Ahroon,

1998; Hamernik and Qiu, 2001; Hamernik et al. , 2002; Hamernik et al. , 2003; , 2007) were 4

re-analyzed statistically. It was shown that the threshold noise level that induces NIHL to chinchillas can be identified by applying statistical methods and clever interpretations of the human guidelines and the noise exposure data. The threshold information was utilized to develop an improved noise guideline for chinchillas as a prototype for a future human guideline. The new guideline has new features that enable recommendations in a frequency-by-frequency and quantitative manner.

I. Introduction to the human auditory system: Structure and functions

The human peripheral auditory system, which has structural and functional similarities to the other mammalian species, encompasses the external ear, the middle ear, the inner ear, and the auditory-vestibular cranial nerve (i.e., sensory CN VIII) as shown in Fig. I-1. Each part plays a unique role in translating acoustic signals into electrochemical signals which the brain can decode. Airborne sound from free-field passes through the outer ear in the mode of pressure vibration. The middle ear changes acoustic vibration into mechanical vibration. The inner ear transforms the mechanical energy into the hydrodynamic motion of the cochlear partition, which generates an electrochemical signal (i.e., neural impulse) to the brain. An alternative mode of sound transmission, called “bone conduction”, conveys airborne sound directly to the cochlea through skull-bone vibration, bypassing the external and middle ear (Homma et al. , 2009a; Homma et al. , 2009b). This chapter provides the anatomical and functional description of the human auditory system engaged in the normal auditory pathway.

Fig. I-1. Schematic of the human ear. (from Flanagan, 1972)

A. External ear

The external ear, the outermost part of the auditory system, is comprised of the pinna (i.e., auricle) and external auditory meatus (i.e., ear canal). The pinna depicted in Fig. I-2 collects and funnels sound waves and helps sound localization. The pinna is shaped like a revered- horn as in Fig. I-3 and pre-amplifies the incoming sound pressure before it enters the earcanal. The auditory canal is an “S”-shaped passage approximately 2.5-3.0 cm in length, whose lateral 1/3 to 1/2 is cartilaginous portion while the medial rest of it tunnels through the temporal bone thus is osseous (i.e., bony) portion (Alvord and Farmer, 1997). The tube- like canal guides acoustic waves up to the tympanic membrane. The resonance frequency of the ear canal is about 3.5 kHz at which the peak pressure gain is about 10 dB in case of

human (Dallos, 1973b).

Fig. I-2. Anatomy of the pinna (from Figure 1 of Alvord and Farmer, 1997)

Fig. I-3. Sectional view of the ear canal (from Figure 2 of Alvord and Farmer, 1997)

B. Middle ear

The middle ear is located between the external and the inner ear, which couples the airborne sound to the fluid-filled cochlear duct (Rosowski and Relkin, 2001). The middle ear is functionally viewed as an impedance transformer buffering the impedance mismatch between the air in the auditory canal (i.e., medium of lower impedance) and the water-like liquid called perilymph in the inner ear (i.e., medium of higher impedance).

1. Tympanic membrane

The tympanic membrane is a cone-shaped thin membrane located at the end of the ear canal thus forms the boundary between the external and middle ear. The membrane has layered architecture (Alvord and Farmer, 1997); the outer epidermal layer of desquamating epithelium continuous with the skin of the auditory canal, the intermediate fibrous layer of collagen fibrils organized in radial and circumferential directions, and the inner mucus membrane layer continuous with the lining of the middle ear. The thick lower portion of the tympanic membrane called “pars tensa” is partially attached to the malleus handle and mechanically reinforced by the radial and circumferential fibrous layers. The thin upper portion of the tympanic membrane is called “pars flaccida”. The tympanic annulus (i.e., a bony ring) stretches the pars tensa while the pars flaccida is slack due to the opening in the annular ring.

The tympanic membrane is excited by the pressure difference on both surfaces, through

which impinging sound is converted into mechanical vibrations in the middle ear. The central portion of the pars tensa is mostly responsible for the mechanical energy transmission thus, different from the anatomical division, the membrane is functionally re- partitioned as the conductive part transmitting the mechanical energy into the ossicular chain and the independent (i.e., residual) part shunting the acoustic energy.

Pars flaccida

Malleus handle (i.e., manubrium)

Umbo Tympanic annulus Pars tensa

Fig. I-4. Tympanic membrane (modified from http://l.yimg.com/us.yimg.com/i/edu/ref/ga/l/910.gif)

2. Ossicular chain

The ossicular chain consists of three ossicles (i.e., malleus, incus, and stapes) and two interconnecting joints (i.e., malleo-incudal joint and incudo-stapedial joint) as shown in Fig.

I-5. The malleus is sustained by the anterior malleal ligament and the superior malleal ligament. The malleo-incudal joint connecting the round head and incus body is a type of ball-socket joint. The long process of the malleus (i.e., manubrium) is attached to the inner layer of the tympanic membrane as well as to the tensor tympani tendon. The incus is supported by the superior incudal ligament and posterior incudal ligament. The lenticular process of the incus and the stapes head are connected by the incudo-stapedial joint. The footplate of the stapes is suspended from the oval window by the annular ligament which restricts the stapes motion. At the head of the stapes, the stapedius muscle is attached.

a) Mechanical transformer

The ossicular chain forms a mechanical transformer through which impedance transition occurs. Since the long process of the malleus is a little bit longer than the long process of the incus, the length ratio exerts a lever effect on the energy transmission. The areal ratio between the effective area of the tympanic membrane and the stapes footplate area causes hydraulic advantage thus boosts up the energy transmission into the inner ear. The combination of the lever and areal ratio makes it possible to reduce the impedance gap existing between the external auditory canal and the inner ear.

b) Acoustic reflex

The acoustic reflex (i.e., middle ear reflex), as a protection mechanism in response to high acoustic stimuli above 85 dB SPL, triggers the contraction of the stapedius muscle which pulls the stapes head along the major axis of the stapes footplate (Bennet, 1984) perpendicular to the direction of piston-like motion. Therefore, the middle ear reflex restricts the stapes movement increasing acoustic stiffness and finally reduces low- frequency acoustic transmission into the inner ear. Activated prior to self-vocalization, it reduces the masking by low-frequency noise thus enhances intelligibility of speech in the presence of environmental noise. The stapedius muscle solely involves the acoustic reflex in case of the human ear; however, both the tensor tympani and the stapedius muscle engages in animal (Møller, 2000). As with other biological responses, the reflex experiences the latency, adaptation, and recovery process.

Fig. I-5. Ossicular chain (from Yost, 2000)

3. Middle ear cavities

Located right behind the tympanic membrane, the human middle-ear air space schematically shown in Fig. I-6 is composed of the tympanic cavity, the aditus ad antrum, the tympanic antrum and the mastoid cells (Voss et al. , 2000). The tympanic cavity contains the ossicular chain system. The aditus ad antrum is a narrow passage which bridges the tympanic cavity and the tympanic antrum. The mastoid cells form a complex network of pneumatic cells attached to the tympanic antrum. It is known that the larger the middle ear air space, the better the low frequency middle ear gain (Rosowski and Merchant, 1995) thus the rodent such as gerbil with larger air cavity space than human shows better

audibility in the low frequency region.

Interacting acoustically with cochlear windows, the air space forms an acoustic transmission pathway whose acoustic resonance characteristic is similar to a Helmholtz resonator. Acoustic communication between the tympanic cavity and the antrum occurs through the narrow passage (i.e., aditus ad antrum). Middle ear air space contributes to the middle ear input impedance peak around 1-3 kHz but is not substantial in other spectral regions.

Fig. I-6. Human middle ear air spaces (from Stepp and Voss, 2005)

4. Eustachian tube

The Eustachian tube connecting the middle ear cavity with the nasopharynx of the throat equalizes the middle ear static pressure to the ambient pressure of the throat (Rosowski,

1996). Since the contraction of the tensor tympani pulls the manubrium toward the middle ear side to increase the middle ear pressure, it also helps open the Eustachian tube.

C. Inner ear

The inner ear is the innermost part of the auditory periphery, which consists of fluid-filled bony labyrinth running through the temporal bone of the skull. The frontal part of the labyrinth is the cochlea, the auditory organ, while the rear part is the semicircular canals, balancing organ. The vestibule is the cavity interconnecting the cochlea and the semicircular canals. It contains the utricle and saccule that contribute to balance and spatial orientation.

The cochlea, a snail-like auditory organ, is the time-frequency analyzer residing in the temporal bones. The cochlear section, as shown in Fig. I-7, has three chambers respectively called scala vestibuli, scala tympani and the scala media. The scala vestibuli and the scala tympani are bony labyrinth chambers filled with perilymph rich in , while the scala Na media is membranous labyrinth filled with endolymph whose ionic concentration is rich in

but low in . Helicotrema is the interconnection of the two bony chambers found at K Ca the apex of the cochlea. The stapes footplate is suspended in the oval window at the lower

extent of the scala vestibuli. The round window is at the basal end of the scala tympani. The round window serves as a pressure valve, that is, it releases the hydraulic pressure built up in the perilymph into the middle ear air space by bulging outward.

The Organ of Corti shown in Fig. I-8 resides in the scala media. It rests on the basilar membrane whose thickness and width varies along the distance from the basal end and generates the impedance gradient. The Organ of Corti has two types of receptor cells; the inner hair cell (IHC) innervated by afferent nerve fibers and the outer hair cell (OHC) innervated by both of afferent and efferent nerve fibers. IHC is basic sensor to convert mechanical signal into electrical spikes. OHC shows somatic motility in which its length changes by the activation of the prestin in the lateral wall and provides active feedback to the cochlear partition response thus is involved in the cochlear amplifier and sharper frequency tuning observed in the live cochlea. Each hair cell has sensory hair bundle called stereocilia whose bending motion opens or closes mechanically-gated ion channels. The tallest row of the OHC stereocilia is embedded in a gelatinous structure called tectorial membrane and is under direct shearing force; however the IHC stereocilia is not in contact with the membrane.

The stapes movement triggers the response of the inner ear. Piston-like motion of the stapes footplate generates the compressive waves in the perilymph of the cochlear duct. As the wave passes, hydraulic pressure difference is introduced across the cochlear partition

(i.e., the basilar membrane). The pressure difference elicits the up-and-down motion of the

cochlear partition thus traveling waves propagate on it. Propagated traveling wave reaches its maximum displacement at specific location of the cochlear partition and is rapidly suppressed. Due to the spatial impedanc e characteristic of the basilar membrane, high frequency waves propagate only at the basal region of the membrane while low frequency waves travel to the apex. The movement of the partition bends the stereocilia on top of hair cells by the relative motion to the tectorial membrane or by the shear flow of the endolymph, which leads to the electrochemical reaction in hair cells by opening or closing transduction channels.

Fig. I-7. Cochlear section (from Kessel and Kardon, 1979)

Fig. I-8. Organ of Corti (from Kessel and Kardon, 1979)

II. Auditory system modeling

In this chapter, modeling techniques of the human ear are discussed. Modeling of the external and middle ear utilizes acousto-electric analogy. Source pressure field is simplified by impedance networks of the head-related diffraction and acoustic radiation. The external ear and the middle ear are represented by several impedance blocks. Typical technique of cochlear modeling are reviewed. The procedure to model the passive cochlea in the frequency domain and obtain the cochlear transfer function is described at the end of this chapter.

A. Sound source and the external ear modeling

1. From the free-field to the concha entrance a) Parallel circuit representation of radiation impedance (Bauer, 1944)

(1) Pulsating sphere

The specific radiation impedance ( ) of a pulsating sphere of radius is: z r

(II-1) k r jkr z = R + jX = ρc + ρc = R + jωL 1 + k r 1 + k r where, and are the resistance and the inductance respectively, which are: R L

(II-2) k r R = ρc 1 + k r

ρr (II-3) L = 1 + k r 19

and the wave number. While the impedance can be represented by the series k = ω c z circuit of and as shown in Fig. II-1(a), the series analogue cannot be used for time- R L domain circuit simulations since and are functions of frequency. R L

The limitation of the series circuit representation can be overcome if the parallel circuit of constant-valued impedances shown in Fig. II-1(b) is used.

Ls Rs

(a) Series circuit (b) Parallel circuit

Fig. II-1. Transformation of the series circuit to equivalent parallel circuit

For the two circuits shown in Fig. II-1 to be equivalent, the total impedance of each circuit should be equal:

1 1 1 (II-4) = + R + jωL R jωL Equating the real and imaginary parts of the two sides of equation (II-4), we can show that:

(II-5) R = ρc (II -6) L = ρr where is the equivalent parallel resistance per unit area and the equivalent parallel R L 20

inertance per unit area. Since the two impedances are frequency independent, the parallel circuit is readily applicable to time-domain circuit simulations.

(2) Circular piston

The radiation impedance of a circular piston set in an infinite baffle can be approximated as that of a pulsating hemisphere of the same surface area. The surface area relationship is

, where is the radius of the hemisphere and the radius of the equivalent 2πr = πr r r circular piston. Therefore:

r (II-7) r = 2 Thus the impedance of the circular piston with an infinite baffle can be approximated as the parallel circuit in Fig. II-1(b) with:

(II -8) R = ρc (II -9) ρr L = ρr = 2 The exact impedance of a circular piston with an infinite flange is given by (Kinsler et al. ,

1999):

z (II-10) = R2kr + jX 2kr ρc where with the first-order Bessel function of the first kind R2kr = 1 − J2kr kr J and with the first-order Struve function. In the low frequency X2kr = H2kr kr H region (i.e., ), it is approximated by the first term of the power series expansion as: kr ≪ 1 21

(II-11) z 1 8 1 ≈ kr + j kr ≈ kr + j0.85 kr ρc 2 3π 2 In the same context, the radiation impedance of an unbaffled circular piston is equal to that of a pulsating sphere having the same surface area with the piston. Since the area relationship is given by , the equivalent radius of the sphere is: 4πr = πr (II-12) r r = 2 Thus the parallel circuit inductance is represented as: L (II-13) L = ρr = 0.5ρr Note that the resistance is invariant regardless of the baffled condition.

Another way to deduce the inductance value of an unbaffled piston is to compare the imaginary part coefficient of equation (II-11) with that of the impedance of a circular piston without flange. The first term approximation of the radiation impedance of an unbaffled piston in low frequency range is:

(II-14) z 1 ≈ kr + j0.6kr ρc 4 Comparing the coefficients from equations (II-11) and (II-14), we can easily conclude that

of the unbaffled piston is given by equation (II-13). L

Fig. II-2 compares the exact and approximate impedances of the circular piston with an infinite baffle. The imaginary part was plotted using Struve function approximation method recently developed (Aartsa and Janssen, 2003). As shown in Fig. II-2, the parallel circuit

acceptably well approximates the exact impedance in overall frequency range, especially well in low frequency region of . kr ≪ 1

real part real 20

0 0 5 10 15 20 25 30 35 40 ka

10 imaginary imaginary part

0 0 5 10 15 20 25 30 35 40 ka

Fig. II-2. Comparison of the impedance of a circular piston ( ) on infinite baffle, solid line: = exact; dashed line: parallel circuit approximation

b) Diffraction-radiation combined sound field model

In the auditory hazard assessment algorithm for the human (AHAAH) model (Price and

Kalb, 2005), the diffraction sound field is approximated as a spherical baffle around which the sound wave diffracts and in which the ear canal is located (Price and Kalb, 1991) using a simple network proposed by Bauer (1967). The radiation sound field by concha entrance is modeled as the radiation by a baffled piston, and also integrated in the AHAAH circuit model.

(1) Equivalent circuit to the massless frictionless piston confronting plane waves

This circuit was proposed by Bauer (1967) to model a microphone exposed to a plane wave field. Considering a diaphragm idealized as a massless and lossless piston mounted on a semi-infinite tube of sectional area with radius as shown in Fig. II-3 (solid line), the A a piston offers an acoustic impedance of to the incoming plane wave from the left ρc A through an imaginary semi-infinitely long tube (dotted line in Fig. II-3) assembled to the original one at the piston location as well as the departing wave to the right. If the pressure sensed at the piston surface is , the driving pressure at the infinite-left end of the p imaginary tube should be and the pressure at the infinite-right end of the original tube 2p be zero. Fig. II-4 shows the preliminary circuit model for description in Fig. II-3 that satisfies the requirements.

massless & lossless piston a incoming A departing plane wave plane wave

p semi-infinite imaginary tube tube

Fig. II-3. Concept drawing of the microphone in plane wave field

piston position

ρc / A ρc / A

2 p p

infinite-left end infinite-right end

Fig. II-4. Preliminary equivalent circuit model

The circuit in Fig. II-4 is not complete from the viewpoint of the piston because the piston experiences radiation impedance by the surrounding medium at the open end of the tube.

Adding an inductance element for the unbaffled piston as shown in Fig. II-5(a) solves this problem. In the absence of the incoming sound wave in the circuit (i.e., without in Fig. 2p II-5(a)), a motion of the piston faces a parallel circuit of the inductance and resistance shown in Fig. II-1(b) in either direction. However, the existence of an incoming sound wave

( in Fig. II-5(a)) makes the volume velocities on both faces of the piston become different. 2p 25

This unreasonable situation can be fixed by blocking the current flow to the inductance as in Fig. II-5(b). By introducing the pressure source in opposite direction to an incoming p sound wave, we can prevent piston volume velocity from leaking into the inductor, and maintain the equal velocity condition on both faces of the piston. Note that the pressure p in Fig. II-4 is not a pressure source as in Fig. II-5(b), but a measured pressure on the piston surface.

ρc / A ρc / A ρc / A ρc / A

5.0 ρ 5.0 ρ 2 p 2p πa πa p

(a) Initial circuit (b) Modified circuit

Fig. II-5. Equivalent circuit model considering sound radiation

Now the circuit shown in Fig. II-5(b), which represents the system described in Fig. II-3 can be modified to model the diffraction field in the inlet of the ear.

(2) Diffraction field model around spherical head

Up to this point, the pressure on the piston surface confronting a sound wave is due to the radiation effect. Assuming the piston is fixed, the radiation field pressure becomes equivalent to one in a diffraction field. When the piston is blocked, the grounded sink of the circuit in Fig. II-5(b) becomes open as in Fig. II-6. Pressure measured on the piston p surface confronting sound wave can be found by the circuit analysis, which is given as

(Bauer, 1967):

(II-15) p 1 + jμ = p 1 + jμ2 where . Equation (II-15) indicates that the pressure on the rigidly fixed piston is μ = 2πa λ dependent not only on the incoming pressure but also on the diffraction effect.

ρc / A

5.0 ρ 2p p1 πa p

Fig. II-6. Equivalent circuit for a blocked piston confronting plane waves

Comparing frequency function with a function for the average sound pressure over an pp area of a 30° angular radius on a sphere of radius confronting a plane wave, a good a agreement is found as shown in Fig. II-7. Thus, if we assume that the head has spherical shape, an equivalent circuit can be obtained by idealizing the sphere as a piston of the equivalent surface area.

1.8

1.6

1.4

1.2

0.8

0.6 Diffraction field ratio pressure 0.4

0.2

0 -1 0 1 10 10 10 ka h

(a) plot; solid line: equation (II-15), asterisks: (b) Average sound pressure plot on a sphere theoretical value (Peterson, 1985) confronting plane wave (Kinsler et al. , 1999)

Fig. II-7. Validity of network model result

(3) Equivalent circuit for a blocked piston with a small hole

Until now, it is assumed that the impedance is distributed uniformly over the piston face confronting sound wave. However, if a relatively small hole exists in the piston, additional impedance components (i.e., a radiation mass component and radiation resistance component) should be inserted into the equivalent circuit of the blocked piston in order to explain the radiation effect, the impedance change over the entrance of the hole.

ah ph

2 Ah = πah (small hole)

infinite baffle (blocked piston)

Fig. II-8. Concept drawing for a blocked piston with a small hole

As the blocked piston network model used to approximate the diffraction field around a spherical shape head in a limited region, the equivalent network model for a blocked piston with a small hole can also be applied for evaluating a combination effect of diffraction and radiation sound field around the entrance of concha. Since the small hole is analogous to a piston in infinite baffle as shown in Fig. II-8, the acoustic impedance values are estimated to be and respectively for radiation mass and radiation resistance component ρc A 0.7ρ πa of the hole, where is the sectional area of the hole and the radius of it.

A a 29

Constructing the network analogue in Fig. II-9, the basic assumption is that the head (or sometimes integrated head and upper torso) confronting a plane wave can be represented by a solid sphere of effective radius , while the concha opening by a small orifice of radius a on the spherical surface. The radius of the orifice corresponds to the effective radius a a of the concha cavity section at the base of the pinna whose flange is excluded in equivalent modeling. Two voltage sources of amplitudes and drive the network in-phase. These p 2p sources, together with elements and , model the head-related sound ρc A 0.5ρ πa diffraction field. The parallel elements and form the equivalent network for ρc A 0.7ρ πa the radiation field around the concha entrance, i.e. the load experienced by a virtual piston located at that position and radiating energy into the surrounding medium. Therefore, the pressure of the network represents the sound pressure at the entrance to the concha p (Giguère and Woodland, 1994a).

ρc / Ah ρc / A

7.0 ρ /πah 2 p 5.0 ρ /πa ph

Fig. II-9. Equivalent circuit for a fixed piston with a small hole

Schematic depicted in Fig. II-10 shows the modeling strategy on which the first part of the

AHAAH program is based.

plane wave Head model (spherical baffle)

ear canal

Fig. II-10. Head-related diffraction and concha entrance radiation modeling schematic

c) Circuit parameters adopted by AHAAH model

In Table II-1, the parameters used in AHAAH program is matched with the equivalent circuit model constructed in Fig. II-9 one by one. Note that impedance components used in equivalent circuit model have specific impedance values, while corresponding acoustic impedance values are used in AHAAH model.

Equivalent circuit parameters AHAAH model Values used in parameter AHAAH

Density of air ( ) 1.15 10 -3 ρ ρ × Speed of sound ( ) 3.52 10 4 c c × Driving source pressure ( ) N/A 2p 2p Sound source pressure ( ) N/A p p Head diffraction field resistance ( ) 1.29 10 -1 ρcA R × Head diffraction field inductance ( ) 2.56 10 -5 0.5ρπa L × Concha entrance radiation resistance ( ) ρcA R 2ρc4.3 Concha entrance radiation inductance ( ) L 3 × 0.85 ρ4.3π 0.7ρπa Table II-1. Circuit parameters matching with AHAAH model

Head-related anthropometric data used in AHAAH is rather underestimated. Calculating the radius of the piston from , it is 9.99cm. Since the radii of the sphere a ρc A = 1.29 × 10 and the piston are related by , this corresponds to the radius of head . a = a 2 a = 5.0 cm The head circumference computed from this is 31.4 cm in circumference. Deducing from

, we obtain different piston size from previous one. The calculated 0.5ρ πa = 2.56 × 10 piston radius becomes 7.15 cm which produces corresponding to 22.5 cm in a a = 3.58 cm circumference. Recalling the typical circumference of the human head is over 50 cm, the head size is quite underestimated in AHAAH model.

Concha entrance area given in AHAAH is 4.3 cm 2, but is not used as it is. The effective area is taken as for , while for . This indicates that the effective piston 4.3 2 cm R 4.3 3 cm L area for the resistance is taken 4.5 times higher than that for the inductance. It seems to be adjusted to consider the contribution of pinna flange which is not included in the circuit model.

Use of 0.85 instead of 0.7 in appears to follow the comment by Bauer (1967) in order to L improve the agreement between the parallel circuit and exact radiation impedance of a pulsating sphere. Bauer (1967) also suggested using 0.6 in place of 0.5 for better approximation of , but it is unclear whether this was followed or not in constructing the R AHAAH model.

d) Comparison with other available models

Table II-2 compares the circuit parameter values of other available models. This part of network is generally omitted in many existing auditory system models; therefore the comparison is made with the three models listed in the table. Coincidentally, they have an identical network for this part, thus a direct comparison of parametric values is possible. In

Model #1 (Giguère and Woodland, 1994a), the radius of the equivalent piston of the spherical baffle to represent the upper torso and head is 25 cm, which was used to calculate

and values. R L

Parameter AHAAH Model #1 Model #2 (Giguère and Woodland, 1994a) (Goode et al. , 1994)

1.29 10 -1 2.03 10 -2 4.50 10 -2 R Ω × × × 2.56 10 -5 7.26 10 -6 1.90 10 -5 L H × × × 1.88 10 1 1.27 10 1 2.30 10 1 R Ω × × × 7.98 10 -4 2.54 10 -4 4.30 10 -4 L H × × × Table II-2. Parametric values used in available network models

Transfer functions defined as the ratio of the free-field sound pressure and the sound pressure of the concha entrance are plotted in Fig. II-11. All transfer functions are similar in trends but 2 – 3 dB difference exists in the frequency range below 2 kHz. Overall trends are fairly close to that of diffraction field pressure transfer ratio shown in Fig. II-7, indicating that the diffraction effect is prevalent in determining the pressure transfer ratio from free- field to the entrance of concha.

3 Pressure ratio [dB] 2

0 2 3 4 10 10 10 frequency [Hz]

Fig. II-11. Pressure ratio comparison; solid line: AHAAH, dotted line: Model #1 (Giguère and Woodland,

1994a), dash-dot line: Model #2 (Goode et al. , 1994)

2. Concha and the ear canal modeling

The output acoustic pressure ( ) from the network in Fig. II-9 is the input to the entrance p of horn-shaped concha followed by cylindrical ear canal.

The AHHAH model employed the ‘two-tube model’ suggested by Wiener et al. (1965) to depict the acoustic pressure field in the human ear canal (Price and Kalb, 1991), showing differences in detail. In this section, the two-tube model and the external ear model of

AHAAH are compared.

a) Two-tube model of the ear canal

Weiner et al. (1965) modeled the external auditory meatus of a cat as two rigid tubes of different cross sectional areas and lengths. In the model, it is assumed that the eardrum is perpendicular to the axis of the tubes and functions as a rigid termination to the meatus.

The first tube of length and cross sectional area was designed to simulate the portion l S from the eardrum to the bend in the external auditory canal. The second tube of and l S described the part of the canal from the bend to a normal plane to the tube axis through the tragus. In this model, the cross sectional ratio is 2.0, and and are 1.5 cm and 0.5 SS l l cm respectively. The schematic of this model is shown in Fig. II-12.

Rigid termination

S2 S1

l2 l1 x = 0 x = l + l 1 2

Fig. II-12. Two-tube model of the external ear with rigid termination condition (Wiener et al. , 1965)

Assuming plane wave incidence without dissipation as well as the rigid termination at the end of the ear canal, the pressure ratio between a position and the auditory canal x entrance ( ) is given by (Wiener et al. , 1965): x = 0

(II-16) p cos kl + l − x = p cos kl cos kl − SS sin kl sin kl

b) Model in AHAAH program

(1) Geometric modeling

Simplified external ear model in AHAAH program is shown in Fig. II-13. The ear canal and the concha are modeled respectively as a cylindrical tube and an exponentially expanding horn which amplifies the sound pressure. The pinna flange is not included in this model.

x1 ∆L

x S2 S1

L L 2 1

Fig. II-13. Concha and ear canal model in AHAAH

Assuming exponentially expanding cross section in concha, the sectional area is given by:

(II -17 ) Sx = Se The sectional area at (i.e., the entrance of the concha) is: x = L + L (II -18 ) SL + L = Se = S Then, we can find:

(II-19) ln SS m = L Integrating equation (II-17) from to , the volume of horn section element for x x x > L and is given by: x > L 38

(II-20) Se − e V = Sxdx = Se = m The volume of cylinder element corresponding to and is: x < L x < L (II -21 ) V = S ∙ ∆L = Sx − x where . For the straddle horn-cylinder boundary region and , ∆L = x − x x < L x > L the volume of an element is:

(II-22) Se − 1 V = SL − x + m

(2) Equivalent circuit model in AHAAH

Due to the geometrical complexity assumed, the whole volume of external ear was considered to consist of 16 sub-volumes. Using the volume computed from equation (II-26),

(II-27) and (II-28), the equivalent capacitance and inductance of a sub-volume are:

(II-23) 1 ρc K = = C V (II -24 ) ρ∆L L = V where subscript means element number, the compliance and the inductance of the i C L element. Each volume element, ignoring the loss, is represented by a parallel circuit shown in Fig. II-14.

