Download the effect of the "third window" on the response of a 2d model

THE EFFECT OF THE "THIRD WINDOW" ON THE RESPONSE
OF A 2D MODEL OF THE COCHLEA
Alice Halpin, Stephen Elliott, Guangjian Ni
ISVR, University of Southampton, Southampton, SO17 3BJ
email: [email protected]
There are multiple mechanisms in which the healthy cochlea can be excited, including air
con-duction, bone conduction and direct non-volumetric excitation of either of the
windows. A 2D box model of the passive cochlea, solved using a finite difference method,
has been used to investigate the basilar membrane response and fluid pressure in the cochlea
due to several of these mechanisms. It is shown that all the investigated mechanisms of
stimulating the cochlea produce a similar pattern of travelling wave along the basilar
membrane, however, the magnitude of this response varied with excitation mechanism. It is
also shown that the cochlea is still responsive to some of these mechanisms when it is
diseased in such a way that the oval window becomes stiff. The effect of including a
cochlear aqueduct and vestibular aqueduct, or ’third window’, in the model was investigated
and shown to have a significant effect to the magnitude of the basi-lar membrane response
when excited by some of these mechanisms when the oval window is assumed to be stiff.
1.
Introduction
It has been shown that the mammalian hearing organ, the cochlea, can be excited by a number
of mechanisms in order to produced a hearing sensation, including the conventional route of air conduction hearing, bone conduction, BC, and excitation at the round window [1], [2]. The cochlea has
a coiled complicated 3D geometry, with many delicate structures inside. It has been suggested that
some of these structure which are fluid outlets from the cochlea, collectively know as the "third window", could have a significant effect on the basilar membrane, BM, response, and therefore hearing
response, when the cochlea is excited by certain mechanisms [3]. A 2D finite difference box model
of the cochlea has been used here to invesitgate a number of mechanisms of exciting the cochlea
and the effect of two of these cochlea structures, the vestibular aqueduct, VA, and cochlear aqueduct,
CA, on the BM response. The simple design of the model allows it to be easily adapted to simulate
the difference cochlear excitation mechanisms as well as to compared the BM response within one
framework.
Air conduction hearing involves a pressure waves travelling down the ear canal, vibrating the
middle ear ossicles and causing the stapes to vibrated the oval window. It is often assumed that
the stapes has a piston-like motion, even though this motion has been shown to be complex at high
frequencies [4]. Therefore, two motions of exiting the cochlea at the oval window have been examined
here, a piston-like motion and a rocking motion of the stapes, which has previously been investigated
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using a finite element model [5]. Bone conduction, BC, hearing is where vibrations of the skull are
transferred by various pathways to the cochlea. BC hearing have been separated into five pathways
[1], the two inertial pathways are simulated here: the fluid inertia and the middle ear component. The
middle ear inertia component is when the vibration of the skull, and hence the temporal bone, cause
a relative motion between the cochlea and the stapes. While the fluid inertia component is due to the
vibrations of the temporal bone imparting inertial forces on the the cochlear fluid.
Conductive hearing loss can be due to diseases such as otoscelerosis, which cause bone to grow
around the oval window area, immobilising the stapes. It is clear that any pathway that would excite
the cochlear via the oval window would not be possible in patients with these conditions, however,
other mechanisms of exciting the cochlea could still be possible. It is a well documented phenomena
that the BC hearing thresholds of patients with otoscelerosis are typical of a healthy ear, apart from
around 2 kHz where the threshold increases, called Carharts notch. It also has been shown that
driving the round window with a transducer that only partially occludes the window, can excite the
cochlea, and is one possible mechanism of exciting a cochlea with an immobile oval window [2].
Here, two mechanisms of exciting the cochlea, when a stiff oval window is defined, are investigated;
the fluid inertia component of BC and local excitation of the round window, where only the middle
section of the round window is driven.
The 2D model was solved and the BM responses due to each of the excitation mechanisms calculated. The VA and CA are modelled as complaint elements in the cochlear walls. The effect of these
aqueducts on the BM frequency response, due to the various cochlear excitation mechanisms, was
investigated.
2.
