Middle Modeling – A Tutorial

[Hello, I am Fereshteh Kalantari. I am a Ph.D student in Electronic Engineering at K. N. Toosi University of Technology, Tehran, Iran. In this Tutorial, I want to explain Modeling, which is my project topic for the Neuro-Muscular System Control course. This course is instructed by Dr. Delrobaei. If you have any questions, please send me an Email ([email protected]). Now let's read and learn.]

ave you ever imagined that you cannot hear your around sounds? Do you know how you hear sounds by your ? H Have your ears ever damaged by loud sounds? Do you know how your ears are safe from annoying sounds? To understand this questions, we need to investigate different parts of an ear and their functions. The middle ear is one of them, which we explain about it in this tutorial.

Where Is the Middle Ear? There are three main parts in an ear, they consist of the , the middle ear and the . As shown in Figure 1, the middle ear lies between the outer ear and inner ear. In process, the middle ear is like an amplifying interface between the outer ear and the inner ear.

Figure 1. The structure of an ear with three main parts. The outer ear, the middle ear and the inner ear.

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What Does the Middle Ear Do? In Figure 2, we illustrated how we hear by an ear step by step and what the middle ear's main job is. According to Figure 2, after sound waves enter the outer ear, they travel through the and make their way to the middle ear. There is the , which is a thin piece of skin stretched tight like a drum. The eardrum separates the outer ear from the middle ear. The middle ear's main job is to take those sound waves and turn them into vibrations and amplify them. The vibrations then move to the inner ear (the ). This is a fluid filled, snail shaped structure that is lined by tiny hair receptors. These hair receptors are attached to nerve endings and as the vibrations wash over the hair receptors, the nerves carry signals to the brain which interprets these stimuli as sound.

How Sound waves enter our outer ear and Our eardrum vibrates with the incoming travel through the ear canal to your sound and sends the vibrations to three tiny we Hear eardrum. bones in our middle ear.

Our auditory nerve carries this The bones in our middle ear amplify the sound vibrations electrical signal to the brain, and send them to our inner ear, or cochlea. The sound which translate it into a sound vibrations activate tiny hair cells in the inner ear, which in you can understand. turn release neurochemical messengers.

Figure 2. The steps of hearing by an ear.

The Structure of the Middle Ear The middle ear consists of an air-filled cavity called the and includes the three and their attaching ligaments; the auditory tube; and the round and oval windows. The ossicles are three small bones that function together to receive, amplify, and transmit the sound from the eardrum to the inner ear. These bones also act as a protective mechanism to prevent very loud sounds from damaging the inner ear. In Figure 3, we shown the different parts of the middle ear and described about every part in the following.

"The Middle Ear has the ossicles, which are three small bones that function together to receive, amplify, and transmit the sound from the eardrum to the inner ear."

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Auditory ossicles

Malleus

Stabilizing ligaments

Oval window External acoustic meatus

Round window Tympanic membrane

Eustachian tube

Tympanic cavity (Middle

Ear) Figure 3. The structure of the middle ear.

The Structure - The Auditory Ossicles

The eardrum is very thin, measures approximately 8-10 mm in diameter and is stretched by means of small muscles. The pressure from sound waves makes the eardrum vibrate. The vibrations are transmitted further into the ear via ossicles that are three bones in the middle ear: the hammer (), the anvil (incus) and the stirrup (stapes). The stapes is the smallest named bone in the body.

These three bones form a kind of bridge, the malleus receives vibrations from sound pressure on the eardrum, where it is connected at its longest part (the manubrium or handle) by a ligament. It transmits vibrations to the incus, which in turn transmits the vibrations to the small stapes bone. The wide base of the stapes rests on the . As the stapes vibrates, vibrations are transmitted through the oval window, causing movement of fluid within the cochlea.

The Structure - The Oval Window

The oval window is a membrane covering the entrance to the cochlea in the inner ear. When the eardrum vibrates, the sound waves travel via the hammer and anvil to the stirrup and then on to the oval window.

When the sound waves are transmitted from the eardrum to the oval window, the middle ear is functioning as an acoustic transformer amplifying the sound waves before they move on into the inner ear. The pressure of the sound waves on the oval window is some 20 times higher than on the eardrum.

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The pressure is increased due to the difference in size between the relatively large surface of the eardrum and the smaller surface of the oval window. The same principle applies when a person wearing a shoe with a sharp stiletto heel steps on your foot: The small surface of the heel causes much more pain than a flat shoe with a larger surface would.

