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A FINITE ELEMENT MODEL OF AN AVERAGE HUMAN EAR CANAL TO
ASSESS CALIBRATION ERRORS OF DISTORTION PRODUCT OTOACOUSTIC
EMISSION PROBES
Makram A. Zebian
Department 1.6 - Sound
Physikalisch-Technische Bundesanstalt (PTB)
Bundesallee 100, D-38116 Braunschweig, Germany
Tel.: +49 531 592 1435, Fax: +49 531 592 69 1435
e-mail: [email protected]
Introduction
Otoacoustic emissions (OAEs) are a class of acoustic signals that are generated in the cochlea
and transmitted backward to the ear canal through the middle ear. This is attributed to the
active nonlinear mechanism in the cochlea [Kemp 1978]. Distortion product otoacoustic
emissions (DPOAEs) are a special type of OAEs generated in response to two pure tone
acoustic stimuli with sound pressure levels L1 and L2 and are best detected when the
following relations are applied [Kummer et. al., 1998]:
f 2 = 1.2 f1
(1)
L1 = 0.4 L2 + 39 dB
(2)
where f1 , f 2 correspond to the frequencies of these stimuli, respectively. The DPOAE travels
backward to the ear canal and can be detected at a frequency of:
f dp = 2 f1 − f 2 .
(3)
These DPOAEs are regarded as providing an attractive objective test of high-frequency
hearing loss of cochlear origin in humans [Siegel and Hirohata, 1994] and are now gaining
importance in assessing hearing ability of newborns. DPOAEs exhibit very small sound
pressure levels and are usually measured by means of an ear canal probe (DPOAE probe)
containing two miniature loudspeakers and one microphone. In principle, the most direct way
to calibrate these stimuli in the ear canal would be to insert a small microphone into the canal
and place it close to the eardrum. This direct measuring procedure has many drawbacks which
will be explained in the next section. It is, however, important to differentiate between the
forward travelling waves of frequencies f1 and f 2 (separately generated by the two
loudspeakers) and the backward travelling DPOAE wave of frequency f dp (generated within
the cochlea).
In this report, an assessment of typical calibration errors was undertaken for a model ear canal
(as a cylindrical tube with rigid walls) with the finite element method (FEM). It was also
differentiated between the calibration of the stimuli which are directed towards the cochlea
and that of the DPOAEs returning from the cochlea.
Calibration difficulties
Despite the intensive interest in the phenomenon of otoacoustic emissions, little attention has
been focused on the problem of specifying the stimulus sound pressure levels (SPLs). It is
generally agreed that the sound pressure at the eardrum is a better measure of the stimulus
than the sound pressure at the DPOAE probe. The accuracy of the estimation of the sound
pressure level at the eardrum position (eardrum SPL) by measurements at the DPOAE probe
position depends on frequency. The commonly accepted reference for the input to the middle
ear can be measured with a probe tube microphone positioned within a few millimeters from
the eardrum. Although the depths of insertion are not always reported in the literature, Siegel
and Hirohata [1994] reported the length of the occluded ear canal to be 21 mm.
At relatively low frequencies, below 3 kHz, the ear canal can be approximated as a straight,
cylindrical tube [Stinson and Daigle, 2005]. The sound pressure throughout the ear canal is
nearly uniform, because the quarter wavelength of the signal is large compared to the
dimensions of the ear canal. This situation is not valid at higher frequencies, where the ear
canal is long enough so that one or more standing wave minima - caused by reflection at the
eardrum - arise. These standing waves can result in a partial cancellation of the sound
pressure measured at certain locations. As a result, the sound pressure measured at the
microphone position of the DPOAE probe could potentially underestimate the sound pressure
at the eardrum for these higher frequencies. In order to reduce these errors, the microphone
should be inserted deeply into the ear canal. Several problems are associated with this kind of
direct measurement, especially because proper insertion and positioning of the microphone
are time consuming and might even cause discomfort, or damage the subject’s eardrum.
Another problem related to the calibration of the DPOAE probes is the fact that the DPOAEs
originate in the cochlea and travel backward with a distinct frequency f dp .
It is therefore probable that the DPOAE probe microphone, calibrated to measure the stimuli,
might not be able to measure the DPOAEs correctly. For these DPOAEs a different
calibration might be advantageous. These problems and suggestions are reassessed in the
following section with an FEM model of an average human ear canal.