The Model
The complex 3D coiled geometry of the cochlea is simplified to a 2D box model, where a passive
BM separates a fluid filled box into two chambers, the scala vestibuli and scala tympani, terminating
at the helicotrema which is located at the apex of the cochlea, such that the BM does not extend to
the end cochlea wall. The oval window and vestibuli aqueduct is located on the upper cochlea wall
and the round window and CA on the lower. The CA is located in very close proximity, 0.2 mm, to
the round window, to reflex the real geometry [6] while the vestibuli aqueduct is located close, but
slightly further away, 0.5 mm, from the oval window, as shown in Fig. 1.
,ǀ
nj KǀĂůtŝŶĚŽǁ
sĞƐƚŝďƵůĂƌ
ƋƵĞĚƵĐƚ
pd x,Ν
mo
vBM x,Ν
Ϭ
Ͳ,ƚ
Ϭ
>
ZŽƵŶĚtŝŶĚŽǁ ŽĐŚůĞĂƌ
ƋƵĞĚƵĐƚ
dž
cBM x
kBM x
vw x,Ν
Figure 1: Left: diagram of the 2D box model of the cochlea, where the thick black line is the basilar membrane, the thick black dotted lines are the cochlear windows and the brown dotted line the
aqueducts. Right: diagram of one single degree of freedom micromechanical model for this passive
basilar membrane.
It is assumed that the cochlear fluid is inviscid and compressible in both directions, and so the
pressure at every location satisfies,
(1)
2
∂2
∂2
p(x,
z)
+
p(x, z) −
∂x2
∂z 2
ω
co
2
p(x, z) = 0,
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Table 1: List of parameters used in the cochlear model.
Variable
L
Hv
Ht
N
M
kow
cow
mow
Aw
co
ωB
mo
madd
l
Qo
kBM (x)
cBM (x)
mm
km
cT
Lr
Parameter
Length of cochlea
Height of scala vestibuli
Height of scala tympani
Number of nodes in x-direction
Number of nodes in z-direction
Stiffness of the oval window
Resistance of the oval window
Mass of the oval window
Area of cochlear windows
Speed of sound in cochlear fluid
Natural frequency at the base of the cochlea
Physical mass of basilar membrane
Inferred fluid mass on the basilar membrane
Frequency distribution characteristic length
Q factor of passive cochlea
Stiffness variation along basilar membrane
Damping variation along basilar membrane
Effective inertial mass of middle ear
Effective stiffness of the middle ear
Effective resistance of the middle ear
Lever ratio
Value
35 mm
1.5 mm
1 mm
295
22
830 N m−1
0.1 N m−1 s
4.5 × 10−6 kg
3.2 × 10−6 m2
1500 ms−1
20 kHz
0.02 kgm−2
0.27 kgm−2
7 × 10−3 m
2.5
−2x
2
(mo + madd )e l
ωBp
Q−1
kBM (mo + madd )
o
5.5 × 10−5 kg
6300N m−1
0.12N m−1 s
1.4
where ω is the angular frequency and co the speed of sound in the cochlear fluid. The model was
separated into 294 elements in the x-direction and 21 in the z-direction, in order to solve this equation,
Eq. 1, where ∆x and ∆z are the sizes of the elements in the x and z directions respectively. The
pressure at location, (n∆x, m∆z), which is written here as [n, m] for convenience, is solved using
the finite difference approximation of Eq. 1,
(2)
p[n+1, m]−2p[n, m]+p[n−1, m] p[n, m+1]−2p[n, m]+p[n, m−1]
+
∆x2
∆z 2
2
ω
−
p[n, m] ≈ 0.
co
The boundary conditions in the x and z directions are imposed by relating the velocity at the
boundaries, vbc , to the pressure gradient in the x and z directions,
(3, 4)
∂p(x, z)
= −iωρvbcz (z),
∂z
∂p(x, z)
= −iωρvbcx (x).
∂x
These equations are used to describe the velocity of the BM, cochlear windows, cochlear walls and
aqueducts. All cochlear walls are assumed to be rigid, and for most cochlear excitation mechanisms
the cochlea is stationary, hence the velocity perpendicular to cochlear wall is equal to zero.
