Fluid-Structure Finite-Element Modelling and Clinical Measurement of the Wideband Acoustic Input Admittance of the Newborn Ear Canal and Middle Ear

Abstract

The anatomical differences between the newborn ear and the adult one result in different input admittance responses in newborns than those in adults. Taking into account fluid-structure interactions, we have developed a finite-element model to investigate the wideband admittance responses of the ear canal and middle ear in newborns for frequencies up to 10 kHz. We have also performed admittance measurements on a group of 23 infants with ages between 14 and 28 days, for frequencies from 250 to 8000 Hz with 1/12-octave resolution. Sensitivity analyses of the model were performed to investigate the contributions of the ear canal and middle ear to the overall admittance responses, as well as the effects of the material parameters, measurement location and geometrical variability. The model was validated by comparison with our new data and with data from the literature. The model provides a quantitative understanding of the canal and middle-ear resonances around 500 and 1800 Hz, respectively, and also predicts the effects of the first resonance mode of the middle-ear cavity (around 6 kHz) as well as the first and second standing-wave modes in the ear canal (around 7.2 and 9.6 kHz, respectively), which may explain features seen in our high-frequency-resolution clinical measurements.

Introduction

Acoustic input immittance provides information about the status of the outer and middle ear by providing a measure of the mobility of the ear components in response to the acoustic excitation. The term immittance refers to either impedance Z or admittance Y. Impedance is calculated by dividing the sound pressure by the volume velocity, and admittance is the reciprocal of impedance. Both quantities are complex numbers, and they are usually reported as magnitudes and phases. The measurement can be done either under ambient pressure or with a pressurized ear canal. The latter type of measurement is referred to as tympanometry. Tympanometry is most often done with a 226-Hz probe tone. Multi-frequency immittance measurements have been shown to improve the test sensitivity in some cases of outer-ear and middle-ear pathology (e.g. Shahnaz and Polka 1997). More information can be obtained quickly over a broad frequency range by using a wideband stimulus (e.g. Keefe et al. 1993).

Although tympanometry provides reasonably easy-to-interpret results for adult ears, and normative data for adult responses are available, it has been shown to produce significantly different results in infants less than 7 months old (e.g. Paradise et al. 1976; Paradise 1982; Holte et al. 1990). Holte et al. (1991), Keefe et al. (1993) and Sanford and Feeney (2008) ascribed the differences to the maturation of the ear canal and middle ear during the first postnatal months. For example, the newborn ear canal is surrounded by soft tissue along all of its length (McLellan and Webb 1950) and the middle-ear air cavity becomes larger due to growth of the antrum and mastoid air cells throughout childhood (Anson and Donaldson 1992, p. 25). Since the canal wall is not ossified in newborns, the canal undergoes large deformations in response to the quasi-static pressure of tympanometry. Qi et al. (2006, 2008) modelled the nonlinear response of the newborn canal wall in response to such pressures.

At low frequencies, because of the long wavelengths of the sound in air (e.g. 170 mm at 2 kHz) in comparison with the canal length (~15 mm) and the dimensions of the middle-ear air cavity, the pressure is distributed almost uniformly inside the canal and the air cavity and across the tympanic membrane (TM). It is therefore reasonable to treat the ear canal and air cavity as discrete (lumped) immittance elements (e.g. Shanks and Lilly 1981; Stinson et al. 1982). Based on this assumption, we developed finite-element models of the newborn ear canal and middle ear and analysed their responses to sound frequencies up to 2 kHz (Motallebzadeh et al. 2013b, 2017). In those studies, sound pressure with an amplitude of 0.2 Pa root mean square (80 dB sound pressure level (SPL)) was applied uniformly on the surfaces of the ear canal and TM; the volumes of air inside the canal and inside the middle-ear cavity were modelled as two compliance elements; and the individual contributions of the ear canal and middle ear to the total immittance response of the ear were investigated.

At higher frequencies, however, the sound pressure is less uniform within the ear canal, across the surface of the TM, and within the middle-ear cavity. Gilman and Dirks (1986), for example, listed the sources of problems occurring in immittance measurements at higher frequencies: (1) standing waves are produced in the ear canal by reflection of sound from the TM; (2) variation of the TM impedance with frequency alters the positions of the standing waves; (3) the geometry of the canal (non-uniform cross-section) and the angle of the TM affect the pressure distribution in very complicated ways at high frequencies; and (4) evanescent waves (waves that decay exponentially within a short distance) are present at both the TM and the probe tip locations. For example, Stinson et al. (1982) measured the sound pressure distribution in 13 adult ear canals between 5 and 10 kHz and reported differences between the SPL at the TM and the SPL at the canal entrance that were greater than 18 dB.

Energy reflectance has been proposed as an alternative to immittance that is insensitive to the probe position (e.g. Stinson et al. 1982; Keefe et al. 1993; Voss and Allen 1994). Normative reflectance responses for adults (e.g. Liu et al. 2008) and newborns (e.g. Merchant et al. 2010) have been reported in the literature. However, energy reflectance measurements are still sensitive to the insertion depth of the probe tip inside the canal in newborns, because the soft tissue surrounding the ear canal absorbs some portion of the energy.

The spatial distribution of sound pressure in the ear canal has been investigated with analytical approaches (e.g. Stinson 1985a; Rabbitt 1988; Stinson and Khanna 1989) and also with 3-D finite-element models that take into account the fluid-structure interactions (FSI). Day and Funnell (1990) presented a finite-element model with a very simplified geometry for the human ear canal and TM. Gan et al. developed more realistic models of the human ear and investigated the pressure distribution in one chamber (ear canal, (2004)), two chambers (canal and middle-ear cavity, (2006)) and three chambers (canal, cavity and the fluid inside the cochlea, (2007) and (2009)). Lee et al. (2010) modelled the effects on the displacement of the umbo when the geometry of the middle-ear cavity was altered. Ihrle et al. (2013) used FSI modelling to study the nonlinear behaviour of the middle ear in response to large static pressures. Volandri et al. (2014) studied the modal frequencies inside the canal with two different FSI approaches, based on finite-element shape functions that were either standard (polynomial) or ‘generalized’ (non-polynomial). Most recently, Wang et al. (2016) presented a finite-element model of a 4-year-old child to evaluate the energy absorbance of the ear. To evaluate the effect of the compliance of the canal, they added a uniform 1.2-mm-thick layer of soft tissue to the bony canal wall. In newborn ears, however, like the one considered here, there is essentially no bony ear-canal wall (e.g. Abdala and Keefe 2012). None of the previous analytical or numerical studies have considered the effects of this condition on the sound pressure distribution and their interaction with the other components of the ear.

