Limitations of present models of blast-induced sound power conduction through the external and middle ear

The effect of blast-like impulses on hearing is a significant concern to the modern military, as deficits in hearing adversely affect communications and pose long-term health risks. Further development of high-powered weapons and explosives has led to an increased awareness of how the ear and hearing can be affected by these devices. At the center of discussions about blast-induced effects on hearing are multiple questions about how to predict the hearing loss produced by different impulsive stimuli as well as how middle-ear pathology or hearing protection schemes may reduce hearing loss. In this paper, we concentrate on two issues: (1) How is the sound power in a blast conducted to the inner-ear sensory apparatus by the external and middle ears? (2) How is the conductive process altered by nonlinear mechanisms with the middle ear?

Past investigations of the effects of the conduction of impulse-related sound power to the inner ear have used measurements and models of normal external- and middle-ear sound conduction (e.g., Tonndorf, 1976; Rosowski, 1991) as well as mathematical descriptions of how the nonlinear processes within the middle ear may limit hearing loss (e.g., Price and Kalb, 1991; Zagadou et al., 2016), though the accuracy of these descriptions is a point of controversy (Price et al., 2017; Zagadou et al., 2017). These analytic approaches have been recently supplemented by measurements of the middle-ear conduction of blasts and intense tones in human cadaveric ears (Greene et al., 2017; Greene et al., 2018; Cheng et al., 2017) and in animals (e.g., Peacock et al., 2018). There are also recent measurements of the effect of perforations of the eardrum (more specifically, the tympanic membrane, or TM) on the conduction of intense sounds by the middle ear (Greene et al., 2018; Cheng et al., 2018).

In this report, we investigate different models that quantify sound energy conduction from the environment to the inner ear. Specifically, (1) we review and revise an older model of sound-power transfer through the human middle ear (Approach 1); (2) we investigate the predicted sound conduction in a class of simple nonlinear middle-ear models (Approach 2); and (3) we compare model outputs with measurements of middle-ear function with high-level sound stimuli, and address shortcomings in the predictions in intact and pathological middle ears (Approach 3). In Sec. IV, we compare and contrast the limited predictions of the simpler models we introduce with the more well-developed models that others use to predict impulse-induced hearing loss, and describe how variations in model parameters may introduce non-monotonicity in the predicted growth of damaging stimuli. We argue that improved models of blast-conduction to the inner ear depend on gathering additional data to describe better the workings of the middle ear at high sound levels. These improved models will provide (a) better predictions of the effects of blast-induced pathology on impulse-induced sensory loss, and (b) better ways to predict the consequences of different protective devices.

Rosowski and co-workers (Rosowski et al., 1986; Rosowski, 1991) introduced a hybrid acousto-electric circuit (Fig. 1) as a framework for a measurement-based quantification of how well the external and middle ears of humans and other animals gathered and transmitted environmental sound power to the inner ear. The analysis of human ear function was based on: measurements of middle-ear input impedance performed in small groups of live subjects over limited frequency ranges (Rabinowitz, 1981; Hudde, 1983), model predictions of the cochlear input impedance (Zwislocki, 1962, 1965), and contemporaneous measurements of sound-induced stapes motion performed in human cadaveric ears (Kringlebotn and Gundersen, 1985; Gyo et al., 1987). Since that time, newer middle-ear input impedance measurements that cover a broad frequency range have been made in a larger number of live humans (Keefe et al., 1993), the cochlear input impedance in human cadavers has been measured by several laboratories (Aibara et al., 2001; Nakajima et al., 2009), and multiple, better-controlled measurements of middle-ear sound transfer in human cadavers have been made (e.g., Aibara et al., 2001; Nakajima et al., 2009). The revised model focuses on the collection and transfer of sound power through the ear, where sound power (with units of watts: symbol W) is the temporal derivative of sound energy.