Lei

Cei

Fig. II-14. Parallel circuit representation of a volume element

In order to construct an equivalent network for the whole external ear, the component i set shown in Fig. II-14 should be connected in series with the inductance of the set i − 1 for . Splitting each inductance into two as shown in Fig. II-15, it is easy to i = 1,2,⋯,16 detect the voltage (or, pressure in acoustic sense) at the geometric center of each sub- V volume element. The pressure is, as previously explained, the pressure introduced at the p entrance of concha and the output pressure acting on the ear drum which locates at the p end of the ear canal. Note that the network in Fig. II-15 was constructed, as Wiener et al.

(1965) did, without resistance component but the middle ear input impedance ( ) Z replaces the blocked condition assumed in Wiener et al. ’s model.

L 2/ Le16 2/ Le 2/ Le 2/ Le 2/ Le 2/ e1 I22 I21 16 15 I 20 15 I7 1 I6

C C C ph V e16 e15 e1 p2 temp 21 Vtemp 20 Vtemp 6

Fig. II-15. Network representation of the human external ear in AHAAH model

(3) Two-tube model pressure transfer ratio for the cat and human external ear

Based on the geometric information listed in Table II-3, the two-tube model pressure transfer ratio given in equation (II-16) between concha entrance ( ) and eardrum x = 0 ( or ) is plotted in Fig. II-16 for the cat and the human external ear. In x = l + l x = L + L case of the cat, the first peak is found in 3.6 kHz, which shows fairly good agreement with the experimental results (Wiener et al. , 1965). In the audible range of human ear, three peaks occur in 1.4, 8.1 and 15.8 kHz frequency region, which means that the pressure reaching the eardrum is much larger than the input pressure on the concha entrance at those frequencies. Beyond the frequency of the first peak, both the cat and human pressure transfer ratios show the band-pass characteristic which heavily dampens the eardrum pressure between two peak frequencies. Note that the blocked end condition at the eardrum location is assumed in these analyses thus the ratio is rather different from the real measurements.

Cat (Wiener et al., 1965) Human (Price and Kalb, 2005)

Parameter Value Parameter Value

(cm) 1.5 (cm) 2.215 l L (cm) 0.5 (cm) 6.962 ×10 -1 l L 2.0 (cm 2) 0.44 SS S (cm 2) 4.3 S Table II-3. Modeling parameters of the cat and human external ear. Geometric values are from Wiener et al.

(1965) for the cat and AHAAH model (2005) for the human external ear.

20 [dB] 0

p/p 10

-10

-20 2 3 4 10 10 10 frequency [Hz]

Fig. II-16. Pressure transfer ratio calculated from two-tube model equation (II-16), where the solid line represents the pressure ratio of the human external ear and the dashed line that of the cat external ear. The rigid-end condition is assumed for both cases.

(4) Pressure transfer ratio from AHAAH model

The pressure transfer ratio simulated in AHAAH is shown in Fig. II-17. The open circuit condition at the eardrum location (i.e., the blocked-end condition as the two-tube model) is assumed for consistent comparison with the prediction by the two-tube model. The pressure transfer ratio is acquired by taking the ratio of the voltage at the concha entrance and the end of ear canal while sweeping the frequency of sinusoidal voltage input.

As predicted in the two-tube model, the concha entrance-eardrum pressure ratio (i.e., dash- dot line in Fig. II-17) shows three peaks in the audible frequency range of human ear but shifted from 1.4, 8.1 and 15.8 kHz to 3.5, 10.3, and 16.3 kHz respectively. The pressure ratio produces higher peaks than those of the free field-eardrum pressure ratio, which is due to the resistance characteristic existing in the diffraction-radiation field. The first peak of 3.5 kHz well explains the highest sensitivity of human ear around the 3-4 kHz frequency region, which also suggests that the second and third peak shown in Fig. II-17 would be suppressed in the middle and/or inner ear since the human audibility becomes less sensitive beyond 4 kHz.

20 [dB] 0

p/p 10

-10

-20 2 3 4 10 10 10 frequency [Hz]

Fig. II-17. Pressure transfer ratio of the human external ear calculated from AHAAH model where solid line represents the pressure transfer ratio of free-field to eardrum and the dot-dashed line that of the concha entrance to eardrum. The dashed line is the pressure ratio calculated from the two-tube model.

B. Middle ear modeling

1. Classic configuration of the human middle ear

Having different sub-structures, classic human middle ear network models (Onchi, 1961;

Zwislocki, 1962; Lutman and Martin, 1979; Shaw and Stinson, 1983; Kringlebotn, 1988;

Giguère and Woodland, 1994a; Goode et al. , 1994; Rosowski and Merchant, 1995; Rosowski,

1996; Pascal et al. , 1998; Zwislocki, 2002a; Price and Kalb, 2005; Price, 2007a) take on the form of ‘the series-configuration’ (Peake et al. , 1992), the series connections of two-port impedance network for the tympanic membrane and its attached ossicular chain Z except stapes and two one-port networks of and respectively for the middle-ear Z Z cavities and the stapes-cochlea complex, as shown in Fig. II-18. Therefore the middle-ear input impedance can be defined in this configuration as: Z

(II-25) P Z ≡ = Z + Z U where indicates ‘the mechanical coupling’ or ‘the ossicular coupling’ (Peake et al. , 1992) Z impedance measurable in the condition of widely-open cavity (i.e., short circuit condition of the network). The coupling impedance in the series configuration is given by: Z

(II-26) P − P Z = U To put it into a transmission matrix form, the mechanical coupling effect can be written as:

(II-27) U A B U = P C D P − P where is the pressure acting on the stapes by the incus. In a macroscopic viewpoint, the P matrix representation can be said to reflect the transmission-line characteristic of

traditional tympano-ossicular chain networks (Puria and Allen, 1998).

ZOC

UTM U ST

P− P P Z PTM TM CAV ZTOC IS SC

UTM UTM U ST ZCAV

P CAV

Fig. II-18. Classic middle ear configuration. Three blocks are reconfigured with reference to Fig. 2 of Peake et al.

(1992). The impedance block is one-port network representation of the middle-ear cavity impedance, the two-port network of the tympanic membrane and the ossicular chain of malleus and incus including ligaments and in-between joints, and the one-port network of the stapes-cochlear complex. The block with dashed-line depicts the middle-ear input impedance when the middle-ear cavity is widely open. Volume velocities are represented by , and exerted acoustic pressures by . The subscript means the tympanic membrane, the middle-ear cavities, the incus-stapes, and the stapes.

In a lumped sense, the pressure difference across the tympanic membrane P − P defines the ossicular chain driving force, where is the pressure from the ear-canal P acting on the eardrum and the pressure inside the middle-ear cavities. The definition P is valid up to the frequency range where the wavelength of the acoustic wave impinging to the eardrum is large enough compared to the diameter of the tympanic membrane since it is grounded on the assumption of constant pressures over the membrane surface (Peake et al. , 1992). Another underlying assumption in this configuration is that the driving force to the inner ear filled with incompressible water-like fluids (i.e., the perilymph) is described

solely by the stapes motion so that the stapes volume velocity should be maintained U across the stapes-cochlear complex block (Peake et al. , 1992). This requires the opposite- phased volume displacement equality of the cochlear windows, which is well-supported by experiments with porcine cadaver ears (Kringlebotn, 1995) and with human temporal bones stimulated by air conduction (Stenfelt et al. , 2004).

Voss et al. (2000) pointed out that there is an implicit constraint in the network topology shown in Fig. II-18. Since the volume velocity is kept throughout the series- U configuration, it requires that:

(II-28) P Z = P Z The implicit constraint of the network indicates that the magnitude of the constraint should be kept unity and its phase angle zero (Voss et al. , 2000):

(II-29) P P Ψω ≡ = 1 Z Z (II -30 ) ∠Ψω = 0 However experimental data of Voss et al. (2000) show that the constraint is only satisfied below 1 kHz and the inconsistency observed above 1 kHz is attributable mainly to the spatial variations in the cavity pressure for which a transmission line model suggested by

Onchi (1961) appears to be more appropriate.

The isolation of the middle-ear cavity impedance block from the stapes-cochlea complex network means that the acoustic interaction between the middle-ear cavities and cochlear 47

windows, namely ‘the acoustic coupling’ effect (Peake et al. , 1992), is largely ignored except the weak coupling with the eardrum in the classic schematic. It leads simulation efforts for a pathological condition of partial acoustic energy transmission to the inner-ear even in case the ossicular path is completely malfunctioning to a fiasco. Peake et al. (1992), in an attempt to integrate the acoustic coupling effect, suggested a generalized model shown in

Fig. II-19. The middle-ear cavity block was modified to have reciprocal two-port network connected both to the tympano-ossicular chain and stapes-cochlea complex block. In addition, mechanical and acoustic coupling effects simultaneously existing in the normal human middle-ear are depicted by inter-coupling of the two blocks in their model. If the model constructed successfully, the partial sound transmission solely through the acoustic coupling pathway can be nicely simulated. Following the modification, the acoustic coupling effect can be expressed in a chain matrix form as:

(II-31) U A B U = P C D P The pressure difference in the cochlear windows is given by: P (II -32 ) P = P − P where is the input pressure to the cochlea at the oval window and is the pressure P P released to the middle-ear cavity at the round window. Transmission-line representation of the middle-ear cavities, such as the cavity model suggested by Onchi (1961), would be one of the best candidates for the modification.

In spite of its physically reasonable implications, the generalized model has an

experimental limitation. For the human middle-ear, the mechanical coupling response is measurable with widely-open cavity. However the experiment only on the acoustic coupling effect cannot be performed without removing the ossicular chain perfectly or damaging the cavities. Thus keeping the middle-ear intact (or preserving the interactions between the two coupling effects) is practically impossible to acquire the chain matrix elements in experiments. For the purpose of analysis of a normal human middle-ear, the use of the classic model is not a compromise in quality since the mechanical coupling effect is reportedly overwhelming in that case (Peake et al. , 1992; Voss et al. , 2007).

UTM U ST

PTM− P CAV P ZTOC IS

UTM U ST

ZSC PTM

UTM U ST

P PCAV ZCAV WD

UTM U ST

Fig. II-19. Generalized middle-ear network including acoustic coupling path. This figure is reconstructed with reference to Fig. 11 from Peake et al. (1992). The dotted box represents the mechanical coupling effect and the dot-dashed box the acoustic coupling effects. The two effects are inter-connected so that they affect the middle ear response simultaneously in a normal condition.

2. Tympanic membrane models

In network modeling, the tympanic membrane is simplified as a two degree of freedom system. Divided into elastically-coupled two sub-parts, the eardrum consists of a conductive part transmitting acoustic energy up to the inner ear through the ossicular chain vibration and a residual part of the membrane. The former is tightly attached to the malleus thus becoming a driving point of the middle-ear mechanical pathway while the latter is moving relatively independent of the motion of the malleus. Through this simplification, it is possible to reduce the whole tympanic membrane transduction characteristics into the equivalent two-port system that converts acoustic properties such as the pressure difference across the membrane and volume velocity of the P − P eardrum into mechanical ones like the force exerted on the malleus and velocity of U F the umbo : V

(II-33) P − P F = T U V where means the two-port transfer matrix. T

Taking the two-port transfer matrix approach, Shera and Zweig (1991) provided well- categorized network schematics of the tympanic membrane. They derived a theoretical transfer matrix for an eardrum model simplified as a mechanically-coupled two pistons.

Considering the transfer characteristics, they showed that ‘the oscillator type’ eardrum model corresponds to the strongly-coupled case of the two-piston model and that ‘the simple type’ is equivalent to the weakly-coupled two-piston model in which two parts of

the eardrum move in nearly an independent way (Shera and Zweig, 1991). They also showed that the eardrum model by Shaw and Stinson (1981), widely referred to as ‘two- piston model’ in terms of network topology, is not exactly the two-piston model but close to the oscillator model. In Table II-4, the eardrum topologies employed in middle ear models are assorted with reference to the Shera and Zweig’s (1991) classification.

Middle -ear model Eardrum network Network p arameter topology (cm 2) (cm 2) (cm 2) Model #1 simple type 0.85 0.55 0.30 (Zwislocki, 1962; A Z Z + Z Zwislocki, 2002a)

Model #2 oscillator type 0.6 N/A ( ) (Kringlebotn, 1988) A A Z L + Z

Model #3 oscillator type 0.6 N/A ( ) (Rosowski and Merchant, A A Z L + Z 1995; Rosowski, 1996)

Model #4 two -piston type 0.6 0.06 0.54 N/A (Goode et al. , 1994) Z, Z + Z

Model #5 simple type 0.85 0.55 0.30 (Pascal et al. , 1998) A Z Z + Z

Model #6 si mple type N/A N/A N/A N/A (Price and Kalb, 2005; Z Z + L Price, 2007a)

Table II-4. Types of the eardrum model employed in middle-ear networks, where represents the tympanic membrane area, and area and impedance of the eardrum rigidly attached to the malleus, and the area and impedance of the residual part, effective eardrum area, coupling impedance, the impedance of the residual and coupling part, impedance of the malleal complex, eardrum inductance, , tympanic membrane suspension impedance, and inductance of the malleus. Values in parentheses are not shown but implicitly assumed.

a) Simple eardrum model

A simple eardrum model is equivalent to the case where mechanical coupling of the two parts is fairly weak. As shown in Fig. II-20(c), it is composed of acoustic shunting impedance of the residual part ( ) and the conductive part impedance ( ) whose Z Z corresponding areas are and respectively, and a mechanical transformer which A A 1: A converts acoustic energy into mechanical energy. In this type of eardrum model, the acoustic flow is divided into the two uncoupled paths. Shera and Zweig (1991) derived its transfer matrix as:

(II-34) 1 0 1 Z 1A 0 = 1Z 1 0 1 0 A Zwislocki (1962) modeled the shunt impedance of the residual part, representing the eardrum loss (Lutman and Martin, 1979), with an R-L-C network (i.e., , , and after R L C his network model) on account of its mass effect at higher frequencies and a complementary R-C circuit (i.e., and after his network model) was added in parallel. R C His recent model (Zwislocki, 2002a) was modified a little to conform to a more extensive experimental data set than the previous. In the model, was integrated to the R-L-C C circuit but separately represented and was connected in parallel to the network. He R remarked that two capacitances and in series describe the elastic suspension of the C C residual part, and that one resistance arises from the membrane viscosity and another R resistance from acoustic energy dissipation in the outer ring (Zwislocki, 2002a). Thus R both networks for the shunt impedance can be characterized to have single resonance but with suppressed higher modes (Zwislocki, 2002a). The malleal complex impedances of

both models were simply described with an R-L-C series circuit (i.e., , , and after his R L C model) which merged the acoustic effects from the malleus, incus, ligaments, and tensor tympani. Since the malleal complex impedance block also has single resonance frequency, together with the shunt impedance network, the eardrum model can be interpreted as two degree of freedom system. Approximating acoustic impedance values, he took the conductive part area given in Table II-4 as the effective area and used it in A A calculating the residual part impedance by assuming the corresponding area to be 0.30 cm 2

(Zwislocki, 1962; Zwislocki, 2002a). This results in a 70% lower value of the shunt impedance than calculated theoretically, given in the transfer matrix, with the residual Z part area . A

With simple modification of parametric values, the tympanic membrane network in the middle-ear model of Pascal et al. (1998) borrowed that of Lutman and Martin (1979) who added parallel R-C circuit (i.e., and after their description) to the shunt branch of R C Zwislocki’s (1962) to make it better fit to the real-ear data (Lutman and Martin, 1979). The other features are same with those of Zwislocki’s (1962) model .

AHAAH model by Price and Kalb (2005) also employs a relatively simple eardrum structure.

The noticeable differences compared with Zwislocki-type eardrum topologies (1962) are that the shunt impedance of malleo-incudal joint (i.e., and after the AHAAH C R diagram). It was assumed as a rigid joint thus omitted in Zwislocki’s (1962) analogue, consequently allowing no relative motion between the two ossicles. Also the mass effect of 54

the incus (i.e., after the AHAAH diagram) is represented separately from the malleal L complex block. Both impedance networks are described as simple R-L-C circuits (i.e., shunt branch by , and , and series impedance by , and after the AHAAH R L C L C R diagram). However, whether the same effective area was used is not clear.

b) Oscillator model

Oscillator model shown in Fig. II-20(a) is the opposite of the simple eardrum model. No acoustic transmission is involved with the conductive part of the tympanic membrane; however the major transmission occurs through the residual part and is partially shunted by the strong coupling. Mechanical energy conversion is carried out by a transformer 1: A whose effective area is that of the residual part. The transfer matrix is approximated by

(Shera and Zweig, 1991):

(II-35) 1 Z 1 0 1/A 0 = 0 1 Z 1 0 A where the coupling impedance (ZC) is represented as a spring-damper combination (i.e.,

). Referring to the transfer matrix, this approach is helpful to construct the Z = R + Kjω tympanic membrane analogue from which the ossicular attachment effect to the eardrum is entirely separated. As an extreme case, if the coupling impedance becomes infinite, the Z shunt flow would become zero. Thus the acoustic energy flows only through the residual part impedance, and the eardrum modeled in two parts behaves like a single piston (i.e., one degree of freedom system) whose transfer matrix is given by (Shera and Zweig, 1991):

(II-36) 1 Z 1/A 0 = 0 1 0 A where represents the piston impedance and the whole area of the tympanic Z A membrane.

Kringlebotn (1988) implicitly implemented the oscillator model of the eardrum in his middle-ear network. As the transfer matrix indicates, his model separated the eardrum suspension block along with eardrum mass block (i.e., shown in the model description) L from the malleo-incudal chain complex (i.e., shown in the model description) by Z positioning the eardrum-manubrium coupling block (i.e., shown in the model Z description) in-between. The eardrum suspension block consists of R-L-C (i.e., , , and R L C shown in the model description) and R-C (i.e., and shown in the model description) R C sub-circuits in parallel. The former sub-block represents the annular ring (or the tympanic annulus), and the latter the mechanical coupling force between the eardrum and the annular ring respectively. There are two mechanical transformers whose transformation ratio is where as an effective area the eardrum area deduced from averaging of 1: A A the projected area data was used instead of the residual part area . This indicates that A single piston theory is partly applied to this model in that is implicitly assumed. A = A One of them is used for converting the mechanical quantities of the eardrum suspension block into acoustic quantities, and the other is for the eardrum-manubrium coupling impedance conversion. Eardrum model by Rosowski and Merchant (1995) borrowed from that of Kringlebotn’s.

c) Two-piston model

Two-piston model has a feature physically more reasonable than the oscillator model or the simple model. The coupling effect is explicitly shown unlike the others, without compromising the characteristics of a two degree of freedom system. The transfer matrix of the model can be derived by considering a mechanically coupled two degree of freedom system, which is represented as (Shera and Zweig, 1991):

(II-37) Z + Z κZZ Z 1 1 0 = Z + Z 1 A Z + 1 + κZ 1 2 + κZ + κ κ 0 1 0 A where defines the area ratio. However, the transfer matrix is not perfectly κ = A/A matched with available network models (Shaw and Stinson, 1983; Goode et al. , 1994) due to the mathematical complexity involved in network analogy transformation. Constructing an equivalent circuit, Goode et al. (1994) employed two parallel transformers of ratio 1: A for the shunt impedance of the residual and coupling part ( ) and for Z, 1: A Z respectively. Although the dual-transformer is not clearly shown in Shaw and Stinson’s model (1981; 1983), the same schematic seems to be employed.

UTM UTM− U OC, (UTM− U OC, ) A M ZM

UO, C

PTM − PCAV 1: AM

ZO, C

(a)

UTM UTM− U C (UTM− U C ) A O ZO

PTM − PCAV :1 AO

(b)

UTM UTM− U O (UTM− U O ) A M ZM

PTM − PCAV 1: AM

(c)

Fig. II-20. Eardrum network schematics following the classification by Shera and Zweig (1991). (a) Two- piston model. Shaw and Stinson’s (1981; 1983) model is close to ‘the oscillator model’ since its effective area for velocity is approximately a real constant (Shera and Zweig, 1991). (b) Oscillator model. This type of model shows a real constant effective area for velocity due to relatively strong mechanical coupling. Thus the movement of the two parts of the eardrum is closely interdependent. (c) Simple eardrum model. A real constant effective area for force is its characteristic because the mechanical coupling between the two parts of the eardrum is so weak that they can move almost independently to each other.

d) Transmission-line model

At higher frequencies, specifically above 3 kHz, the tympanic membrane shows complex vibrational behavior (Fay et al. , 2006) that cannot be described with conventional two degree of freedom models. The two-piston model was initially designed in an attempt to explain the complex deformation pattern at high frequencies (Shaw and Stinson, 1983) however it was successful only up to 8kHz (Puria and Allen, 1998). A transmission-line model becomes more appropriate choice since it provides spatial characteristics. Puria and

Allen (1998) introduced a transmission line model for feline eardrum and inserted the model into their middle-ear model as in a two-port transmission matrix form, but the human version of it is not available to date.

3. Ossicular-chain and middle-ear transformer

As previously mentioned, two types of coupling exist in the middle-ear, ‘the ossicular coupling’ which transmits the acoustic energy to the inner ear through its vibratory conduction and ‘the acoustic coupling’ by the acoustic interactions between the middle-ear cavities and cochlear windows. The ossicular coupler defines the mechanical transforming action in the middle-ear, an impedance matching mechanism by the combination of a mechanical lever and a pressure amplifier, usually modeled in the network as an ideal transformer whose features are perfect flux coupling and zero power loss. Middle-ear acoustic transmission has been known to be hinged largely on the mechanical action. On the other hand, due to its minor effect on the normal middle-ear transmission, the acoustic

coupler is conventionally ignored in network modeling.

Upon the classical assumption that the malleus and incus are rotating as one unit around a fixed axis to form a lever system (Wever and Lawrence, 1954), the lever ratio is simply defined by the ratio of the lengths of the ossicular arms between the malleus (or length of the handle of the malleus) and the incus (or length of the long process of the incus) as: l l

(II-38) l l = l Strictly speaking, the malleal lever arm length indicates the distance from the umbo to the rotational axis of the malleus-incus chain and incudal arm length the distance from the incudo-stapedial joint (i.e., lenticular process) to the axis (Wever and Lawrence, 1954).

According to this definition, if the ossicles translate without any rotational motion, the ratio becomes unity because both of the arm lengths are infinite. Therefore the definition is applicable in a limited way to the pure rotation of the rigidly-connected ossicles around the fixed axis running through the anterior process of the malleus and the end of the short crus of the incus. As a result, the two ossicles are constrained to the condition of the same angular displacement or velocity. Since the lever length difference is equivalent to the displacement ( ) or velocity ( ) transformation in the system, it is re-defined as: Δ V

(II-39) Δ V l = = Δ V where is the velocity at the effective attack point of the malleus assumed to be located at V the umbo (Kringlebotn, 1988) and the velocity of the incus at the incudo-stapedial joint V (I-S joint). The attack point velocity is often given by the umbo velocity , and the V 60 V

lenticular process velocity is replaced by that of the stapes if the I-S joint compliance V V in the direction of the stapes movement is ignorable (Ravicz et al. , 2004):

(II-40) V l = V Since those definitions are based on the fixed arm lengths, the ratio is a constant which is generally adopted in modeling the middle-ear network.

However the actual lever ratio is a frequency-dependent quantity since it is commonly observed in an experimental data that the displacement difference between the umbo and stapes exists, which can be expressed in terms of the measured displacement of the umbo and the stapes (Goode et al. , 1994):

(II-41) Δω lω = Δ ω where is angular frequency. Recently reported frequency-dependent velocity transfer ω function (TF) between the umbo and lenticular process (Willi et al. , 2002) or observed the complex motion of the malleus (Decraemer et al. , 1991; Decraemer and Khanna, 1994) and incus (Schön and Müller, 1999) strongly supports the concept of the variant lever ratio in that the location of the rotational axis is largely dependent on the excitation frequency.

Physically, the area difference between eardrum and stapes footplate is responsible for the pressure transformation through the ossicular chain. It is conventional to define the area difference as the ratio of the effective tympanic membrane area to that of the stapes A footplate : A 61

A (II-42) A = A Borrowing Shera and Zweig’s idea (1991), the theoretical effective area of the tympanic membrane can be the effective area for force or the effective area for velocity A A defined as (Shera and Zweig, 1991):

P − P (II-43) A ω = F (II -44 )

U A ω = V where is a force exerted by the eardrum on the manubrium. The corresponds to the F A condition of ossicular fixation while the to the condition of ossicular detachment. The A effective area defined in this way is naturally frequency-dependent quantities so that the area ratio also shows frequency characteristic. However, somewhat confusing due to its ill- defined loading condition of the or (Shera and Zweig, 1991), the classical concept of V F effective area based on the comparison with sweeping volume by the eardrum (Wever A and Lawrence, 1954) is widely used in middle-ear model as shown in Table II-4. Adopting the classical concept, the area ratio has a constant value thus the time-domain simulation on the middle-ear becomes more convenient.

The mechanical transformer ratio is given by the product of the area ratio and the lever ratio:

(II -45 ) T = A × l The physical meaning of the transformer ratio is interpreted as the middle-ear pressure 62

gain which is defined by:

(II-46) P G ≡ P where is the cochlear input pressure. In case of human, the gain is 28 dB based on P Wever and Lawrence’s (1954) anatomical measurements (Puria et al. , 1997). This can be interpreted as an impedance transformer (i.e. the impedance matcher) to make the acoustic input impedance of the middle-ear close to that of the cochlear fluid near the stapes Z footplate (i.e., the cochlear input impedance). The impedance transformer ratio is Z Z given by (Dallos, 1973b):

(II-47) Z Z ≡ = A × l Z Since the area and length ratios are real quantities, no phase transform is involved in those definitions. Comparing the values of the both ratios shown in Table V, it is easily found that principal transformation occurs through the area change between the tympanic membrane and the stapes footplate. For the modeling convenience, both of the ratios are assumed to be constant in general.

One of controversial issues in ossicular chain modeling is whether the malleo-incudal (M-I) joint can be treated as functionally rigid joint or not. Traditional concept of the M-I joint is that it is a rigid junction of the two ossicles thus the malleus and incus are vibrating as one unit. However, this concept is not well conforming to the known mechanical characteristics of the M-I joint entirely composed of elastic tissues (Willi et al. , 2002). Implying the flexible

M-I junction, Hato et al . (2004) inferred the slippage of the joint from the measurement of 63

time lag between umbo and stapes footplate. Nakajima et al . (2005) indicated indirect evidence for flexibility of the ossicular joints in human temporal bones with the umbo and stapes velocity measurement under ossicular fixation. Under the assumption of compliance-dominant character in low frequencies, the mechanical model of the ossicular chain allowing flexible I-M junction by Ravicz et al . (2004) well explains the higher velocity ratio observed in human temporal bone experiments than the classical lever ratio. V/V The model predicts that the more flexibility in the joint the less efficient in sound transmission.

The typical network topology of the ossicular chain takes the form of a transmission-line as shown in Fig. II-21, which enables two-port matrix representation of the mechanical transmission in the middle-ear. Ossicular joints consuming transmitted energy are represented in shunt blocks of R-C network (i.e., spring-damper combination in mechanical representation) and mass effects of the ossicles are represented in series blocks of inductor.