In this study, we extended our previously developed linear finite-element models of the newborn ear canal and middle ear by taking into account the interactions between the ear structures and the non-uniform sound pressures in the two chambers bounded by (a) the probe, the ear-canal wall and the TM, and (b) the TM and the walls of the middle-ear cavity. This model enables us to study the wideband input immittance as well as the spatial distribution of the sound pressure in the canal and the middle-ear cavity. We also performed clinical measurements partially reported in Motallebzadeh et al. (2017) on a group of 23 infants with ages between 14 and 28 days, for frequencies from 250 to 8000 Hz with 1/12-octave resolution. These data and the data reported by Keefe et al. (1993) were used to adjust some parameters and validate the model. The model is capable of replicating and helping to interpret the key characteristic features of the frequency-domain responses of both clinical data sets, such as the canal and middle-ear resonances, the first and second modes of standing waves inside the canal, and the middle-ear cavity resonance. The influence of the material properties, the measurement position and geometrical variations were explored using the model.

Methods

3-D Geometry

The geometries used here for the ear-canal and middle-ear models were almost the same as in Motallebzadeh et al. (2017), the only differences being small changes in the geometry of the canal at its medial end where it connects to the TM (to make a smoother connection between these two components), and the fact that an explicit model of the middle-ear cavity was added to the model. The middle-ear cavity includes a set of air-filled and inter-connected spaces within the temporal bone consisting of the tympanic cavity, aditus, antrum and mastoid air cells. However, according to Cinamon (2009), at least for the first few days after birth, only the tympanic cavity and the antrum are air-filled and the mastoid air cells are filled by mesenchymal tissue. As in our earlier paper, the models were reconstructed from a clinical X-ray CT scan (GE LightSpeed16, Montréal Children’s Hospital) of a 22-day-old newborn’s right ear. The scan had a pixel size of 0.187 mm and a slice thickness of 0.625 mm with a 0.125-mm overlap, resulting in a slice spacing of 0.5 mm. Fie, Tr3 and Fad (three locally developed programs, available at http://www.audilab.bme.mcgill.ca/sw/) were used to generate a surface model for each structure. Gmsh (http://www.geuz.org/gmsh/) was then used to generate a 3-D solid model with tetrahedral elements, and the solid models of the various structures were combined using Fad. The complete model (Fig. 1) consists of the soft tissue surrounding the lumen of the ear canal; the TM, malleus, incus, anterior mallear ligament (AML) and two bundles of the posterior incudal ligament (PIL); and the middle-ear cavity. More detail about the model, including the thickness distribution assumed for the TM, can be found in Motallebzadeh et al. (2017).

FIG. 1.

Meshed geometry of the finite-element model. a Superior-to-inferior view of the overall model including the ear canal, surrounding soft tissue, middle ear and middle-ear cavity. (The cavity is presented as partially transparent to provide better visualization of other parts.) b Expanded medial-to-lateral view of the middle-ear model, with the TM annulus almost parallel to the page. (ME middle ear, PIL posterior incudal ligament, AML anterior mallear ligament, PT pars tensa, PF pars flaccida, S superior, I inferior, M medial, L lateral, A anterior, P posterior) c Expanded lateral view of the middle-ear cavity, with the TM annulus almost parallel to the page.

Material Properties

The intensity of the probe tone in immittance measurements is low enough that linear elastic material properties can be used for the ear. As described in Motallebzadeh et al. (2017), in the absence of precise experimental data, ranges of plausible a priori values were adopted for the material properties. The material parameters of the ear components, listed in Table 1, are the same as in Motallebzadeh et al. (2017). As in that study, three models each for the canal and the middle ear were generated: (a) a low-impedance model (i.e. a model with the lowest stiffness, density and damping values for all components), (b) a baseline model (i.e. a model with baseline parameter values) and (c) a high-impedance model (i.e. a model with the highest stiffness, density and damping values for all components). The air enclosed in the ear canal and middle-ear cavity was modelled as a compressible, inviscid medium with a density of 1.22 kg/m³ and a speed of sound within it of 340 m/s.

TABLE 1.

Material properties

	Minimum	Baseline	Maximum
Young’s modulus (MPa)
Pars tensa (E pt)	2	6	10
Pars flaccida (E pf)	0.4	1.2	2
Soft tissue around canal (E st)	0.02	0.21	0.4
Ossicles (E os)	4000	10,000	16,000
Ligaments (E lig)	2	5	8
Poisson’s ratio
Soft tissues (around canal and in middle ear) (ν st)	0.485	0.49	0.495
Ossicles (ν os)	0.3
Density (kg/m3)
Soft tissues (ρ st)	1000	1100	1200
Ossicles (ρ os)	1600	1800	2000
Damping ratio (ζ)	0.1	0.25	0.4
Cochlear load
Spring (N/m) (K c)	200	600	1000
Dashpot (N.s/m) (C c)	0.2	0.45	0.7
Stapes mass (kg) (M s)	2 × 10−6	3 × 10−6	4 × 10−6

Boundary and Loading Conditions

The ear-canal surface was clamped where it is in contact with the probe tip, because the probe tip is assumed to be securely held in the canal. The peripheral border of the TM and the ends of the AML and PIL were clamped at the places where they connect to the temporal bone, and the temporal bone surfaces, which delimit both the soft tissues and the middle-ear cavity, were also clamped. Translational spring and damping elements were attached to the tip of the long process of the incus to represent the stapedial annular ligament and the cochlear load, respectively, and the mass of the stapes was represented by a nodal mass at that location, as in Motallebzadeh et al. (2017). The incus and malleus were modelled as isotropic and elastic components and the incudomallear joint (IMJ) is assumed to be fused so there is no relative motion between the malleus and incus.