Figure 1 describes the acousto-electric circuit. The acoustic input on the left is the diffuse-field sound intensity with units of W/m², where “diffuse” describes a sound-field that is omni-directional (Shaw, 1979, 1988; Kuhn, 1979; Rosowski et al., 1988). A fraction of the sound power in this stimulus is collected by an equivalent receiver area of the external ear loaded by the middle ear and conducted to the TM at the entrance of the middle ear. The variables of interest¹ at the TM are the sound pressure and volume velocity just lateral to the TM, P_T, and U_T, whose ratio describes the middle-ear input impedance Z_T = P_T/U_T or admittance Y_T = U_T/P_T = 1/Z_T. The variables of interest at the entrance to the inner ear are the sound pressure P_V in the vestibule (the perilymph-filled entrance of the inner ear) and the volume velocity of the stapes U_S, which is generally computed from measurements of the stapes translational velocity and the area of the stapes footplate. The ratio P_V/U_S is the cochlear input impedance Z_C, with an admittance Y_C = 1/Z_C. In the sinusoidal steady state, the time averaged sound power² absorbed at the entrance of the middle ear is Π_T = (1/2) |P_T|² Real{Y_T}, and the time-averaged power that is absorbed by the inner ear is Π_C = (1/2) |P_V|² Real{Y_C}. The volume velocities, sound pressures and impedances use to define the circuit are chosen because they have all been measured in either live or cadaveric human ears. Also of interest is the radiation impedance looking out the ear canal from the TM, Z_E (Shaw, 1988; Rosowski et al., 1988).

In this revision of the earlier analysis, we use more modern data including the input impedance Z_T measured in adult ear canals over a broad frequency range by Keefe et al. (1993), and temporal bone measurements of the middle-ear pressure gain P_V/P_T and the cochlear input impedance Z_C made by Nakajima et al. (2009). These later measurements are very similar to other temporal-bone measurements made by Aibara et al. (2001). The use of scala vestibuli sound pressure measurements to describe middle-ear sound transfer has the benefit that these sound pressures are not affected by possible complex three-dimensional (3D) motions of the stapes. However, the effects of such complicated motion may be small, as Decraemer et al. (2007) found a strong correlation between the measured one-dimensional (1D) piston-like motion of the stapes and scala vestibuli sound pressure in live gerbils.

At the heart of the power analysis are two quantitative descriptions of power collection and transmission by the external and middle ear. The first of these is the Diffuse-Field Absorption Cross Sectional Area, A_DF (Shaw, 1979, 1988; Rosowski et al., 1988; Keefe et al., 1994) that quantifies the equivalent area over which diffuse-field sound-intensity (with units of W/m²) is collected to produce the sound power (with units of W) that enters the middle ear. The A_DF depends on the wavelength of sound λ and estimates of the middle-ear input impedance Z_T and the radiation impedance Z_E looking out the ear canal from the tympanic membrane³

where η is the radiation efficiency through the external ear (Shaw, 1988; Keefe et al., 1994), and λ, the wavelength of sound in air, equals the ratio of the speed of sound in air c_S and the sound frequency f.

The A_DF computed in 1988 and the new A_DF computed using the more recent and more extensive Z_T measurements of Keefe et al. (1993), including a calculation of η, are illustrated in Fig. 2(A), along with the ideal receiver cross-section (A_DFideal = λ²/4π) that describes the power-collection function of an ideal acoustic receiver (Shaw, 1979, 1988), and is inversely proportional to the square of the frequency. The two non-ideal receiver cross-sections are similar below 2 kHz, where they show a large decrease in receiver area relative to the ideal. The large non-idealities below 2 kHz result from the significant mismatch between the high-impedance stiffness dominated Z_T and the low-impedance inertance (acoustic-mass) dominated Z_E at frequencies below 1 kHz (Shaw, 1979, 1988; Rosowski et al., 1988). Between 2 and 8 kHz, both computations are closer to the ideal, where the computation based on the newer measurements generally suggests higher A_DF and better performance as a receiver between 4 and 6 kHz, with poorer sound-power collection above 7 kHz.

The second descriptor of sound power flow through the auditory periphery is the Middle-Ear Efficiency, MEE (Rosowski et al., 1986; Rosowski, 1991), that quantifies the fraction of the sound power entering the middle ear that reaches the inner ear. If we describe the average sound power entering the middle and inner ears in terms of the sound pressures at the entrances, P_T and P_V, respectively, and the acoustic admittances at the entrances Y_T and Y_C (Fig. 1), then

Figure 2(b) compares the MEE computed in 1986 with the computation based on the newer data of Nakajima et al. (2009) and Keefe et al. (1993). While there are differences, there are also many similarities. Both the new and old MEE suggest that, even at low-frequencies where MEE is highest, only about 10% of the sound energy that enters the human middle ear is conducted to the inner ear. Both calculations demonstrate a significant decrease in power transfer at frequencies above 2 kHz. Much of the loss in power is probably due to energy loss and shunting within the incudo-mallear joint of the human middle ear (Willi et al., 2002; Gerig et al., 2015), which appears to be adapted to protect the inner ear from large low-frequency motions of the eardrum associated with changes in air pressure, swallowing, etc. (Hüttenbrink, 1988). The use of alternative modern datasets, e.g., the middle-ear pressure gain and cochlear impedance data of Aibara et al. (2001), have only minor effects on the computed MEE.

