Download The traveling wave in the human inner ear studied by means of a

23rd International Congress on Sound & Vibration
Athens,
Greece
10-14 July 2016
ICSV23
THE TRAVELING WAVE IN THE HUMAN INNER EAR STUDIED BY MEANS OF A FINITE-ELEMENT MODEL INCLUDING
MIDDLE AND OUTER EAR
Johannes Baumgart
Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
email: [email protected]
Mario Fleischer
Technische Universität Dresden, Department of Medicine Carl Gustav Carus,
Otorhinolaryngology, Dresden, Germany
Charles Steele
Mechanical Engineering, Stanford University, Stanford, CA, USA
The mammalian hearing organ is a remarkable biophysical system, which converts the incoming
sound into a neural signal. The main components in the signal chain are the middle ear, the
basilar membrane, and the organ of Corti. The middle ear matches the impedance of the liquid
filled inner ear to the surrounding air. If a tone reaches the ear, a traveling wave builds up along
the basilar membrane, which peaks at a frequency dependent location due to multiple gradients
in stiffness and geometry. Atop of the basilar membrane resides the organ of Corti wherein the
mechanotransduction process takes place.
Here we study the mechanics of the traveling wave by a detailed three-dimensional finiteelement model. The model is based on geometry data from µCT-scans of a human ear and the
mechanical properties of the basilar membrane are adjusted to provide the known tonotopic map.
Furthermore, we ensure that the orthotropic material of the basilar membrane has a compressibility similar to water. In the current version of the model we omit the organ of Corti with its various
cell types but add a model of the middle ear and the ear canal filled with an acoustic fluid and
attached infinite elements to represent the outer ear.
This model allows us to compare different load cases and we analyzed for them forces and
motions. First we studied the case of normal passive hearing by exciting our model by an oscillating pressure at the entrance of the ear canal to validate our model. In a second scenario we
applied an oscillatory force locally on the basilar membrane and analyzed the pressure in the ear
canal. This provides some estimate how much energy is dissipated of the active process inside the
organ of Corti on the way to the outer ear canal.
1.
Introduction
The mechanics of the traveling wave was studied in detail experimentally by Békésy [1]. This
pioneering work was followed by models to capture the mechanics of the traveling wave and taking
care of the three dimensional nature of the problem, e.g. [2, 3]. But the approach with analytical
functions has limitations with representing the geometrical details of the coiled cochlea and with the
oval window, which connects the cochlear to the middle ear (see Fig. 2). Due to the development
of numerical methods it was later possible to study the mechanics of the traveling wave on arbitrary
1
The 23rd International Congress of Sound and Vibration
Figure 1: Finite-element model of the outer, middle, and inner ear. The outer ear canal (orange)
has a half sphere attached at the ear canal outlet. The ossicles are schematically represented by bar
structures. The geometry of the inner ear (light brown) is from µCT-scans of a human ear and has a
simplified geometry for the coupling region to the oval window.
geometries [4]. Based on the research questions different strategies were used, e.g. [5, 6, 7, 8]. Here
we focus on a model, which captures the known geometry of the human cochlea including the threedimensional nature of the basilar membrane with orthotropic and spatial dependent properties and
the liquid is modeled as compressible, viscous and inert. We employ the finite-element method to
discretize the differential equations of mass and force balance in space. This model of the cochlea
we couple to an available finite-element model of the middle and outer ear [9]. This model is able to
capture the known geometry and to make predictions about the energy transfer for local excitations
on the basilar membrane.
2.
Model
2.1
Geometry of the model
The geometry data are obtained from µCT-scans of a human ear. For simplification, in the current
version of the model we omit the organ of Corti with its various cell types for the geometry to simplify
the meshing. As we tune the properties of the model to experiments with an organ of Corti the
properties of the basilar membrane are lumped properties of the actual basilar membrane and organ
of Corti. The model of the inner ear is coupled to a model of the middle ear and the ear canal filled
with an acoustic fluid and with attached infinite elements to represent the outer ear (see Fig. 1).
The basilar membrane is about 3 cm long and the volume of the inner ear liquids measures in total
about 107 pL. The ear canal has a diameter of about 1 cm and a length of about 3 cm. At the ear
canal outlet a half sphere is attached to provide a basis for the infinite elements. The entire model is
discretized and modeled by using the finite-element software Ansys (v. 15.0, 2014).
2.2
Mechanical properties of the model
The inner ear is filled with endolymph and perilymph. The mechanical properties are close to
water in terms of density, bulk modulus, and viscosity. For the model here we use a dynamic viscosity
of η = 1 mPa·s, a bulk modulus of K = 2 GPa, and a density of ρ = 1 pg/µm3 .