A harmonic velocity source with a constant amplitude of 0.15 mm/s was applied normal to the medial surface of the probe tip (which had a surface area of 13.2 mm²) to represent an acoustic driver delivering the sound energy into the ear canal. This particular velocity value was set in order to generate a pressure of 80 dB SPL (typical of clinical measurements) at 250 Hz in the baseline model. The same input velocity was applied at frequencies from 25 to 10,000 Hz in 25-Hz steps and the input admittance of the model was calculated for each frequency.

Finite-Element Mesh

The volume elements of the mesh consisted of second-order TETRA10 tetrahedra and the interface elements (between fluid and structural elements) consisted of second-order TRIA6 triangles. The ear-canal model consisted of 45,991 elements (17,972 and 28,039 elements for the air in the canal and the surrounding soft tissues, respectively). The middle-ear mesh consisted of 29,430 elements (23,102, 440, 5888 and 11,393 elements for the TM, the ligaments, the ossicles and the volume of the middle-ear cavity, respectively). As described in Motallebzadeh et al. (2017), the newborn TM is too thick to be modelled as a shell; it was meshed with 3-D tetrahedral elements and modelled as an isotropic elastic material. We have again performed mesh-convergence tests to assess the adequacy of the mesh resolution. The resonance magnitude (at location (2) of Figure 8) and its frequency changed by less than 1.2 and 1 %, respectively, when the mesh resolution was doubled.

FIG. 8.

Results for model with adjusted parameters and comparison with two sets of clinical data. Admittance magnitudes (a) and phases (b) are presented for the 1-month-old data of Keefe and Levi (1996) (solid black lines and error bars) and for data from this study (thick grey lines for the mean, thin grey lines for the responses of individual subjects), and for the output of the model with adjusted parameters (blue lines). The inset shows a magnified view of the admittance magnitudes at frequencies between 150 and 2000 Hz. Features indicated by arrows and numbers are discussed in the text.

There were at least 25 nodes along the canal length; for a wavelength of 34 mm at 10 kHz, this more than satisfies the recommendation for at least ten nodes per wavelength (e.g. Ihrle et al. 2013).

Computational Methods

Finite-Element Solver

Code_Aster (http://www.code-aster.org) version 11.5 was the finite-element solver in this study. It is free (libre) and open-source software. The complex linear dynamic pressure responses of the models were obtained using the DYNA_LINE_HARM module, which calculates the steady-state response of a model for a harmonic excitation. Simulations were performed on the supercomputer Guillimin of McGill University. Guillimin is a part of the Compute Canada national High Performance Computing platform. It is a cluster of Intel Westmere EP Xeon X5650 and Intel Sandy Bridge EP E5-2670 processors running under the CentOS 6 Linux distribution. The frequency range of 25–10,000 Hz was divided into four jobs, each consisting of 100 frequencies in 25-Hz steps. We ran a maximum of 12 jobs at a time on nodes with 16 processors each, each job on a single processor. The run times were about 450 min per job for a total of 1800 min for the complete frequency range for one simulation scenario.

Implementation of Fluid-Structure Interaction

Fluid-structure interactions are modelled by coupling the constitutive equations of the two domains, fluid (air) and structure (ear components), on their interface surfaces. At the interface, there are two conditions that should be satisfied: (1) the continuity of the normal stresses and (2) the continuity of the normal velocities. The simultaneous satisfaction of these two conditions couples the two domains. The detailed mathematical formulation can be found elsewhere (e.g. Greffet 2013).

Admittance Calculation

The impedance components (resistance R and susceptance X) were calculated by dividing the real and imaginary components of the pressure (usually taken from a node on the medial surface of the probe tip) by the input volume velocity at each frequency. Admittance is the reciprocal of impedance and has two components, conductance G and susceptance B. The admittance magnitude and phase were calculated as $\sqrt{G^2{+B}^2}$ and Tan⁻¹(B/G), respectively.

Sensitivity Analyses

In this study, we performed four sensitivity analyses. First, all nine combinations of the three ear-canal scenarios and three middle-ear scenarios (low-impedance, baseline and high-impedance for each) were simulated to provide estimates of the contributions of the canal and middle-ear responses to the total ear model.

Second, a traditional one-parameter-at-a-time analysis was performed to investigate the effects of these parameters. As stated in the ‘Material Properties’ section, plausible ranges were established for the material parameters of the ear components. The range between the minimum and maximum values for each parameter (shown in Table 1) was divided into four intervals so each parameter had five evenly spaced values.

Third, we evaluated the effect of the location of the pressure measurement point by examining the pressures at five nodes in the canal, between 0.0 and 6.0 mm from the medial surface of the probe tip. This procedure was suggested by the fact that Keefe et al. (1993) extended the microphone probe approximately 3 mm beyond the surface of the foam eartip to minimize the effects of evanescent waves between the source and the receiver.

Fourth, to provide an estimate of the effects of anatomical variability, the overall geometry of the model was scaled by −10, −5, +5 and +10 % in the x, y and z directions separately and also in all three directions simultaneously. The scaling affects the dimensions of all model components simultaneously. The x, y and z directions represent the lateral-medial, posterior-anterior and superior-inferior directions, respectively.

Clinical Data

As in our previous study, two sets of clinical data were used for comparison with the model. The first set consisted of the impedance measurements obtained by Keefe et al. (1993) under ambient pressure in a group of 1-month-old infants, for frequencies from 125 to 10,700 Hz with a 1/3-octave resolution. The second set consisted of admittance measurements that we performed in parallel with another study (Pitaro et al. 2016), on a group of 23 infants with ages between 14 and 28 days, for frequencies from 250 to 8000 Hz with 1/12-octave resolution. That study was approved by the Institutional Review Board of the McGill University Health Centre. The measurements were made with a wideband tympanometry research system (WBTymp 3.2, Interacoustics Inc.). All measurements except one were performed in the Otolaryngology out-patient clinic of the Montreal Children’s Hospital. More details about the measurement procedure can be found in (Pitaro et al. 2016). The low-frequency admittance measurements (up to 2 kHz) were reported in our previous paper (Motallebzadeh et al. 2017).

