Buenos Aires – 5 to 9 September, 2016 Acoustics for the 21st Century…

PROCEEDINGS of the 22nd International Congress on Acoustics

Biomedical Acoustics: Paper ICA2016-284

The variable cochlear hydro-mechanical inertance Santos Tieso(a), Lucas Fantini(a), Francisco Messina(a), Nahuel Cacavelos(a), Gilda

Farelli(a), Leonardo Zavala(a), Maria Tieso(a), Sebastian Iezzi(a), Nicolás Casco Richiedei(a) (a) Universidad Nacional de Tres de Febrero, Argentina, valentinolucasfantini@gmail.com

Abstract

In this paper, the way in which the cochlear changes its inertance when stimulated by different frequencies is studied and explained. This phenomenon allows the human’s ear to have an eleven octave frequency range. The cochlear hydro-mechanical inertance is defined as the density of the liquid contained inside the cochlea, set in motion with a particular geometry. This geometry is modified by histo-anatomic structures in response to the different stimulus frequency. The study of the fluid dynamics inside the cochlea will allow a deeper understanding of the way the mammal's ears work as well as providing new ways to treat diseases or injuries.

Keywords: cochlear mechanics, fluid dynamics, theoretical model, physiology.

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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The variable cochlear hydro-mechanical inertance

1 Introduction When applied a certain sinusoidal pressure difference to the windows of an harmonic hydromechanical system (H. H. S.), it is transferred to its liquid fluid a heterogeneous pressure field, which generates in this an inertial movement that we named hydromechanical flow (Qh) whose magnitude depends of the hydromechanical impedance that the system presents to this stimulus.

This is because the liquid fluid that is part of a (H. H. S.) it behaves like an inertance, it means a density with geometry (ρ.l/A), which is the inelastic component and lacking of resistance that conform the hydromechanical impedance.

Be able to clarify the liquid fluid movement of the inner ear, is an essential step to decode the entire complex cochlear performance because it is the responsible of the ciliary movement, which depends on how much of chemical mediator that the ciliated cell releases into the synapse with the bipolar neuron of the Organ of Corti.

The inertance modification by the inner ear in response to the different frequencies, it occurs through of the Basilar Membrane whose structured anatomy attends to two restrictions of design, length and elasticity.

Regarding to the length restriction we know that in an average adult it is 3.5 cm, while as regards to the elasticity coefficient varies from 1 to 100 at that distance from the apex to the base.

The windows position respect to the elasticities distribution in the (B.M) determines the decrease of the inertance value with every increase of the stimulus frequency in the order of 1 to 10 from the apex to the base.

2 Methods With the purpose of simplifying the analysis of the harmonic hydro-mechanical motion, the simple harmonic hydro-mechanical system S.H.H.S. is defined as a rectangular recipient, with rigid walls, two circular windows (oval and round) with equal areas placed in opposite walls from each other and closed by membranes with equal elasticity and resistance, only if it meets the five restrictions listed below [2]:

• The distance between windows must be much smaller than the stimulus’s wavelength. 𝑙 ≪ 𝜆

• The square root of the window’s area must be much smaller than the stimulus’s wavelength. 𝐴 ≪ 𝜆

• The cube root of the 95% of the liquid in motion must be much smaller than the stimulus’s wavelength. 𝑉! ≪ 𝜆

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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• The windows area must be smaller than the area of the walls in which they are carved.

• The displacement or volumetric elongation must be small.

Source: (S. Tieso et al., 2013)

Figure 1: Two-dimensional graphic representation of: (a) of a simple harmonic hydro-mechanical system and (b) its electrical analogy

It is essential to describe the liquid’s motion in this system, in order to understand how the cochlea works. But this description is not enough to decipher the totality of the complex cochlear functionality.

The inner ear is not related with the acoustic wave through the characteristic impedance of its liquid, because its small elasticity in relation to the air is compensated with the elasticity of the windows. The windows motion produces a laminar flow to the liquid.

The stimulus of any (H. H. S.) is the harmonic pressure difference (Δp) applied to their windows. This stimulus not changes the density of the liquid. We define the hydromechanical impedance (Zh) as the relation between the pressure difference applied to the system and the hydromechanical transferred flow (Qh) as is shown in equation (1).

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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𝑍! =∆𝑝𝑄!

(1)

𝑍! = 𝑅! + 𝑗 𝑚!𝜔 −𝑘!𝜔

(2)

In the equation 2, the hydromechanical inertance is defined by the equation 3.

𝑚! =𝜌. 𝑙𝐴 (3)

Where ρ is the fluid density, l is the distance between the windows and A the average area of the moving liquid.

In any (H. H. S.) the pressure difference transferred to the liquid (∆pmh) relates to the volumetric acceleration of it through of the mh as is shown in equation 4.