Model ear canal
a) Ear canal modelled as a simple cylindrical tube
The sound field inside a model human ear canal was computed, to show longitudinal
variations along the canal length. An FEM model was implemented to compute the full threedimensional sound field (using COMSOL Multiphysics 3.5). For simplicity, a canal diameter
of 8 mm and a simulated occluded ear canal length of 20 mm were used, in conjunction with a
theoretical eardrum impedance ( Z eardrum = ∞ ) corresponding to a rigid boundary condition.
Other calculations (not shown) have been performed to confirm the accuracy of our model for
a simple test case for which analytical solutions are available. An example of these analytical
solutions included the prediction of sound field in a uniform cylinder of 20 mm length, of
small diameter, and with a rigid termination perpendicular to the walls of the cylinder. The
modified horn equation after [Khanna and Stinson, 1985] is:
d
dp( z )
( A( z )
) + k 2 A( z ) p( z ) = 0
dz
dz
(4)
where z is the length coordinate, k = w / c is the wave number, w is the angular frequency, c is
the speed of sound in air (340 m/s) and A is the cross-sectional area. Equation (4) was solved
for a constant uniform cross section (constant area) using MATLAB. The two approaches gave
good agreement and the FEM model was thus used for the more complex cases where the
one-dimensionality of the wave equation vanishes.
For a plane piston source, a sound pressure of p(z=0) = 1 Pa is assumed at the entrance plane
(at z = 0). Note that this is a theoretical assumption, especially that 1 Pa corresponds to 94 dB
and is by no means acceptable for calibrating in human ears. The sound pressure level along
the z-axis of the ear canal was computed for frequencies up to 20 kHz. Standing waves in the
ear canal are evident and become increasingly complex as the frequency increases. In Fig. 1,
for 16 kHz (dashed line), two minima are apparent. For calibration in the vicinity of the
minima, for example at z = 4 mm, the difference between the calibrated stimulus level at the
microphone position and the actual eardrum SPL would be about 35 dB.
Fig. 1: Sound pressure level (simulation) plotted over the center axis of the cylindrical ear canal (of length
20 mm denoted by the z-coordinate) for two frequencies: 4 kHz (solid line) and 16 kHz (dashed line). At 16
kHz, two minima are observed (z = 0.004 m and z = 0.0145 m) with SPL levels differing from the actual
level at the eardrum position (z = 0.02 m) by more than 35 dB.
b) Ear canal modelled with an oblique eardrum
In reality, the human ear canal is not uniform. For most ear canals the cross-sectional area
decreases towards the medial end and the eardrum terminates the ear canal obliquely. Because
of this tapering, there is an increase in sound pressure at the end of this taper relative to the
maximum sound pressure in the lateral portion of the ear canal [Stevens et al., 1987]. In the
following, a more realistic ear canal geometry was assumed for the FEM model. The ear canal
was modelled by a uniform cross-sectional area over the lateral portion of its length, whereas
the inner 6 mm of the canal length were tapered to form a gradually narrowing cross-sectional
area. The length of the ear canal along the central z-axis was set to about 25 mm. A plane
piston source (p(z=0) = 1 Pa at the entrance plane) was used to stimulate the ear canal as
described above in a). The mesh elements that were used to represent the ear canal geometry
are shown in Fig. 2. The rule of thumb for FEM calculations that element dimensions should
be less than 1/6 of the wavelength, was satisfied up to the maximum frequency considered (20
kHz).
In this simulation, transverse variations through cross-sectional slices became obvious at high
frequencies. By studying the frequency range from 1 kHz up to 20 kHz, it could be noted that,
for this model, the one-dimensional nature of the sound field in the ear canal disappeared at
frequencies higher than 14 kHz. As a result, large transverse modes were apparent in the FEM
results above this frequency. In Fig. 3 (f = 14 kHz), two standing wave minima are seen but
only small transverse variations of about 10 dB could be observed across the eardrum.
However, in Fig. 4 (f = 18 kHz), transverse variations across the eardrum of about 40 dB,
along with two apparent standing wave minima are observed. Note that the colors used to
visualise the SPL values were scaled in a unified manner for all figures. This means that in all
figures the color dark red corresponds to the same maximum SPL (110 dB) and dark blue
corresponds to the minimum SPL (20 dB). This makes a direct comparison between the
different figures possible. The differences given in dB below each figure were estimated by
visualising the wave form on each of the eardrum surfaces. For practicability, the color bar is
only included in Fig. 4 but is valid for all other figures.
Fig. 2: Mesh elements representing the geometry
of the ear canal with an oblique eardrum.
Element size was chosen to be at least 1/6 of the
wavelength.
Fig. 3: Simulation at 14 kHz. SPL of the
modelled ear canal with an oblique eardrum.
Piston source is located at z = 0 (as in Fig. 4).