Results

Sound Pressure and Admittance Responses

Figure 2 shows the pressure distributions inside the canal and middle-ear cavity for frequencies of 6.1, 7.2 and 9.6 kHz, representing the first acoustic resonance of the middle-ear cavity and the first and second standing-wave modes inside the canal, respectively, for the baseline model. The descriptions of these features are provided later in this section. Figure 3 shows the pressure (panel a), admittance magnitude (panel b) and admittance phase (panel c) frequency responses for a measurement point at the medial face of the probe tip. (Note that, since the admittance is proportional to the reciprocal of the pressure, minima and maxima of pressure correspond to maxima and minima of the admittance, respectively.) The solid black curves represent the responses of the overall model with baseline parameters. The other curves represent the responses obtained when one or more parts of the model are removed or made rigid. The dashed black curves represent the baseline model with open middle-ear cavity (i.e. the model without the cavity effects). The red curves represent the responses of the simplest case, namely, a rigid canal with a rigid TM. The green curves represent the behaviour of the non-rigid canal only—the TM is rigid and so the middle ear does not contribute to the responses. The blue curves represent the responses of models with a rigid canal and either an open middle-ear cavity (dashed curves) or a closed cavity (solid curves). In Figure 3a, b, the upper limits of the pressure and admittance magnitude axes are limited to 1 Pa and 225 mm³/s/Pa, respectively, clipping some of the curves at some frequencies in order to provide better visualization of other parts of the curves. An inset shows the admittance magnitudes with an expanded vertical scale for frequencies below 1 kHz. Comparisons among these models reveal the contributions of the different components to the overall response of the newborn ear. The important features are pointed out by numbered arrows in the figure and can be summarized in order from low to high frequencies as follows, with the paragraph numbers corresponding to the arrow numbers. (The focus is on panel b unless otherwise stated.)

1. A broad resonance is seen at frequencies around 650 Hz (green curve, inset figure) for a model with a rigid TM and compliant canal wall. In other models with compliant canal (i.e. the baseline with open and closed cavity, represented by dashed and solid black curves, respectively), this resonance appears but merges with the next peak and appears as a smooth shoulder. The feature disappears when the canal wall is rigid (red curve).

2. A resonance peak is seen at 1.05 kHz for the open-cavity models (dashed blue and dashed black curves). Changing the canal wall from rigid (dashed blue curve) to compliant (dashed black curve) does not alter the curve significantly in this frequency region (less than a 4 % increase in magnitude and 25-Hz shift of the peak). When the middle-ear cavity is closed, the resonance magnitude decreases by 5 % and shifts upward to 1.2 Hz (solid blue and black curves).

3. A sharp resonance occurs at 5.1 kHz in the models with a rigid TM, whether the canal walls are rigid or compliant (solid red and green curves, respectively). At this frequency, the first standing-wave mode of the model occurs inside the canal; the pressure has a node (minimum) at the entrance (as seen at 5.1 kHz in panel a) and an anti-node (maximum) at the rigid medial end of the canal.

4. Another resonance, which is very sharp, happens at 6.1 kHz in the models with closed cavity (solid blue and black curves), representing the first resonance of the middle-ear cavity. The pressure distribution of this mode is shown in Figure 2a.

5. Admittance maxima are seen at 7.2 kHz for the models with compliant canal wall and TM, whether the cavity is open or closed (blue and black, dashed and solid curves). This represents the first standing-wave mode in the canal, which occurs at 5.1 kHz when the TM is rigid (see item 3 above). The pressure distribution at this frequency is presented in Figure 2b for the baseline model; there is a pressure node at the entrance of the canal and an anti-node at the medial end. The models with open cavities have slightly higher magnitudes (around 10 %) for this feature (dashed blue and black curves).

6. Admittance minima are observable at frequencies in the vicinity of 8.2 kHz in the models with a rigid TM, whether the canal walls are rigid or compliant (solid red and green curves, respectively). At this frequency, the second standing-wave modes occur inside the canal, resulting in pressure anti-nodes at both the entrance and the medial end of the canal. The pressures in these two models have sharp peaks at this frequency (panel a).

7. Admittance minima are observable at frequencies in the vicinity of 9.6 kHz for the models with compliant canal wall and TM, whether the cavity is open or closed (blue and black, dashed and solid curves). This represents the second standing-wave mode in the canal, which occurs at 8.2 kHz when the TM is rigid. The models with open cavities have slightly lower magnitudes (a difference of less than 10 %) at this frequency (dashed blue and black curves). The pressure distribution at 9.6 kHz is presented in Figure 2c for the baseline model; there are pressure anti-nodes at both the entrance and the medial end of the canal.

FIG. 2.

Posterior view of pressure distribution map inside the ear canal and middle-ear cavity in response to a harmonic velocity source with a constant amplitude of 0.15 mm/s, normal to the medial surface of the probe tip. a First resonance mode of the middle-ear cavity (6.1 kHz). b First standing-wave mode inside the ear canal (7.2 kHz). c Second standing-wave mode inside the ear canal (9.6 kHz).

FIG. 3.

Pressure and admittance responses of the ear models at the medial surface of the probe tip. a Pressure magnitudes. b Admittance magnitudes. c Phases. The inset shows a magnified view of the admittance magnitudes at frequencies between 150 and 2000 Hz. Features indicated by arrows and numbers are discussed in the text.

Sensitivity Analysis

Combinations of Low-Impedance, Baseline and High-Impedance Models

Admittances for all nine combinations of the three ear-canal and three middle-ear scenarios (low-impedance, baseline and high-impedance for each) are presented in Figure 4. In this figure, each pair C _i + M _j corresponds to the combination of the canal model i and the middle-ear model j, and the indices i, j = 1, 2 and 3 represent the low-impedance, baseline and high-impedance models, respectively. In all models with the low-impedance canal model (green curves), a peak with a magnitude of ~27 mm³/s/Pa is visible at ~250 Hz. A drop and a subsequent rise of the phase response between 100 and 1000 Hz are also observable for these models. As the canal becomes stiffer (red and blue curves), the local admittance peak at low frequencies merges with that of the middle-ear resonance in the frequency range of 1–2 kHz. The resonances of the models at frequencies in the vicinity of 1.5 kHz are not affected significantly by the conditions of the ear canal, so they may be taken to mainly represent the middle-ear resonances. Around 7.2 kHz (the occurrence of the first standing-wave mode inside the canal), the stiffer canals have increased admittance magnitudes but the stiffer middle ears have decreased magnitudes.