𝑚! =∆𝑝!!𝑎!

(4)

Then the hydromechanical inertant reactance 𝑋!! is defined by the equation 5.

𝑋!! = 𝑗𝑚!𝜔 =∆𝑝!!𝑄!

(5)

A S.H.H.S. has a single resonant frequency. When the system is stimulated at that frequency, all the particles oscillate in phase with the Δp applied to the system.

In a S.H.H.S. the value of the hydromechanical inertance does not change. Then, the inertant reactance is proportionally to the frequency of stimuli.

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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In a S.H.H.S. stiffness is provided by the windows (oval and round). The hydromechanical elastic reactance Xkh is inversely proportional to the frequency of stimuli and is defined by the equation 6.

𝑋!! = −𝑗𝑘!𝜔 (6)

The phase and module of the flow inside the system in response to the stimulus depends on the quality factor of the system while the inertance and the stiffness of the system remain constants.

In the harmonic movement of a S.H.H.S. all of its acoustic particles oscillate in the same phase, which with their respective modules and directions determine the geometry of the inertial movement or flow. This geometry is the inertance geometry.

The natural frequency f0 of a S.H.H.S. is defined by equation 7.

𝑓! =12𝜋

𝑘!𝑚!

(7)

When the frequency of the stimulus is not 𝑓! a phase shift between the acoustic particles and the pressure difference of the stimulus occurs.

A complex hydromechanical harmonical system C.H.H.S. is defined like a S.H.H.S. but containing a basilar membrane (b. m.) with different elasticities distributed on its length positioned between the two windows as is shown in figure 2.

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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Source: (S. Tieso et al.,2016) Figure 2: Complex Hydromechanical Harmonical System (C.H.H.S.).

The differential pressure stimulus arrives in phase to the entire membrane. The b.m. segments has different elasticities (with a relation of 100:1 from base to apex). Then the flow occurs through the segment have different modules and phase, but only in the resonant segment, the Qh transferred to the b.m. is in phase with the stimuli. The particles that oscillate in phase with each other form the mh that corresponds to the resonance frequency. The figure 3 shows an Qh in resonance with a segment of the b. m. that forms the mh.

Source: (S. Tieso et al.,2016)

Figure 3: C.H.H.S. with a low frequency resonance and its mh (the geometry of the Qh).

For those segments that have a higher frequency resonance (closer to the base), the Qh phase is advanced respect to the stimuli, and for those that have a lower frequency resonance (farther to the base) the Qh phase is delayed. Due to this, acoustic particles of the liquid oscillate with different phases.

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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The cochlea is very similar to a C.H.H.S defined in this work. The oval and round windows positions in relation to the distributed elasticities on the b.m. allow the mh decrease when the stimulus frequency increases (see Figure 4).

Source: (S. Tieso et al., 2016)

Figure 4: Two-dimensional graphic representation of a C.H.H.S.

The flow containing the acoustical particles that oscillate with the same phase, it cannot be considered as a mass vibration. In a Qh the particles oscillate with different velocities and directions while in a mass vibration all the particles oscillate with the same velocity and direction.

3 Conclusions In the present work a theoretical model of the liquid motion inside inner ear is presented. When the inner ear is stimulated with a given frequency and sound pressure level, a differential pressure arrives in phase to the entire b.m. Then, an oscillatory movement is defined by the elasticity and resistance of the different segments and the hydromechanical inertance of the flow as well.

The oscillatory movement of the entire b.m. is similar to a wave that travels from the base to the apex, with a maximum displacement in the resonant segment.

References [1] Tieso, S. Harmonic Hydro-mechanical Movement. Journal of the Acoustical Society of America,

volume (19), 2013.

[2] Paparella, M. Otorrinolaringology. W B Saunders Co, NY (USA)??, 2nd edition, 1980.

[3] Tieso, S. Fisiologia auditiva. Corrales, Bs As (ARG), 1st edition, 2006.

[4] Withe, F. Fluid Mechanics. McGraw Hill, NY (USA),1st edition, 1979.

[5] Resnick, R. Fisica Volumen 1. Continental S.A, Bs As (ARG), 4th edition, 2001.

[6] Recuero, M. Ingenieria acustica. Paraninfo S.A., NY (USA), 4th edition, 1995.

[7] Boylestad, L. Introductory Circuit Analysis. Prentice Hall, NY (USA), 2nd edition, 2010.

22nd International Congress on Acoustics, ICA 2016 Buenos Aires – 5 to 9 September, 2016

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[8] Mataix, C. Mecánica de fluidos y maquinas hidráulicas, Alfaomega, Bs As (ARG), 2nd edition, 2010.

[9] Churchill, R. Complex Variables and applications, Mc Graw Hill, NY (USA), 2nd edition, 1960.