The velocity of the complainant structures in the cochlea are related to the admittance of the
structure. The BM is modelled as a series of locally reacting single degree of freedom systems, as
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shown in Fig. 1, where the stiffness, kBM , and resistance, cBM , varies along its length, but the physical
mass, mo , remains constant. The values and variation of the parameters are shown in Table 1. The
admittance of the BM, at location x, which is equal to,
YBM (x) =
(5)
iω
,
iωcBM (x) + kBM (x) − ω 2 mo
which is related to the BM velocity, vBM , relative to the horizontal cochlear wall velocity, vw , by,
vBM (x) = −YBM (x)pd (x) + TBM (x)vw (x).
(6)
pd (x) is the pressure difference across the BM at location x and, TBM (x), is the transmissibility of
the wall velocity to the relative BM velocity, which is given by,
TBM =
(7)
ω 2 mo
.
iωcBM (x) + kBM (x) − ω 2 mo
The admittances of the cochlear windows and aqueducts are calculated as if looking out from
the cochlea, and represented here a series of single degree of freedom systems, similar to the BM.
The admittance of the oval window is dependant of the effect mass, stiffness and resistance of the
middle ear looking out from the cochlea while the round window admittance is modelled to only have
a stiffness, with a value of a tenth of the stiffness of the oval window, [7],
(8, 9)
Yow =
iAw ω
,
(kow + iωcow − ω 2 mow )
Yrw =
10Aw iω
.
kow
The parameters used to determine the admittance of the oval window were calculated from those
measured by Puria [8], and can be found in Table 1. The velocities of the oval window, vow , and
round window, vrw are related to the their respective admittances by,
(10, 11)
vow = vw + Yow p(x, Hv ),
vrw = vw − Yrw p(x, Ht ).
The CA and VA are modelled as fluid filled cylindrical tubes, where their admittances, YAi , are
dependant on the dimensions of the tube. The values can are the inverse of the impedances defined
by Stenfelt [3]. The velocities at the VA, vAV and CA vAC are defined as,
(12, 13)
2.1
vAV = vw + YAv p(x, Hv ),
vAC = vw − YAc p(x, Ht ).
Methods of Exciting the Model
Described above is the complete cochlear model with all cochlear structures outlined. The model
is excited by applying a velocity at specific locations, either cochlear windows or walls, in order to
simulate five different mechanisms of cochlear excitation. When a velocity is applied to a cochlear
window, in order to excited the cochlea, the velocity is no longer proportional to the admittance of
the window but is now a defined input condition of the model.
A piston-like motion of the stapes is simulated here by applying uniform velocity at the oval
window locations in the model. This is a volumetric excitation mechanism because it produces a net
volume velocity through the oval window. A rocking-types motion of the stapes, pivoting about its
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short axes has been examined here, which has also been considered by Edom et al. [5]. The rocking
motion of the stapes is modelled here by applying a graded velocity to the oval window, such that
there is a zero net volume velocity through the oval window, creating a non-volumetric excitation
mechanism.
The middle ear inertia BC component is modelled by applying a piston-like velocity at the oval
window. The magnitude of this velocity is frequency dependant and was calculated from a lumped
parameter model of the middle ear. The middle ear is represented by a damped lever connected by
springs and dampers to a surrounding box, which represents the temporal bone. The velocity of the
stapes, vs , due to the displacement of the surrounding box, vE , is calculated by,
vs =
(14)
ω 2 mm
vE
,
Lr km − ω 2 mm + iωcT
where the values of these parameters can be found in Table 1. The fluid inertia component of BC is
simulated by assuming that the cochlear model is harmonically vibrated in the z-direction. This is
modelled by defining an equal upper and lower cochlear wall velocity. This mechanisms is examined
with both a flexible and stiff oval window. When a stiff oval window is assumed, the stiffness, kow , is
1030 times greater than that of the flexible oval window.
Local excitation at the round window was modelled by keeping the 0.6 mm at either edge of the
window complaint but forcing the middle 1.1 mm with a proscribed velocity. This excitation method
is modelled with an immobile oval window. The effect of the CA and VA on all of the mechanisms
described above is investigated here by comparing the BM response with and without the aqueducts
present.