FIG. 4.

Admittance responses of the combinations of three ear-canal and three middle-ear models. a Admittance magnitudes. b Phases. Each pair C _i + M _j corresponds to the combination of the canal model i and the middle-ear model j; the indices i, j = 1, 2 and 3 represent the low-impedance, baseline and high-impedance models, respectively.

The phases of the models with stiffer canals (red and blue curves) remain close to 90° over a broader frequency range below 1 kHz. Between 1 and 3 kHz, the canal condition does not affect the phase response significantly, but the stiffer middle ears have higher phases (e.g. blue curves). Between 3 and ~7.8 kHz the stiffer canals show higher phase values, and above 7.8 kHz they have decreased phases. The stiffness of the middle ear works in the opposite direction above 3 kHz: the stiffer middle ears show lower phase values between 3 and ~7.8 kHz and higher values above 7.8 kHz.

Effects of Material Parameters

Since the main goal of clinical admittance measurements is to characterize the middle-ear response, the features that we looked into were at frequencies in the vicinity of the middle-ear resonance (between 1 and 2 kHz). The results of the one-parameter-at-a-time sensitivity analysis are presented in Figure 5, for two features of the admittance (maximum magnitude in panel a and frequency of the maximum in panel b).

FIG. 5.

One-parameter-at-a-time sensitivity analysis, showing the influence of the material parameters on the maximum admittance magnitudes (a) and the corresponding frequencies (b) for the ear models. E _pt = Young’s modulus of the pars tensa, ζ _pt = damping ratio of the pars tensa, ρ _st = density of all soft tissues in the model, E _st = Young’s modulus of soft tissue surrounding the canal, ν _st = Poisson’s ratio of all soft tissues.

All material parameters of the ear-canal and middle-ear models (i.e. stiffness, damping ratio, density, Poisson’s ratio, stapes mass and cochlear load) were included in this analysis. Here, only parameters with effects greater than 1 % on either criterion are reported. The parameters shown in Figure 5a, in order of decreasing influence on the maximum admittance magnitude of the ear model, are the damping ratio ζ _pt and Young’s modulus E _pt of the pars tensa, the density ρ _st of all soft tissues in the model, the Young’s modulus E _st of the soft tissue surrounding the canal, and the Poisson’s ratio υ of the soft tissues. The parameters with the greatest effects on the maximum admittance magnitude are ζ _pt and E _pt with maximum deviations of 80 and 46 %, respectively. The effects of the other parameters are less than 7 %. The coefficients of determination R ² for these parameters (from left to right in the figure) are 0.88, 0.82, 0.99, 0.51 and 0.97, respectively; this suggests that E _st has a nonlinear effect, but its overall effect is quite small. (As described in detail in Motallebzadeh et al. (2017), R ², the coefficient of determination of a linear regression, is an indication of how well a straight line fits a set of data and can be used as a measure of the linearity or nonlinearity of the effect of a parameter. R ² is a number between 0 and 1, and the larger it is, the more linear the parameter effect is.)

Most of the material parameters have very small effects on the frequency of the maximum admittance in the frequency range of interest, as shown in Figure 5b. The most important parameter is the pars - tensa Young’s modulus E _pt, shifting the maximum frequency by 250 Hz. The pars - tensa damping ratio ζ _pt and the soft - tissue density ρ _st shift the maximum frequency by 50 Hz. The coefficients of determination are 0.96, 0.78 and 0.94 for the pars - tensa Young’s modulus E _pt, damping ratio ζ _pt and soft tissue density ρ _st, respectively, indicating that ζ _pt is the only parameter with a notably nonlinear effect. E _pt has an almost linear effect at values higher than 4 MPa.

Effect of Measurement Location

To investigate the effects of the distance between the source (i.e. the speaker) and the measurement point (i.e. the microphone), which was estimated to be 3 mm by Keefe et al. (1993), we computed the admittance data at five nodes inside the canal at distances of 0.0, 1.5, 3.0, 4.5 and 6.0 mm from the medial surface of the probe tip (Fig. 6).

FIG. 6.

Admittance responses of the measurements at five nodes along the ear canal: nodes N1 to N5 are at 0.0, 1.5, 3.0, 4.5 and 6.0 mm, respectively, from the medial surface of the probe tip. a Admittance magnitude. b Phase.

At frequencies below 1 kHz, the maximum difference between the admittance magnitudes calculated at 0.0 and 6.0 mm is less than 15 %. The first resonance peak not only decreases in magnitude (from 52.6 mm³/s/Pa at 0.0 mm to 46.2 at 6.0 mm (a change of about 12 %) but also becomes broader and shifts from 1300 to 1400 Hz. The admittance peak due to the second standing-wave mode inside the canal is also significantly affected by the choice of the measurement point: as the distance from the driving point increases from 0 to 6 mm, the admittance peak in the vicinity of 7.2 kHz decreases in magnitude (by up to 35 %) and shifts to 8.2 kHz.

Effect of Geometry Variations

In Figure 7, panels a, b and c compare the geometry-variation effects in x, y and z, respectively, and panel d presents the effects of simultaneous variation in all directions, for −10, −5, +5 and +10 % scaling of the baseline geometry. The scaling affects the dimensions of all model components together. Since the model components extend in all directions, defining a clear contribution of each component to the total response is rather difficult. However, based on the major orientations of the different components, we have provided some explanation of their possible geometrical effects. Since no significant deviations are observed between 2.5 and 4 kHz, this frequency range was excluded in order to provide better visualization of the other parts of the curves. As described in the ‘Sensitivity Analyses’ section, the x, y and z directions mainly represent the lateral-medial, posterior-anterior and superior-inferior directions, respectively. The most influential dimension is x. As the model expands in the x direction, the middle-ear resonance shifts to lower frequencies (e.g. from 1.4 to 1.1 kHz for elongations from −10 to +10 %). The resonance magnitude is increased by 30 %, from 46.0 to 59.7 mm³/s/Pa. The resonance of the middle-ear cavity (which extends mainly in the x direction) also shifts to lower frequencies by about 10 % (from 6.2 kHz at −10 % elongation to 5.6 kHz at +10 %). The elongations in the other two directions (i.e. y and z) do not affect the admittance response significantly, with a maximum magnitude change of 15 % and a maximum frequency shift of 25 Hz for elongations in the range of ±10 %. The combined scaling in all directions simultaneously (panel d) cancels out the effects of elongations in each individual direction to some extent at the middle-ear resonance. Scaling up in the x direction increases the main resonance and shifts it to lower frequencies, while scaling up in the z direction decreases the main resonance with little shift of its frequency (dashed red curves in panels a and c); for simultaneous x, y and z elongations from −10 to +10 %, the resonance magnitude changes by less than 1 % (from 52.6 mm³/s/Pa to 52.1) and the corresponding frequency shifts by about 14 % (from 1.4 to 1.2 kHz). However, the effect on the middle-ear-cavity resonance is significantly enhanced for simultaneous scaling from −10 to +10 % in all directions simultaneously, as the corresponding frequency shifts by about 18 % from 6.5 to 5.3 kHz.