For the two regions of the basilar membrane, the zona pectinata and zona arcuata (see Fig. 2),
2
ICSV23, Athens (Greece), 10-14 July 2016
The 23rd International Congress of Sound and Vibration
Oval window
Scala vestibuli
Scala media
Zona arcuata
Basilar membrane
Zona pectinata
Round window
Scala tympani
Figure 2: Sketch of the cross-section of the cochlea. There are two fluid channels: the upper one with
scala vestibuli and scala media and the lower one with the scala tympani. The upper one is connected
to the middle ear by the oval window and the lower one by the round window. The basilar membrane
consists of the two regions zona arcuata and zona pectinata. Atop the basilar membrane resides the
organ of Corti (not shown).
we use different material descriptions as the ultrastructure differs. We use here a similar material
description as used by Fleischer et al. [10]. The zona arcuata we model with a Young’s modulus of
E = 30 kPa and a bulk modulus of K = 2 GPa over the entire length. The zona pectinata we model
with a transverse isotropic material to account for the fibers in the radial direction. The compliance
matrix reads


νrs
νrs
1
−
−
Er
Er
 Er
1
νst 
−
(1)
C = − νErsr
Es
Es  ,
νst
1
νrs
− Er − Es
Es
where Er is the Young’s modulus in the radial direction and Es in the transverse directions (a long the
length and normal to the basilar membrane). The Poisson ratio for the directions normal to the radial
direction is set to
Es (Er + 3 K)
νst = 1 −
,
(2)
6 Er K
where K is the bulk modulus. The Poisson ratios combining the radial with the other directions are
νrs =
1 1 Er
−
.
2 6 K
(3)
By this we ensure that if we put a pressure load on a block of material the effective bulk modulus will
be K. The Young’s modulus for the transverse direction is set to Es = 30 kPa and the bulk modulus
to K = 2 GPa over the entire length. The Young’s modulus in the radial direction is a function of
position and decays logarithmical from Erbasal = 1 MPa to Erapical = 30 kPa. For both zones we use
a density of ρ = 1 pg/µm3 and damping is implemented by a damping matrix, which is proportional
to the stiffness matrix with the coefficient β = 0.01. The geometry and properties of the outer and
middle ear were taken without modifications from [9].
2.3
Validation of the model
For the inner ear model we use for the liquid a user-defined element as described in [11] and
employed for similar problems [12, 13]. For the basilar membrane we use the solid186 element as
ICSV23, Athens (Greece), 10-14 July 2016
3
The 23rd International Congress of Sound and Vibration
10
9
F /a (ng)
x
10 6
x
F/a (ng)
F y/a x (ng)
F z/a x (ng)
z
y
F y/a y (ng)
F z/a y (ng)
10 3
F z/a z (ng)
x
10 0
0.1
1
Frequency (kHz)
10
Figure 3: Left: Force with respect to acceleration at the footplate as a function of the frequency for
different excitation directions and resulting acceleration components. Right: The force F excites the
traveling wave by acting either normal to the plane of the oval window (z-direction), along the major
axis (x-direction), or along the minor axis (y-direction) on the footplate plate
provided by Ansys (v. 15.0, 2014). This is a higher order 3-D 20-node solid element that exhibits
quadratic displacement behavior. The mesh is fine enough to resolve the viscous boundary layer for
frequencies up to 10 kHz.
With our model we can reproduce a tonotopic map as known from experiments for human [14].
If we block the motion at the oval window and enforce a piston like motion on the round window, we
excite a pure compression mode of the liquids and the basilar membrane. Estimating a quasi-static
response yields an effective bulk modulus of around K = 2.2 GPa, a value close to the prescribed
value of 2.0 GPa.
3.
Results
3.1
Excitation at the footplate
At first we excite the inner ear model with a force acting on the footplate in all three directions
(see Fig. 3). We analyze the effective mass for all 6 possible combinations by computing the ratio of
force to acceleration. Only for a force acting normal to the oval window a significant part of the fluid
mass of the inner ear gets excited in the same direction. All other modes are by order of magnitudes
smaller.
3.2
Excitation at the outer ear
To model the complete ear we added the middle and outer ear. With this model we excited with
an oscillatory pressure at the ear canal outlet and analyzed the response of the basilar membrane (see
Fig. 4). The peak is maximal for the 1-kHz excitation. In all cases the amplitude builds up slowly
until the peak and decays afterwards rapidly in the region with steep phase changes.