In many middle-ear models, flexibility of the ossicular chain is ascribed entirely to the I-S joint (Zwislocki, 1957; , 1962; Lutman and Martin, 1979; Shaw and Stinson, 1983;

Kringlebotn, 1988; Giguère and Woodland, 1994a; Rosowski and Merchant, 1995; Rosowski,

1996; Pascal et al. , 1998; Zwislocki, 2002a) while Price and Kalb (2005; Price, 2007a) reflected flexible characteristic of the M-I joint in their AHAAH model. Møller (1961) analogously described the flexibility by allowing the mobility of the axis passing through the malleal head and short process of the incus, which is an approach very similar to what was taken by Goode et al . (1994) that allows translational movement of the rotation axis. 64

Assuming that the M-I joint is rigid enough to guarantee a unit rotation of the malleus-incus complex, the masses of the malleus and incus are lumped together. This eliminates the shunt block for the joint from network model as shown in Fig. II-21(a) since no consumption of the transmission energy takes place in the joint. When it comes to the flexible M-I joint, the mass effect of the malleus and the incus should be represented separately and the shunt block that represents energy loss in the joint is to be positioned in-between as shown in Fig. II-21(b). Goode et al. (1994) incorporated the influence of

“malleus-head axis wobble”, the translational motion of the rotation axis, into the middle- ear network in an attempt to explain the lever ratio increase above 1000 Hz. Attached at the tail of the ossicular chain network, it has the same network structure as that of the ossicular chain (i.e., series inductor and shunt R-C combination) so that the additional mass effect by the translation of the ossicular chain accompanying transmission loss is equivalently modeled.

Lm+ L i Ls Lm Li Ls

Cis Cmi Cis

R R is Rmi is

(a) (b)

Fig. II-21. Typical structures of the middle-ear ossicular chain network: (a) Network based on rigid malleo- incudal (M-I) joint assumption. (b) Network based on flexible M-I joint assumption. Impedance elements by the other anatomical structures attached to the ossicular chain such as ligaments and conductive portion of the tympanic membrane are not represented in this configuration, which are to be included in the whole middle-ear network models. The subscript corresponds to the malleus, the incus, and the stapes. Inductors describe the effective mass of each ossicles while the malleo-incudal (M-I) joint and incudo-stapedial (I-S) joint are analogized to R-C networks.

The parametric values for the ossicular chain network are listed in Table II-5. The inertial effect of the malleus including a partial mass of the eardrum and the mass of the incus L are generally lumped together as other than the AHAAH model (Price and Kalb, L L + L 2005; Price, 2007a) which adopts the configuration introduced in Fig. II-21(b). Mass effect of the stapes is considered negligible (Zwislocki, 2002a), lumped into the cochlear L complex in some analogues (Zwislocki, 1962; Kringlebotn, 1988; Goode et al. , 1994), or described as a separate entity (Rosowski and Merchant, 1995; Rosowski, 1996; Pascal et al. ,

1998; Price and Kalb, 2005; Price, 2007a). Depending upon whether the rigidity assumption is used or not, the compliance of the M-I joint is given as infinity or specific 66

value. In case of the rigid M-I joint model, the accompanying joint resistance is not available, but zero value is implicitly assumed since no energy loss is expected in the junction. I-S joint, unlike the M-I joint, is commonly assumed to be flexible; therefore it has specific compliance and resistance values. Differences in parametric values of the models are not so substantial except Model #4 (Goode et al. , 1994) which employed a dual-piston model for eardrum to describe the mechanical pathway more clearly by taking conductive portion of the membrane as the effective area for acoustic energy transmission.

Middle -ear model Network parameter

L + L L C R C R Model #1 A 40 lumped into cochlear ∞ N/A 0.25 3000 (Zwislocki, 1962) complex (2.5 mg)

Model #1 B 40 Ignored ∞ N/A 0.18 200 (Zwislocki, 2002a)

Model #2 22 lumped into c ochlear ∞ N/A 0.3 6000 (Kringlebotn, 1988) (8 mg) complex

Model #3 7.9 mg 3.0 mg ∞ N/A 4. 9×10 -4 m/N 3.6 N-s/m (Rosowski and Merchant, (22*) (4.9*) (0.3*) (5917*) 1995; Rosowski, 1996)

Model #4 10.8 mg lumped into cochlear ∞ N/A 0.44 ×10 -6 cm/dyne 500 dyne -s/cm (Goode et al. , 1994) (3000*) complex (0.003*) (82182*)

Model #5 40 8 ∞ N/A 0.03 170 (Pascal et al. , 1998)

Model #6 : 22 2440 4.33×10 -4 839100 4.57×10 -4 47500 (Price and Kalb, 2005; L: 2440 (6.1*) (6.1*) (0.17*) (2097*) (0.18*) (118*) Price, 2007a) L

Table II-5. Ossicular chain network impedance parameters. Not being specified, the units of the network impedance parameters are given in [µF], [mH], and [Ω] respectively for capacitance, inductance, and resistance. The values in parentheses are deduced or borrowed from available data. Asterisks (*) represent the parameter values converted by considering transformer effect.

In itself an impedance scaling device, the ideal transformer whose voltage and current transforming action is equivalent to the pressure and volume velocity converting in acoustic terms is generally used to model the mechanical conduction by the ossicular chain.

In network analogue, it is either suppressed (Zwislocki, 1962; Lutman and Martin, 1979;

Pascal et al. , 1998; Zwislocki, 2002a), or included as a single-stage transformer (Price and

Kalb, 1991; Price and Kalb, 2005; Price, 2007a) or multi-stage transformer (Kringlebotn,

1988; Goode et al. , 1994; Rosowski and Merchant, 1995; Rosowski, 1996). Not shown in the middle-ear network models (Zwislocki, 1962; Pascal et al. , 1998; Zwislocki, 2002a), the ideal transformer exists in implicit form with specific ratios listed in Table II-6 in order to merge the cochlear-complex into the equivalent circuit constructed through the suppression of the transformer. The multi-stage transformer concept implemented in several electrical analogues (Kringlebotn, 1988; Goode et al. , 1994; Rosowski and Merchant,

1995; Rosowski, 1996) classifies the energy transmission in the middle-ear into the acoustic energy transmission up to the eardrum and middle-ear cavities, mechanical energy transmission through the ossicular chain, and re-converted acoustic energy transmission into the inner ear fluid. Following the phase changes, the first transformer by the eardrum,

, the second by the ossicular lever system, , and the third by the stapes footplate, 1: A l: l , are used in order. The first and the third transformer represent the area change A : 1 while the second describes the lever ratio. The parametric values between the first and the third transformer are given in the unit of mechanical impedance thus, once the first transformer is suppressed then they are converted into acoustic impedances. AHAAH model (Price and Kalb, 2005; Price, 2007a) adopted single-stage transformer located 69

between the malleus and the incus by fusing the dimensionless lever and area ratios not clearly defined. Therefore, all the impedance elements after the transformer are given in acoustic impedance unit. Referring to Table II-6, as in the case of impedance values, substantial model-wise discrepancy is not observed other than in the Model #4 which considers much smaller area of 0.06 cm 2 as its effective eardrum area for the mechanical conduction (Goode et al. , 1994).

Middle -ear model Transformer Network parameter type

l A A A T Model #1 suppressed (1.3) 0.55 0.032 (17) (22) (Zwislocki, 1962; Zwislocki, 2002a)

Model #2 3-stage 1.3 0.6 0.032 (18.8) (24.4) (Kringlebotn, 1988)

Model #3 3-stage 1.3 0.6 0.032 (18.8) (24.4) (Rosowski and Merchant, 1995; Rosowski, 1996)

Model #4 3-stage 1.3 0.06 0.0315 (1.9) (2.48) (Goode et al. , 1994) with a dual transformer

Model #5 suppressed 1 0.55 0.032 17 (17) (Pascal et al. , 1998)

Model #6 single -stage N/A N/A 0.021 N/A 20 (Price and Kalb, 2005; Price, 2007a)

Table II-6. Transformer parameters and types in middle-ear networks. The unit for areas is [cm 2], and others are dimensionless. Values in parentheses are calculated with reference to available data.

4. Nonlinear elements

The human middle-ear shows linear response characteristic for moderate sound pressure level (SPL) in which the superposition principle is valid. Many network models were constructed based on the linear assumption (Zwislocki, 1962; Shaw and Stinson, 1983;

Kringlebotn, 1988; Rosowski and Merchant, 1995; Rosowski, 1996; Zwislocki, 2002a).

Exposed to a highly intensive sound however, the response characteristic of the middle ear becomes nonlinear. If the response remains linear, severe damage would occur in the auditory system. Nonlinear mechanism restricting the SPL-proportional response is indispensable to keep the system intact. In other words, the impedance matching nature of the middle-ear is moderated by the nonlinear behaviors. Two nonlinearities generally considered in simulating the responses of human middle-ear are the acoustic reflex (AR) of the stapedius muscle and stapes displacement-restricting action by the annular ligament

(AL). Based on its stiffness-controlling nature, the acoustic reflex has been modeled as a pressure-dependent capacitor (Lutman and Martin, 1979; Giguère and Woodland, 1994a;

Pascal et al. , 1998), time-variant capacitor (Giguère and Woodland, 1994b), or time- dependent impedance gain controller (Price and Kalb, 2005; Price, 2007a). Mechanical characteristics of the annular ligament are equivalently described by the combination of a variable capacitance and its accompanying nonlinear resistance (Pascal et al. , 1998; Price and Kalb, 2005; Price, 2007a).

a) Nonlinear effect of the acoustic-reflex

The functional effect of the AR is to undermine any impedance matching advantage of the middle ear by attenuating the acoustic energy transmission into the inner ear at low frequencies. Innervated respectively by the facial nerve and trigeminal nerve, the ‘stapedius muscle’ and ‘tensor tympani’ restrict the ossicular vibrations through muscular contraction in response to impinging sounds above specific acoustic reflex threshold (ART). It is widely accepted that the AR of animal subjects of clinical studies is involved with both muscles

(Northern and Gabbard, 1994; Geisler, 1998; Møller, 2000) while that of the human middle ear solely with the stapedius (Northern and Gabbard, 1994; Geisler, 1998; Møller, 2000;

Finch, 2004). The reflex of tensor tympani in humans is known to be attributed to the non- acoustic stimulus (Northern and Gabbard, 1994) or to self-vocalization (Borg et al. , 1984).

Referring to experimental results conducted on cats, the stapedius contraction does not lead to detectable displacement in the incus or malleus but increases the stapes impedance through altering the mechanical strain in the annular ligament (Pang and Peake, 1985a;

1985b). Considering the anatomical similarity among mammalian auditory systems, a parallel situation is feasible in the human middle-ear such that the reflexive contraction of the stapedius limits stapes movement toward the cochlea to control the conduction of sound waves to the inner ear. From the deflection viewpoint, the AR mechanism is regarded as a middle-ear compliance controller. A resistance regulator is also an acceptable functional description. The contraction of the stapedius muscle pulls the stapes-head along the major axis of the footplate approximately perpendicular to the direction of the stapes motion, consequently leading to the relative deflections on the anterior and posterior 73

portion of the annular ligament (AL) thereby results in the acoustic impedance variation on the corresponding AL areas (Bennet, 1984). Since the impedance provided by interactions with the perilymph has the resistance-dominant characteristic (Zwislocki, 1962; Zwislocki,

2002b), it makes sense to say that the AR contributes to the acoustic resistance variation

(Bennet, 1984) as well as the control of the middle-ear compliance.

The AR influence on the acoustic transmission into middle-ear network has been incorporated in various ways. Assuming the fixed stapes condition by the stapedius muscle contraction in which the ossicles rotate around the incudo-stapedial joint, Møller (1961) estimated the AR effects through pathological condition (i.e., immobile stapes condition) modeling in all-or-nothing fashion (Møller, 1961). Lutman and Martin (1979) characterized the main function of the AR in the middle-ear as a stiffness enhancer thus incorporated the variable capacitor concept into their network model to assess the AR influence on the middle-ear sound transmission (Lutman and Martin, 1979). However, the parametric values of the variable capacitor , which is not really a nonlinear component in a strict C sense, were not determined through automatic feed-back loop but artificially changed in a linear network to have infinite (i.e., no reflex), 0.75 µF (i.e., moderate reflex), and 0.25 µF

(i.e., near-maximal reflex) respectively corresponding to 0, 100, and 120 dB SPL acoustic stimuli at the eardrum. The model predicted the main AR effect to attenuate the middle-ear transmission below 1 kHz, and to increase phase angle up to 60 degrees between around

0.2 and 2 kHz. Kringlebotn (1988) studied the stapes muscle contraction effect on the middle-ear sound transmission through a parametric study on the muscle compliance; 74

however did not employ a nonlinear capacitor. Shown in Fig. II-22, Pascal et al . (Pascal et al. ,

1998) mathematically refined the AR functional model in terms of a compliance dependent on the sound pressure at the tympanic membrane . Utilizing Lutman and Martin’s (1979) P network parameters and experimental data, they formulated mathematical relationship between given in ‘Pa’ and the AR capacitance as (Pascal et al. , 1998): P

(II-48) 1 C = 1675 × 20 log P P + 33000 × 20 log P P where is given as 0.2 Pa corresponding to 80 dB SPL close to the acoustic reflex P threshold (ART) for broadband stimulus (Dallos, 1964). Prediction of this model shows not much discrepancy from that of Lutman and Martin (1979). Giguère and Woodland (1994a;

1994b) implemented the AR effect in dual pathways: static operation during which stapedius compliance value remains constant in ascending path model (Giguère and

Woodland, 1994a) and dynamic operation in their descending branch model (Giguère and

Woodland, 1994b). In their static AR effect study, they considered the compliance value to be 0.1 µF at maximal contraction, predicting approximately 10-15 dB middle-ear transmission loss below 1 kHz and 0-2 dB gain above 1.5 kHz. For dynamic AR simulation, they devised a feedback regulator whose role is to instantaneously update the stapedius compliance on the basis of the averaged firing rate of entire inner hair cell (IHC) afferent nerves. The AR effect in AHAAH model is described implicitly as an impedance gain controller in time-domain as shown in Fig. II-24. The AR block in AHAAH, unlike the other network models focusing on the compliance controlling influence of AR, does not exist separately but fused into the AL impedance block to update both of the resistance and

capacitance during time-domain simulations (Price and Kalb, 2005), which is quite conforming to the functional description of AR previously mentioned. The AHAAH model provides two options for the acoustic reflex application to the middle-ear response simulation (Price and Kalb, 2005): unwarned (i.e., if the subject is unaware of upcoming burst of sound) and warned (i.e., if the subject is aware of the burst of sound beforehand) case simulation. In the unwarned case, the stapedius contraction starts right after latency time, passes through exponentially-rising period and reaches its saturation. On the other hand, the AR function is pre-activated before the burst of stimulus in case of warned condition. Typical time-domain impedance gain obtainable in the AHAAH model is G plotted in Fig. II-23.

-6 x 10 8

(F) 4 AR C

0 80 90 100 110 120 Tympanic membrane SPL (dB)

Fig. II-22. Variable capacitance represented by AR model in the middle-ear network of Pascal et al . (1998) in which the ART was assumed to be 80 dB SPL at the eardrum.

15 15

10 10

5 5 Impedance gain Impedance gain Impedance

0 0 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Time (sec.) Time (sec.)

(a) Unwarned case (b) Warned case

Fig. II-23. Typical impedance gain profile obtained in the AHAAH model (Price and Kalb, 2005): (a) Unwarned case, and (b) Warned case. Note that the gain values are used for updating both of the annular

ligament impedance values of and in time-domain.

There are some functional limitations of currently available AR models embedded in the middle-ear network. Nonlinear AR compliance model by Pascal et al . (1998) lacks adaptive and frequency-dependent features of AR while the model reflects temporal characteristic of input signal due to its intensity-dependency. Giguère and Woodland (1994a) modeled AR operation in the middle-ear with feedback control algorithm but without consideration of growth functions and intensity-dependent dynamic properties. Although AHAAH model

(Price and Kalb, 2005) provides variable compliance and resistance gain values along the simulation time, the gain value profiles shown in Fig. II-23 have no adaptation or recovery property (i.e., Once saturated, it remained in the steady-state whatever the input signal variation is.). Triggering mechanism for the AR function, an intensity-frequency-dependent

ART, is also missing thus the AR effect is always activated regardless of input signal 77

characteristics and simulation option. It is partially due to that the AHAAH program was originally developed for the assessment of auditory damage from high intensity noise with impulsive characteristics.

In the absence of incorporated complex AR mechanisms, middle-ear response simulations to complex noise inputs in which intermittent transient sound signals and long-lasting steady-state acoustic stimuli coexist would produce significant errors on its final result of the simulated stapes displacement. Therefore, for more reliable simulation, commonly ignored dynamic characteristics such as adaptation property (Dallos, 1964; Wilson et al. ,

1978; Wilson et al. , 1984; Longtin and Derome, 1986) observed when middle-ear is exposed to prolonged stimulus and recovery process (Borg and Ödman, 1979; Longtin and

Derome, 1986) due to intermittent or repeated exposures have to be added. In addition, the nonlinear AR will have to be modeled as a function of both the intensity and frequency of the acoustic signals in order to enhance simulation accuracy.

AR has intrinsic limitations as a protective mechanism. The AR response is known to increase with intensity of eliciting stimulus and saturate approximately 40 dB above the

ART (Lutman and Martin, 1977). Over a certain SPL e.g. 120 dB, adopted as the AR operational limit by several middle-ear models (Lutman and Martin, 1979; Pascal et al. ,

1998) for a wide-band noise exposure, the strength of the stapedius contraction remains saturated so that the AR protection against stimuli with higher intensity become less effective. Another limitation is rooted in its feedback characteristic. The AR latency is 78

reportedly varying from 25 ms for high-intensity sound to 100 ms or longer for the SPL near ART (Møller, 2000). Due to its relatively long delay in feedback response, the AR does not provide timely protective reactions against highly impulsive sounds lasting only for a short duration e.g. gunfire or air-bag explosion sound. Therefore mechanically-induced protection mechanism becomes necessary for such noises.

b) Stapes displacement limiting by the annular ligament

Above the specific SPL at which the saturation of stapedius contraction occurs, a complementary nonlinear mechanism mainly controls the energy transmission into the inner-ear. The AL plays the complementary role by restricting excessive stapes displacement beyond the functional limit of the AR. Bounded by the AL, the stapes displacement response becomes nonlinear, which modifies the cochlear input pressure and eventually influences the basilar membrane response. The mechanical function of the AL is to suspend the stapes footplate. In addition, it consumes mechanical energy otherwise transferred lossless to the inner ear as the M-I joint or I-S joint does, thus the analogue is composed of R-C combination whose parametric values are variables. The impedance nature of the AL was experimentally examined by comparing the drained stapes-cochlea

(SC) complex impedance with the intact SC complex impedance at frequencies Z Z below 0.5 kHz where the AL compliance and resistance are dominant (Merchant et al. ,

1996). However, unlike ossicular joints, the AL does not shunt the acoustic energy flowing into the inner ear thus the impedance block of AL is connected in series with the cochlear

input impedance analogue.

(1) Mechanical modeling of nonlinear annular ligament

Somewhat limited amount of modeling efforts toward nonlinear AL in the human middle- ear network are found. In the early work of Price (1974), it was estimated that the linear limit of the human middle-ear is around 10 µm of the peak-to-peak stapes displacement, which corresponds to the free-field SPL of 110 to 120 dB. He also estimated that the maximum stapes displacement would be less than 30 µm. Hypothesizing the AL is primarily responsible for nonlinear responses of anesthetized cat’s middle-ear above 130 dB (Guinan and Peake, 1967), Price and Kalb (1991) proposed a mechanical model in which the AL is described as a nonlinear spring suspending the stapes-footplate whose displacement was assumed to asymptotically approach up to 20 µm. Damping proportional to the increase of the spring stiffness was artificially added to suppress resonance in their middle-ear analogue.

Assuming the operational similarity of the AL mechanisms of the human to that of cats,

Pascal et al . (1998) modeled R-C block representing nonlinearity of the AL. With initial best-fit values of and , they constructed linear circuit model from which the linear C R pressure transfer function (PTF) was found. The linear PTF was defined by the ratio H ω of the cochlear input pressure and tympanic membrane pressure : P P

(II-49) Pω H ω = P ω From the acoustic impedance definition, the linear displacement of the stapes footplate can be calculated as:

(II-50) H ω ∙ P δ ω, P = jωZω ∙ A where is the cochlear input impedance found in the linear circuit. Conversion of the Z linear displacement into nonlinear quantity was performed utilizing an empirical δ equation:

(II-51) δ ω, P δ ω, P = 1 + δ ω, P δ The limit displacement was given as that of the cat (i.e., 40 µm in peak-to-peak sense). δ Utilizing the nonlinear displacement, backward-calculation on the nonlinear AL impedance was conducted and curve-fitted to empirical functions of (i.e., the tympanic membrane P pressure normalized by 1 Pa) to find:

(II -52 ) . R = 72 P − 251 P + 442

. (II-53) C = −3.33 × 10 e + 4.64 × 10 P + 0.54 where is given in µF and in Ω. Since nonlinear AL elements modeled by Pascal et al . C R (1998) are dependent on , the AL behavior can reflect the temporal characteristics P (even together with frequency characteristics) of the acoustic pressure without time-delay.

However, the formulation for the nonlinear AL resistance seems to be mistaken because its plot in Fig. II-25(a) is quite different from the plot in ‘Fig. 8’ of Pascal et al .’s (1998) paper.

This needs to be carefully reexamined. In terms of application limit, the AL compliance

model is valid up to 160 dB of (i.e., 2000 Pa), which is sufficiently high for any purposes. P

Mechanical model developed for nonlinear AL of cats (Price and Kalb, 1991) is implemented in AHAAH (Price and Kalb, 2005) in which the AL network embraces the AR functional block controlling temporal gains of the and as shown in Fig. II-24. The C R stress-strain relationship in AL area describing the asymptotical characteristic of the stapes displacement near the breaking point was assumed as:

ε π (II-54) σ = Ctan ∙ ε 2 where is mechanical stress in the AL, the strain, the rupture strain of the AL, and a σ ε ε C constant. Nonlinear mechanical stiffness relates the stapes footplate displacement and k the mechanical force acting on it:

(II-55) F = k ∙ δ The acoustic stiffness calculated from the mechanical stiffness and stapes volume K k displacement consists of two parts i.e., linear acoustic stiffness which is constant and K nonlinear amplification factor varying with the stapes footplate displacement: K (II -56 ) K = K ∙ K Different from the eardrum SPL-based AL model that Pascal et al . (1998) constructed, the

AL stiffness in AHAAH is updated on the basis of the stapes displacement through feedback loop. Assumed to be proportional to the AL stiffness, the nonlinear acoustic damping is R given by a linear equation of the amplification factor:

(II -57 ) R = CK + C 82

where and are constants initially given. Incorporating the temporal gain effects on C C stiffness and resistance exerted by AR, the nonlinear stiffness and damping Gt Gt K of the AL become: R (II -58 ) K = Gt ∙ K (II -59 ) R = Gt ∙ R Temporal variations of the two impedance parameters are equivalently modeled as controlled voltage source in electrical sense.

'Gr' 'Gr' 'Ral' 'Gk' 'Kamp'

Acoustic Reflex 'Ral' calculation

'Gk' 'Kal' 'Ral' 'Kal' Sig. Sig. 'Kamp' 'Kamp' Curr. Curr. Curr. 'Kal' calculation 'Cal' Voltage Signal 'Kamp' calculation 'Ral' Voltage Signal

Sig. in Curr. Sig. in Curr.

1 in out in out 2 in out Annular Ligament Annular Ligament Capacitance Resistance

Fig. II-24. Nonlinear AL impedance block of AHAAH model regenerated in Simulink. The AR block is incorporated into the AL block as a gain controller and the capacitance and resistance of the AL are equivalently described by voltage-source models. Refer to the appendix.

5 x 10 0 0.4

0.35 -1

0.3

-2 0.25 ) F) Ω µ ( -3 ( 0.2 AL AL R C

0.15 -4

0.1

-5 0.05

-6 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 P P TM TM

(a) (b)

Fig. II-25. Nonlinear AL impedance values of (a) Resistance which seems to be misrepresented in Pascal et al .’s (1998) paper and (b) Compliance along the variation of the normalized tympanic membrane pressure whose validity limitation is 160 dB SPL.

(2) Allowable SPL predicted by stapes displacement limit

Stapes is the entrance to the inner ear; therefore provides the input to introduce the pressure difference across the cochlear partition, which generates traveling waves along the basilar membrane. The motion of the stapes is assumed to the purely translational (i.e., the piston-like motion), which seems to be valid up to 2 kHz (Heiland et al. , 1999; Voss et al. ,

2000; Stenfelt et al. , 2004). However the actual stapes motion is a combination of translation and rocking motions due to the three-dimensional geometry of the middle-ear, unevenly distributed mechanical properties and irregular geometry of the annular ligament.

It was hypothesized that the AL limits the response of the stapes by clipping its peak

displacement up to 20 µm (Price, 1974; Price and Kalb, 1991). If this hypothesis is valid, the linear middle-ear models are valid up to that point. The limit can be found in terms of the maximum tympanic membrane pressure ( ) is computed by: P

(II-60) D P = H ω where is the limit displacement of the stapes given as 20 µm and is the D H ω stapes displacement-TM pressure transfer function (DPTF). The allowable limits of the tympanic membrane pressure of each linear model are plotted in Fig. II-26, which has V- shaped distribution along the frequency variation. Taking the lowest tympanic membrane pressure found in the vicinity of 1 kHz, the valid range of the linear models is up to around

140 dB as shown in Fig. II-26. The pressure limit of 140 dB SPL based on the stapes displacement is much larger than the threshold of the auditory nonlinearity of 80 dB SPL at which the acoustic reflex is assumed to start to involve in middle-ear transmission (Pascal et al. , 1998). The pressure limit is determined by imposing the biological constraint (i.e., the stapes displacement limit) to linear models thus is not equivalent to the pressure limit showing linearity in the human middle-ear response.

250 250 250

200 200 200 (dB) (dB) (dB) max max max  150  150  150 TM TM TM P P P    100 100 100 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) (b) (c)

250 250 250

200 (dB) 200 200 (dB) (dB) max max max   150 

TM 150

150 TM TM P P P    100 100 100 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(d) (e) (f)

250 250

200 (dB) 200 (dB) max max  150  TM 150 TM P P   100 100 Frequency (Hz) Frequency (Hz)

(g) (h)

Fig. II-26. Applicable SPL limit based on the maximum displacement of the stapes. (a) Model #1-A (Zwislocki,

1962), (b) Model #1-B (Zwislocki, 2002a), (c) Model #2 (Kringlebotn, 1988), (d) Model #3 (Rosowski and

Merchant, 1995; Rosowski, 1996), (e) Model #4 (Goode et al. , 1994), (f) Model #5 (Pascal et al. , 1998), (g) Model

#6 (Price and Kalb, 2005; Price, 2007a; 2007b), and (h) The range of SPL limit, the shaded area, about the average shown as the dashed line. Model #5 and model #6 are linearized.

5. Stapes-cochlea (SC) complex network model

Equivalent circuit models covering the tympanic membrane and ossicular chain assumes the two-port network terminated with load impedance corresponding to one port network of the stapes-cochlea (SC) complex. The underlying assumptions are that the mechanical stimulations to the cochlea is transmitted solely through stapes and the volume velocity of the stapes is exactly equal to that at the round window (Peake et al. , 1992). This is not always valid due to the non-pistonlike motion of the stapes (Heiland et al. , 1999; Hato et al. ,

2003), vibrational mode of the round window and other complications (Bance et al. , 2004).

Therefore these complications are largely neglected in network modeling. The cochlear interaction with middle-ear cavity is also entirely ignored, thus there is no acoustic coupling between the tympanic membrane and the cochlea.