FIG. 7.

Sensitivity analysis showing the effects of variations in x, y and z (panels a, b and c, respectively) and simultaneous variation in all directions (panel d) for −10, −5, +5 and +10 % scaling of the baseline geometry. The left and right panels represent the frequency ranges of 0.5 to 2.5 kHz and 4 to 10 kHz, respectively. Frequencies from 2.5 to 4 kHz are excluded because no significant deviations occur in that range.

Comparison with Clinical Data

The sensitivity analyses in the ‘Combinations of Low-Impedance, Baseline and High-Impedance Models’ and ‘Effects of Material Parameters’ sections showed that the canal response has a pronounced effect on the total admittance of the ear at frequencies below 1 kHz. Comparing the behaviour of the ear model for the nine combinations of canal and middle-ear parameters with two sets of clinical data (described in the ‘Clinical Data’ section), it was concluded that the main feature of the clinically measured low-frequency response (a peak at frequencies below 1 kHz due to the canal resonance) could be matched by assigning an intermediate impedance to the canal parameters (an intermediate stiffness of E = 80 kPa together with a high damping ratio of ζ = 0.4). These are the same adjusted parameters that we adopted in our previous model (Motallebzadeh et al. 2017). A comparison of such an adjusted model with the clinical data is presented in Figure 8. As shown in the inset figure, the resonance of the ear canal for the adjusted model happens at ~500 Hz (arrow 1) as a smooth shoulder, merging into the middle-ear resonance. The corresponding peak in the mean data of Keefe and Levi (1996) is around 400 Hz, and the first peaks of our individual subjects are spread over the frequency range of 250–750 Hz, with most being around 250 Hz.

As with our previous model (Motallebzadeh et al. 2017), the high-impedance middle-ear model generated a reasonable match for middle-ear resonance magnitudes, so it was combined with the adjusted canal model. The resonance of the middle ear of the adjusted model results in a magnitude peak of 34.7 mm³/s/Pa at 1550 Hz (arrow 2), and the width of the peak (as defined by the frequencies at which the magnitudes are 90 % of the peak value) is 450 Hz. (90 % is the same cutoff value as used in Motallebzadeh et al. 2017.) The mean curve of Keefe and Levi shows a resonance peak of 30.0 mm³/s/Pa at around 2 kHz. The resonances of our individual subjects are spread from about 1 to 2.5 kHz. Averaging the individual curves smears their peaks, resulting in a broad resonance peak (with about three small local peaks) between 1 and 2 kHz. This frequency range covers the corresponding frequency of the adjusted model, with a good match in magnitude values.

The adjusted model predicts low admittance magnitudes between the frequencies of 3 and 5 kHz. This local minimum is visible in both sets of clinical data. In both sets of clinical data, the mean magnitude of this minimum is higher than that in the model, but the model response is still within the range of the individual responses.

The resonance of the middle-ear cavity of the model causes a sharp peak of admittance magnitude at 6.1 kHz (arrow 3), followed by a minimum, and then there is another local peak at around 7.2 kHz (arrow 4) due to the second standing-wave mode inside the ear canal (described in the ‘Sound Pressure and Admittance Responses’ section). The individual subject responses show two local peaks, one between 4.5 and 5.5 kHz and another one between 5.6 and 7.2 kHz. The mean curve has two peaks in this frequency range, one at 5.0 kHz (arrow 3′) and one at 6.4 kHz (arrow 4′). The data of Keefe and Levi, however, simply show a constantly increasing admittance at frequencies higher than 4 kHz, presumably because the frequency resolution of their measurements was not fine enough to exhibit such sharp features.

The frequencies of the minima and maxima in the phase of the model closely match those of the mean phase data of Keefe et al. (1993), but differences in the actual phase values are seen. The differences are within about 20° up to about 2.5 kHz but they increase to 75° at 5 kHz (the second maximum). The model phase shows a sharp notch ~6.1 kHz and rises at frequencies above 8.2 kHz. This notch is not observable in the data of Keefe and Levi, where the phase continues to decrease after the maximum at 5 kHz. The phase of the model is within the range of phases of the individual subjects of our clinical data at all frequencies, except for the sharp notch at 6.1 kHz. The first rather sharp minimum of our mean phase occurs at 300 Hz; the corresponding minimum of the model is much more shallow and happens at 550 Hz. The first maximum of the mean phase of our clinical data has a broad peak at 800 kHz. The corresponding frequency in the model is 900 Hz, at a phase that is 10° lower. The second minimum of the mean phase of the clinical data occurs at 2.1 kHz. The corresponding frequency in the model is 2.4 Hz at a phase that is 18° lower. Two peaks of the mean measured phase response are seen at 4.3 and 5.7 kHz with a shallow notch between them. The corresponding features in the model are two maxima at 5.2 and 6.4 kHz, with a sharp notch between them. The phases of these maxima for the model are 8° lower and 3° higher than those of the mean phase of our clinical data, respectively; in both model and clinical data, the second maximum is lower than the first.

