A FINITE ELEMENT MODEL OF AN AVERAGE HUMAN EAR CANAL TO ASSESS CALIBRATION ERRORS OF DISTORTION PRODUCT OTOACOUSTIC EMISSION PROBES

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1 A FINITE ELEMENT MODEL OF AN AVERAGE HUMAN EAR CANAL TO ASSESS CALIBRATION ERRORS OF DISTORTION PRODUCT OTOACOUSTIC EMISSION PROBES Introduction Makram A. Zebian Department Sound Physikalisch-Technische Bundesanstalt (PTB) Bundesallee 100, D Braunschweig, Germany Tel.: , Fax: 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 L 1 and L 2 and are best detected when the following relations are applied [Kummer et. al., 1998]: f 2 = 1. 2 f1 (1) L1 = 0.4 L2 + 39dB (2) where f 1, 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 f2. (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 f 1 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

2 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 = ) corresponding to a rigid boundary condition. eardrum 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 dz dp( 2 ( A( ) + k A( p( = 0 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

3 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 = m and z = 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 kh. 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 kh, 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 kh, 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

4 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:

5 Z probe 3 = Z = ρ c, with the density of air: ρ = 1.25 kg / m and the speed of sound in air: air c = 340m / 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 =. Two standing wave minima could be probe 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 f 1 = 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 f 1 = 11.7 kh, 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( f2 /1.2) f2 ) 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

6 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 Kemp, D. T. (1978). Stimulated acoustic emissions from within the human auditory system, J. Acoust. Soc. Am. 64, Khanna, S. M., and Stinson, M. R. (1985). Specification of the acoustical input to the ear at high frequencies, J. Acoust. Soc. Am. 77(2), Kummer, P., Janssen, T., and Arnold, W. (1998) The level and growth behavior of the 2f1-f2 distortion product otoacoustic emission and its relationship to auditory sensitivity in normal hearing and cochlear hearing loss, J. Acoust. Soc. Am. 103(6), Siegel, J. H., and Hirohata, E. T. (1994). Sound calibration and distortion product otoacoustic emissions at high frequencies, Hear. Res. 80, Stevens, K. N., Berkovitz, R., Kidd, Jr., G., and Green, D. M. (1987). Calibration of ear canals for audiometry at high frequencies, J. Acoust. Soc. Am. 81, Stinson, M. R., and Daigle, G. A. (2005). Comparison of an analytic horn equation approach and a boundary element method for the calculation of sound fields in the human ear canal, J. Acoust. Soc. Am. 118,

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