The SC impedance block ( ) is at the right-most portion in the middle-ear analogues. It is Z composed of the stapes inertia ( ), annular ligament resistance ( ) and compliance L R ( ), and the inner-ear impedance block containing the volume effect of fluidic column in C the vestibule ( ), input impedance of the cochlea ( ) which occasionally accompanies L Z the effect of the helicotrema ( ), and the compliance of the round window ( ). All of Z C them are connected in series, thus definition of the SC complex impedance is represented as:

(II-61) P ω Z ω ≡ = Z + Z + Z + Z + Z U ω where is the sound pressure acting on the stapes surface seen from the middle-ear side P 87

and the stapes volume velocity with reference to the effective footplate area ( ). The U A definition implies that the sound pressure acting on the round window from the middle-ear cavities is negligible (Merchant et al. , 1996) thus the interaction with the middle-ear cavities is ignored.

Typical one-port analogues of the SC complex are shown in Fig. II-27 and related parametric values are listed in Table II-7. Type I adopted in various middle-ear networks

(Zwislocki, 1962; Lutman and Martin, 1979; Kringlebotn, 1988; Goode et al. , 1994) has basic R-L-C analogue whose and values were fitted to averaged eardrum impedance R C and was deduced from the mass of effective perilymph column in the vestibule and L L the stapes (Zwislocki, 1962). The cochlear input impedance was ascribed mainly to L R but not much to . Not explicitly represented, the acoustic effects of the AL both on L C and , ST on , and RW on were incorporated into the analogue in a lumped way. In R L C the type II (Zwislocki, 2002a), the simplified form of the type I, the inertia component is L ignored due to its limited effect only at low frequencies especially below 100 Hz (Zwislocki,

1962; Zwislocki, 2002a). Detail descriptions are available through type III (Rosowski and

Merchant, 1995; Rosowski, 1996), type IV (Pascal et al. , 1998), and type V (Price and Kalb,

2005; Price, 2007a) into which the cochlear input impedance and the other sub- Z components are separately incorporated. Following Zwislocki’s (1962) earlier model, type

III assumes the cochlear input impedance to be an L-R combination. The AL is modeled as

R-C combination, and the stapes inertial impedance ( ) is separated from the cochlear L impedance block. Type IV and type V characterize the cochlear input impedance as purely 88

resistive ( ), but add the helicotrema effect significant at low frequencies. The round R window impedance is omitted in the type III and type IV on account of its little contribution to , but represented in the type V as compliance ( ). The vestibular volume effect Z C ( ) is separately embodied in the type IV and type V. L

a) Human cochlear input impedance

Among the subcomponents of , the cochlear input impedance represents the load Z Z impedance to the middle ear (i.e., the load that the inner-ear presents to the ossicular conductive pathway) thus serves as an impedance boundary condition which dampens standing waves that would propagate in the middle ear in the absence of the cochlear load such cases as drained cochlea or disarticulated stapes (Puria and Allen, 1998). Therefore accurate representation of is the most crucial task in modeling the SC complex since Z middle-ear response is significantly influenced by it. In reality, it also influences on the sound transmission through the outer-ear by affecting as well as on the cochlear Z transfer function. The cochlear input impedance is important not only because it influences the acoustic transmission but also it is the only measurable cochlear parameter in live human ear (Zwislocki, 2002b) indirectly from the middle ear input impedance ( ) Z measurement.

The cochlear acoustic input impedance is defined by the complex ratio of the scala Z vestibule ( ) sound pressure at the basal end of the cochlea (i.e., , the location sv P x = 0

near the stapes footplate) and volume velocity of the stapes footplate : U

(II-62) P ω, x = 0 Pω Pω Zω ≡ = = U ω A ∙ V ω A ∙ jωD ω where and respectively represent the stapes velocity and displacement. The scala V D vestibule pressure at the cochlear base, , is replaced by , the intra- P ω, x = 0 Pω cochlear pressure at the same location. Strictly speaking, the input impedance should be defined in terms of the perilymph volume velocity ( ) at the entrance of the scala U vestibule as:

(II-63) P ω, x = 0 Pω Zω ≡ = U ω, x = 0 V ω, x = 0 ∙ S x = 0 where represents the sectional area of the scala vestibule. However, by assuming the S mass conservation of incompressible perilymph, the fluid volume velocity at the scala vestibule entrance is approximately equal to that at the stapes footplate location (Puria and

Allen, 1991):

(II -64 ) U ω, x = 0 ≈ U ω Therefore, the previous definition adopting instead of is valid. U U

Theoretical and experimental evaluation of the cochlear input impedance for human ear has shown that the resistive nature is prevalent over wide range of frequency. Assuming the negligible mass and resistance of cochlear partition near the basal end, Zwislocki calculated cochlear input impedance from the characteristic impedance of the cochlea evaluated at the stapes footplate location (i.e., ) in terms of Hankel functions as x = 0 (Equation 4.71 from Zwislocki, 2002b): 90

/ (II-65) ρ jH a ω/2π Z ω = ∙ / SC H a ω/2π where is effective cross sectional area of the cochlear duct at the base, which is given by: S

(II-66) S x ∙ S x Sx = S x + S x and represents the acoustic compliance of the cochlear partition ( ) at the base. For C CP above 1 kHz where approximation is valid, the cochlear input impedance is H ≈ jH simplified by (Equation 4.74 from Zwislocki, 2002b):

ρ (II-67) Zω = SC Clearly understood from equations (II-65) and (II-67), the acoustic input impedance of the cochlea is a real constant thus the resistive character is dominant above 1 kHz. Based on the cochlear dynamic solution, Zwislocki (1950) concluded that would be essentially Z frictional and have specific resistance of 9000 Ω approximately which was converted into acoustic resistance of 540 Ω and employed in his primitive middle-ear circuit model

(Zwislocki, 1957). Later, the acoustic resistance value was adjusted to 600 Ω to fit measured

and adopted into updated version of network model (Zwislocki, 1962). Recently Z published result (Zwislocki, 2002b) for equation (II-65) is plotted in Fig. II-29(a) indicating that the magnitude of converges into around 800 Ω at frequencies above 1 kHz, which Z was implemented as in his recent middle-ear analogue model (Zwislocki, 2002a; R 2002b). Succeeding Zwislocki’s reasoning, Dallos (1973a) theoretically reaffirmed the resistive attribute of . The calculated specific impedance was 11200 Ω, somewhat larger Z 91

value than that of Zwislocki (1950). Independent of Zwislocki’s theory, Geisler and Hubbard

(1972) modified the RW boundary condition (i.e., zero pressure at the RW) of the Peterson-

Bogert cochlear model (Peterson and Bogert, 1950) and reported that the impedance is approximately resistive between 0.8 kHz and 8 kHz. Merchant et al . (1996), by examining the measured impedance difference between drained SC complex and intact SC complex, reported that the cochlear input impedance of human is resistance-dominant between 0.1 kHz and 5 kHz. The cochlear input impedance they experimentally estimated was not exactly but , which is: Z Z (II -68 ) Z ≡ Z + Z By analyzing measured and , Puria et al . (1997) suggested that the input impedance Z U is resistive between 0.3 kHz and 2 kHz. Aibara et al . (2001) measured two quantities, Z middle-ear sound pressure gain (GME) and velocity transfer function (SVTF), from which they derived the cochlear input impedance in 12 human temporal bones. Their estimation revealed that the mean cochlear input impedance is resistive between 0.5 kHz and 5 kHz

(Aibara et al. , 2001). The resistive character implicates the absorption of all acoustic energy entering the cochlea (Zwislocki, 2002b), which means that the acoustic energy flow in human middle-ear is unidirectional (i.e., no backward reflection from the cochlea into the middle-ear). In addition, the resistive property suggests that the BM displacement at the basal end leads the stapes displacement by 90° if the cochlear partition impedance is controlled by its compliance (Zwislocki, 2002b).

However, the resistive nature of the cochlear impedance has often been questioned. 92

Although not for the human cochlea, frequency-dependent input impedance characteristic was mentioned. In their WKB (Wentzel-Kramers-Brillouin) approach to analyze the cat cochlea, Koshigoe et al . (1983) represented the mass and resistance as frequency- dependent quantities. Puria and Allen (1991) showed that the human cochlear input impedance is frequency-dependent (i.e., increases approximately by 4 dB/octave below 1 kHz and decreases by -6 dB/octave between 1.2 and 10 kHz), using their nonuniform transmission line cochlear model. Consideration of the apical reflection of the traveling waves due to scalae areal change makes difference from the Zwislocki’s model shown in equation (II-67) derived from the characteristic impedance near the basal region implicitly assuming no reflection within the cochlea (Puria et al. , 1997). Through experimental analysis, Puria et al . (1997) concluded that the cochlear input impedance including the effect of is a function of frequency in the frequency range above 2 kHz. Experimental Z works by Aibara et al . (2001) also shows the frequency-dependent characteristic of the cochlear input impedance above 5 kHz.

b) Helicotrema modeling and its influence on the cochlear input impedance

The helicotrema influence on the cochlear input impedance is known to be confined to the low frequency range. Dallos (1970) regarded the helicotrema as a tube of very small diameter whose impedance is characterized by a simple L-R network (Beranek, 1954):

(II-69) 8ηl 4ρl Z ω = R + jωL = + jω πa 3πa where and are the length and radius of the helicotrema tube respectively. and l a η ρ individually mean the viscosity coefficient and the density of the perilymph. The frequency limit of this approximation given in cgs unit is (Beranek, 1954):

(II-70) 0.2 a < f which shows that the validity of the helicotrema impedance model is restricted to very low frequencies. The effect can be incorporated to block if it is designed so that the Z influence of is subdued by the cochlear input resistance at higher frequencies. The Z R L-R analogue is appended as a shunt network to (Dallos, 1970), which is deducted from R the wave propagation up to the apical end of the cochlea at very low frequencies. In such a case, the volume velocity of the stapes is assumed to be close to that through the helicotrema (i.e., ) thus the driving point volume velocity would be shunted U ≈ U significantly through the helicotrema but relatively small through the cochlear partition.

The volume velocity shunting through the helicotrema attenuates the BM displacement induced by pressure difference across the cochlear partition thus degenerates the low- frequency sensitivity of the cochlea (Cheatham and Dallos, 2001; Marquardt et al. , 2007).

Type IV (Pascal et al. , 1998), and type V (Price and Kalb, 2005; Price, 2007a) use the 94

approximated model of the helicotrema, in which magnitude of is bounded by two Z asymptotic resistances given as (Dallos, 1970; Dallos, 1973a):

(II-71) Z = R

(II-72) RR Z = R + R where and individually represent the upper and lower limit impedances. It is Z Z seen that the upper limit is determined solely by while the lower limit depends on R HT . R Shown in Fig. II-28(a), the helicotrema effects are appreciable below 1 kHz, especially below 100 Hz, where the cochlear input impedance is almost identical to the helicotrema impedance envelop. It indicates that the AHAAH model (Price and Kalb, 2005) conforms the low-frequency prediction by the equivalent tube model of helicotrema, whereas the model by Pascal et al . (1998) shown in Fig. II-28(b) reveals discernible discrepancy between Z and at low frequencies. This is due to the much smaller value which Pascal et al . (1998) Z assigned to than Price and Kalb (2005) did, which makes roughly . R Z R2

However, in many equivalent auditory system networks, the influence of the helicotrema on the cochlear input impedance has been largely ignored. That is, the circuit was modeled to be open by taking , such that no shunting phenomena occur through the Z = ∞ helicotrema because the cochlea is believed to behave as a pure resistance against the ossicular system except at very low frequencies (Zwislocki, 1950; Dallos, 1970; Dallos,

1973a; Cheatham and Dallos, 2001; Marquardt et al. , 2007). Based on the mathematical model (Zwislocki, 1950) predicting that acoustic waves with frequencies above 100 Hz do

not reach the helicotrema, Zwislocki (1962; 2002a) characterized the essential property of cochlear input impedance as purely resistive. The cochlear input resistance ( ) was fused R into , and his notion is reflected in type I and type II with which type III is in the same R context although is apparently shown. R

In Fig. II-29, cochlear input impedances from the mathematical model for the passive human cochlea (Zwislocki, 2002b) and network models (Pascal et al. , 1998; Price and Kalb,

2005) into which helicotrema effects are incorporated are compared. The magnitudes of

represented in Fig. II-29(a) show that Zwislocki’s model has lower values at frequencies Z over 1 kHz than the others that incorporated the helicotrema, and that the mass effect is persistent at frequencies below 100 Hz where network models predict . In the Z frequency region between 0.2 and 1 kHz, network models closely approximate the mathematical results. Exceptionally higher cochlear input impedance of AHAAH model at high frequencies implies the suppression of high frequency resonance in the middle-ear. Fig.

II-29(b) shows that phase of is controlled by the mass of cochlear fluid, indicated by the Z positive phase angles, although differences among individual predictions are noticeable.

Zwislocki’s model predicts gradually decreasing phase lead as the frequency increases while estimations of the equivalent network models start from near-zero phase angle to reach the maximum lead by 65° around 1.5 kHz in case of AHAAH and by 25° around 2 kHz in the model of Pascal et al .(1998).

The SC complex impedances are compared in Fig. II-30. The two nonlinear models Z 96

(Pascal et al. , 1998; Price and Kalb, 2005) are linearized for comparison, that is, the and C are fixed to their linear values. Since the linear values are not clearly indicated in Pascal R et al .’s (1998) model, those of AHAAH model are employed after applying the middle-ear transformer ratio of 17. The magnitude plot shows that the compliance effect is dominant in the frequency range below 0.1 kHz, and mass effect become observable around 1 kHz except in type II (Zwislocki, 2002a) which is the only SC block without inertial element. The compliance effects in the AHAAH model and the model by Pascal et al .’s (1998) are higher than those of the others because they assumed relatively low linear compliance of AL (i.e.,

0.196 µF for AHAAH and 0.141 µF for Pascal et al .’s model). Referring to the phase plot in

Fig. II-30(b), the transition from negative to positive phase that represents the compromise of compliance-mass effects on is found around 1 kHz except Zwislocki’s model (2002a) Z whose impedance is close to resistance above 1 kHz. The nonlinear AL impedance effects on are plotted in Fig. II-31 for the AHAAH (2005) model and Fig. II-32 for Pascal et al .’s Z (1998) model. As expected, the influence of the compliance on the magnitude is noticeable below 1 kHz but resistance makes little difference. If the linearized compliance is increased by two-fold, their impedance characteristics become close to predictions of the others, which means that the linearized AL is modeled stiffer than other models. Increase of R reduces the phase lead or lag, and widens the transition range as shown in Fig. II-31(d) and

Fig. II-32(d). On the other hand, as shown in Fig. II-31(b) and Fig. II-32(b), higher value C makes the phase transition occur toward lower frequencies.

LST CAL RAL

CSC L CSC C0

LSC

RSC RC0 RSC

(a) Type I (b) Type II (c) Type III

LST CAL RAL LVB LST CAL RAL LVB

LHT LHT

RC0 RC0

RHT RHT

(d) Type IV (e) Type V

Fig. II-27. Typical network structures for the cochlear-complex employed in middle-ear models. The subscripts , , , , , and represent the stapes-cochlea complex, cochlear input, stapes, vestibule, helicotrema, and round window respectively. The AR-related element shown in several network models is omitted since it is assumed to affect overall middle-ear transmission.

Middle -ear model Network parameter

L R C L R L R L C R L C Model #1 -A 20 600 0.6 N/A N/A N/A N/A N/A N/A N/A N/A N/A (Zwislocki, 1962)

Model #1 -B N/A 800 0.52 N/A N/A N/A N/A N/A N/A N/A N/A N/A (Zwislocki, 2002a)

Model #2 46 330 0.56 N/A N/A N/A N/A N/A N/A N/A N/A N/A (Kringlebotn, 1988)

Model #3 N/A N/A N/A 40.4 336.6 N/A N/A 4.9 0.56 0 N/A N/A (Rosowski and Merchant, 1995; Rosowski, 1996)

Model #4 19.9 1631 0.50 N/A N/A N/A N/A N/A N/A N/A N/A N/A (Goode et al. , 1994) (12.2 ×10 3) (1 ×10 6) (0.82 ×10 -9)

Model #5 N/A N/A N/A N/A 1211 150 850 8 NL NL 21 N/A (Pascal et al. , 1998)

Model #6 N/A N/A N/A N/A 6600 131 450 6.1 NL NL 15.6 5.52 (Price and Kalb, 2005; Price, 2007a)

Table II-7. Impedance parameters in the SC block. The unit used are [µF], [mH], and [Ω] for compliance, inductance, and resistance respectively. The ‘NL’ means nonlinear element conditionally varying . Middle-ear transformer effects are not applied to values in parentheses.

7000

) 6000 Ω 5000

4000

3000

2000

acoustic impedance ( acoustic impedance 1000

0 2 3 4 10 10 10 Frequency (Hz)

(a)

7000

) 6000 Ω 5000

4000

3000

2000

acoustic impedance ( acoustic impedance 1000

0 2 3 4 10 10 10 Frequency (Hz)

(b)

Fig. II-28. Cochlear input impedances and helicotrema effects. (a) AHAAH model (Price and Kalb, 2005) where solid line represents the cochlear input impedance and dashed line the helicotrema impedance . Two dotted lines in red color represent asymptotes of the impedance and . (b) Pascal et al .’s model (1998)

100

7000

6000

5000 )

Ω 4000 ( 

C0 3000 Z  2000

1000

0 2 3 4 10 10 10 Frequency (Hz)

(a)

(degree) 40 C0 Z φ 20

0 2 3 4 10 10 10 Frequency (Hz)

(b)

Fig. II-29. Cochlear input impedance . (a) Magnitude plot of the cochlear input impedance where the solid line represents the estimation from the mathematical model of Zwislocki (2002b) to which middle-ear transformer ratio of 22 is applied, intermittent line from AHAAH (Price and Kalb, 2005), and dash-dot line from Pascal et al.

(1998) (b) Phase plot

101

4 x 10 6

) 4 Ω (

 3 SC Z  2

0 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

(a)

100

0 (degree) SC Z φ -50

-100 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

(b)

Fig. II-30. Stapes-cochlea complex impedance . (a) Magnitude plot of the SC complex impedance where the dashed line is from linearized AHAAH model (Price and Kalb, 2005), and dash-dot line from linearized Pascal et al . (1998)’s. Solid lines are from other models. (b) Phase angle plot

102

4 x 10 10 100

8 50 )

Ω 6 (

 0 (degree) SC

Z 4 SC  Z φ -50 2

0 -100 1 2 3 4 5 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

4 x 10 5 100

4 50 )

Ω 3 (

 0 (degree) SC Z

2 SC  Z φ -50 1

0 -100 1 2 3 4 5 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

Fig. II-31. AL impedance effects on SC complex impedance in AHAAH model. (a) Magnitude and (b) phase for change of , and (c) magnitude and (d) phase for changes. The gain factor of 0.5 (blue line), 1.0 (green line), and 2.0 (red line) are multiplied to or .

103

4 x 10 12 100

10 50

) 8 Ω (

 6 0 (degree) SC Z SC  Z

4 φ -50 2

0 -100 1 2 3 4 5 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

4 x 10 6 100

5 50

) 4 Ω (

 3 0 SC (degree) Z SC  2 Z φ -50 1

0 -100 1 2 3 4 5 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

Fig. II-32. AL impedance effects on SC complex impedance in Pascal et al .’s model. (a) Magnitude and (b) phase for change of , and (c) magnitude and (d) phase for changes. The gain factor of 0.5 (blue line), 1.0 (green line), and 2.0 (red line) are multiplied to or .

104

6. Network analogues for the middle-ear cavities

The middle-ear cavities acoustically interact both with a part of the tympanic membrane and with the round window of the cochlea. In anatomical terms, it consists of tympanic cavity just behind the tympanic membrane, the aditus ad antrum (i.e., a narrow passage to the antrum) separating the tympanic cavity from the antrum, and complex of the antrum and air-cells (i.e., pneumatic cells or mastoid cells) of the mastoid bone. Simplifying the intricate structure of the air-cell system, they can be acoustically modeled as two air cavities connected through a narrow passage. Electro-acoustically, two parallel capacitors, in which one of them representing the air volume in the mastoid cells, are linked in series with inductor-resistor combination of the aditus ad antrum. This may be the simplest analog (i.e., four-element-topology) of the system.

Typical one-port network representations of the middle-ear cavities are shown in Fig. II-33, and their component values are listed in Table II-8. Type I is modeled first by Zwislocki

(1962). Inductance depicts the mass effect of the narrow passage of the aditus ad L antrum, which is included in all models (i.e., and ). Resistance and respectively L L R R represent the resistance in the aditus ad antrum, and the sound absorption in the walls of the tympanic cavity, the Eustachian tube and in the pneumatic cell complex. Compliance C is the low frequency description of the air volume in the antrum and pneumatic cells. The other compliance represents the compliance effect of the air volume in the tympanic C cavity. Type II is a variation of the type I, which is also suggested by Zwislocki (2002a).

Resistance represents the multiple connections of the pneumatic cell system, and R R 105

has nearly the same representation as that of the . Three compliances , , and R C C C are respectively model the effect of the air volume in pneumatic cells, antrum, and tympanic cavity. Type III, named as ‘four-element topology’ by Voss et al . (2000), has the simplest structure. In the model, represents the resistance of the tube-like aditus ad antrum, R C the compliance of the mastoid air space and the tympanic cavity air volume compliance C (Rosowski and Merchant, 1995). AHAAH has the same network structure. All the parametric values in the type III are constant except the frequency-dependent resistance R of Voss et al .’s (2000) model (i.e., model 7), which is evaluated based on the middle-ear air space impedance measurement. In the audible frequency range (i.e., between 20 and 20000

Hz), the resistance value varies from 2 to 70 Ω in case of ‘Bone 24L’ ear. Cavity R impedance measured with mostly-removed mastoid air-cell tract shows fairly good Z agreement with the four-element topology model predictions and compliance dominant property below 1 kHz, whose contribution to the middle-ear input impedance is minor at most frequencies but peaks between 2 and 4 kHz can have significant effects on it (Voss et al. , 2000). However, impedance measurement with intact mastoid air-cell systems reveals the compliance-dominant nature below 0.5 kHz but sample-dependent complex character between 0.5 and 6 kHz, which limits validity of the four-element topology (Stepp and Voss,

2005).

Fig. II-34 and Fig. II-35 show the magnitude and phase angle of the impedance of each network model of the middle ear cavity. Two-extrema character in impedance magnitude, which comes from the series inductance-capacitance combination and its parallel 106

connection to another capacitance, is observed in common. Impedances of Type I and type

II show similar impedance characteristics (i.e., band-pass feature around 0.5 kHz, high-pass property and similar trend in phase angle), indicating that they have the same origin. Type

III shows no zero in their magnitude. While the AHAAH model (i.e., the model 5) shows smooth concavity in the low frequency region and a peak, model 6 does not have any noticeable zeros or peaks in its impedance. Type I, type II and AHAAH model have their peaks at around 2-3 kHz. Impedance of the model 7 has much higher magnitude than the others up to 1 kHz, and has a constant phase angle of other than the range where the − π 2 peak and zero exist. It shows lower impedance relative to other models at high frequencies.

The discrepancies between the model 7 and the other models is due to the fact that the model 7 was constructed based on the direct impedance measurement at the middle ear cavities while the others referred to the middle ear impedance data acquired at the tympanic membrane. Thus the parameters of the model 7 need more adjustment than the other models to minimize errors of overall middle ear impedance. However, all these three types of analog structure fail to regenerate multiple resonances in high frequencies due to oversimplification or exclusion of the acoustic complexity existing in the air-cell tract.

Employing the transmission-line concept, Stepp and Voss (2005) suggested three types of feasible network models for the air-cell system.

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Network Type I Type II Type III element Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7

(mH) 14 14 12 - - - - L - - - 20 - - - L - - - - 14 1 8.9 L (µF) 5.1 5.1 3.6 - - - - C 0.35 0.35 0.35 - - - - C - - - 5.0 - - - C - - - 0.35 - - - C - - - 0.35 - - - C - - - - 5.1 3.9 0.63 C - - - - 0.35 0.4 0.63 C (Ω) 10 10 20 - - - - R 390 500 420 - - - - R - - - 20 - - - R - - - 300 - - - R - - - - 91.85 60 R (2.23-70)0.5 × f

Table II-8. Parameter values in network model of the human middle ear cavities. Model 1 (Zwislocki, 1962; Lutman and Martin, 1979; Giguère and Woodland, 1994a), Model 2 (Goode et al. , 1994), Model 3 (Pascal et al. , 1998), Model 4

(Zwislocki, 2002a), Model 5 (Price and Kalb, 2005), Model 6 (Kringlebotn, 1988; Rosowski and Merchant, 1995),

Model 7 (Voss et al. , 2000). The parametric values of the model 7 refers to ‘Bone 24L’ ear data of Voss et al . (2000).

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R21 C21

L2 L1 R11 C11 L3 R3 C31 C22

R12 R22

C12 C23 C32

(a) (b) (c)

Fig. II-33. Network model structures for the human middle ear cavities. (a) Type I (Zwislocki, 1962; Lutman and

Martin, 1979; Giguère and Woodland, 1994a; Goode et al. , 1994; Pascal et al. , 1998) (b) Type II (Zwislocki,

2002a) (c) Type III (Kringlebotn, 1988; Rosowski and Merchant, 1995; Voss et al. , 2000; Price and Kalb, 2005)

3 3 10 3 10 10

2 10 2 2 10 10 Magnitude (Ohms) Magnitude (Ohms) Magnitude Magnitude (Ohms) Magnitude

1 10 2 3 4 2 3 4 10 10 10 10 10 10 2 3 4 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) (b) (c)

Fig. II-34. Magnitude of the middle ear cavity impedance. (a) Type I (solid line: model 1, intermittent line: model

2, solid-dotted line: model 3), (b) Type II (model 4), (c) Type III (solid line: model 5, intermittent line: model 6, solid-dotted line: model 7)

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2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0

-0.5 -0.5 -0.5 Phase angle Phase (rad) angle Phase angle (rad) angle Phase Phase angle Phase (rad) angle -1 -1 -1

-1.5 -1.5 -1.5 -2 2 3 4 -2 -2 2 3 4 2 3 4 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) (b) (c)

Fig. II-35. Phase angle of the middle ear cavity impedance. (a) Type I (solid line: model 1, intermittent line: model 2, solid-dotted line: model 3), (b) Type II (model 4), (c) Type III (solid line: model 5, intermittent line: model 6, solid-dotted line: model 7)

Shown in Fig. II-36, the location of the middle-ear cavity impedance block in the middle ear network is different. As previously mentioned, the middle-ear cavity block can be represented as a one-port network connected in series with the rest part of the middle-ear

(i.e., in network) because it is isolated from the mechanical coupling pathway. Z Therefore the middle ear input impedance can be simplified as a series of the cavity Z impedance and the input impedance of the rest parts measured with ‘open Z Z cavities’ condition (Rosowski, 1996):

(II -73 ) Z = Z + Z Zwislocki (1962) well supported the impedance block separation with his short comment that the placement of the cavity block in front of the eardrum block is due to the fact that a displacement of any part of the eardrum compresses or rarefies the air enclosed in the cavities. Since he put the block at the input port of , it naturally affects the responses of Z

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the rest of the middle ear model. Rosowski and Merchant (1995) pointed out a sequence problem in that sound waves in the ear canal moves the tympanic membrane and the air volume inside the cavities is compressed or rarefied through motions of the membrane. The location of the impedance block does not matter in estimating the input impedance of the middle ear and TM driving pressure ( ) whether it is placed in upper-leftmost or P − P lower-leftmost of the middle ear network. However, Zwislocki’s (1962) block arrangement shown in Fig. II-36(b) faces pressure drop (i.e., voltage drop in the network) by the P cavity impedance before sound waves reach the tympanic membrane. Referring to the network ground (i.e., zero pressure point), it means that the sound pressure at the end of the ear-canal does not reach the TM as it is. Thus, it makes more sense to connect P Z block to the output port of (i.e., lower-leftmost of the middle ear network) as shown in Z Fig. II-36(a). It also more clearly defines the cavity pressure influenced by the relative motion among the eardrum, the oval and round window (Shera and Zweig, 1992; Rosowski and Merchant, 1995). Network models of AHAAH (Price and Kalb, 2005) and Rosowski and

Merchant (1995; Rosowski, 1996) follow such an arrangement as shown in Fig. II-36(a).