The Structure - The

The round window in the middle ear vibrates in opposite phase to vibrations entering the inner ear through the oval window. In doing so, it allows fluid in the cochlea to move.

The Structure - The

The Eustachian tube is also found in the middle ear, and connects the ear with the rearmost part of the palate. The Eustachian tube’s function is to equalize the air pressure on both sides of the eardrum, ensuring that pressure does not build up in the ear. The tube opens when you swallow, thus equalizing the air pressure inside and outside the ear.

In most cases the pressure is equalized automatically, but if this does not occur, it can be brought about by making an energetic swallowing action. The swallowing action will force the tube connecting the palate with the ear to open, thus equalizing the pressure.

Built-up pressure in the ear may occur in situations where the pressure on the inside of the eardrum is different from that on the outside of the eardrum. If the pressure is not equalized, a pressure will build up on the eardrum, preventing it from vibrating properly. The limited vibration results in a slight reduction in hearing ability. A large difference in pressure will cause discomfort and even slight pain. Built-up pressure in the ear will often occur in situations where the pressure keeps changing, for example when flying or driving in mountainous areas.

First Mechanical Model of the Middle Ear The human middle ear is a tiny mechanical structure consisting of the tympanic membrane, three ossicles (malleus, incus, and stapes), middle ear ligaments and muscle tendons, and the middle ear cavity. In the meantime, investigations on modeling the middle ear have been developed for a better understanding of the sound transmission mechanism in the human ear. In the Figure 4, you can see a lumped model of an ear, which is proposed by Feng and Gan (2004). This model has been drawn from external ear canal to cochlea. This lumped parametric model consisting of 6 messes connected by several pairs of spring and dashpot is proposed for mechanical analysis of the ear, including the external ear canal, tympanic membrane, middle ear ossicles, and cochlea. The air inside the external ear canal was represented by the mass M1, which coupled the mass M2, the tympanic membrane (TM), through the spring K2 and dashpot C2. Spring K1 and dashpot C1 represented the TM annulus. The three ossicular bones (malleus, incus, and stapes) were represented by masses M3, M4, and M5, respectively. The malleus-incus joint and the incus-stapes joint, which connect the three ossicles and form the ossicular chain, were represented as two pairs of springs and dashpots: K5, C5 and K6, C6, respectively. The malleus (M3) was attached to the TM (M2) through K3 and C3. The middle ear suspensory ligaments and intra aural muscles supported the ossicles. Two major ligaments suspending the malleus and incus were simulated as dashpots C4 and C7. M6 represented cochlear fluid supported by dashpots C9 and C10. The stapes coupled with the cochlear fluid through the stapedial annulus (K8 and C8). All unknown parameters are identified based on the governing equations of the system and the determined through optimization and parameter-perturbation process. This lumped model serves as the starting stage for understanding the relation between the middle ear components for sound transmission.

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The middle Ear Tympanic Malleus membrane

M2 M3 Incus Stapes M5 M4

Figure 4. Lumped parametric model of the human ear. As shown in Figure 4, you can see the model of the middle ear in red dash and its different parts are represented by using different colors.

In a damped mass-spring system, the equation of motion can be represented as

푚푥̈̈ + 푐푥̇̇ + 푘푥 = 푓(푡) (1) 풙̈ : Acceleration 풙̇ : Velocity Mass Stiffness 풙 : Displacement Damping parameter f(t) : Externally applied force

푥 = 푋̂ 푒푗휔푡 (2)

푥̇ = 푗푤푋̂ 푒푗휔푡 (3)

푥̈ = −푤2푋̂ 푒푗휔푡 (4)

Where m is the mass, c is the damping parameter, k is the stiffness, x is the displacement, is the acceleration, is the velocity of the system matrices, and f (t) denotes an externally applied force.

Therefore, a mass-spring system includes the basic components of an oscillating system: mass, spring (stiffness), and damping. When a force acts on the system, the resulting equations of motion are

휔 = 2휋푓 (5)

푘 휔 = √ (6) 푚

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1 푘 푓 = √ (7) 2휋 푚

Where 휔 is the angular frequency of the system in radians per second and f is the frequency in Hz.