Standing wave minima and transverse variations
across the eardrum of about 10 dB are observed.
z [m]
Fig. 4: Simulation at 18 kHz. SPL of the modelled ear canal with an oblique eardrum. Piston source is
located at z = 0. Transverse variations across the eardrum are about 40 dB.
Eardrum acting as a piston source (simulating the DPOAEs)
In this section, the oblique eardrum as a source piston (of elliptical form) was simulated in
order to assess calibration errors of the backward travelling DPOAEs. The simulations were
performed for two conditions representing the impedance of the probe: a match impedance
condition and a rigid boundary condition. In Fig. 5, it was assumed that the impedance of the
probe is equal to the characteristic impedance of air (i.e., impedance match condition:
Z probe = Z air = ρ c , with the density of air: ρ = 1.25 kg / m 3 and the speed of sound in air:
c = 340 m / s ), so that no reflections from the probe (at z = 0) towards the eardrum occur. In this
case, variations of 6 dB were noted across the eardrum at 14 kHz, but no standing wave
pattern could be observed. In Fig. 6, the probe (at z = 0) was assumed to be rigid (i.e., rigid
boundary condition) with an impedance of Z probe = ∞ . Two standing wave minima could be
noticed at this certain frequency along with transverse variations across the eardrum of about
30 dB.
Fig. 5: Simulation at 14 kHz. DPOAE Probe
located at z = 0 (as in Fig. 4). Boundary
condition of the probe: impedance match.
Oblique eardrum acts as the source of DPOAEs.
Transverse variations across the eardrum are
about 6 dB.
Fig. 6: Simulation at 14 kHz. DPOAE Probe
located at z = 0 (as in Fig. 4). Boundary
condition of the probe: rigid. Oblique eardrum
acts as the source of DPOAEs. Transverse
variations across the eardrum are about 30 dB.
However, a DPOAE with f dp = 14 kHz corresponds to f1 = 17.5 kHz and f 2 = 21 kHz
(calculated using equations (1) and (3) stated above), which is a mere theoretical case. A more
realistic case was therefore applied for f 2 = 14 kHz (and thus f1 = 11.7 kHz), as a response to
the stimulation presented in Fig. 3. For this stimulation, and with the help of equations (1) and
(3), a DPOAE of 9.3 kHz ( f DPOAE = 2 ( f 2 / 1.2) − f 2 ) was obtained. This was simulated for both
boundary conditions and is shown in Fig. 7 and Fig. 8. No standing wave minima and no
transverse variations could be observed for the impedance match condition (Fig. 7), however,
for the rigid boundary condition, one standing wave minimum but no transverse variations
were observed (Fig. 8).
Fig. 7: Same impedance match condition as in
Figure 5. Simulated at 9.3 kHz as a response to
the stimuli f 1 = 11.7 kHz, f 2 = 14 kHz ( f 2
depicted in Fig. 3): No standing wave minima
and no transverse variations across the eardrum
are observed.
Fig. 8: Same rigid condition as in Figure 6.
Simulated at 9.3 kHz as a response to the stimuli
f 1 = 11.7 kHz, f 2 = 14 kHz ( f 2 depicted in Fig.
3): One standing wave minimum but no
transverse variations across the eardrum are
observed.
Conclusion
Generally, it could be seen that the eardrum SPL, when estimated from a measurement point
in the ear canal (at a certain distance from the eardrum) can be underestimated when
measuring in the vicinity of a standing wave minimum. Above a certain frequency (14 kHz
for the model applied in this study) the simple one-dimensional wave equation is not valid
anymore and transverse variations occur. The FEM model with the oblique eardrum showed
that, when calibrating the probes, huge discrepancies could arise between the forward and the
backward travelling signals, and the locations of their minima also vary for the same
frequency (compare Fig. 3 and Fig. 6) as well as for the stimulus and its DPOAE response
(compare Fig. 3 and Fig. 8). The reflectance of the probe also plays an important role in
determining the sound field in the ear canal with the eardrum acting as a source. For a rigid
probe, significant transverse variations (up to 30 dB) across the eardrum appeared at about 14
kHz (Fig. 6). However, when the probe was assumed to fulfil the impedance match condition,
variations of only 6 dB occurred at 14 kHz (Fig. 5) and even at lower frequencies. This leads
to the conclusion that the calibration procedure for the stimuli sent into the cochlea should
differ from that for the DPOAE which comes back from the cochlea. In future studies, it is
intended to measure the reflection coefficient of the probe by means of an impedance tube.
This should give more insight into the impedance of the probe and can be helpful when
choosing the boundary conditions represented in our FEM model.
Bibliography
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