Equation 2, along with the constraints of the boundary conditions and input conditions for a specific mechanisms, was solved using a matrix equation that relates the vector containing the pressure
at each node, p, to the vector defining the external excitation, q,
Ap = q,
(15)
so that
p = A−1 q,
for the pressure distribution in the cochlea. The BM velocity, due to the excitation mechanism, was
then determined using Eq. 6. In order to be able to compare the BM responses due to the different
cochlea excitation mechanisms, the BM velocity was divided by the excitation velocity. This was
either the maximum window input velocity or, for the BC components, the assumed excitation velocity of the surrounding bone. The frequency response of the BM due to the different excitation
mechanisms was calculated by determining the peak BM velocity for a range of frequencies.
3.
Results
It was found that every mechanism of exciting the cochlea investigated produced a travelling wave
along the BM, even when an immobile oval window was assumed. An instantaneous BM velocity
was found for each mechanism, which for ease of comparison, was normalised to its peak value
and its phase altered to so that the peaks all aligned, which in shown in Fig. 2. The shape of the
travelling wave is very similar at locations towards the apex of the cochlea, for all mechanisms of
excitation. However, when excited by the rocking motion of the stapes or the local excitation of the
round window there is a small deviation. This is due to the relatively high near field pressure produced
at the excitation window for these mechanisms. The BM response is shown at 1 kHz here, however,
a similar shapes of response is produced at other frequencies but the location of the peak velocity is
dependant on the excitation frequency.
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1
Fluid Inertia
Piston−like Stapes Motion
Rocking Motion of the Stapes
Local Excitation at the Round Window
Fluid Inertia With a Stiff Oval Window
Normalised Basilar Membrane Velocity
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
5
10
15
20
25
Distance along Cochlea, mm
30
35
Figure 2: A comparison of the instantaneous basilar membrane velocity, normalised to the peak
velocity, for multiple mechanisms of exciting the cochlea, all at 1 kHz
3.1
Basilar Membrane Response when Excited at the Oval Window or via the Inertia
Components of Bone Conduction
Fig. 3 shows the frequency response of the peak relative BM response due to excitation via the
oval window by a piston-like motion and by a rocking motion, as well as the two inertia components
of BC, with and without the inclusion of the aqueducts in the model. It was found that the presence
of the aqueducts had little effect on the overall BM response for all the mechanisms at frequencies
above 500 Hz. However, at frequencies less than 500 Hz, the aqueducts increased the BM response
by a maximum of 15 dB when the cochlea was excited by the rocking of the stapes and 20 dB when
excited by the fluid inertia component.
20
Middle Ear Aqueducts
Middle Ear No Aqueducts
Piston−like Stapes, No Aqueducts
Piston−like Stapes, with Aqueducts
Rocking Stapes, Aqueducts
Rocking Stapes, No Aqueducts
Fluid Inertia, No Aqueducts
Fluid Inertia, Aqueducts
Peak Relative Basilar Membrane Velocity, dB
10
0
−10
−20
−30
−40
−50
−60
−70
0
500
1000
1500
2000 2500 3000
Frequency, Hz
3500
4000
4500
5000
Figure 3: Comparison of the peak value along the cochlea of the relative BM velocity for multiple
excitation mechanisms, with and without the inclusion of the aqueducts.
3.2
Basilar Membrane Response with an Immobile Oval Window
Fig. 4a shows the BM frequency response, when an immobile oval window is assumed, due to the
excitation by the fluid inertia component of BC and the local excitation of the round window, when
both aqueducts are included, only the CA and when no aqueducts are modelled.