In the present study, the mass-spring model of the ear was composed of six masses - M1, M2, M3, M4, M5, and M6 - representing, respectively, the masses of the external auditory canal, tympanic membrane, malleus, incus, stapes, and cochlear fluid. The mass of the external auditory canal corresponds to the volume of air that fills the canal. Masses M1, M2, M3, M4, M5, and M6 are suspended by massless springs and dashpots that simulate the ligaments, supporting muscles, and interfaces, and are represented in Figure 5, respectively, by K1, C1, K2, C2, K3, C3, K5, C5, K6, C6, K8, C8, C4, C7, C9, C10.

k c

m

f(t) x(t)

Figure 5. Diagram of a mass-spring model.

According to Equation (1), the following matrices for a damped mass-spring system are obtained:

Mass Matrix (8)

Damping (9) Matrix

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Stiffness (10) Matrix

From Equations 1, 2, 3, and 4 the following equation can be obtained:

푥 = −휔2(푙) × [푚] + 푗휔(푙) × [푐] + [푘] (11)

Where (l) is length, which varies from 1 to f. The value of f is between 250Hz and 8000Hz.

In Feng's work, A force equivalent to 90dB (SPL) was applied to this system. To obtain the force in Newtons (N) from the intensity in decibels (dB), we used Equation 12 to convert the 90dB into sound pressure. Then, the sound pressure value was substituted in Equation 13 to yield the force in N.

푃 푑퐵 = 20 log ( ) (12) 푃0

퐹 푃 = (13) 퐴

–5 2 P0 = 2 x 10 N/m

F = Force (N)

A = Area (m2)

P = Pressure (N/m2)

Equation 13 leads to the force in (N) based on the surface area of the tympanic membrane. In the literature, the total surface area of the tympanic membrane is considered to be 85mm2. However, only approximately 56mm2 of the total surface has motility. For that reason, surface area of 56mm2 was considered.

In Table 1and 2, we reported mass and stiffness value and damping value in the discrete mass-spring model representatively, from Feng and Gan's work (2004).

Table 1. Mass and stiffness value in the discrete mass-spring model. Mass (mg) Stiffness (N/m) M1 1.55 K1 1175 M2 2.70 K2 20001 M3 4.00 K3 9474 M4 4.00 K5 1000017 M5 1.78 K6 167 M6 25.50 K8 623

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Table 2. Damping value in the discrete mass-spring model. Damping Values (Ns/m) C1 0.00007 C2 0.5 C3 1.74 C4 0.122 C5 0.0216 C6 0.00036 C7 0.020 C8 0.00004 C9 0.1 C10 0.1

Second Mechanical Model of the Middle Ear In Figure 6, Liu and Neely(2010)'s model is summrasied. This Figure shows the mechanical model of the ear. The model of the middle ear is specified by red dash. This model, the equations are similar to first presented model.

The Middle

Ear

Figure 6. The mechanical model of the ear.

The eardrum–malleus–incus system is modelled by parameters {Mm, Rm, Km}. The malleus–incus lever ratio is 푔 ≤ 1. The joint between the incus and the stapes is modelled by parameters {Ri, Ki}. The stapes and its surrounding structures are modelled by parameters {Ms, Rs, Ks}. Pd and P(0) are coupled to each other via the following equations:

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Dampin Stiffnes s g Mass 푀푚 푣̇푚 = −퐾푚 푥푚 − 푅푚 푣푚 + 푔 푓푖 + 푃푑 퐴푒 (14)

(푀푠 + 푀푟)푣̇푠 = −(퐾푠 + 퐾푟)푥푠 − (푅푠 + 푅푟)푣푠 − 푓푖 − 푃(0)퐴푠 (15)

푓푖 = 퐾푖(푥푠 − 푔 푥푚) + 푅푖(푣푠 − 푔 푣푚) (16)

As is the effective area of the stapes footplate; xs and vs denote the displacement and the velocity of the stapes. Parameters {Mr, Rr, Kr} represent the round window.

Electrical Model of the Middle Ear

푒 This model proposed by Shera and Zweig (1992). As shown in Figure 7, a transfer function .푇̂푢 from ear canal to umbo relates the mechanical force Fu and velocity Vu at the umbo to the acoustical pressure Pe and volume velocity Ue within the ear canal just in front of the eardrum by

푃푒 − 푃푡푐 푒 퐹푢 ( ) = .푇̂푢 ( ) (17) 푈푒 푉푢

푢 The transfer function .푇̂표푤 from umbo to oval window relates the acoustical pressure Pow and volume velocity Uow just inside the cochlear at SV to the mechanical force and velocity at the umbo by

퐹푢 푢 푃표푤 − 푃푡푐 ( ) = .푇̂표푤 ( ) (18) 푉푢 푈표푤

Middle Ear Eardrum (Ossicular Chain) Inner Ear

Figure 7. Transfer matrix framework for the middle ear model.