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ZOC

UTM U ST

P− P P Z PTM TM CAV ZTOC IS SC

UTM UTM U ST ZCAV

P CAV

(a)

ZOC U U TM TM U ST ZCAV

P− P P Z PTM TM CAV ZTOC IS SC

UTM U ST

P CAV

(b)

Fig. II-36. Middle-ear cavity impedance block connected to the middle-ear network. (a) located on the left-lower position (i.e., input port of the ). With reference to the ground, the pressure arrives at the TM is (b) located on the left-upper position (i.e., output port of the ) after Zwislocki’s (1962) arrangement. The pressure reaches the TM is given by . −

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C. Cochlear modeling

The cochlea is a time-frequency analyzer that utilizes the resonance characteristic of the cochlear partition vibrating in traveling wave pattern. The stapes vibration, the input to the cochlea, generates the compressive wave in cochlear fluid. The compressive wave induces hydraulic pressure difference across the cochlear partition. The pressure difference creates motion of the cochlear partition in a form of traveling wave. The motion of the partition mechanically opens or closes the ion channels of the outer hair cells to generate the electrochemical signal communicable with the brain. In modeling the cochlear macro- mechanics, the effect of the cochlear geometry (i.e., the spiral shape of the cochlear bony wall) is generally assumed to be negligible, which enables to simplify the cochlea as an uncoiled two-channel fluid-structure interaction system. Fluid in the cochlear chambers is assumed to be incompressible and inviscid, thus has primarily only the mass effect. The cochlear partition is simplified as a membrane structure with longitudinally-varying impedance. In this section, cochlear modeling procedure introduced by Zwislocki is encapsulated and transfer functions of Zwislocki’s model and AHAAH model are compared.

1. Passive cochlear model by Zwislocki (2002b)

Zwislocki’s cochlear model is a one-dimensional, long-wave based, passive model. The model is equivalently depicted as a transmission line (Zweig et al. , 1976). The governing differential equation for passive cochlea with the long-wave assumption (i.e., the wavelength of a wave on the cochlear partition is larger than the height of the cochlear duct)

113

is:

(II-74) + = where, is the pressure difference across the cochlear partition, is the spatial coordinate along the cochlear partition in the longitudinal direction from the base to the apex, and are respectively the density and damping property of the cochlear fluid (i.e., perilymph), is effective sectional area of the cochlear channel and is the cochlear partition impedance. The effective area is given by:

(II-75) ∙ = = ∙ + where and are the sectional area of the scala vestibule and scala tympani respectively, and is the effective sectional area at the location of the stapes (i.e., ). = 0

As Zweig et al. (1976) pointed out, the dominant impedance component of the incompressible cochlear fluid is its inertia and can be assumed to be small enough to be negligible. The governing differential equation can be simplified as:

(II-76) =

The acoustic impedance of the cochlear partition ( ) can be represented as:

(II-77) 1 = + + where resistance ( ), compliance ( ) and mass ( ) of the partition are respectively given by exponential function: 114

(II -78 ) = ∙ (II -79 ) = ∙ (II -80 ) = ∙

Since the compliance effect is dominant in the low frequency region, the partition impedance can be approximated by ignoring the effect of the partition resistance ( ) and mass ( ) as:

(II-81) 1 ≈ = Thus the differential equation becomes:

(II-82) =

The exact solution of equation ((II-82) has a form of Bessel function, specifically the Hankel function of the second kind of zero-th order ( ): (II-83) / , = where is the pressure difference at the stapes location (i.e., ) and is the excitation = 0 frequency of the stapes. The other parameters are given by:

(II -84 ) = +

/ (II-85) / 4 =

115

/ / (II-86) / 2 / 1 = Substituting (II-86) into (II-83), the pressure difference across the cochlear partition is given by:

/ / (II-87) 2 / 1 , = Note that equation (II-87) is based on the assumption of negligible and in the low frequency region, that is, the basal region of the cochlea.

Expansion of the solution with the generalized partition impedance given in equation (II-77) requires the replacement of the compliance ( ) with the partition impedance ( ). Thus, the expanded solution is given by:

/ / (II-88) 2 / 1 , =

2. Cochlear transfer function

The cochlear transfer function is a linear function defined by the ratio of the excitation of the stapes and the response of the cochlear partition. Use of different cochlear modeling methodology does not make much difference in the formulation procedure of the transfer function.

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a) Volume displacement transfer function

In terms of the volume displacement, the cochlear transfer function can be defined by:

(II-89) , , = where and are volume displacements of the cochlear partition and the stapes respectively. The volume displacement at a specific location on the cochlear partition can be calculated from its steady-state volume velocity of the partition ( ):

(II-90) , , = The partition volume velocity is related with the cochlear partition impedance ( ) by:

(II-91) , , , = = , , The cochlear characteristic impedance at the stapes location (i.e., the cochlear input impedance evaluated at ) is: = 0

(II-92) = = Thus, the pressure at the stapes position is:

(II -93 ) = ∙ The transfer function in equation (II-89) can be rewritten as:

(II-94) , , =

In case of the Zwislocki’s cochlear model, the pressure field is given by a form of zero-th order Hankel function of second kind (Zwislocki, 2002b). Substituting equation (II-88) into

117

(II-94),

/ / (II-95) 2 / 1 , = The characteristic impedance at the position of the stapes ( ) is also given by Zwislocki (2002b):

/ (II-96) 4 = / ∙ / where is first order Hankel function of second kind.

b) Displacement transfer function of the cochlea

Displacement transfer function of the cochlea is defined by the ratio of the displacement of the cochlear partition ( ) and the displacement of the stapes footplate ( ):

(II-97) , , = The displacement of the cochlear partition and the stapes footplate are respectively represented as follows, using the partition area ( ) and the stapes footplate area ( ):

(II-98) , , =

(II-99) = Finally, the displacement transfer function of the cochlea is represented as:

(II-100) , = , ∙

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c) Spatial profile of the displacement transfer function

The parametric values of the cochlear model by Zwislocki and AHAAH program are listed in

Table II-9, and each impedance component along the spatial coordinate of the basilar membrane is plotted in Fig. II-37. Two noticeable differences are in the resistance and mass component of the cochlear partition impedance, which is partly attributed to the response of the cochlear partition. The reason for those discrepancies is not clear since the AHAAH model does not provide full description for the impedance components.

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Parameter Zwislocki ’s model AHAAH model

Stapes footplate area [cm 2] [cm 2] ( ) 3.2 × 10 2.1 × 10 Resistance of the [dyne -sec/cm 4] [dyne -sec/cm 3] . cochlear partition ( ) 3.0 × 10 91 .2 Compliance of the [cm 4/dyne] 1.0 × 10 . [cm 3/dyne] . cochlear partition ( ) 0.538 × 10 Mass of the cochlear 0.4. [g/cm 3] 5.8 × 10 [g/cm 2] partition ()

Width of the cochlear 1.3 × 10 . [cm] 0.8 × 10 . [cm] partition ( )

Scala sectional area ( ) 1.16 × 10 . [cm 2] 1.25 × 10 . [cm 2]

Characteristic 1.085 × 10 . [Hz] 2.0 × 10 . [Hz] frequency ( )

Table II-9. Cochlear modeling parameters. Components of the cochlear partition impedance are represented per unit length (Zwislocki’s model) or per unit area (AHAAH model).

120

-7 ) 4

x 10 4 x 10 3 2.5

) 0.35 2.5 /dyne) 3 4 2 0.3 2 1.5 0.25 1.5 1 0.2 1

0.5 Mass (g/cm 0.15 0.5

Compliance (cm Compliance 0 1 2 3 0 1 2 3 0 1 2 3 Resistance (dyne-sec/cm Distance from the BM base (cm) Distance from the BM base (cm) Distance from the BM base (cm)

(a)

-7 -3 ) x 10 x 10 3 )

3 3000 )

1.5 2 4 2000 1 2 1000 0.5 Mass (g/cm 0 0 0 Compliance (dyne/cm Compliance 0 1 2 3 0 1 2 3 0 1 2 3 Resistance (dyne-sec/cm Distance from the BM base (cm) Distance from the BM base (cm) Distance from the BM base (cm)

(b)

Fig. II-37. Magnitude of the cochlear partition impedance along the BM length from the base to the apex, represented per unit length (Zwislocki’s model) or per unit area (AHAAH model): (a) Zwislocki’s model (b)

AHAAH model where two vertical lines represent the BM region of consideration in AHAAH

121

The displacement transfer function profiles obtained based on the Zwislocki’s cochlear model along the basilar membrane location for several input frequencies (i.e., stapes excitation frequency) are shown in Fig. II-38. For the purpose of comparison, the transfer function profiles of the AHAAH program are plotted in Fig. II-39. The most obvious difference between the two spatial transfer function profiles is the variation of the peak level, which is nearly constant in the Zwislocki model but increases exponentially in the

AHAAH model as the excitation frequency increases. It can be seen that the AHAAH model has frequency-dependent response amplitude characteristic while Zwislocki’s model has nearly constant response amplitude characteristic. The relative locations of the peaks are shifted toward the basal direction in Zwislocki’s model. The envelopes of Zwislocki model show no noticeable change of spatial bandwidth as the excitation frequency increases. On the other hand, those of AHAAH show decreases in spatial bandwidth. Although the AHAAH cochlear model is linear model, its response bears the sharp tuning features of the active cochlea.

Zwislocki’s cochlear model fairly well approximates the experimental results of Békésy

(Zwislocki, 2002b). However the cochlea model does not include the active mechanism of the live cochlea. Therefore, the transfer functions calculated from the model have limited applicability. The cochlear model of AHAAH program is not free from the limitation either.

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8 1

D 4 0.5 H (normalized) (normalized) (normalized)

2 D H H

0 0 0 1 2 3 4 0 1 2 3 4 BM location [cm] BM location [cm]

(a) Displacement transfer function (b) Normalized displacement transfer function

Fig. II-38. Spatial displacement transfer function of Zwislocki’s cochlear model (Zwislocki, 2002b) (blue solid line:

500 Hz, green dashed line: 1000 Hz, red dash-dot line: 2000 Hz , cyan dotted line: 4000 Hz)

3 1

D 0.5 H 1 (normalized) D H

0 0 0 1 2 3 4 0 1 2 3 4 BM location (cm) BM location (cm)

(a) Displacement transfer function (b) Normalized displacement transfer function

Fig. II-39. Spatial displacement transfer function of AHAAH cochlear model (Price and Kalb, 2005) (blue solid line:

500 Hz, green dashed line: 1000 Hz, red dash-dot line: 2000 Hz , cyan dotted line: 4000 Hz)

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III. Comparative study of human middle ear models based on the frequency response solutions

Numerous human ear models have been proposed, which have different scopes and degrees of simplification. As a part of a comparative study of ear models, several popular human middle ear models are selected and their transfer functions are compared in this chapter. Variations of the transfer function, defined as the ratio of the stapes displacement and the pressure input to the tympanic membrane, when a part of the given model is replaced by that used in different models are examined to roughly estimate accuracy of the models.

A. Middle ear network models

Middle ear is a mechano-acoustic apparatus located between the external ear and the inner ear, which can be viewed as an impedance matching device. Ignoring the bone conduction, sound transmission into the inner ear is determined by the interactions of the ossicular coupling, acoustic coupling and stapes-cochlear input impedance (Merchant et al. , 1997).

The ossicular coupling represents a mechanical conduction through the ossicular chain.

The acoustic coupling is the direct acoustic stimulation through the middle ear cavity to the two cochlear windows. The stapes-cochlear input impedance is the impedance boundary condition applied to the end of the middle ear. Due to its functional significance in the auditory sound transmission, numerous models have been developed by taking different approaches.

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Compared to continuous models (Wada et al. , 1992; Eiber, 1999; Koike et al. , 2002), lumped parameter models are relatively simple and easy to modify for diverse pathological conditions. Network models can be obtained by applying electro-acoustic analogy to the lumped parameter model, which are versatile analysis tools that produce solutions in both the frequency and time domains.

There are several intrinsic limitations in network models such as the applicable range of frequency and sound pressure level (SPL). On account of the underlying one-dimensional

(1-D) concept in network models, although three-dimensional (3-D) model (Hudde and

Weistenhöfer, 1997) exists in a restricted sense, the calculated stapes displacement is interpreted as a translational motion. The 1-D description of the stapes displacement becomes less accurate at higher frequencies as the rocking motion comes into play. The network model of the tympanic membrane (TM) cannot properly represent complex vibration modes observed in the high frequency. For example, the validity of the two-piston model is limited only up to 4-6 kHz (Puria and Allen, 1998; O'Connor and Puria, 2008). Also, linear network models become inaccurate as the input sound pressure increases due to nonlinearities of the middle ear mechanism ignored in the model.

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1. Classic configuration of the middle ear network model

Fig. III-1 shows the schematic of a typical full-ear network model. The two pressure sources

(2P and P ) and the head-related impedance ( Z ) describe the diffraction and radiation effects of the head and concha entrance viewed by the impinging sound waves. The two- port network of the external ear ( Z ) describes the acoustic propagation through the pinna, concha and ear canal represented by a transmission-line. Z represents the middle ear network into which the cochlear input impedance ( Z in Fig. III-2) is integrated as a boundary condition. A detailed analysis of the cochlear response can be conducted by using a separate cochlear model (Z) using the stapes response ( P or U shown in Fig. III-2) calculated from the middle ear model as the input.

While details vary among the models, the middle ear network ( Z ) in Fig. III-1 can be represented as the classic configuration (Peake et al. , 1992), the impedance block model shown in Fig. III-2. The middle ear model is composed of the two-port network of the conductive path (Z ), the middle ear cavity impedance ( Z ), and the stapes-cochlear complex impedance ( Z ). Z describes the function of the tympanic membrane that transforms the incident acoustic pressure ( P ) into the mechanical vibration and the mechanical energy transmission through the ossicular chain up to the incudo-stapedial joint. Z depicts the impedance boundary condition imposed to the middle ear by the cochlear input impedance (Z) and the stapes complex impedance ( Z ) in which the stapes inertia, the annular ligament impedance, vestibular volume effect and the round window compliance are lumped together. The impedance of the middle ear cavity (Z ) is 126

connected in series with the rest of the middle ear network, indicating that the acoustic interaction between the cavity and the cochlear windows is largely ignored in this configuration.

U EE UTM U ST ZHD

Z PEE Z EE PTM ZME PST C 2PSF PSF

U U EE TM U ST

Fig. III-1. Schematic of full ear model.

ZOC ZSC

UTM U ST ZST

Z PTM PTM− P CAV ZTOC PST C0

ZCAV UTM UTM U ST

P CAV

Fig. III-2. Classic configuration of the middle ear network model(Peake et al. , 1992).

127

2. Selected network models for comparison

Human middle ear network models selected for comparison in this section are listed in

Table III-1. In accordance with the classification of the classic configuration, impedance blocks of Z , Z , and Z were identified for each reference model. Model #6 (Price and

Kalb, 2005) has nonlinear parameters for annular ligament compliance and resistance, which were linearized by using their initial values for consistent comparison with other linear models. Model #5 (Pascal et al. , 1998) was also linearized by borrowing the parametric values from model #6 due to the lack of published information on the parameters.

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Reference Authorship Conductive path Stapes -cochlear model network ( ) complex network ( )

Model #1A Zwislocki (1962) Suppressed Implicit stapes transformer complex

Model #1B Zwislocki (2002a) Suppressed Implicit stapes transformer complex

Model #2 Kringlebotn (1988) Three -stage Implicit stapes transformer complex

Model #3 Rosowski and Merchant Three -stage Separate stapes (1995) transformer complex

Model #4 Goode et al. (1994) Three -stage Implicit stapes transformer, Two- complex piston model for TM, Malleus-head axis wobble effect

Model #5 Pascal et al. (1998) Suppressed Separate stapes transformer complex, Nonlinear acoustic reflex and annular ligament model, Helicotrema effect

Model #6 Price and Kalb (20 05) Single -stage Separate stapes transformer, Separate complex, Nonlinear malleo-incudal joint acoustic reflex and annular ligament model, Helicotrema effect

Table III-1. Selected reference network models of the human middle ear

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3. Impedance characteristics of reference network models

Fig. III-3 compares network block impedances of reference models. The overall middle ear

input impedance ( Z ) represented in Fig. III-3(g) and (h) is similar to Z , which implies

that the ossicular conduction is the dominant transmission mode. Comparing Z and Z

plots, the former is heavily influenced by the latter both in magnitude and phase although

the latter has a wider variation. The characteristics of Z do not affect Z except around

the resonance frequency of the cavity, which reassures that the middle ear cavity is a minor

transmission path in a normal middle ear.

5 5 10 10

) 100 ) 100 Ω Ω

0 0

Magnitude Magnitude ( -100 ( Magnitude -100 Phase (degree) Phase (degree) Phase (degree)

0 0 10 10 2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) magnitude (b) phase (c) magnitude (d) phase

5 5 10 10 ) ) 100 100 Ω Ω

0 0 Magnitude ( Magnitude Magnitude Magnitude ( -100 -100 Phase Phase (degree) Phase (degree) Phase (degree) Phase (degree)

0 0 10 10 2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) magnitude (f) phase (g) magnitude (h) phase

Fig. III-3. Impedance of reference models

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B. Middle ear transfer functions of network models

Transfer functions can be obtained from the circuit models either in the time domain or frequency domain. In the time-domain, a sinusoidal pressure wave input (i.e., alternating voltage input) of a specific frequency to the tympanic membrane can be used while sweeping the frequency over the range of interest. In this case, the simulation has to be conducted until the transient response decays out. Alternatively, a unit impulse pressure input can be applied to the tympanic membrane, then apply the Fourier transform to the response. The latter has an advantage in terms of computation time over the former. In the time domain analysis, the time step size to be small enough to ensure the Nyquist frequency is higher than the highest frequency of interest. The frequency-domain approach obtains transfer functions analytically from the circuit.

1. Pressure transfer function

The pressure transfer function (PTF) of the middle-ear is defined as:

P(ω) P (ω) H(ω) ≡ = (III-1) P (ω) P(ω) where P represents the cochlear input pressure approximately equivalent to the pressure acting near the stapes footplate ( P ). As shown in Table III-2, the cochlear input pressure is acquired through voltage measurement across corresponding impedance elements. The

concept of PTF can be equivalently extended to the middle-ear pressure gain G (Puria et al. , 1997) given by:

131

P (ω) G (ω) ≡ (III-2) P (ω) where P is the vestibule pressure near the stapes footplate and P is the ear canal pressure near the tympanic membrane.

PTFs for the network models in Table III-1 are plotted in Fig. III-4. PTFs have the maximum value of 20-30 dB at around 1 kHz. A second peak is seen in model #1-A, model #4, model

#5 and model #6. Positive PTF in 0.1-10 kHz range indicates that pressure is amplified by the middle ear ossicular pathway in this range. Fig. III-5 implies that the phase angle of P leads that of P approximately by 90 degrees in low frequencies while transition to the phase lag occurs in frequencies around 1 kHz.

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Reference model Impedance element over which cochlear input pressure is measured

Model #1A R (Z )

Model #1B R (Z )

Model #2 R (Z )

Model #3 R-L combination (Z)

Model #4 R (Z )

Model #5 R (Z)

Model #6 R (Z)

Table III-2. Cochlear input pressure measurement location in the equivalent networks. or in parenthesis represents the stapes-cochlea complex impedance or cochlear input impedance block over which the measurement conducted.

133

20 20 20 (dB) (dB) (dB)    

TM 0 TM 0 TM 0 /P /P /P ST ST ST P P P     -20 -20 -20 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) (b) (c)

20 20 20 (dB) (dB) (dB)   

TM 0 TM TM

/P 0 0 /P /P ST ST ST P P P    -20 -20 -20 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(d) (e) (f)

20 20 20 (dB) (dB) (dB)    TM 0 TM 0 TM 0 /P /P /P ST ST ST P P P    -20 -20 -20 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(g) (h) (i)

Fig. III-4. Magnitude of PTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model

#5, and (g) Model #6. The shaded area indicates: (h) The range of magnitude of PTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

134

200 200 200

0 0 0

-200 -200 -200 PTF (degree) phase PTF (degree) phase PTF phase (degree) 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) (b) (c)

200 200 200

0 0 0

-200 -200 -200 PTF (degree) phase PTF (degree) phase PTF(degree) phase 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(d) (e) (f)

200 200 200

0 0 0

-200 -200 -200 PTF (degree) phase PTF phase (degree) PTF (degree) phase 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(g) (h) (i)

Fig. III-5. Phase of PTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model #5, and

(g) Model #6. The shaded area indicates: (h) The range of phase of PTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

135

2. Middle-ear transfer admittance

Transfer function defined by the ratio of the stapes volume velocity ( U ) to the eardrum pressure ( P ) in steady-state is called ‘middle-ear transfer admittance’ (Rosowski and

Merchant, 1995) or ‘volume velocity-pressure transfer function (UPTF)’ representing how much stapes volume velocity is generated by a unit sinusoidal pressure input to the middle ear:

U (ω) H (ω) ≡ (III-3) P (ω)

In Fig. III-6, magnitudes of the UPTFs are plotted. Peaks of the transfer functions, at which the most efficient pressure-motion transformation occurs through the human middle-ear, are seen around 1 kHz. The peaks are more conspicuous than those of PTFs. A Second peak appears around 3-5 kHz in model #1-A, model #4, model #5 and model #6. Phase of UPTF shown in Fig. III-7 indicates that U leads the middle-ear pressure input approximately by

90 degrees in low frequencies while starts to lag from below 1 kHz.

136

-4 -4 -4 x 10 x 10 x 10 1 1 1 /dyne-sec)

5 0.5 0.5 0.5 (cm  UP

H 0 0 0 2 4 2 4 2 4  10 10 10 10 10 10 Frequency (Hz)

(a) (b) (c)

-4 -4 -4 x 10 x 10 x 10 1 1 1

0.5 0.5 0.5

0 0 0 2 4 2 4 2 4 10 10 10 10 10 10

(d) (e) (f)

-4 -4 -4 x 10 x 10 x 10 1 1 1

0.5 0.5 0.5

0 0 0 2 4 2 4 2 4 10 10 10 10 10 10

(g) (h) (i)

Fig. III-6. Magnitude of UPTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model

#5, and (g) Model #6. The shaded area indicates: (h) The range of magnitude of UPTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

137

200 200 200

0 0 0

-200 phase (degree) -200 -200 UP H 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz)

(a) (b) (c)

200 200 200

0 0 0

-200 -200 -200 -200

2 4 2 4 2 4 10 10 10 10 10 10

(d) (e) (f)

200 200 200

0 0 0

-200 -200 -200 -200

2 4 2 4 2 4 10 10 10 10 10 10

(g) (h) (i)

Fig. III-7. Phase of UPTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model #5, and (g) Model #6. The shaded area indicates: (h) The range of phase of the UPTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

138

3. Displacement-pressure transfer function

A transfer function derived from H is ‘displacement-pressure transfer function (DPTF)’, defined as the ratio of the stapes displacement ( D ) and the input pressure at the tympanic membrane ( P ) :

D (ω) H (ω) ≡ (III-4) P (ω)

It represents the transduction efficiency of the middle ear transforming acoustic energy into mechanical energy. D can be calculated from the stapes volume velocity ( U ) and the footplate area ( A ) as:

U (ω) D (ω) = (III-5) jωA

Therefore, the transfer function can be rewritten as:

U 1 H (ω) H (ω) = ∙ = (III-6) jωA P jωA

Magnitudes of DPTFs for reference middle-ear network models are plotted in Fig. III-8. The peaks shown in Fig. III-6 are significantly suppressed or disappeared by high frequency effect represented in Eq. (III-6). With reference to Fig. III-10 where magnitude of DPTFs is represented in logarithmic scale, the magnitude of the transfer function is nearly constant up to about 1 kHz thus the stapes velocity will increase linearly in this range with

6dB/octave slope in terms of sound energy. Beyond 1 kHz, DPTFs decrease approximately by 11dB/octave, which corresponds to 17 dB/octave decrease of the velocity.

139

The phase of DPTF reveals that D is nearly in-phase with P in low frequencies but lags behind P in high frequencies. This suggests that the translational displacement of the stapes regenerates the temporal profile of P fairly well in the low frequency region but not in high frequencies due to the increased rocking motion of the stapes and delay in the ossicular transmission. Near-zero phases in the low frequency region also indicate that the middle ear transmits the acoustic energy so that the inner ear receives the input signal applied to the tympanic membrane with little temporal distortion.

140

-7 -7 -7 x 10 x 10 x 10 6 6 6

/dyne) 4 4 4 3

(cm 2 2 2  DP H

 0 0 0 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz)

(a) (b) (c)

-7 -7 -7 x 10 x 10 x 10 6 6 6

4 4 4

2 2 2

0 0 0 2 4 2 4 2 4 10 10 10 10 10 10

(d) (e) (f)

-7 -7 -7 x 10 x 10 x 10 6 6 6

4 4 4

2 2 2

0 0 0 2 4 2 4 2 4 10 10 10 10 10 10

(g) (h) (i)

Fig. III-8. Magnitude of DPTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model

#5, and (g) Model #6. The shaded area indicates: (h) The range of magnitude of DPTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

141

200 200 200

0 0 0

phase (degree) -200 -200 -200 DP H 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz)

(a) (b) (c)

200 200 200

0 0 0

-200 -200 -200

2 4 2 4 2 4 10 10 10 10 10 10

(d) (e) (f)

200 200 200

0 0 0

-200 -200 -200

2 4 2 4 2 4 10 10 10 10 10 10

(g) (h) (i)

Fig. III-9. Phase of DPTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model #5, and (g) Model #6. The shaded area indicates: (h) The range of phase of DPTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

142

-6 10

-8 10 /dyne) 3 (cm

 -10

DP 10 H 

-12 10 2 3 4 10 10 10 Frequency (Hz)

Fig. III-10. DPTF magnitude of middle ear network models. The dashed thick line represents the mean DPTF and dotted lines are DPTFs of reference models.

143

4. Volume velocity transfer function

Volume velocity ratio of the stapes to the eardrum, volume velocity transfer function (UTF), provides another measure of the human middle-ear transmission characteristic:

U (ω) H(ω) ≡ (III-7) U (ω)

Shown in Fig. III-11, smooth peaks of H magnitude appear around 1 kHz in several models but relatively flat in low frequencies. 30-40 dB reduction in the volume velocity which is much higher than that by the middle-ear ideal transformer represents additional reduction by the ossicular joint and cochlear input boundary. The reduction ratio sharply increases above 1 kHz, suggesting that the stapes movement is heavily suppressed in higher frequencies. Shown in Fig. III-12, near-zero phase angles in the low frequency region indicate that the stapes and tympanic membrane have motions of the same phase. Phase dips occur in the 1-4 kHz region while phase peaks are not so prominent. Although the phase is negative (i.e., stapes movement lagging behind the tympanic membrane motion) in high frequencies, model-wise variations are very large.

144

-20 -20 -20

-40 -40 -40 (dB)  U

H -60 -60 -60 

-80 -80 -80 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz)

(a) (b) (c)

-20 -20 -20

-40 -40 -40

-60 -60 -60

-80 -80 -80 2 4 2 4 2 4 10 10 10 10 10 10

(d) (e) (f)

-20 -20 -20

-40 -40 -40

-60 -60 -60

-80 -80 -80 2 4 2 4 2 4 10 10 10 10 10 10

(g) (h) (i)

Fig. III-11. Magnitude of UTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model

#5, and (g) Model #6. The shaded area indicates: (h) The range of magnitude of UTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

145

200 200 200

0 0 0

-200 -200 -200 (degree) phase U H 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz)

(a) (b) (c)

200 200 200

0 0 0

-200 -200 -200

2 4 2 4 2 4 10 10 10 10 10 10

(d) (e) (f)

200 200 200

0 0 0

-200 -200 -200

2 4 2 4 2 4 10 10 10 10 10 10

(g) (h) (i)

Fig. III-12. Phase of UTF. (a) Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model #5, and (g) Model #6. The shaded area indicates: (h) The range of phase of UTFs about the average shown as the dashed line, and (i) One standard deviation in either direction from the average value shown as the dashed line.

