Physics of middle ear

 

Principle

 

The vibration of the tympanic is transmitted by the three middle ear ossicles (consecutively malleus or hammer, incus (anvil), stapes or footplate; Fig. 1). The middle ear (also called meatus internus) with the three bones acts nearly perfectly as a pressure to displacement to pressure transducer with a sensitivity which is at quantum mechanical level. The acoustic power P  which enters the ear through the eardrum at the threshold of hearing (12 μPa at 4 kHz) is very small, about 12 aW (1 attowatt = 10–18 watt) and equivalent to an IR photon of 1660 nm in one second or the fluorescence of one chromophore transition/s from e.g. 527 to 400 nm.

This can be calculated from the relation between the intensity I (W/m2) and the pressure p:

I = pv = p(p/Z) = p2/Z                                                                                                                            (1a)

And since power is intensity times area, it follows that::

P = AI = πr2p2/Z.                                                                                                                                    (1b)

Completing yields 11.7 aW. For 1 kHz and 20 μP 32.5 aW is found.

 

 

Fig. 1    Middle ear.

 

The tympanic membrane is characterized by a large surface and low impedance (or vice versa), the oval window by a small surface and high impedance. Generally speaking, going from a low to high impedance, there wouldn't be hardly any transfer at all: the sound wave reflects. However, the ear has a very high sensitivity, up to the level which is set by the physics of quantum mechanics as described above. The problem is solved by impedance adjustment which must take place in the middle ear (sometimes called the meatus internus). Impedance adjustment might be realized by increasing the pressure acting on the membrane of the oval window. Here, this pressure is transformed to displacement in the cochlear partitions.

Three principles are used to adjust the impedance in the middle ear:

1) The first principle, the area ratio tympanum/oval window is by far the most important one. The surface of the tympanic membrane is much larger than the surface of the oval window, which is practically the surface of the stapes. Knowing from physics, the force (F) produced by sound pressure (p) working on an area (A) is given by the formula:

F = pA,                                                                                                                                                                   (2a)

and assuming no losses, the sound pressure on the oval window (ow) of the scala vestibuli pow is:

pow = F/Aow.                                                                                                                                                            (2b)

The result is that pow is considerably larger than the pressure at the tympanic pt since Aow < At. Since F is preserved the gain in pressure is:

g = pow/pt = At/Aow.                                                                                                                                                (2c)

The pressure at the tympanic membrane must be multiplied by a factor g in order to obtain the value for the pressure at the cochlea. Finally, g, the “gain” in pressure on the window is ca. 28 dB.

The gain is in pressure rather than in displacement since the ossicles can enlarge displacements only marginally.  Simulations with a physical model showed that the displacement of the malleus and the stapes are rather the same, be it that for different frequencies with the same pressure at the tympanum the displacements differ more than a factor 10 (Hudde and Weistenhofer, 1906) .

2) A second factor is the lever action of the middle ear bones. The arm of the malleus is longer than the arm of the incus. Normally a lever is in balance, i.e. the sum of the moment (M = F∙L) of both arms is zero:

ΣMn = 0, or                                                                                                                                                         (3a)

│Fmalleus ∙ Lmalleus = │Fincus ∙ Lincus│,                                                                                                 (3b)

where L is length.

This causes a gain with a factor 1.3 (1 dB) at the stapes.

3) The third factor depends upon the conical shape of the tympanic membrane. Less surface moves due to the elasticity of the membrane when the stapes pushes the oval window in and out. This also increases the pressure, finally transformed to displacement of the oval window.

The first system is by far the most important one. This contributes to a gain of approximately 28 dB to the overall displacement gain of about 30 dB.

 

Another approach is to study qualitatively the action of the middle ear with a mechanical model (Fig. 2a), which can also be calculated through, although many parameters are numerically not known. Therefore its action is hard to evaluate and has hardly practical value. However, we learn from it that the implicit assumption of the above description that all is frequency independent cannot hold. Systems with a mass or spring dictate that there is frequency dependency. That the transfer is frequency dependent is obvious from the model presented in Fig. 2a and also from the experimental results given in Fig. 2b. The later shows that only for a frequency of nearly 1 kHz the theoretical gain of about 30 dB is approximated.