The BM response was found to be very low, due to excitation via both of these mechanisms, when
no aqueducts were modelled. For this condition, the only outlet in the cochlea is the round window
and therefore no volumetric flow can be generated, only non-volumetric. It has already been shown
that a non-volumetric excitation mechanism, the rocking motion of the stapes, produces a low BM
response which is of similar magnitude to the BM response in Fig. 3, without the aqueducts, since
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Total Excitation
Volumetric Excitation
Non−Volumetric Excitation
10
10
0
0
Peak Relative Basilar Membrane Velocity
Peak Relative Basilar Membrane Velocity, dB
Local Excitation at Round Window, Both Aqueducts
Local Excitation at Round Window, No Aqueducts
Local Excitation at Round Winodw, Only Cochlear Aqueduct
Fluid Inertia, Both Aqueducts
Fluid Inertia, Only Cochlear Aqueduct
Fluid Inertia, No Aqueducts
−10
−20
−30
−40
−50
−60
−70
−80
−10
−20
−30
−40
−50
−60
−70
0
500
1000
1500
2000 2500 3000
Frequency, Hz
3500
4000
4500
(a)
5000
−80
0
500
1000
1500
2000 2500 3000
Frequency, Hz
3500
4000
4500
5000
(b)
Figure 4: (a) Comparison of the relative BM velocity due to the fluid inertia and local round window
excitation mechanisms, with no aqueducts, only the CA and both aqueducts. (b) Comparison of
the total excitation due to local excitation at the round window and the separate non-volumetric and
volumetric components.
these excitation are due to the near field pressure distribution [5], [2], [9]. The CA is on the same
side of the cochlea as the round window and including it in the model does not significantly alter the
magnitude of the BM response in Fig. 4a. However, when both aqueducts were included, the BM
response is significantly changed, for the both excitation mechanisms modelled here. It was found
that the BM response also has a very similar shape and magnitude to those in Fig. 4a when only the
VA was modelled, however, for clarity this has not been shown. The VA is on the opposite side of
the cochlea, creating a complaint structure either side of the BM, and hence allowing for a volumetric
excitation of the cochlea to be generated.
The BM response due to local excitation at the round window is increased by more than 50 dB at
low frequencies by the inclusion of the VA, however, at around 2 kHz, the BM response is reduced
by up to 10 dB and then at higher frequencies the response becomes comparable to a model with no
aqueducts present. This suggests that the excitation mechanism is volumetric at low frequencies and
non-volumetric at high frequencies. The total BM response, when both aqueducts are present, was
separated into its volumetric and non-volumetric components, which is shown in Fig. 4b. At low
frequencies the volumetric component dominates, while at high frequencies, this switches and the
non-volumetric dominates. Between about 1 kHz and 3 kHz the magnitude of the components are
similar, however, there is a phase different between the BM responses due to each component, which
causes the combined response to be lower.
The BM response, due to the fluid inertia component, could not be separated into a volumetric
and non-volumetric component, in the way the local round window excitation can be. However, it
was concluded that the introduction of a volumetric component of excitation, via the addition of the
VA, significantly increased the BM response due to this excitation mechanism.
4.
Conclusion
A simple 2D finite difference model of the cochlea was used to investigate multiple mechanisms of
cochlear excitation and, in particular, the effect of the aqueducts on the BM response was considered.
The simplicity of the model allowed the model to be easily adapted to simulate the various excitation
mechanisms; the piston-like and rocking motion of the stapes, the two inertia components of BC
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hearing and local excitation at the round window, as well as simulating a diseased ear, which has
an immobile oval window. Because all the mechanisms were simulated on the same model, the BM
responses could be easily calculated and justifiably compared.
It was found that all mechanism of exciting the cochlea produced a similar travelling wave along
the BM. When the cochlea was excited at the oval window or by the fluid inertia component of BC
with a flexible oval window, the presence of the aqueducts made little difference to the peak BM
frequency response, apart from a low frequencies. Below 500 Hz, the BM response was increases
by the addition of the aqueducts when excited by the rocking motion of the stapes or by fluid inertia
component. However, when the oval window was assumed to be immobile, and local excitation
at the round window or the fluid inertia component of BC were simulated, the addition of the VA
had a considerable effect on the BM response. When no aqueducts or just the CA was modelled
the model could only be excited by non-volumetric flow, however, the inclusion of the VA in the
model allowed for volumetric excitation of the cochlea. The increase in BM response when excited
by the fluid inertia component was due to the ability to generate a volumetric excitation. The BM
response, due to the local excitation at the round window, was separated into its volumetric and nonvolumetric components. It was shown that the response at low frequencies was mainly volumetric,
while at high frequencies the non-volumetric component dominated, however, at mid frequencies the
two components added destructively producing a low BM response.
Acknowledgement
This research was supported by the EU project SIFEM Grant No. FP7-600933.
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