146

C. Comparison of frequency-domain stapedial responses of middle ear network models (Song and Kim, 2008)

Variations in the middle ear models arise from differences in degree of simplifications and system parameters. A numerical study was conducted to understand the effect of these variations. For each model shown in Table III-1, DPTFs were calculated from the original model used as the reference model, and from the six variations of the model constructed by replacing Z , Z or Z of the reference model by the one used by the other models.

Fig. III-13 shows the effect of variations of Z on DPTF. For example, Fig. III-13(a) compares DPTF of model #1A with DPTFs of varied models constructed by replacing the cavity impedance block of model #1A (Z, ) with one of those of the other six models #

( , , , , or ). It is seen that the effect is limited only Z,# Z,# Z,# Z,# Z,# Z,# in the mid-frequency range close to the resonance frequency of the middle ear cavity. This supports the classic assumption that the middle ear cavity plays a minor role in the middle ear transmission. Fig. III-14 and Fig. III-15 show the comparison in terms of the ratio of , D where the variation ranges between -6 and 10 dB in magnitude, and -65 and 50 degrees in phase.

147

-6 -6 -6 -6 10 10 10 10 /dyne)

3 -8 -8 -8 -8 10 10 10 10 (cm 

-10 M -10 -10 -10 T TM 10 10 10 10 / P T S ST

D -12 -12 -12 -12

 10 10 10 10 2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

-6 -6 -6 10 10 10 /dyne)

3 -8 -8 -8 10 10 10 (cm 

-10 -10 M -10 T TM 10 10 10 / P T S ST

D -12 -12 -12

 10 10 10 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-13. DPTF variation by replacing of reference models. The thick dotted line represents the transfer function of the reference model and thin dotted lines are those of replaced models. Caption under each plot indicate the reference model.

148

20 20 20 20 (dB) 

0 0 0 0 ST(ref) /D ST

D -20 -20 -20 -20 

2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

20 20 20 (dB) 

0 0 0 ST(ref) /D ST

D -20 -20 -20 

2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-14. Ratio of with regard to reference model prediction by replacing . Caption under each plot ( ) indicate the reference model. Thick solid line represents the mean value, and the thin dotted lines are ratios with respect to the reference model.

149

100 100 100 100 (degree) 0 0 0 0 ST(ref)

/D -100 -100 -100 -100 ST D

2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

100 100 100 (degree) 0 0 0 ST(ref)

/D -100 -100 -100 ST D

2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-15. Phase difference of with regard to reference model prediction by replacing . Caption ( ) under each plot indicate the reference model. Thick solid line represents the mean value, and the thin dotted lines are ratios with respect to the reference model.

Fig. III-16 shows the effect of replacement on DPTF. Since the middle-ear transformer is Z located explicitly or implicitly in front of and each reference model has a different Z transformer ratio, parametric values of of the other models were adjusted properly by Z utilizing the ratio of the reference model before the replacements. Variation of Z influences DPTF more significantly than variation of does, over a wider frequency Z range. Fig. III-17 and Fig. III-18 show that the ratio of varies between -17 and 17 dB in D magnitude, and -90 and 90 degrees in phase. However, the pattern of the variation is hard 150

to be characterized, probably because the cochlear input impedance ( ) that dampens the Z stapes motion is over-simplified.

-6 -6 -6 -6 10 10 10 10 /dyne)

3 -8 -8 -8 -8 10 10 10 10 (cm  -10 -10 -10 -10 TM 10 10 10 10 / P ST

D -12 -12 -12 -12

 10 10 10 10 2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

-6 -6 -6 10 10 10 /dyne)

3 -8 -8 -8 10 10 10 (cm  -10 -10 -10 TM 10 10 10 / P ST

D -12 -12 -12

 10 10 10 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-16. DPTF variation by replacing of reference models. The thick dotted line represents the transfer function of the reference model and thin dotted lines are those of replaced models. Caption under each plot indicate the reference model.

151

20 20 20 20 (dB)  ) f e r 0 0 0 ( 0 T ST(ref) S /D T ST S

D -20 -20 -20 -20 

2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

20 20 20 (dB) 

0 0 0 ST(ref) /D ST

D -20 -20 -20 

2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-17. Ratio of with regard to reference model prediction by replacing . Caption under each plot ( ) indicate the reference model. Thick solid line represents the mean value, and the thin dotted lines are ratios with respect to the reference model.

152

100 100 100 100 (degree) 0 0 0 0 ST(ref)

/D -100 -100 -100 -100 ST D

2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

100 100 100 (degree) 0 0 0 ST(ref)

/D -100 -100 -100 ST D

2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-18. Phase difference of with regard to reference model prediction by replacing . Caption under ( ) each plot indicate the reference model. Thick solid line represents the mean value, and the thin dotted lines are ratios with respect to the reference model.

Fig. III-19 through Fig. III-21 show how the replacements of affect the stapes Z responses. Notable changes in DPTF occur in the mid and high frequency ranges as shown

in Fig. III-19. The same trend is observed both in the magnitude and phase ratio as seen in

Fig. III-20 and Fig. III-21. The insensitivity of the stapes response to replacement in Z the low frequency region indicates that the network blocks of the models were designed to

have practically same characteristics agreeing well with the common observations in the

frequency region; the tympanic membrane moves in a single piston mode(Decraemer et al. ,

153

1989; Wada et al. , 2002) and the ossicular chain has little relative motion without much

change in the rotational axis(Decraemer et al. , 1991; Decraemer et al. , 1994; Willi et al. ,

2002; Nakajima et al. , 2005). The larger variations of the response in the mid and high

frequency ranges can be attributed to the differences in the tympanic membrane network

topologies and details of the ossicular chain networks.

-6 -6 -6 -6 10 10 10 10 /dyne)

3 -8 -8 -8 -8 10 10 10 10 (cm  -10 -10 -10 -10 TM 10 10 10 10 / P ST

D -12 -12 -12 -12

 10 10 10 10 2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

-6 -6 -6 10 10 10 /dyne)

3 -8 -8 -8 10 10 10 (cm 

-10 M -10 -10 TM 10 T 10 10 / P T ST S

D -12 -12 -12

 10 10 10 2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-19. DPTF variation by replacing of reference models. The thick dotted line represents the transfer function of the reference model and thin dotted lines are those of replaced models. Caption under each plot indicate the reference model.

154

20 20 20 20 (dB) 

0 0 0 0 ST(ref) /D ST

D -20 -20 -20 -20 

2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

20 20 20 (dB) 

0 0 0 ST(ref) /D ST

D -20 -20 -20 

2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-20. Ratio of with regard to reference model prediction by replacing . Caption under each plot ( ) indicate the reference model. Thick solid line represents the mean value, and the thin dotted lines are ratios with respect to the reference model.

155

100 100 100 100 (degree) ) f

0 0 0 e 0 r ( T ST(ref) S

/D -100 -100 -100 -100 T ST S D

2 4 2 4 2 4 2 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

(a) Model #1A (b) Model #1B (c) Model #2 (d) Model #3

100 100 100 (degree) 0 0 0 ST(ref)

/D -100 -100 -100 ST D

2 4 2 4 2 4 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) Frequency (Hz)

(e) Model #4 (f) Model #5 (g) Model #6

Fig. III-21. Phase difference of with regard to reference model prediction by replacing . Caption under ( ) each plot indicate the reference model. Thick solid line represents the mean value, and the thin dotted lines are ratios with respect to the reference model.

156

IV. Comparison of human middle ear models based on their temporal responses to impulsive waveforms

Auditory responses can be obtained from network models either in the time or frequency domain. Most network models adopting acousto-electric analogy have been developed based on steady-state experimental data; therefore responses obtained from the network models have more often been interpreted in the frequency domain. However, time domain analysis is more convenient to study responses to impulsive stimulus (i.e., short duration signals such as sound from gunshot, airbag explosion, high-power pneumatic tools, etc.).

Time-domain analysis of the middle ear responses to the impulsive sounds has not been performed in a systematic manner to date. There have been previous studies (Eiber, 1999;

Price, 2007a; 2007b) using network models to obtain middle ear responses to the noises generated by rifles, cannons, or shockwave generators. By using real wave forms, which is difficult to be characterized in a few parameters, the studies could not yield useful information. In this chapter, simplified waveforms described by only a few parameters are used for a more effective parametric study.

157

A. Methodology

The network models listed in Table III-1 are used for this comparison study. The models are regenerated in a graphical language of MATLAB (Simulink ®, 2007a) to obtain temporal responses. Two nonlinear models, model #5 (Pascal et al. , 1998) and model #6 (Price and

Kalb, 2005), are linearized to be compared with other linear models.

Reference m odel Authorship Model type

Model #1A Zwislocki (1962) Linear

Model #1B Zwislocki (2002a) Linear

Model #2 Kringlebotn (1988) Linear

Model #3 Rosowski and Merchant (1995) Linear

Model #4 Goode et al. (1994) Linear

Model #5 * Pascal et al. (1998) Nonlinear

Model #6** Price and Kalb (2005) Nonlinear

Table IV-1. Selected network models of the human middle ear. *Linearized by borrowing the parametric values from model #6 **Linearized by using initial values of the annular ligament compliance and resistance.

Two types of simplified pressure waveforms shown in Fig. IV-1 (a) and (b) are used for the parameter study. Type 1 impulse is an idealized impulse with a very short rise time ( ) and A-duration ( ). Type 2 waveform is the same as type 1 impulse except it is followed by a depression. The rise time and A-duration of the waveforms shown in Fig. IV-1 (c) are selected as the parameters to be controlled. The effect of the parameters on the stapes 158

displacement and the cochlear input pressure are investigated.

Ppk P 1000 1000

500 500

0 0 t t0 1 Sound PressureSound (Pa) -500 -500 t 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Tr Time (sec.) Time (sec.) T A

(a) Type 1 (b) Type 2 (c) Impulsive noise parameters

Fig. IV-1. Types of impulsive waveform and definition of impulsive noise parameters (Chan et al. , 2001)

B. Comparison of temporal responses of human middle ear models

1. Unit impulse responses of human middle ear models

Prior to the parametric study, the unit impulse response of various middle ear models are examined. As a testing stimulus, a square wave of 1 Pa with duration of 1/50000 second is applied to the tympanic membrane. The stapes displacement and cochlear input pressure for 0.01 second are plotted in Fig. IV-2 and Fig. IV-3. The peak of the pressure response occurs almost at the same time as the peak of the impulse input while the peak of the stapes displacement appears with some time-delay. Undershooting responses are seen in both responses except the stapes displacement of the model #4 in Fig. IV-3(e). It seems to be due to the relatively high damping in the cochlea ( ) used in the model. In Fig. IV-4, 159

variation of the impulse response due to the variation of the cochlear input resistance are plotted. It is seen that a higher resistance causes a smaller stapes displacement and less oscillations in the response. Frequency domain analysis shows that the middle ear has a band-pass filter property for the cochlear input pressure in the mid-frequency range and a low-pass filter property for the stapes displacement.

Nonlinear effects on temporal response for the impulse are shown in Fig. IV-5. As explained previously, AHAAH model (i.e., model #6) provides two options, warned and unwarned simulation, to apply non-linearity to the middle-ear. In the unwarned simulation, the middle-ear acoustic reflex is not activated before the outburst of an acoustic stimulus; thus a time-delay precedes the activation of the reflex. Therefore, if the duration of an acoustic stimulus is short enough, the middle-ear protection mechanism is not activated and the temporal response will be similar to that of the linear model. Comparing Fig. IV-2(g) and

Fig. IV-3(g) with Fig. IV-5(a) and (c), the difference is hardly detected. However, in the warned case simulation in which the acoustic reflex is assumed to be pre-triggered, impulse responses are much smaller as shown in Fig. IV-5(b) and (d).

160

3 2.5 0.6

2.5 2 0.4 2 1.5

1.5 0.2 1 [Pa] [Pa] [Pa] 1 C0 C0 0.5 C0 P P 0.5 P 0 0 0 -0.2 -0.5 -0.5

-1 -1 -0.4 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec] Time [sec] (a) (b) (c)

2 1 4

0.8 1.5 3 0.6

1 2 0.4 [Pa] [Pa] [Pa]

C0 C0 0.2 C0 P 0.5 P P 1

0 0 0 -0.2

-0.5 -0.4 -1 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec] Time [sec] (d) (e) (f)

1.5

0.5 [Pa] C0 P 0

-0.5

-1 0 0.002 0.004 0.006 0.008 0.01 Time [sec] (g) Fig. IV-2. Response of the cochlear input pressure when 1 Pa acoustic impulse with 1/50000 second duration is applied at the tympanic membrane. (a)

Model #1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model #5, and (g) Model #6. Model #5 and model #6 are linearized as previous.

161

-3 -3 -4 x 10 x 10 x 10 3 2.5 15

2.5 2 10 2 1.5 1.5 m] m] m] µ µ µ [

[ 1

[ 5 ST ST 1 ST D D D 0.5 0.5 0 0 0

-0.5 -0.5 -5 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec] Time [sec] (a) (b) (c)

-4 -4 -4 x 10 x 10 x 10 15 6 15

5 10 10 4 m] m] m] µ µ µ

[ 3 [ 5

[ 5 ST ST ST D D D 2 0 0 1

-5 0 -5 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec] Time [sec] (d) (e) (f)

-4 x 10 12

6 m] µ [

ST 4 D 2

-2 0 0.002 0.004 0.006 0.008 0.01 Time [sec] (g) Fig. IV-3. Response of the stapes displacement when 1 Pa acoustic impulse with 1/50000 second duration is applied at the tympanic membrane. (a) Model

#1-A, (b) Model #1-B, (c) Model #2, (d) Model #3, (e) Model #4, (f) Model #5, and (g) Model #6. Model #5 and model #6 are linearized as previous.

162

-3 -4 x 10 x 10 1.5 8

1 6

0.5 4 m] m] µ µ [ [ ST ST D 0 D 2

-0.5 0

-1 -2 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec]

(a) (b)

-4 -4 x 10 x 10 6

4 2 m] m] µ µ

[ 3 [ ST ST D D 2 1

0 0 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec]

Fig. IV-4. Cochlear input resistance effects on the stapes displacement in the model #4. (a) , (b) , (c) , and (d) . × . × ×

163

1.5 0.4

0.3 1 0.2

0.5 0.1 [Pa] [Pa] C0 C0 0 P 0 P

-0.1 -0.5 -0.2

-1 -0.3 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec]

(a) (b)

-4 -4 x 10 x 10 12 2.5

10 2

8 1.5

6 m] m] µ µ

[ 1 [ ST ST 4 D D 0.5 2

0 0

-2 -0.5 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec]

Fig. IV-5. Nonlinear effects on impulse response when 1 Pa acoustic impulse with 1/50000 second duration is applied at the tympanic membrane. (a) Unwarned case cochlear input pressure, (b) Warned case cochlear input pressure, (c) Unwarned case stapes displacement, and (d) Warned case stapes displacement.

164

2. Parametric study on the responses of network middle ear models to impulsive inputs

The peak pressures ( ) are set to 1000 Pa (i.e., 154 dB) in all cases while the A-duration

() and rise time () of input impulses are varied:

(1) With the rise time fixed at 0.0001 second, the A-duration changes from 0.001 to 0.005 second by 0.001 second increment as in Fig. IV-6 (a) and (b).

(2) With the A-duration fixed at 0.001 second, the rise time changes from 0.0001 to 0.0005 second by 0.0001 second increment as in Fig. IV-6 (c) and (d).

The maximum translational displacement of the stapes ( ) and the highest cochlear input pressure ( ) toward the cochlea side and middle ear side are compared for the selected models for type I and type II of impulsive inputs.

165

1000 1000

500 500 [Pa] [Pa] [Pa] in in P P P 0 0

-500 -500 0 0.002 0.004 0.006 0.008 0.01 0 0.01 0.02 0.03 0.04 0.05 Time [sec] Time [sec]

(a) Type 1 impulses with A-duration change (b) Type 2 impulses with A-duration change

1000 1000

500 500 [Pa] [Pa] in in P P 0 0

-500 -500 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 Time [sec] Time [sec]

Fig. IV-6. Impulses used in parametric study.

166

a) Influence of A-duration change on the stapes displacement

The maximum displacement of the stapes toward the cochlear side increases as the A- duration increases as shown in Fig. IV-7 (a) and (c). Model #1A shows the largest stapedial displacement toward the cochlea. Model#4 estimates the smallest displacement, showing approximately 30 μm difference from model #1A. The maximum displacement of the stapes toward the middle ear side, shown in Fig. IV-7 (b) and (d), slightly decreases as the A- duration increases except the cases of type 2 exposure to model #1A and model #4.

In case of type 1 stimulus which has no pressure depression, small displacements toward the middle ear are detected in model #2, model #3 and model #6 for all cases. Model #1A and model #5 show the middle ear side displacement only for the A-duration of 0.001 second. Model#1B and model#4 suppress the backward displacement efficiently.

167

60 14 50 12 Model #1A Model #1A 40 10 Model #1B Model #1B 8 30 Model #2 Model #2 6 20 Model #3 Model #3 Model #4 4 Model #4 Stapes displacement (μm) 10 Stapes displacement (μm) Model #5 2 Model #5 0 Model #6 0 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.)

(a) Type 1: toward the cochlea (b) Type 1: toward the middle ear

60 14 50 12 Model #1A Model #1A 40 10 Model #1B Model #1B 8 30 Model #2 Model #2 6 20 Model #3 Model #3 Model #4 4 Model #4

Stapes displacement (μm) 10 Stapes displacement (μm) Model #5 2 Model #5 0 Model #6 0 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.)

Fig. IV-7. Maximum translational displacement of the stapes by A-duration changes

168

b) Influence of A-duration on the cochlear input pressure

As shown in Fig. IV-8 (a) and (c), the maximum forward input pressure slightly increases as the A-duration increases and becomes saturated above the A-duration of 0.003 second.

Model #4 and model #5 respectively show the highest and the lowest cochlear input pressure, the highest pressure being more than two times of the lowest. The order of the pressures from all models is almost identical at all values of the A-duration.

Shown in Fig. IV-8 (b) and (d), the maximum backward pressure decreases as the duration increases. Model #5 estimates the pressures to the lowest level.

169

1.60E+04 1.60E+04 1.40E+04 1.40E+04

1.20E+04 Model #1A 1.20E+04 Model #1A 1.00E+04 Model #1B 1.00E+04 Model #1B 8.00E+03 Model #2 8.00E+03 Model #2 6.00E+03 Model #3 6.00E+03 Model #3 4.00E+03 Model #4 4.00E+03 Model #4

Cochlear Cochlear input pressure (Pa) 2.00E+03 Model #5 Cochlear input pressure (Pa) 2.00E+03 Model #5 0.00E+00 Model #6 0.00E+00 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.)

(a) Type 1: toward the cochlea (b) Type 1: toward the middle ear

1.60E+04 1.60E+04 1.40E+04 1.40E+04 1.20E+04 1.20E+04 Model #1A Model #1A 1.00E+04 Model #1B 1.00E+04 Model #1B 8.00E+03 Model #2 8.00E+03 Model #2 6.00E+03 Model #3 6.00E+03 Model #3 4.00E+03 Model #4 4.00E+03 Model #4

Cochlear input pressure (Pa) 2.00E+03 Model #5 Cochlear input pressure (Pa) 2.00E+03 Model #5 0.00E+00 Model #6 0.00E+00 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.)

Fig. IV-8. Maximum cochlear input pressure by A-duration ( ) changes.

170

c) Influence of rise time on the stapes displacement

Fig. IV-9 (a) and (c) show that the maximum forward displacement of the stapes increases slightly as the rise time increases for all models with almost the same rate. Backward stapedial displacement shown in Fig. IV-9 (b) increases as the rise time increases except model #1B and model #4 that do not give any backward displacements.

60 14 50 12 Model #1A Model #1A 40 10 Model #1B Model #1B 8 30 Model #2 Model #2 6 20 Model #3 Model #3 Model #4 4 Model #4

Stapes displacement (μm) 10 Stapes displacement (μm) Model #5 2 Model #5 0 Model #6 0 Model #6 0.0001 0.0002 0.0003 0.0004 0.0005 0.0001 0.0002 0.0003 0.0004 0.0005

Rising time (sec.) Rising time (sec.)

(a) Type 1: toward the cochlea (b) Type 1: toward the middle ear

60 14 50 12 Model #1A Model #1A 40 10 Model #1B Model #1B 8 30 Model #2 Model #2 6 20 Model #3 Model #3 Model #4 4 Model #4 Stapes displacement (μm) 10 Stapes displacement (μm) Model #5 2 Model #5 0 Model #6 0 Model #6 0.0001 0.0002 0.0003 0.0004 0.0005 0.0001 0.0002 0.0003 0.0004 0.0005

Rising time (sec.) Rising time (sec.)

Fig. IV-9. Maximum translational displacement of the stapes by rise time () changes

171

d) Influence of rise time on the cochlear input pressure

Fig. IV-10 shows the influence of the rise time on the maximum cochlear input pressure. As rise time increases, the maximum forward input pressure to the cochlea increases. Partial deviation from the trend is found in the cases of model #1A and model #4 subject to type 2 inputs. Model #1A and model #5 give the highest and the lowest forward pressure estimated among the network models. The maximum backward input pressure generally increases as the rise time increases, which is opposite to the effect of the A-duration. Model

#5 predicts the lowest backward pressure for both types of input.

172

1.60E+04 1.60E+04 1.40E+04 1.40E+04

1.20E+04 Model #1A 1.20E+04 Model #1A 1.00E+04 Model #1B 1.00E+04 Model #1B 8.00E+03 Model #2 8.00E+03 Model #2 6.00E+03 Model #3 6.00E+03 Model #3 4.00E+03 Model #4 4.00E+03 Model #4

Cochlear Cochlear input pressure (Pa) 2.00E+03 Model #5 Cochlear input pressure (Pa) 2.00E+03 Model #5 0.00E+00 Model #6 0.00E+00 Model #6 0.0001 0.0002 0.0003 0.0004 0.0005 0.0001 0.0002 0.0003 0.0004 0.0005

Rising time (sec.) Rising time (sec.)

(a) Type 1: toward the cochlea (b) Type 1: toward the middle ear

Cochlear input pressure (Pa) 2.00E+03 Model #5 Cochlear input pressure (Pa) 2.00E+03 Model #5 0.00E+00 Model #6 0.00E+00 Model #6 0.0001 0.0002 0.0003 0.0004 0.0005 0.0001 0.0002 0.0003 0.0004 0.0005

Rising time (sec.) Rising time (sec.)

Fig. IV-10. Maximum cochlear input pressure by rise time () changes.

173

e) Influence of impulse parameters on the time delay of the middle ear transmission

The time difference between the peak pressure and the maximum forward stapes displacement defines the time delay of the middle ear transmission. The delays by A- duration change shown in Fig. IV-11 are found to be about 0.5-ms for both types of stimulus, which means that the stapes reaches maximum displacement approximately 0.5-ms after the pressure peak. Model #4 shows the largest delay among the network models and also shows a proportional increase to A-duration. Model #5 produces the smallest middle ear delay.

As shown in Fig. IV-12, increase of the rise time in input pressure wave has the effect of reducing the delay of the middle ear transmission. Type 2 impulses incur slightly lower delays than type 1 impulses.

174

0.0020 0.0020

0.0015 Model #1A 0.0015 Model #1A Model #1B Model #1B 0.0010 Model #2 0.0010 Model #2 Model #3 Model #3 Time delay (sec.) 0.0005 Model #4 Time delay (sec.) 0.0005 Model #4 Model #5 Model #5 0.0000 Model #6 0.0000 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.)

(a) Type 1 (b) Type 2

Fig. IV-11. Time delay between two peaks of the input pressure and the stapes displacement by A-duration ( ) changes.

0.0010 0.0010

0.0008 0.0008 Model #1A Model #1A 0.0006 Model #1B 0.0006 Model #1B Model #2 Model #2 0.0004 0.0004 Model #3 Model #3 Time delay (sec.) Model #4 Time delay (sec.) Model #4 0.0002 0.0002 Model #5 Model #5 0.0000 Model #6 0.0000 Model #6 0.0001 0.0002 0.0003 0.0004 0.0005 0.0001 0.0002 0.0003 0.0004 0.0005

Rising time (sec.) Rising time (sec.)

(a) Type 1 (b) Type 2

Fig. IV-12. Time delay between two peaks of the input pressure and the stapes displacement by rise time () changes.

175

f) Influence of impulse parameters on time delay of pressure wave transmission

The time difference between the peak pressure and the maximum forward cochlear input pressure defines the time delay of the pressure waves in the middle ear. Shown in Fig. IV-13, the delay is almost constant for all network models when A-duration of input pressure is above 2-ms. Model #5 has no time delay in pressure transmission and the other models have delays less than 0.3-ms.

Fig. IV-14 shows the influence of the rise time upon the delay of a pressure wave in the middle ear, indicating that the increase of the rise time decreases the time delay. For both types of impulses, model #5 and model #6 show negative delay (i.e., the cochlear input pressure wave reaches the peak faster than the middle ear input pressure wave) when the rise time is 0.4- and 0.5-ms.

176

0.0005 0.0005

0.0004 0.0004 Model #1A Model #1A 0.0003 Model #1B 0.0003 Model #1B Model #2 Model #2 0.0002 0.0002 Model #3 Model #3 Time delay (sec.) Model #4 Time delay (sec.) Model #4 0.0001 0.0001 Model #5 Model #5 0.0000 Model #6 0.0000 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.)

(a) Type 1 (b) Type 2

Fig. IV-13. Time delay between two peaks of the input pressure and the cochlear input pressure by A-duration

() changes.

0.0005 0.0005

0.0003 0.0003 Model #1A Model #1A 0.0001 Model #1B 0.0001 Model #1B Model #2 Model #2 -0.0001 -0.0001 Model #3 Model #3 Time delay Time delay (sec.) Model #4 Time delay (sec.) Model #4 -0.0003 -0.0003 Model #5 Model #5 -0.0005 Model #6 -0.0005 Model #6 0.0001 0.0002 0.0003 0.0004 0.0005 0.0001 0.0002 0.0003 0.0004 0.0005

Rising time (sec.) Rising time (sec.)

(a) Type 1 (b) Type 2

Fig. IV-14. Time delay between two peaks of the input pressure and the cochlear input pressure by rise time () changes.

177

C. Effect of impulse parameters on the inner ear responses

Parameters of impulsive noise affect the cochlea responses. Parametric study to understand the effect of impulse parameters on the basilar membrane (BM) motion is also conducted.

A-duration effect of only type 1 impulse is considered.

In order to extract time history of the BM motion, the cochlear model of AHAAH program

(Price and Kalb, 2005) was combined with the middle ear network models. Time-domain responses of the stapes obtained from the network model simulations are directly applied to the cochlear model. The cochlear model is slightly modified to pick up vibration signals at six BM locations corresponding to 0.5, 1, 2, 4, 8 and 16 kHz.

Distribution of the maximum displacement of the BM caused by type 1 impulse is shown in

Fig. IV-15 and Fig. IV-16. The upward displacement has a peak at 1 kHz for A-duration of 1- ms but suppressed as A-duration increases. The peak of downward displacement is found at 1- or 2-kHz location and kept nearly constant without regard to the variation of A- duration. Model #4, which has the smallest stapes displacement as in Fig. IV-7 and Fig. IV-9, results in the smallest BM displacement in both directions. For frequency region less than 4 kHz, model #1A predicts the largest downward displacement caused by the largest forward movement of the stapes. Network models producing the largest upward displacement are respectively model #1A at 0.5 kHz and model #2 at the others.