Comparison with Non-FSI Model

As mentioned above, in the present study the geometry of the canal has been modified slightly from that in our previous study (Motallebzadeh et al. 2017). The modification resulted in differences of less than 1 % in both the magnitude and phase responses of the baseline model (i.e. the C ₂ + M ₂ curves of Figure 8 of that study). In Figure 9, we compare the non-FSI model and the FSI model, with the new geometry for both. Three non-FSI scenarios (dashed lines), with the low-impedance, baseline and high-impedance models, are compared with the corresponding FSI models (solid lines) at frequencies up to 2.5 kHz, somewhat beyond the nominal 2-kHz frequency range of the non-FSI model.

FIG. 9.

Comparison between FSI and non-FSI models. Admittance magnitudes (a) and phases (b) are presented for low-impedance (green), baseline (red) and high-impedance (blue) parameters.

At frequencies up to 800 Hz, the non-FSI models show a good match with the corresponding FSI models. However, at higher frequencies the FSI models start to deviate from the non-FSI models. In the low-impedance FSI model (solid green curve), a small peak and a main peak are seen with magnitudes of 63 and 130 mm³/s/Pa at ~925 and 1200 Hz, respectively. In the corresponding non-FSI model (dashed green curve), these features are shifted to 950 and 1400 Hz with magnitudes of 51 and 114 mm³/s/Pa, respectively, about 19 and 12 % lower, respectively. In addition, in the FSI model the resonance peak decreases smoothly up to 2.5 kHz, but in the non-FSI model two shoulders appear on the downslope of the main peak, at 1650 and 2200 Hz. The baseline FSI model (solid red curve) shows a simple peak of 52 mm³/s/Pa at 1300 Hz, whereas in the non-FSI model (dashed red curve), a broad peak is seen with a maximum of 44 mm³/s/Pa at 2000 Hz and a shoulder around 1600 Hz. The peak of the high-impedance FSI model occurs at 1500 Hz with a magnitude of 32 mm³/s/Pa, while that of the non-FSI model reaches its maximum of 29 mm³/s/Pa at 2100 Hz.

A good match between the phase responses of the FSI and non-FSI models remains up to 800 Hz (maximum of 8° difference between the low-impedance models at 800 Hz). At higher frequencies, however, although the sequences of rises and drops are similar in the corresponding models, the phase values differ considerably. In the low-impedance models, the minimum at 950 Hz and maximum at 1050 Hz in the FSI model are shifted by 50 and 120 Hz, respectively, in the non-FSI model, and the phases are very different at higher frequencies. In the stiffer models, the corresponding models deviate by up to 60° (baseline models at 1.8 kHz) and 45° (high-impedance models at 2 kHz).

The inconsistencies are presumably due to the treatment of the air spaces as lumped admittance elements in the non-FSI models. In those models, it was assumed that the wavelength of the sound at frequencies below 2 kHz is long enough (in comparison with the model dimensions) that we could assume a uniform pressure distribution throughout the ear canal and middle-ear cavity and across the TM surface. However, the pressure is not uniform in spite of the long wave length, because of the interaction of the incident and reflected sound waves: at 2 kHz, the pressure at the entrance of the canal and the pressure at the TM already differ by more than 30 % in the FSI model.

Discussion

Pressure Distribution Inside the Canal and Middle-Ear Cavity

To the best of our knowledge, no experimental measurements have been reported for the spatial sound pressure distribution within the newborn ear canal, but there have been several such studies of older humans and of other species (e.g. Stinson et al. 1982; Stinson 1985a, b; Gilman and Dirks 1986; Stinson and Khanna 1989; Bergevin and Olson 2014; Khaleghi and Puria 2017). Wang et al. (2016) simulated the effects of the compliant canal on the energy absorbance response, but they did not report the pressure distribution within the canal and middle-ear cavity. Since the ear canal is longer in adults (~25 mm (Anson and Donaldson 1992, p. 146)) than in newborns (~16 mm from the entrance to the umbo, in our model), the onset of standing waves happens at lower frequencies in adult canals. For example, Gilman and Dirks (1986) found a frequency of ~3.6 kHz for a standard ear simulator. In our model with rigid canal wall and rigid TM, the first standing-wave mode, with one node (minimum pressure) at the entrance and one anti-node (maximum pressure) at the medial end of the canal, occurred at 5.1 kHz. The second standing-wave mode is observed at 8.2 kHz, with anti-nodes at both the entrance and the medial end of the canal.

The first standing-wave mode of a cylindrical pipe, driven at the entrance and closed at the end, happens at 5.1 kHz if its length is 16.6 mm (1/4 of the wavelength). The second standing-wave mode for such a pipe occurs at 10.2 kHz, twice the first mode, much larger than the 8.2-kHz frequency of the corresponding mode in our model with a realistically shaped but rigid ear canal and a rigid TM. To explore the discrepancy further, we also modelled waveguides shaped like truncated cones with entrance diameters of 4.4 mm (approximately the diameter of the canal in newborns) and closed terminations of diameter 9 mm (approximately the diameter of the TM), and different lengths. We observed a standing-wave pattern at 5.1 kHz for a length of 12.4 mm, but the second standing-wave mode for such a cone with that length happens at 14.4 kHz, again much higher than 8.2 kHz. These simulations indicate that a realistically shaped but rigid newborn ear canal cannot be well approximated by rigid cylindrical or conical waveguides. The differences can be attributed to the variations of the cross-sectional area, the curvature along the length of the canal, and the angle of the TM at the termination. These features presumably also affect the behaviour of the newborn canal when its walls are not rigid.

The cavity in our model can be enveloped in a box of dimensions 23 × 18 × 15 mm. Treating such a box-shaped cavity as being driven at one end of the longest dimension and closed at the other end, its first standing-wave mode happens at 3.7 kHz. Since the middle-ear cavity is terminated by the TM impedance, it can be expected to have its first resonance at a higher frequency (i.e. around 6.1 kHz in our model). Due to the limited resolution of the CT images that we used for reconstructing the geometry of our model, the fine structures of the cavities (especially the mastoid air cells) and their narrow connections were modelled only very approximately. Moreover, since the air was modelled as an inviscid fluid, the viscous losses associated with sound pressure propagation inside the connecting passages were not modelled. Any energy absorption at the walls of the cavities was also not modelled. Although damping would not be expected to shift the resonance frequency significantly, the absence of damping does result in a resonance that is probably unrealistically sharp. The complex structure of the middle-ear cavity, including the tympanic cavity, aditus, antrum and mastoid air-cell system (e.g. Keefe 2015), makes the exploration of its sound pressure distribution very challenging.