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In Figure 8, the model of the middle ear is specified and its transfer matrix is obtained. The ossicular 푢 model for .푇̂표푤 in Equation (18) is based on a one-dimensional model of vibration transmission. This transfer matrix is evaluated by concatenating seven transfer matrices of its constituent parts from left to right in the circuit shown in Figure 8 as follows: 1 푗휔푚 + (푗휔퐶 )−1 + 푅 푢푇̂ = ( 푚 푚푚 푚푚) . 표푤 0 1

−1 푘2 0 1 0 × ( ) ( −1 −1 ) 0 푘2 [(푗휔퐶푚푚푖) + 푅푚푚푖] 1

1 푗휔푚푖 1 0 × ( ) ( −1 −1 ) 0 1 [(푗휔퐶푚푖푠) + 푅푚푖푠] 1 1 푗휔푚 + (푗휔퐶 )−1 + 푅 × ( 푠 푚푎푙 푚푎푙) 0 1

푘−1 0 × ( 3 ) (19) 0 푘3 The first matrix contains a series mechanical oscillator impedance to represent the malleus attached to the TM with malleus mass mm, mechanical compliance Cmm, and mechanical resistance Rmm.

The second matrix is a transformer with transformer ratio k2, which is interpreted as an ossicular lever ratio between malleus and incus. The third matrix contains a shunt admittance composed of the series combination of a mechanical compliance Cmmi and resistance Rmmi to represent the mechanics of the malleolar-incudo joint. The fourth matrix represents the impedance for the incus mass mi. The fifth matrix contains a shunt admittance composed of the series combination of a mechanical compliance Cmis and resistance Rmis to represent the mechanics of the incudo-stapedial joint.

The sixth matrix is a series mechanical oscillator with elements ms to represent stapes mass, and a mechanical compliance Cmal and resistance Rmal to represent the annular ligament. The seventh matrix is a transformer from force variables back to acoustical variables within the cochlear fluid with transformer ratio k3 defined by

−1 푘3 = 푆푠 (20)

k3 is the inverse of the stapes footplate area Ss.

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Figure 8. Transfer matrix model of middle ear for eardrum and ossicular chain. Boundary condition are represented at the tympanic cleft for the eardrum, and oval and round windows.

Summary The function of the human is to transform the mechanical sound wave collected from the external environment into electrical impulses that are sent to the brain, where the auditory information is interpreted. The human ear has a complex micro-structure with several components whose purpose is difficult to characterize in terms of the overall function of the ear. One of the parts of the ear is the middle ear. The human middle ear consists of three ossicles: the malleus, the incus, and the stapes. The ossicles form a sound conduction system that transmits sound from the external ear to the fluids of the inner ear. The ossicles are connected to each other by the incudo-mallear and incudo-stapedial joints. The ossicular chain is supported by two muscles: the , attached with its tendon to the handle of the malleus, and the , attached to the stapes neck or posterior crus. The malleus is also firmly connected to the tympanic membrane, whereas the stapes is attached to the bony walls of the oval window by an annular ligament forming a stapedio-vestibular junction. The eardrum (tympanic membrane), and the neighboring cavity with three tiny bones (ossicular chain) comprise the middle ear. The sound waves travelling through the ear canal causes the vibrations of the tympanic membrane, and these vibrations are further transmitted through the chain of bones towards the inner ear. For the eardrum to function optimally, it should not be bent either inwards or outwards in the absence of the sound waves. This means that the air pressure should be the same in the outer and middle ear. The pressure equalization is achieved through the eustachian tube. A tube which connects the middle ear cavity with the back of the throat. Normally this tube is closed, but it opens with swallowing or shewing, thus working as a pressure equalizer. The middle ear bones (malleus, incus and stapes) are the smallest bones in the body. They work together as a lever system, to amplify the force of the vibrations. The malleus is attached to the tympanic membrane, the stapes enters the oval window of the inner ear, and the incus lies in between.

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References

 Feng B, Gan RZ. Lumped parametric model of the human ear for sound transmission. Biomechanics and modeling in mechanobiology. 2004; 3(1):33-47.

 Liu Y-W, Neely ST. Distortion product emissions from a cochlear model with nonlinear mechanoelectrical transduction in outer hair cells. J Acoust Soc A