Frequency-wise variation of BM displacement with regard to A-duration is shown in Fig. 178

IV-17 and Fig. IV-18, respectively for upward and downward displacement. For all frequencies, A-duration exerts the opposite effects on upward and downward movement of the cochlea partition. A-duration up to around 3-ms boosts the maximum downward displacement, especially at low frequency locations. As A-duration increases, the maximum upward movement of the BM decreases. Significant suppression of the BM displacement occurs at 1- and 2-kHz locations.

179

16 16 16 14 14 14 12 Model #1A 12 Model #1A 12 Model #1A 10 Model #1B 10 Model #1B 10 Model #1B 8 Model #2 8 Model #2 8 Model #2 6 Model #3 6 Model #3 6 Model #3 4 Model #4 4 Model #4 BM displacement BM (μm) BM displacement BM (μm) 4 Model #4 2 Model #5 2 Model #5 displacement BM (μm) 2 Model #5 0 Model #6 0 Model #6 0 Model #6 0.5 kHz 1.0 kHz 2.0 kHz 4.0 kHz 8.0 kHz 16.0 kHz 0.5 kHz 1.0 kHz 2.0 kHz 4.0 kHz 8.0 kHz 16.0 kHz 0.5 kHz 1.0 kHz 2.0 kHz 4.0 kHz 8.0 kHz 16.0 kHz Frequency Frequency Frequency

(a) A-duration of 0.001 second. (b) A-duration of 0.002 second. (c) A-duration of 0.003 second.

16 16 14 14

12 Model #1A 12 Model #1A 10 Model #1B 10 Model #1B 8 Model #2 8 Model #2 6 Model #3 6 Model #3 4 Model #4 4 Model #4 BM displacement BM (μm) displacement BM (μm) 2 Model #5 2 Model #5 0 Model #6 0 Model #6 0.5 kHz 1.0 kHz 2.0 kHz 4.0 kHz 8.0 kHz 16.0 kHz 0.5 kHz 1.0 kHz 2.0 kHz 4.0 kHz 8.0 kHz 16.0 kHz

Frequency Frequency

(d) A-duration of 0.004 second. (e) A-duration of 0.005 second.

Fig. IV-15. Maximum upward BM displacement by the exposure to type 1 impulse.

180

(a) A-duration of 0.001 second. (b) A-duration of 0.002 second. (c) A-duration of 0.003 second.

16 16 14 14

Frequency Frequency

(d) A-duration of 0.004 second. (e) A-duration of 0.005 second.

Fig. IV-16. Maximum downward BM displacement by the exposure to type 1 impulse.

181

16 16 16 14 14 14

12 Model #1A 12 Model #1A 12 Model #1A 10 Model #1B 10 Model #1B 10 Model #1B 8 Model #2 8 Model #2 8 Model #2 6 Model #3 6 Model #3 6 Model #3 4 Model #4 4 Model #4 4 Model #4 2 Model #5 2 Model #5 2 Model #5 BM displacement BM (μm): 1.0kHz displacement BM (μm): 2.0kHz BM displacement BM (μm): 0.5kHz 0 Model #6 0 Model #6 0 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.) A-duration (sec.)

(a) 0.5 kHz (b) 1 kHz (c) 2 kHz

16 16 16 14 14 14

12 Model #1A 12 Model #1A 12 Model #1A 10 Model #1B 10 Model #1B 10 Model #1B 8 Model #2 8 Model #2 8 Model #2 6 Model #3 6 Model #3 6 Model #3 4 Model #4 4 Model #4 4 Model #4 2 Model #5 2 Model #5 2 Model #5 BM displacement BM (μm): 4.0kHz displacement BM (μm): 8.0kHz BM displacementBM (μm): 16.0kHz 0 Model #6 0 Model #6 0 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005

A-duration (sec.) A-duration (sec.) A-duration (sec.)

(d) 4 kHz (e) 8 kHz (f) 16 kHz

Fig. IV-17. Maximum upward displacement of the BM at 6 frequency points.

182

16 16 16 14 14 14

A-duration (sec.) A-duration (sec.) A-duration (sec.)

(a) 0.5 kHz (b) 1 kHz (c) 2 kHz

16 16 16 14 14 14 12 Model #1A 12 Model #1A 12 Model #1A 10 Model #1B 10 Model #1B 10 Model #1B 8 Model #2 8 Model #2 8 Model #2 6 Model #3 6 Model #3 6 Model #3 4 Model #4 4 Model #4 4 Model #4 2 Model #5 BM displacement BM (μm): 4.0kHz 2 Model #5 2 Model #5 BM displacement BM (μm): 8.0kHz 0 Model #6 displacement BM (μm): 16.0kHz 0 Model #6 0 Model #6 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005 0.001 0.002 0.003 0.004 0.005 A-duration (sec.) A-duration (sec.) A-duration (sec.)

(d) 4 kHz (e) 8 kHz (f) 16 kHz

Fig. IV-18. Maximum downward displacement of the BM at 6 frequency points.

183

V. Time-domain stapes responses obtained from reconstructed transfer functions

The mathematical model of the auditory system has limited validity in many aspects. These limitations are due to the ‘black-box design concept’ in which parametric values are conditionally adjusted to acoustic measurements, such as volume velocity, pressure or impedance, in conjunction with anatomical or biological considerations. Therefore, in constructing a mathematical model where measured transfer functions are implemented as objective functions or constraint conditions, the curve-fitting and/or optimization processes are essential. However, because of limited number of parameters, no more than a locally-optimized model is achievable.

In the current chapter, the transfer function from the free field to the cochlear input is reconstructed using transfer functions experimentally obtained in order to acquire time- domain stapes responses of the human and chinchilla that would be applicable to various auditory researches. Not affected by the modeling artifacts within the measured frequency range, the experimentally-obtained transfer functions are expected to provide accurate results over a wider frequency range than mathematical auditory models do.

As an application of the reconstructed transfer function, the time-domain stapedial responses are calculated and used to formulate noise metrics. The metric values are investigated to detect inter-species differences in the responses. Furthermore, the 184

correlations between the metrics and an auditory damage indicator are checked to determine the applicability of the stapes response-based metrics to NIHL study.

A. Transfer functions from measurements

Measured transfer function data shown in Table V-1 are provided by researchers in the U.S.

Army Research Laboratory (ARL) and the National Institute for Occupational Safety and

Health (NIOSH). The magnitude data of the transfer functions are available for four mammalian species (i.e., human, chinchilla, guinea pig and cat); however the phase data of the guinea pig and cat are unavailable. Thus the two species were ruled out from transfer function reconstruction.

Species Human Chinchilla Guinea pig Cat

External ear Shaw (1974) * Bismarck Sinyor and Wiener et al. pressure Mehrgardt and (1967)* Laszlo (1973)* (1965)* transfer Mellert (1977) Bismarck and function Pfeiffer (1967)* Murphy and Davis (1998)

Middle ear Kringlebotn and Ruggero et al. Nuttall (1974) Guinan and transfer Gundersen (1990) Peake (1967) admittance (1985)

Table V-1. Measured transfer function data. * Phase data are unavailable.

185

1. External ear pressure transfer function

The pressure transfer function (PTF) of the external ear ( ) is defined as the ratio of the free-field sound pressure ( ) and the tympanic membrane input pressure ( ):

() (V-1) () ≡ () The transfer function represents the pressure gain obtained while the free-field sound wave is passing through the external ear. The PTFs of external ear of different species are plotted in Fig. V-1 and Fig. V-2.

186

20 20 (dB) (dB) FF FF 10 10 /P /P TM TM 0 0 CH: P CH: CH2: P CH2:

-10 -10 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (a) (b)

20 20 (dB) (dB) FF 10 FF 10 /P /P TM TM 0 0 CT: P GP: P

-10 -10 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (c) (d)

20 20 (dB) (dB) FF FF 10 10 /P /P TM TM 0 0 HM: P HM2: HM2: P

-10 -10 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (e) (f)

Fig. V-1. Magnitude of external ear PTF with reference to azimuth of 90° and elevation of 0°. (a) Chinchilla

(Bismarck, 1967; Bismarck and Pfeiffer, 1967) where sound wave comes from azimuth of 0° and elevation of 0°,

(b) Chinchilla (Murphy and Davis, 1998), (c) Cat (Wiener et al. , 1965) (d) Guinea pig (Sinyor and Laszlo, 1973),

(e) Human (Shaw, 1974), and (f) Human (Mehrgardt and Mellert, 1977)

187

2 0.5

0 0 (period) (period) FF FF -2 -0.5 /P /P TM TM -4 -1 CH2: P CH2: HM2: HM2: P -6 -1.5 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

Fig. V-2. Phase of the external ear PTF. (a) Chinchilla (Murphy and Davis, 1998) and (b) Human (Mehrgardt and

Mellert, 1977)

2. Middle ear volume velocity-pressure transfer function

Transfer function defined by the ratio of the stapes volume velocity ( ) to the eardrum pressure ( ) is called “middle-ear transfer admittance” (Rosowski and Merchant, 1995) or middle ear volume velocity-pressure transfer function (UPTF). The transfer function represents how much stapes volume velocity would be generated if unit sinusoidal pressure wave is applied on the tympanic membrane:

() (V-2) () ≡ () The middle ear UPTFs of several mammalian species are plotted in Fig. V-3 and Fig. V-4, where the stapes volume velocity commonly bears stiffness-controlled characteristic at low frequencies (Rosowski, 1994) but the peak response is species-dependent.

188

-4 -4 x 10 x 10

1 1 sec) sec) ⋅ ⋅ /dyne /dyne 5 0.5 5 0.5 (cm (cm CT UP CH UP H H 0 0 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (a) (b)

-4 -4 x 10 x 10

1 1 sec) ⋅ sec) ⋅ /dyne 5 /dyne 5 0.5 0.5 (cm (cm HM UP GP UP H H 0 0 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(d) (c)

-4 x 10

1 sec) ⋅ /dyne 5 0.5 (cm HM2 UP H 0 1 2 3 4 10 10 10 10 Frequency (Hz) (e)

Fig. V-3. Magnitude of the middle ear UPTF. (a) Chinchilla (Ruggero et al. , 1990), (b) Cat (Guinan and Peake,

1967), (c) Guinea pig (Nuttall, 1974), (d) Human (Kringlebotn and Gundersen, 1985) based on the mean volume velocity measurement over 30 cadaver ears (Rosowski, 1994), and (e) Human (Kringlebotn and Gundersen,

1985) based on the median volume velocity measurement over 68 cadaver ears (Rosowski, 1991)

189

0.4 0.4

0.2 0.2 0 0 (period) (period) P -0.2 T CH UP C CT UP G H H -0.2 -0.4

-0.6 -0.4 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (a) (b)

0.3 0.4

0.2 0.2 0 0.1 (period) (period) -0.2 HM UP GP UP H H 0 -0.4

-0.1 -0.6 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (c) (d)

0.4

0.2

(period) -0.2 HM2 UP H -0.4

-0.6 1 2 3 4 10 10 10 10 Frequency (Hz) (e)

Fig. V-4. Phase of the middle ear UPTF. (a) Chinchilla (Ruggero et al. , 1990), (b) Cat (Guinan and Peake, 1967), (c)

Guinea pig (Nuttall, 1974), (d) Human (Kringlebotn and Gundersen, 1985) based on the mean volume velocity measurement over 30 cadaver ears (Rosowski, 1994), and (e) Human (Kringlebotn and Gundersen, 1985) based on the median volume velocity measurement over 68 cadaver ears (Rosowski, 1991)

190

B. Reconstruction of the transfer functions

The transfer function describing the integrated transmissibility of the external and middle ear is defined by the ratio between the free-field sound pressure ( ) and the stapes volume velocity ( ):

∗ () (V-3) () ≡ = () ∙ () ()

In the process of this reconstruction, within measured frequency range, the transfer function is approximated with a spline function passing through the measured values.

Curve fitting is carried out with the available data sets to extend the transfer function beyond the measured frequency range. The curve fitting of and are separately performed for the low and high frequency regions, during which the magnitude is kept greater than zero and DC component are suppressed. Curve fitting results are represented in TableV-2 , TableV-3 and Fig. V-5 through Fig. V-9.

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Human (Mehrgardt and Mellert, Chinchilla (Murphy and Davis, 1977) 1998)

Magnitude Gauss2 (0.96, f<1 kHz) Sin4 (0.97, f<1 kHz)

Poly1 (0.91, f > 8 kHz ) Fourier6 (0.92, f>15 kHz)

Phase Gauss2 (0.95, f<1 kHz) Sin8 (0.99, f<1 kHz)

Power2 (0.98, f>1 kHz) Power2 (0.99, f>15 kHz)

TableV-2. Curve-fitting results for the external ear PTFs where fitted functions are represented with R- square values (first elements in parentheses) and the frequency subset of measured data used for curve- fitting (second elements in parentheses). The fitted functions are provided by MATLAB ® curve fitting toolbox (MATLAB ®, 2007a).

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7 20 Curve fitting Curve fitting 6 Extrapolated 10 Extrapolated Raw data Raw data 5 0 4 (dB) (dB)

FF FF -10

/P 3 /P TM TM P P -20 2

1 -30

0 -40 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

Fig. V-5. Curve fitting of the human external ear PTF data (Mehrgardt and Mellert, 1977) . (a) low frequency magnitude (b) high frequency magnitude (c) low fre quency phase (d) high frequency phase

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6 0 Curve fitting Curve fitting 5 Extrapolated -1 Extrapolated Raw data Raw data 4 -2

3 -3 (dB) (dB) FF FF

/P 2 /P -4 TM TM P P 1 -5

0 -6

-1 -7 0 1 2 3 4.2 4.3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

0.04 -4 Curve fitting 0.02 Extrapolated -4.5 Raw data 0

-0.02

(period) (period) -5 FF -0.04 FF /P /P TM TM P -0.06 P -5.5 Curve fitting -0.08 Extrapolated Raw data -0.1 -6 0 1 2 3 4.2 4.3 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

Fig. V-6. Curve fitting of the chinchilla external ear PTF data (Murphy and Davis, 1998). (a) low frequency magnitude (b) high frequency magnitude (c) low frequency phase (d) high frequency phase

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Human* (Kringlebotn Human** (Kringlebotn Chinchilla (Ruggero et and Gundersen, 1985; and Gundersen, 1985; al., 1990) Rosowski, 1994) Rosowski, 1991)

Magnitude Poly7 (0.99, f<1 kHz) Poly4 (0.99, f<1 kHz) Poly5 (0.99, f<1 kHz)

Exp2 (0.99, f>1 kHz) Power2 (0.99, f>1 kHz) Power2 (0.84, f>1 kHz)

Phase Poly4 (0.99, f<1 kHz) Poly6 (0.99, f<1 kHz) Gauss6 (0.99, f<1 kHz)

Power2 (0.99, f>1 kHz) Power2 (0.98, f>1 kHz) Power2 (0.98, f>1 kHz)

TableV-3. Curve-fitting results for the middle ear UPTFs where fitted functions are represented with R-square values (first elements in parentheses) and the frequency subset of measured data used for curve-fitting (second elements in parentheses). The fitted functions are provided by MATLAB ® curve fitting toolbox (MATLAB ®, 2007a).

* Based on the mean volume velocity measurement over 30 cadaver ears (Rosowski, 1994). ** Based on the median volume velocity measurement over 68 cadaver ears (Rosowski, 1991).

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-5 -5 x 10 x 10 5 5 Curve fitting Curve fitting Extrapolated Extrapolated 4 4 Raw data Raw data

3 3 /dyne-sec.) /dyne-sec.) 5 5 2 2 (cm (cm UP UP H H 1 1

0 0 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

0.25 0 Curve fitting 0.2 -0.1 Extrapolated Raw data 0.15 -0.2

0.1 -0.3

(period) 0.05 (period) -0.4 UP UP H H 0 -0.5 Curve fitting -0.05 Extrapolated -0.6 Raw data -0.1 -0.7 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

Fig. V-7. Curve fitting of the human middle ear UPTF data (Kringlebotn and Gundersen, 1985; Rosowski, 1994).

(a) low frequency magnitude (b) high frequency magnitude (c) low frequency phase (d) high frequency phase

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-5 -5 x 10 x 10 5 5 Curve fitting Curve fitting Extrapolated Extrapolated 4 4 Raw data Raw data

3 3 /dyne-sec.) /dyne-sec.) 5 5 2 2 (cm (cm UP UP H H 1 1

0 0 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

0.4 0 Curve fitting -0.1 Extrapolated 0.3 Raw data -0.2

0.2 -0.3

(period) (period) -0.4 UP 0.1 UP H H -0.5 0 Curve fitting Extrapolated -0.6 Raw data -0.1 -0.7 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

Fig. V-8. Curve fitting of the human middle ear UPTF data (Kringlebotn and Gundersen, 1985; Rosowski, 1991).

(a) low frequency magnitude (b) high frequency magnitude (c) low frequency phase (d) high frequency phase

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-5 -5 x 10 x 10 6 6 Curve fitting Curve fitting 5.5 5 Extrapolated Extrapolated Raw data 5 Raw data 4 4.5

3 4 /dyne-sec.) /dyne-sec.) 5 5 3.5 (cm 2 (cm UP UP

H H 3 1 2.5

0 2 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz) (a) (b)

0.3 0 Curve fitting 0.25 Extrapolated -0.2 Raw data 0.2

-0.4 0.15

(period) 0.1 (period) UP UP -0.6 H H 0.05 -0.8 0

-0.05 -1 0 1 2 3 3 4 5 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

Fig. V-9. Curve fitting of the chinchilla middle ear UPTF data (Ruggero et al. , 1990). (a) low frequency magnitude

(b) high frequency magnitude (c) low frequency phase (d) high frequency phase

Reconstructed ∗ of the human and chinchilla ear are compared in Fig. V-10. Compared with Fig. V-1 and Fig. V-3, the magnitude of ∗ is mostly affected by the magnitude of rather than by that of . This suggests that the external ear plays a critical role in determining the auditory transmissibility of the external and middle ear. The first peak of the transfer function is found at 1 kHz region in both species. The chinchilla ∗ shows a wider and higher second peak than that of the human at the second resonance frequency,

198

covering 3-7 kHz region, while the phase lags significantly behind that of the human beyond

4 kHz.

-4 x 10 1 U /P  ST FFHM 0 U /P  ST FFCH -1 2 -2 /dyne-sec.) 5 -3

-4

1 Phase (period) -5 Magnitude (cm Magnitude (U /P ) ST FF HM -6 (U /P ) ST FF CH 0 0 1 2 3 4 5 -7 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

Fig. V-10. Comparison of reconstructed ∗ of the human and chinchilla (a) magnitude, and (b) phase. The human

is from Mehrgardt and Mellert (1977) and from Kringlebotn and Gundersen (1985; Rosowski, 1994), while the chinchilla is from Murphy and Davis (1998) and from Ruggero et al. (1990).

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C. Time-domain responses of the stapes

From equation (V-3), derivative transfer functions represented by the stapes velocity ( ) or the stapes displacement ( ) are:

1 (V-4) ∗ () ∙ () ≡ = ()

1 ∗ () ∙ (V-5) () ≡ = () where is the stapes footplate area, the effective area for translational motion of the stapes assumed to be constant.

Referring to equations (V-4) and (V-5), the stapes response in frequency-domain can be represented as:

∗ (V-6) () = () ∙ ()

∗ (V-7) () = () ∙ ()

Once the stapes responses are obtained in spectral form, time-domain responses are available through the inverse fast Fourier transform (IFFT):

(V-8) () = ℱ [ ()]

(V-9) () = ℱ [ ()] where and are time-domain stapes velocity and displacement, respectively. () ()

Typical stapes responses of the human and chinchilla to two types of input waveform, the

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impulsive and complex noises shown in Fig. V-11, are given in Fig. V-12 and Fig. V-13. The impulsive sound in Fig. V-11 (a) has a peak frequency of 300 Hz and the complex noise in

Fig. V-11 (b) has embedded impulses of frequency bands centered at 1, 2, and 4 kHz. The stapes footplate areas of 3.2×10 -2 cm 2 for human (Békésy, 1960) and 1.9×10 -2 cm 2 for chinchilla (Fleischer, 1973) are used for the response calculation.

The time-domain profile of the stapes responses to the complex noise makes little difference compared to the pressure wave, as shown in Fig. V-12. In case of the impulsive noise exposure, the displacement profiles are similar to that of the input pressure wave but the velocity profiles are quite different, indicating that the stapedial displacement is more a conforming indicator of the impulsive pressure input. Comparing the stapes volume velocity profiles in Fig. V-13 (a) and (c), the maximum forward (i.e., toward the inner ear side) stapedial velocity of the human to the impulsive noise is smaller than that of the chinchilla as expected from the reconstructed transfer functions. On the other hand, the maximum backward (i.e., toward the middle ear side) stapedial velocity of the human stapes is comparable to that of the chinchilla, suggesting that the human stapes moves backward as fast as the chinchilla stapes right after the rise time of the impulse. The temporal profile of chinchilla stapes displacement shows a reverberant tail while the displacement of human stapes decays out with little fluctuation. Therefore, the temporal response of the stapes to highly impulsive sounds requires careful investigation since the response of stapes is highly variable.

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The maximum stapes displacement by the impulsive noise in Fig. V-13 exceeds 200 μm for human and 300 μm for chinchilla. These values are 10 and 15 times of the displacement limit set by the mechanical model of Price and Kalb (1991), in which the stapes-footplate displacement was assumed to asymptotically approach up to 20 µm. This is because the non-linearity of the stapes displacement response that is not captured in the linear transfer function. Therefore, in a high pressure noise such as the blast overpressure, stapes displacement should be evaluated with considering nonlinearities.

4 x 10 150 8

100 6 ] ] 2 2 50 4

0 2 [dyne/cm [dyne/cm FF FF -50 0 P P

-100 -2

-150 -4 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Time [sec.] Time [sec.]

(a) (b)

Fig. V-11. Noise examples of: (a) complex noise (G-44 provided by SUNY Plattsburgh collaborators) and (b) impulsive noise (test impulse provided by ARL collaborators)

202

-5 x 10 3

0.5 2

0 0 [cm]: HM [cm/sec]: HM ST -1 d ST v -0.5 -2

-3 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time [sec.] Time [sec.]

(a) (b)

-5 x 10 3

0.5 2

0 0 [cm]: CH ST [cm/sec]: CH

d -1 ST v -0.5 -2

-3 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time [sec.] Time [sec.]

Fig. V-12. Stapes response to complex noise exposure: (a) of the human, (b) of the human, (c) of the chinchilla, and (d) of the chinchilla

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300 0.04

0.03 200

0.02 100

[cm]: HM 0.01 [cm/sec]: HM ST d ST

v 0 0

-100 -0.01 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Time [sec.] Time [sec.]

(a) (b)

300 0.04

0.03 200

0.02 100

[cm]: CH 0.01 [cm/sec]: CH ST d ST

v 0 0

-100 -0.01 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Time [sec.] Time [sec.]

Fig. V-13. Stapes response to impulsive noise exposure: (a) stapes volume velocity of the human, (b) stapes displacement of the human, (c) stapes volume velocity of the chinchilla, and (d) stapes displacement of the chinchilla.

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D. Noise metric based on the stapedial responses

The stapes displacement can be used to calculate the noise metric. Such a metric may have better correlations with the NIHL because the characteristics of the external and middle ears are reflected in the metric. Using the time-frequency decomposition capability of the

AWT (Zhu and Kim, 2006; Zhu, 2008), the time-domain stapes displacement is described as:

(V-10 ) , = AWT [ ] (V-11 ) , = AWT [ ] where AWT is the analytic wavelet transform operator. and represent , , the time-frequency domain stapes velocity and displacement.

1. Design of stapes response based noise metric

A conventional noise metric is defined as: L

1 (V-12) L ≡ 10 log In a similar way, the stapes velocity-based metric ( and displacement-based metric L ( ) can be defined as: L

1 , (V-13) L ≡ 10 log

(V-14 ) 1 , L ≡ 10 log where and are the reference stapes velocity and displacement, which are the stapes velocity and displacement responses to the reference sound pressure input of = 20 × 205

at 1 kHz. Stapes responses to the reference pressure are plotted in Fig. V-14. The 10 Pa values of the reference stapes responses at 1 kHz are listed in Table V-4.

-6 -6 x 10 x 10 3 8

2.5

ref 6

2 ref

1.5 4

1 [cm]: P=P ST [cm/sec.]: P=P d 2 ST v 0.5

0 0 0 5 0 5 10 10 10 10 Frequency [Hz] Frequency [Hz]

(a) (b)

-6 -6 x 10 x 10 1.5 1

0.8 ref

1 ref 0.6

0.4 0.5 [cm]: P=P ST [cm/sec.]: P=P d ST

v 0.2

0 0 3 4 3 4 10 10 10 10 Frequency [Hz] Frequency [Hz]

Fig. V-14. Stapes responses of human (blue solid line) and chinchilla (green dotted line) when the reference sound pressure is applied as a source: (a) stapes velocity, (b) stapes displacement, (c) stapes velocity at 6 frequency points and (d) stapes displacement at 6 frequency points.

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Human Chinchilla

(cm/sec) 5.1255 ×10 -7 1.1512 ×10 -6 (cm) 4.0787 ×10 -7 9.1610 ×10 -7

Table V-4. Reference stapes responses of human and chinchilla evaluated at 1 kHz.

2. Comparison of the stapes response metrics of human and chinchilla

Stapes displacement metrics are obtained for the 23 exposure noises in Table V-5, used for chinchilla exposure studies in chapter VI. Fig. V-15 represents values of 1/3 octave L bands of 6 frequency components (0.5, 1, 2, 4, 8 and 16 kHz) of the 23 noises. The same noises are used as the free-field pressures for computation of the stapes responses by using the reconstructed transfer functions.

1 23 100 2 22 3 80 21 4 60 0.5 kHz 20 5 40 1.0 kHz 19 20 6 2.0 kHz 0 18 7 4.0 kHz

17 8 8.0 kHz

16 9 16.0 kHz 15 10 14 11 13 12

Fig. V-15. Equivalent sound pressure level ( ) of 23 noises used in chinchilla exposure tests ( ). =

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Exposed Chinchilla Number of Overall SPL (dBA) Noise type noise index group index chinchilla subjects G-44 244 12 100 Complex G-49 249 12 100 Complex G-50 250 11 100 Complex G-51 251 11 100 Complex G-52 252 11 100 Complex G-53 253 12 100 Complex G-54 254 11 100 Complex G-55 255 12 100 Complex G-59 259 12 100 Complex G-60 260 16 100 Complex G-61 261 15 100 Gaussian G-63 263 11 100 Complex G-64 264 12 100 Complex G-65 265 12 100 Complex G-66 266 12 100 Complex G-68 268 11 100 Complex G-69 269 9 100 Complex G-70 270 12 100 Complex G-47 247 12 90 Gaussian G-48 248 11 90 Complex G-56 256 11 90 Complex G-57 257 15 95 Gaussian G-58 258 12 95 Complex

Table V-5. Noise and chinchilla group data used in the metric study.

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The and metrics, L eq metrics based on the stapes motion, are shown in Fig. V-16 L L and Fig. V-17. Compared with the equivalent sound pressure level ( ) in Fig. V-15, the L stapedial metrics show higher variations among frequency components. The changes are more noticeable in high frequency components, 8 kHz and 16 kHz components, due to the filtering effect of the external-middle ear. Human stapes responses are most significantly reduced at 16 kHz while the responses of the chinchilla are suppressed mostly at 8 and 16 kHz.