Model Validation

Holte et al. (1991) and Keefe et al. (1993) found that the resonance of the ear canal in infants less than 1 month old occurs at frequencies around 450 Hz. The corresponding frequencies in our clinical data for individual subjects are spread over the frequency range of 250 to 750 Hz. By adjusting the parameters of the model within the proposed plausible ranges, we reproduced the resonance of the canal at 500 Hz, consistent with the clinical data.

Due to their limited frequency range, Holte et al. (1991) did not observe the middle-ear resonance, but they suggested that it is beyond 900 Hz. Keefe et al. (1993) found that the overall ear resonance occurred in the vicinity of 1.8 kHz. In our clinical measurements of individual subjects, the middle-ear resonance occurs in the range of 1 to 2.5 Hz. Our model with the adjusted parameters presents a clear resonance peak at 1.8 kHz, well placed among those of our individual subject responses and the finding of Keefe et al. (1993).

Keefe et al. (1993) found an increase in the admittance magnitudes at frequencies higher than 4 kHz. They suggested that this might be because of the resonance of the middle-ear cavity. However, their frequency resolution was not high enough to distinguish this effect from the effect of the first standing-wave mode in the canal, which also results in a peak of admittance magnitude. In our clinical data with 1/12-octave resolution, both the individual responses and the mean curve show two peaks, at 5.1 and 6.3 kHz. These features, which may be attributable to the resonance of the middle-ear cavity and the occurrence of the first standing-wave mode inside the ear canal, occurred in our model at 6.1 and 7.2 kHz, respectively. As shown in Figure 3, the sharp notch of 6.1 kHz is due to the presence of the standing waves within the middle-ear cavity. We expect that this notch and its associated peak would be smoother if the attenuation effects due to the viscosity of the air in the middle-ear cavity were taken into account. In addition, as Cinamon (2009) reported, in newborns the bony surface of the cavity is covered by mesenchyme as well as by a mucosal layer, and they also absorb some portion of the acoustic energy.

Significance

Previous clinical reports (e.g. Holte et al. 1991; Keefe et al. 1993; Keefe and Levi 1996; Sanford and Feeney 2008) concluded that the canal contribution to the total admittance response of the ear is substantial at frequencies below 1 kHz and that traditional low-frequency tympanometry at single-probe tones of 250, 650 and 1000 Hz does not reflect the middle-ear response. Our model provides a description of the canal contribution to the overall admittance response of the newborn ear and shows quantitatively that at frequencies around the middle-ear resonance (around 1.8 kHz), the admittance of the newborn ear is mainly dominated by that of the middle ear, similar to our previous low-frequency model (Motallebzadeh et al. 2017). The present model also predicts the features of the first resonance mode of the middle-ear cavity (around 6 kHz) as well as the first and second standing-wave modes in the ear canal (around 7.2 and 9.6 kHz, respectively). Wideband immittance measurements with higher frequency resolution will be required to further investigate these features.

In this study, due to the small deformations in response to acoustical loads, all materials are assumed to be linearly elastic. To investigate the effects of the static pressure in pressurized immittance measurements (tympanometry), it will be important to include nonlinear and time-dependent behaviour of the soft tissues, especially the canal and TM (Qi et al. 2006, 2008; Motallebzadeh et al. 2013a; Charlebois et al. 2013) in future studies.

Limitations

At frequencies up to about 1 kHz, only the total volume of the adult mastoid air-cell system is important in determining the cavity admittance (e.g. Stepp and Voss 2005). At higher frequencies, however, the fine structure of the air-cell network adds multiple peaks because the air cells act like local resonators with viscous damping (e.g. Stepp and Voss 2005; Keefe 2015). There is no mastoid air-cell structure in our model beyond what is indicated by the irregular surfaces seen in Figures 1 and 2. Because of the low resolution of the clinical CT images, the fine structures of the middle-ear cavity may not be captured. However, according to Cinamon (2009), ‘At birth, there are usually no additional air cells’ (other than the antrum), so our geometry may be realistic.

In this study, the fluid (air) was assumed to be inviscid so viscous damping was not taken into account. Since the model does not contain narrow passages, the viscosity effects are not expected to affect the main characteristics of the model response except by smoothing the sharp standing-wave-related features in Figures 3 and 8.

As Ruah et al. (1991) reported, the TM is thicker in newborns and infants than that in adults, especially in the first year of life. The spatially varying thickness of the TM of this model is based on real histological data for one 3-week-old newborn. We expect that the model results would be quite sensitive to TM thickness, as other studies have pointed out (e.g. Funnell and Laszlo 1978; Maftoon et al. 2015). The non-uniform thickness distribution causes non-uniformity of both the local stiffness and the local inertia. The thickness can be expected to be quite variable from one individual to another (e.g. Kuypers et al. 2006). In future studies, sensitivity analyses of the effects of the TM thickness could be done.

Due to the low CT image resolution, the shape of the stapes was not discernible and it was not included in the model. Thus, neither the complex motion of the stapes nor the frequency dependence of the cochlear load was incorporated in this study. We represented the stapes inertia, the annular ligament and the cochlear load by nodal loads. As stated in our previous paper (Motallebzadeh et al. 2017), the stapes moves in a piston-like manner at frequencies below 2 kHz. However, its motion becomes more complicated at the higher frequencies that are considered in this paper (e.g. Decraemer and Khanna 2004). The malleus and incus were large enough to be seen and included in the geometry reconstruction. The incus in our model serves both (1) as an attachment for the posterior incudal ligament, which is important in determining the ossicular axis of rotation, and (2) as a place to attach the springs and dashpots representing the stapes and cochlea. It is commonly believed that the IMJ is not rigid and that its behaviour depends on both intensity and frequency (Willi et al. 2002; Nakajima et al. 2005). It has been reported that this behaviour could also be age-dependent (Willi et al. 2002). With a geometry constructed from higher-resolution images, the effects of the ossicular joints and annular ligament on the immittance response of the newborn ear could be included in future work, but overall the TM contributes more to the overall middle-ear admittance response than other components do (e.g. Feldman 1974).