 

 

 

Fig. 2 (a)   Mechanical model of the middle ear. The incus is fixed to the stapes, such that there is no cantilever function. The ineffective or unused motion is also modeled by the components indicated by “loss”. (b)  Top: Ratio of fluid pressure in the scala vestibuli to pressure in the ear canal in decibels (dB). The inset shows the click response. Bottom: phase characteristic. Numbers 1-4 denote asymptotes (see More Info).

 

 

Application

 

Knowledge of the biomechanics of the middle ear is of importance for e.g. optimizing replacement of ossicles (ossiculoplasty).

In terrestrial mammals, the high-frequency hearing limit (fH) is roughly inversely proportional to the cubic root of the mass of the vibrating structures, i.e. fH ~ W ─3, where W is the summed mass of malleus and incus. This means that an ossiculoplasty should be done with material as light as possible.

 

 

More Info

 

Frequency dependency of middle ear transduction

The gain characteristic of Fig. 2b can very roughly, as a first approximation, be interpreted as being caused by a band-pass second order filter, and indicated by the three asymptotes 1 to 3. At low frequencies it gives rise to an asymptotic phase lead of 90o and at the high end of a 90 o phase lag. But above about 5 kHz three poles (and possible more) are added. There action is indicated by the asymptote number 4, resulting in a final lag of 360o as the phase characteristic suggests.

Comparative aspects and a functional model of the mammalian outer and middle ear

Morphometric measurements have shown that there is a relation between the radius of the idealized tympanum rt and the radius of the idealized middle ear volume rv:

                rv = 0.05 + 1.685 rt.                                                                                                                              (4a)

Similarly there is a relation between the wavelength λ of the ear canal resonance frequency fd and rt.:

                λ = - 0.45 + 2.718 rt.                                                                                                                             (4b)

The basic assumptions of a generalized model of the middle ear are (Plassmann and Brändle, 1992):

1.     the middle ear behaves as a Helmholz resonator with resonance frequency fb. This is a hollow solid sphere filled with a gas at much higher pressure than the ambient pressure. Via a small tube with a valve the sphere can be depressurized. When the valve is opened very fast, the process of depressurizing is not continuous but intermittent with a certain frequency, i.e. the gas outflow resonates. This is based at adiabatic expansion (see Adiabatic compression and expansion)

2.     both structures resonate with the same frequency, i.e. they are coupled:

fd = fb.                                                                                                                                 (5)

Both frequencies can be expressed by equations comprising anatomical parameters and material constants. With some assumptions about the tympanum (thin, uniform circular fixed at the rim, negligible stiff, ideal elastic and uniform tension) it is found that:

                fd = H∙M/(2πrt) where M = √(T/m)                                                                                                      (6)

H is a constant (= 2.405) holding for membranes like the one defined. M is a material constant derived from the tension T per unit length and mass m per unit area of the membrane.

For fb of the Helmholtz resonator it holds that:

                fb = (c/2π)√(Sm/L’V), L’ = 1.5∙rm + ln,                                                                                                  (7)

where c is the sound velocity of air (352.9 m/s at 36 oC), rm the radius of the neck opening, Sm the neck opening area into the meatus, L’ the effective length of the resonator neck, ln, the morphological neck length and V the middle ear volume.

By applying the model of Eq. 5 to a number of species, some general constants and species specific parameters could be estimated. By comparing them with measured parameters the model could be verified. For Homo sapiens it was found that fd = 2.9 kHz, rv = 1.93 mL, meatus cross area 44 mm2, meatus length 22.5 mm and rt = 3.22 mm. These values are rather close to the empirical values.

 

References

Hudde H, Weistenhofer C. Key features of the human middle ear. ORL J Otorhinolaryngol Relat Spec. 2006;68:324-8. Review.

Mammano F. and Nobili R.,  http://147.162.36.50/cochlea/index.htm and http://147.162.36.50/cochlea/cochleapages/theory/index.htm

Plassmann W and Brändle K. A functional model of the auditory system in mammals and its evolutionary implications. In: The evolutionary biology of hearing, Webster DB, Fay RR and Popper AN (eds.), 1992, pp 637-653. Springer, New York.