209

1 1 23 100 2 23 100 2 22 3 22 3 80 80 21 4 21 4 60 60 0.5 kHz 20 5 0.5 kHz 20 5 40 40 1.0 kHz 1.0 kHz 19 20 6 19 20 6 2.0 kHz 2.0 kHz 0 0 18 7 18 7 4.0 kHz 4.0 kHz 17 8 8.0 kHz 17 8 8.0 kHz 16.0 kHz 16 9 16.0 kHz 16 9 15 10 15 10 14 11 14 11 13 12 13 12 (a) Human (b) Chinchilla

Fig. V-16. Stapes velocity metric values ( ) for 23 noises, where − (cm/sec) for human and = . × − (cm/sec) for chinchilla. = . ×

1 1 23 50 2 23 50 2 22 3 22 3 21 0 4 21 0 4 0.5 kHz 0.5 kHz 20 5 20 5 -50 1.0 kHz -50 1.0 kHz 19 6 19 6 2.0 kHz 2.0 kHz -100 -100 18 7 4.0 kHz 18 7 4.0 kHz

17 8 8.0 kHz 17 8 8.0 kHz

16 9 16.0 kHz 16 9 16.0 kHz 15 10 15 10 14 11 14 11 13 12 13 12 (a) Human (b) Chinchilla

Fig. V-17. Stapes displacement metric values ( ) for 23 noises, where − (cm) for human = . × and − (cm) for chinchilla. = . ×

210

Inter-species comparison can be made for the stapes metric, either for the velocity or the displacement metric. For both of and , low frequency response of 0.5, 1 and 2 kHz L L are almost identical between the human and the chinchilla. High frequency responses show approximately 8-, 5- and 19-dB difference at 4, 8 and 16 kHz. The chinchilla has higher response at 4 and 16 kHz while the human has higher stapes velocity at 8 kHz. This suggests that chinchilla has higher sensitivity in 4 and 16 kHz frequency bands.

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1 1 23 100 2 23 100 2 22 3 22 3 80 80 21 4 21 4 60 60 20 5 20 5 40 40 19 20 6 19 20 6 Chinchilla Chinchilla 0 18 7 0 Human 18 7 Human

17 8 17 8

16 9 16 9 15 10 15 10 14 11 13 12 14 11 13 12 (a) 0.5 kHz (b) 1 kHz

1 1 23 100 2 23 100 2 22 3 22 3 80 80 21 4 21 4 60 60 20 5 20 5 40 40 19 20 6 19 20 6 Chinchilla Chinchilla 0 0 18 7 Human 18 7 Human

17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12

1 1 23 100 2 23 100 2 22 3 22 3 80 80 21 4 21 4 60 60 20 5 20 5 40 40 19 20 6 19 20 6 Chinchilla Chinchilla 0 0 18 7 Human 18 7 Human

17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12

(e) 8 kHz (f) 16 kHz

Fig. V-18. Inter-species comparison of the stapes velocity metric ( ) values, where − = . × (cm/sec) for human and − (cm/sec) for chinchilla. = . ×

212

1 1 23 50 2 23 50 2 22 3 22 3 21 4 21 0 4 0 20 5 20 5 -50 -50 19 6 19 6 Chinchilla Chinchilla -100 -100 18 7 Human 18 7 Human

17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12 (a) 0.5 kHz (b) 1 kHz

1 1 23 50 2 23 50 2 22 3 22 3 21 4 0 21 0 4 20 5 20 5 -50 -50 19 6 19 6 Chinchilla Chinchilla -100 -100 18 7 Human 18 7 Human

17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12 (c) 2 kHz (d) 4 kHz

1 1 23 50 2 23 50 2 22 3 22 3 21 0 4 21 0 4 20 5 20 5 -50 -50 19 6 19 6 Chinchilla Chinchilla -100 -100 18 7 Human 18 7 Human

17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12 (e) 8 kHz (f) 16 kHz

Fig. V-19. Inter-species comparison of the stapes displacement metric ( ) values, where − = . × (cm) for human and − (cm) for chinchilla. = . ×

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3. Correlation study

The correlations between three metrics (i.e., , and ) and average permanent L L L threshold shift (PTS) at each frequency point are compared in Fig. V-20. The human exposure test results are not available thus current comparison is confined to chinchilla.

The stapes response metrics have little influence on the correlation at each frequency point because the transfer functions just consistently vary the metric values at individual frequency as shown in Fig. V-22. The overall correlations shown in Fig. V-21 are worse than those between the L eq and PTS. The stapes displacement metric even produces negative overall correlation, as shown in Fig. V-21 (d), due to large shifting from metric values at L high frequencies. Therefore, the noise metric based on the stapes responses does not provide a useful option.

214

1 1 1

0.5 0.5 0.5

Correlation: 0.5 kHz Correlation: 0 1.0 kHz Correlation: 0 2.0 kHz Correlation: 0 1 2 3 1 2 3 1 2 3 Metric Number Metric Number Metric Number

1 1 1

0.5 0.5 0.5 Correlation: 4.0 kHz Correlation: 8.0 kHz Correlation: 0 0 16.0 kHz Correlation: 0 1 2 3 1 2 3 1 2 3 Metric Number Metric Number Metric Number

Fig. V-20. Correlation between metrics and averaged PTS at 6 frequency points. The metric number 1, 2 and 3 are respectively corresponding to , and .

215

0.6 80

60 0.4

40 0.2 20 Mean Mean PTS (dB)

Overall Correlation 0 0

-0.2 -20 1 2 3 60 70 80 90 100 Metric Number L (dB) eq (a) (b)

80 80

60 60

40 40

20 20 Mean Mean PTS (dB) PTSMean (dB) 0 0

-20 -20 40 50 60 70 80 90 100 -60 -40 -20 0 20 L (dB) L (dB) eqV eqD (c) (d)

Fig. V-21. Overall correlation between metrics and averaged PTS: (a) overall correlation where the metric number

1, 2 and 3 are respectively corresponding to , and , (b) scatter plot of metric data with regression line, (c) scatter plot of metric with regression line, and (d) scatter plot of metric with regression line.

216

1 1 23 100 2 23 100 2 22 3 22 3 21 4 21 50 4 50 20 5 20 5 0 0 19 6 Leq 19 6 Leq -50 Leq(Dst) -50 Leq(Dst) 18 7 18 7 Leq(Vst) Leq(Vst) 17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12 (a) 0.5 kHz (b) 1 kHz

1 1 23 100 2 23 100 2 22 3 22 3 21 50 4 21 50 4 20 5 20 5 0 0 19 6 Leq 19 6 Leq -50 Leq(Dst) -50 Leq(Dst) 18 7 18 7 Leq(Vst) Leq(Vst) 17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12 (c) 2 kHz (d) 4 kHz

1 1 23 100 2 23 100 2 22 3 22 3 21 50 4 21 50 4 20 5 20 5 0 0 19 6 Leq 19 6 Leq -50 Leq(Dst) -50 Leq(Dst) 18 7 18 7 Leq(Vst) Leq(Vst) 17 8 17 8

16 9 16 9 15 10 15 10 14 11 14 11 13 12 13 12

(e) 8 kHz (f) 16 kHz

Fig. V-22. Comparison of the three metric values ( , , and ) of chinchilla, where − (Pa), = × − (cm/sec) and − (cm). = . × = . ×

217

VI. Development of a prototype of advanced noise guideline by using noise induced hearing loss threshold level of chinchillas extracted from existing exposure test data

A. Reprocessing of the exposure data

1. Re-analysis of exposure data

Correlations of ω and PTS of 275 chinchillas are studied at 6 frequency points (i.e., 0.5, L 1, 2, 4, 8 and 16 kHz). Fig. VI-1 shows six scatter plots of ω -PTS data of the 275 L chinchillas of 23 groups. ω –PTS ω correlation study implicitly assumes that a given L octave frequency component of the noise induces the NIHL primarily at the same frequency band, which is supported by the position theory of the cochlea.

The data points of each scatter plot are fitted with a linear function as follows:

ω ω (VI -1) PTS = ∙ L + Table VI-1 shows the values of parameters and (see Eq. (VI-1)) and R 2 values obtained by the regression analysis. R 2 values at 0.5 kHz and 16 kHz are very low, indicating poor correlations at these frequencies. These correlations may improve if the exposure study data can be expanded by adding data obtained with noises of a broader range of SPL. In Fig.

VI-1, the solid line is the regressed function according to Eq. (VI-1) and dotted lines represent the upper and lower prediction bounds of corresponding to the given PTS of L 84% confidence level (CL). The reason to choose an 84% CL will be evident in the following

218

section.

80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 PTS (dB): 0.5 kHz -20 PTS (dB): 1.0 kHz -20 PTS (dB): 2.0 kHz -20 60 80 100 60 80 100 60 80 100 Leq (dB): 0.5 kHz Leq (dB): 1.0 kHz Leq (dB): 2.0 kHz

80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 PTS (dB): 4.0 kHz PTS (dB): 8.0 kHz -20 -20 PTS (dB): 16.0 kHz -20 60 80 100 60 80 100 60 80 100 Leq (dB): 4.0 kHz Leq (dB): 8.0 kHz Leq (dB): 16.0 kHz

Fig. VI-1. Regression analysis results of -PTS data of six frequency components. In each plot, data points are represented by hollow circle, the regressed line is the solid line, upper and lower prediction bound of PTS are dotted lines estimated with 84% of confidence level.

219

Frequency

0.5 kHz 1.0 kHz 2.0 kHz 4.0 kHz 8.0 kHz 16.0 kHz

0.224 1.234 1.502 2.491 1.982 0.9703 a b -6.745 -96.43 -112.9 -199.7 -154.5 -60.75

0.0192 0.2798 0.1774 0.2919 0.1427 0.0671 R

Table VI-1. The result of regression analysis of -PTS relationship measured in past chinchilla exposure studies

B. Estimation of NIHL threshold and its applications

1. Permissible exposure level (PEL) and NIHL threshold

Many human noise guidelines, for example NIOSH 98-126 (NIOSH, 1998) and ISO-1999

(ISO, 1990), recommend 85-dBA as the permissible exposure level (PEL) for 8 hour daily exposure to be used with the 3-dB exchange rule. 8 hour exposure is allowed to the noise environment of 85-dBA, 4 hour exposure to 88-dBA, and so on. PEL was determined mainly based on population studies.

In noise guidelines, PEL is defined as the 8 hour time weighted average that will cause L 6% (ISO, 1990) – 8% (Prince et al. , 1997; NIOSH, 1998) excess risk of hearing impairment after 40 years of exposure. Excess risk is defined as the difference in the percentages of the hearing impaired among the group exposed to occupational noise and the group not exposed to occupational noise (control group) (NIOSH, 1998). Hearing impairment is

220

defined as having 25-dB or higher binaural average of hearing threshold level (HTL) averaged for 1, 2, 3, 4, kHz (NIOSH, 1998).

PEL is not known for chinchillas; however a similar concept can be defined and estimated from the regression result of the measured -PTS data. Fig. VI-2 illustrates the procedure L using the -PTS scatter plot of 2 kHz component. The intersection of 25-dB PTS line and L the upper prediction limit of the -PTS is at = 76-dB as shown in Fig. VI-2. Therefore, L L 76-dB may be considered as the level at 2 kHz that will cause PTS of 25-dB or higher to 8%, the same percentage as in the NIOSH definition of excessive risk, of chinchillas exposed to the noise. Applying the same method to all six frequencies, the levels are estimated to be

78, 86, 76, 81, 76, 58-dB at 0.5, 1, 2, 4, 8, 16 kHz. These NIHL threshold levels may be considered as the PEL of chinchillas obtained as a function of frequency. The threshold levels obtained by considering different excess risks, 8% in NIOSH 98-126 and 6 % in ISO-

1999 (NIOSH, 1998), are represented in Fig. VI-3.

There are a few differences between the NIHL threshold levels identified by the above- explained procedure and the PEL of human used in noise guidelines . The NIHL threshold represents the levels of 0.5, 1, 2, 4, 8, 16 kHz components of the noise that induce L hearing impairment to 8% of chinchillas after 5 days of continuous (24 hours a day) exposure, while the PEL is the result of a long-term, 40 years exposure. As the exposure is short-term (5 days) and chinchillas are prescreened for any pre-existing hearing loss, the percentage of hearing impairment is essentially the same as the excess risk. Therefore, 221

hearing impairment in chinchillas is defined as having 25-dB or higher mono-aural PTS.

Upper bound of 84% C.I. 100 100 8 % 8 50 50 25 dB PTS (dB) 0 0 84% PTS PTS (dB): 2 kHz

-50 -50 60 70 80 90 100 110 Leq (dB) 0 -100

Threshold SPL 0.02 0.01 0.025 0.015 0.005

Density

Fig. VI-2. Procedure to find NIHL threshold level illustrated with 2 kHz data

70 Leq (dB) Leq

50 3 4 10 10 Frequency (Hz)

Fig. VI-3. NIHL threshold level evaluated by the excess risk of 8% in NIOSH 98-126 (circular marks) and 6% in

ISO-1999 (rectangular marks).

222

2. Applications of NIHL threshold data

The NIHL threshold levels identified at the six frequency points according to the procedure explained above will serve as useful information in planning future chinchilla exposure tests. For example, exposure level may be determined by comparing the frequency components of the exposure noise with the threshold levels.

The threshold level also can be used to estimate necessary noise reductions for a given noise to prevent chinchillas from the hearing impairment in a frequency by frequency manner. Fig. VI-4 (a) shows the NIHL threshold level at the six frequency points plotted with the ω calculated for one of the complex noises of 100-dBA used in this study. By L comparing ω with the threshold levels, it is seen that the SPL of the noise is higher L than the threshold levels by 5, 13, 18, 11, 9, 21-dB at the six frequency points as shown in

Fig. VI-4 (b). If it is limited to the hearing impairment in the frequency range of human speech and communication as most human noise guidelines do, 5, 13, 18 and 11-dB reductions of the SPL of the noise will be necessary respectively at 0.5, 1, 2, 4 kHz. These reductions may be achieved by combinations of the noise reduction and hearing protection.

The above example suggests that the NIHL threshold identified as a function of frequency can be used as the core part of an advanced noise guideline capable of making frequency by frequency, quantitative recommendations. Identifying the NIHL threshold level for human as a function of frequency will require the relationship between of the noise L environments and the HTL induced among workers obtained at each octave frequency. 223

Such a task will not be easy, however certainly possible if studies are properly planned and conducted.

120 25 110

100 20

90 15 80 Leq (dB) Leq 70 10 60

50 (dB): NRL Recommended (G263) 3 4 5 3 4 10 10 10 10 Frequency (Hz) Frequency (Hz)

(a) (b)

Fig. VI-4. NIHL Threshold Levels of chinchillas and its use as a noise guideline. (a) NIHL threshold levels of chinchillas identified (solid line with filled circle); of the given complex noise. (b) Necessary noise reduction estimated by comparing with the threshold level.

224

Summary and Conclusions

A systematic comparison study of diverse simulation models to understand their limitations and proper use and development of a prototype of an advanced noise guideline based on chinchilla exposure data are two major contributions of this study.

Various simulation methods of human auditory system were studied and a prototype of an improved noise guideline was developed in this study. This work was a part of the long- term approach to develop an improved noise guideline that is accurate for all types of occupational noises. Many researchers believe that current noise guidelines underestimate risk of exposure to complex noises in that impulsive noises are embedded in a continuous, broadband background noise (Roberto et al. , 1985; Ahroon et al. , 1993; Hamernik and Qiu,

2001; Harding and Bohne, 2004; Hamernik et al. , 2007). Because response characteristics and damage mechanisms of the auditory system are highly complex, a noise guideline will have to be developed relying heavily on the data obtained from animal noise exposure tests and demographic studies. However, well-developed simulation techniques can provide guidance in designing experimental studies and interpreting measured results necessary for the development.

Operating principles and biology of the human auditory system were discussed to provide understanding necessary to develop simulation techniques. The external and middle ear were modeled by an electrical network by using analogies between the electrical

225

components and the mechanical or acoustic components of the system. The circuit model was programmed using SimPowerSystems™ (Simulink ®, 2007a), a graphics-based program language of MATLAB (MATLAB®, 2007b). For the inner ear, the cochlear simulation model was developed using a simplified mathematical description of the wave propagation in the cochlea. The cochlear model was solved in the frequency domain. The Fourier and inverse

Fourier transform were used to calculate the response of the whole-ear, e.g., the basilar membrane response to the free space pressure disturbance, using the time domain model of the external-middle ear and the frequency domain model of the cochlea.

Middle ear transfer functions obtained from seven popular human middle ear network models were cross-compared. To understand expected accuracy of auditory system response simulation, the stapes responses of seven human middle ear network models were also obtained by replacing one of three impedance blocks of the reference model by the same block used by the other models. The three impedance blocks were impedances of the middle ear cavity, the conductive path, and the stapes-cochlear complex. The stapes displacement in response to the pressure input to the tympanic membrane was used in the comparison. The range of the predicted values from these varied models reflects the expected accuracy of the middle ear models. The comparison showed that the simulation results showed similar qualitative results; however with fairly large quantitative differences. The results showed that differences in the estimated response amplitude can easily reach 20-dB. This suggested that proper uses of network human middle ear model are for qualitative purposes such as interpreting experimental results or comparing 226

solutions in a relative sense in conjunction with the experimental data.

The network middle ear models were used for parameter study of the temporal response of the stapes to impulsive noises in Chapter IV. The waveforms were described in simplified forms only by two parameters, A-duration and the rise time. It was seen that the maximum displacement of the stapes toward the cochlear side and the maximum forward input pressure increased but the maximum displacement toward the middle ear showed slight decrease as the A-duration increased. The trend became saturated when the A-duration increased to 3-ms or more. Stapes motion showed about 0.5-ms delay to the start of the impulse. Increase of the rise time had an effect to reduce the delay of the middle ear transmission. The pressure wave delay did not increase any more when the A-duration was more than 2-ms. Increase of the rise time reduced the pressure wave delay in the middle ear.

Experimentally obtained transfer functions were also used to obtain the stapes responses.

Use of the measured transfer functions removes the artifacts of mathematical models while measurement errors are introduced. Measured transfer functions of human and chinchilla were reconditioned to have the same frequency range by extrapolation of the functions to compare the responses of the two species. Using the re-conditioned transfer functions, stapes responses of the human and chinchilla to selected impulsive and complex type noises were computed in the frequency domain. The responses were transformed into the time domain and compared with each another. The maximum stapes displacement response to an impulsive noise of 170–dB peak pressure showed 10 and 15 times of the 227

displacement limit suggested by Price and Kalb (1991). Due to non-linearity of the middle ear, which was not captured in the transfer function, such a large displacement will not occur in reality. Therefore, the measured transfer function will have to be used only in the linear range.

It was attempted to define noise metrics using the stapes responses. The idea was tested for the human and chinchilla. The two species had similar stapes response characteristics in the low frequency range, at 0.5, 1 and 2 kHz. Compared to human, chinchilla had higher responses approximately by 8- and 19-dB at 4 and 16 kHz and smaller stapes velocity by about 5-dB at 8 kHz. However, stapes response based metrics showed poor correlations with PTS observed in the chinchilla exposure data, indicating that the stapes response based noise metric is not a good option to be used in noise guidelines.

A set of chinchilla noise exposure study data obtained in previously conducted research was reanalyzed. The NIHL threshold level, which can be considered as the chinchilla version of the permissible exposure level (PEL) in an approximate sense, was obtained as a function of frequency by studying statistical relationship between and PTS. NIHL L threshold level of chinchillas identified in this way has potentially important applications.

The threshold level will be useful information in designing a chinchilla noise exposure study; for example to determine proper exposure levels to induce intended PTS levels in chinchillas. The most important application of the NIHL threshold level will be to the development of an advanced noise guideline that enables quantitative, frequency by 228

frequency recommendation to prevent hearing impairment. This will be an interesting subject of future research, which will require new population studies designed to obtain the noise level–hearing threshold level relationship as a function of frequency.

229

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Appendix: Variable Impedance component modeling using Simulink

Since the SimPowerSystems (Simulink ®, 2007a) module of Matlab Simulink does not provide nonlinear circuit elements, it is not possible to model the variable impedance components frequently appearing in a non-linear circuit model. Thus, it is necessary to take an indirect approach to use Simulink in non-linear circuit simulation.

A simple strategy to model nonlinear resistors and capacitors is introduced, which can be conveniently implemented to construct Simulink model of variable impedance components.

The validity of the strategy is checked with simple network models whose results are close to those of original RLC branch model with variable electric component. Thus it is expected that the modeling approach is readily applicable to the equivalent network model by

Simulink for nonlinear elements in an auditory system.

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A. Equivalent model of a nonlinear electric component

1. Equivalent modeling of a variable resistor

Considering a simple circuit shown in Fig. A-1, where is the initial resistance of a variable resistor, is the current flowing through the resistor, and is the measured voltage-drop across the resistor.

R o I0

V 0

Fig. A-1. Circuit with a variable resistor at time

At time , the resistance at time becomes as shown in Fig. A-2. With + ∆ + ∆ the change of the resistance, the current over the circuit and voltage-drop across the resistor are also changed into and respectively.

Ro + ∆ R I

Fig. A-2. Circuit with a variable resistor at time + ∆

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From Fig. A-2, it is easily found that:

(1) = ∙ + ∆ Assuming the variable resistor in Figure 2 can be equivalently modeled with a resistor and controlled current supplier as depicted in Fig. A-3, the total current in the circuit is given by:

(2) = + ∆ where ′ is the current flowing through and is the feed-back current compensation by ∆ the supplier.

Since the ideal current supplier has no internal resistance, the voltage drop across the equivalent resistor is:

(3) = In equations (2) and (3), the total current and voltage-drop are kept same with those in equation (1) because the circuit in Figure 3 is the equivalent to the one in Fig. A-2.

∆I

Ro I′ I

Fig. A-3. Equivalent circuit of the network shown in Fig. A-2.

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From equation (1), (2) and (3), the necessary current fed back by the supplier is:

− ∙ ∆ (4) ∆ = + ∆ Equation (4) shows that if the resistance change ( ) and the voltage-drop ( ) over the ∆ variable resistor are known, can be calculated to implement the equivalent circuit in Fig. ∆ A-3. Since in an equivalent circuit of mechano-acoustic system is usually determined by ∆ one or more mechanical relationships, it can be considered as a known variable. If we place a voltmeter across the circuit, the voltage difference ( ) over the equivalent resistor can be measured to become a known variable. Therefore, once the amount of a feed-back current

( ) is determined, the variable resistor can be modeled just with a constant resistor ( ) ∆ integrated with a controlled current source.

An alternative equivalent network represented in Fig. A-4 is made of a resistor ( ) and a voltage source ( )instead of the controlled current source in parallel. ∆

R0 ∆V I

Figure A-4. Alternative way to model a variable resistor

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In this case, the feed-back voltage is simply given by:

(5) ∆ = ∆ Once the current is measured, the controlled source voltage is determined by equation ∆ (5).

2. Equivalent modeling of a variable capacitor

A similar strategy can be employed to model a variable resistor to construct an equivalent model for a variable capacitor ( ). The original circuits at time and are shown in + ∆ Fig. A-5 (a) and (b), respectively. The equivalent circuit of Fig. A-5 (b) is shown in Fig. A-6 with a controlled voltage source ( ). ∆

Co I0 Co + ∆ C I

V V 0

(a) At time (b) At time + ∆

Fig. A-5. Circuit with a variable capacitor

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Co ∆V I

Fig. A-6. Equivalent circuit of Fig. A-5 (b)

From Fig. A-5 (b) and Fig. A-6, voltage drops are expressed respectively as:

1 (6) = + ∆

1 (7) = ∆ + Thus the feed-back voltage ( ) is determined by: ∆

1 1 (8) ∆ = − + ∆

Similar to the case of a variable resistor circuit, and are known. ∆

An alternative model is to use a current source. In this case, the feed-back current ( ) from ∆ the source is represented as:

(9) ∆ = ∆ As in the previous equivalent model that used the voltage source, is known and is ∆ calculated from the voltage measurement.

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Comparing equation (8) and (9), the obvious difference is that the unknown takes on an ∆ integration form but has a derivative form which is more sensitive to numerical error ∆ than integration. Therefore, as an equivalent model for a variable capacitor, the controlled voltage source model is preferable to the current source model.

B. Validity check of equivalent network models

In order to check the validity of the equivalent models shown in Fig. A-7, a simple RLC branch composed of a variable resistor ( ), capacitor ( ) and inductor ( ) or a + ∆ variable capacitor ( ), resistor ( ) and inductor ( ) is considered. + ∆

R0+deltaR L0 C0 C0+deltaC R0 L0

(a) RLC branch with a variable resistor (b) RLC branch with a variable capacitor

Fig. A-7. RLC branch

Equivalent networks are constructed with Simulink according to the procedure explained previously, which are represented in Fig. A-8, A-9, A-10 and A-11.

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v + - deltaR

s -1/(R0*(R0+deltaR)) + -

Fig. A-8. Simulink equivalent circuit with a controlled current source for a variable resistor

deltaR

s + i - + - R0

Fig. A-9. Simulink equivalent circuit with a controlled voltage source for a variable resistor

- + v

du/dt

deltaC s + -

Fig. A-10. Simulink equivalent circuit with a controlled current source for a variable capacitor

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1/(C0+deltaC)-1/C0

1 s s + i - + - C0

Fig. A-11. Simulink equivalent circuit with a controlled voltage source for a variable capacitor

For each circuit, their current and voltage output differences from the output current and voltage of the original RLC branch are plotted in Fig. A-12, A-13, A-14, and A-15, where

, , , and are used. The inductance value is = 1Ω Δ = 0.1Ω = 1μF Δ = 0.1μF = 1mH assumed to be fixed. The input voltage signal is 1000 Hz sine-wave with sampling rate of

50000 per second and amplitude of 100.

-13 -13 x 10 x 10 1 1

0.5 0.5 0 0 -0.5 -0.5 -1 current difference current [A] voltage difference [V] -1 -1.5 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 time [sec] time [sec]

(a) Current difference (b) Voltage difference

Fig. A-12. Result comparison between Fig. A-7 (a) and Fig. A-8.

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-15 -15 x 10 x 10 2 4

2 0 0 -2 -2 current difference current [A] voltage difference [V] -4 -4 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 time [sec] time [sec]

(a) Current difference (b) Voltage difference

Fig. A-13. Result comparison between Fig. A-7 (a) and Fig. A-9.

0.2 4

0.1 2

0 0

-0.1 -2 current current difference [A] voltage difference [V] -0.2 -4 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 time [sec] time [sec]

(a) Current difference (b) Voltage difference

Fig. A-14. Result comparison between Fig. A-7 (b) and Fig. A-10.

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-15 -13 x 10 x 10 4 1

2 0.5

0 0

-2 -0.5 current difference current [A] voltage difference [V] -4 -1 0 0.002 0.004 0.006 0.008 0.01 0 0.002 0.004 0.006 0.008 0.01 time [sec] time [sec]

(a) Current difference (b) Voltage difference

Fig. A-15. Result comparison between Fig. A-7 (b) and Fig. A-11.

The result from the equivalent variable resistor model shows little difference whether it is based on a current source model or a voltage source model. As seen in Fig. A-16, the voltage source model is slightly more stable than the current source model due to the simplicity of the former. However the difference is prominent in the equivalent variable capacitor model as shown in Fig. A-17. The voltage source model is much more stable than the current source model. As was explained, the difference comes from the different sensitivity to the numerical errors associated with the differentiation and integration.

257

-13 x 10 1

0.5

current difference current [A] -0.5

-1 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 time [sec]

-13 x 10 1

0.5

-0.5 voltage difference [V] -1

-1.5 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 time [sec]

Fig. A-16. Variable resistor model comparison (solid line: current source model, dotted line: voltage source model)

258

0.15

0.1

0.05

-0.05

-0.1 current difference current [A]

-0.15

-0.2 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 time [sec]

-1

voltage difference [V] -2

-3

-4 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 time [sec]

Fig. A-17. Variable capacitor model comparison (solid line: current source based model, dotted line: voltage source based model)

C. Application of the non-linear component models to the AHAAH

As shown in Fig. II-24, the annular ligament network model of the AHAAH is composed of a variable resistance ( ) and a variable capacitance ( ). Using the approach introduced in previous section, it can be modeled as a Simulink nonlinear subsystem block. In this block, the resistance and capacitance change are calculated according to the annular ligament’s mechanical model suggested by Dr. Kalb, then used to calculate voltage and current source input values.

259