3.3
Excitation on the basilar membrane
Finally, to estimate the sound produced by an active process in the cochlea we studied how much
energy travels outwards and compared the ratio of the two (see Fig. 5). The supplied power we
estimated based on the force we applied locally and the computed local velocity. We excited at
locations, which were identified based on the location of the maxima in the previous computation.
The output was estimated by extracting from the computation the pressure at the ear canal outlet and
4
ICSV23, Athens (Greece), 10-14 July 2016
The 23rd International Congress of Sound and Vibration
10 1
10 0
Ang(uBM/pIN) (Cycles)
|u BM/pIN| (mm"s -1"Pa-1)
0
10 -1
10 -2
10 -3
-2
-4
-6
-8
10 -4
0.25
0.5
1
2
4
-10
10
20
Distance on BM (mm)
30
10
20
Distance on BM (mm)
30
Figure 4: Amplitude and phase of the velocity of the basilar membrane along the length. Left:
Amplitude normalized by the pressure amplitude at the ear canal outlet. Right: Unwrapped phase of
the same quantity. Colors indicate excitation frequency in kHz.
Ratio Pout/P in
10 -3
10 -4
0.5
1
2
3
4
6
8
10 -5
10 -6
10 -1
10 0
Frequency (kHz)
10 1
Figure 5: The acoustic power estimate at the ear canal outlet is compared to the power estimate of the
locally on the basilar membrane applied force. The place of excitation corresponds to the respective
best frequency location. Colors indicate excitation frequency in kHz.
ICSV23, Athens (Greece), 10-14 July 2016
5
The 23rd International Congress of Sound and Vibration
assuming a plane wave in air. In this model only a small fraction of the power travels outwards. So
far the location of major losses could not be identified and needs further investigation.
REFERENCES
1. Békésy, G. v., Experiments in hearing, American Institut of Physics (1989).
2. Lighthill, J. Energy flow in the cochlea, Journal of fluid mechanics, 106, 149–213, (1981).
3. Steele, C. R. and Taber, L. A. Comparison of wkb calculations and experimental results for threedimensional cochlear models, The Journal of the Acoustical Society of America, 65 (4), 1007–1018, (1979).
4. Givelberg, E. and Bunn, J. A comprehensive three-dimensional model of the cochlea, Journal of Computational Physics, 191 (2), 377–391, (2003).
5. Steele, C. R., de Monvel, J. B. and Puria, S. A multiscale mode of the organ of corti, Journal of Mechanics
of Materials and Structures, 4, 755–778, (2009).
6. Edom, E., Obrist, D., Henniger, R., Kleiser, L., Sim, J. H. and Huber, A. M. The effect of rocking stapes
motions on the cochlear fluid flow and on the basilar membrane motion., J Acoust Soc Am, 134 (5), 3749–
3758, (2013).
7. Cheng, L., White, R. D. and Grosh, K. Three dimensional viscous finite element formulation for acoustic
fluid structure interaction., Comput Methods Appl Mech Eng, 197 (49-50), 4160–4172, (2008).
8. Cai, H., Shoelson, B. and Chadwick, R. S. Evidence of tectorial membrane radial motion in a propagating
mode of a complex cochlear model, PNAS, 101 (16), 6243–6248, (2004).
9. Bornitz, M., Hardtke, H.-J. and Zahnert, T. Evaluation of implantable actuators by means of a middle ear
simulation model, Hearing research, 263 (1), 145–151, (2010).
10. Fleischer, M., Schmidt, R. and Gummer, A. W. Compliance profiles derived from a three-dimensional
finite-element model of the basilar membrane, The Journal of the Acoustical Society of America, 127 (5),
2973–2991, (2010).
11. Baumgart, J., The Hair Bundle: Fluid-Structure Interaction in the Inner Ear, Ph.D. thesis, Technische
Universität Dresden, (2010).
12. Ni, G., Elliott, S. J. and Baumgart, J. Finite-element model of the active organ of corti, Journal of The
Royal Society Interface, 13 (115), (2016).
13. Kozlov, A. S., Baumgart, J., Risler, T., Versteegh, C. P. C. and Hudspeth, A. J. Forces between clustered
stereocilia minimize friction in the ear on a subnanometre scale., Nature, 474 (7351), 376–379, (2011).
14. Greenwood, D. D. A cochlear frequency-position function for several species—29 years later, The Journal
of the Acoustical Society of America, 87 (6), 2592–2605, (1990).
6
ICSV23, Athens (Greece), 10-14 July 2016