The computed A_DF demonstrates that with sound of frequencies less than 2 kHz, much of the sound power that enters the external ear is reflected at the TM, while the computed MEE suggests that middle-ear power losses increase at frequencies above a few kHz. The combination of the two processes limits sound power transmission to the inner ear at both low and high frequencies. This band-pass limited power conduction is illustrated in Fig. 2(C), which compares the power-collection functions of: (a) an ideal acoustic receiver, (b) the computed human A_DF, and (c) the combination of the human A_DF and MEE. The acoustic receiver area defined by the combined A_DF and MEE is from 3 to 7 orders of magnitude smaller than the ideal at frequencies between 0.1 and 10 kHz. Figure 2(D) shows how these non-idealities affect the transfer of impulse related sound power to the inner ear. The figure compares the frequency distribution of the sound power in a Friedlander blast wave that mimics that of a rifle shot from an M16 (with a 1 kPa peak and a zero crossing at 0.4 ms, e.g., Price et al., 1989), with the distribution of the sound power absorbed at the TM (the entrance to the middle ear) and the power absorbed at the oval window (OW, the entrance to the inner ear). The analysis suggests that a significant fraction of the sound power is lost by reflection at the TM or absorption within the middle ear before it reaches the inner ear. Based on this result, we would predict that while blasts and impulses produce most sound energy at frequencies below 0.5 kHz, it is the sound energy between 0.5 and 3 kHz that dominates the inner-ear stimulus. The integrated sound power defined by the area under the OW curve in Fig. 2(D) is roughly equivalent to the integrated sound power metric use by Zagadou et al. (2016) to quantify impulse-induced hearing loss.

While the data-based power analysis described above provides quantitative descriptions of sound-power conduction to the inner ear, it has multiple limitations: (1) Since the average absorbed sound power depends on the temporally averaged (root mean square) sound pressure and volume velocity, the analysis ignores phase and is limited to frequency-domain representations of the stimulus and conducted sound power. This limitation prevents its use in investigations of how the time-varying sound power in impulses is conducted through the ear. (2) It is a linear analysis, and the effects of known nonlinear processes in the middle ear (e.g., Kobrak, 1948; Guinan and Peake, 1967; Greene et al., 2017) are not taken into account.

Predictions of impulse transfer and effects of nonlinear elements may be better assessed using structure-based models of the middle ear that can be analyzed in the time domain, e.g., the nonlinear circuit models of Price and Kalb (1991) or Zagadou et al. (2016). Those models use instantaneous nonlinearities to describe the conduction of high-intensity sound through the middle ear; here, we take a simpler approach that allows the development of intuition regarding the processes involved.

The basis of our approach is to introduce static-nonlinearities (nonlinearities that change with average response level but not with time) into an existing linear circuit model of human middle-ear function, that of O'Connor and Pura (2008). That linear model was designed to fit recent measurements of middle-ear gain and stapes displacement made in human temporal bones [Fig. 3(A) and 3(B)]. We add an external-ear model based on the work of Siebert (1973), Rosowski et al. (1986), and Keefe et al. (1994), and use the extended model to predict the power-collected from impulsive sounds that are transmitted through the middle ear. Figure 3(C) compares the previously described data-based estimate of MEE with that predicted by the linear O'Connor and Puria (2008) circuit model. While the circuit appears to overestimate the efficiency of power conduction through the human middle ear (especially at frequencies above 2 kHz), the low frequency MEE of about 0.2 that is predicted by the circuit and the significant roll-off in power transmission at frequencies above 2 kHz are consistent with the data-based model of Fig. 1.

The temporal response of the linear model of the external and middle ear, as it is stimulated by the Friedlander blast wave used in the computation of Fig. 2(D) is illustrated in Fig. 3(D). The input impulse (the thick line) is assumed to pass the entrance of the ear canal at a grazing incidence so no correction for diffraction about the head is necessary. The thin solid line in Fig. 3(D) is the resultant sound pressure in the vestibule of the inner ear predicted by the linear model. This prediction is based on the inverse Fourier Transform (iFFT) of the linear multiplication of the forward Fourier Transform of the input impulse with model transfer functions that relate the sound pressure at the entrance to the ear with either the sound pressure in the vestibule or stapes displacement. The predicted sound pressure in the vestibule has a peak pressure of about three times that of the input, and the peak in the vestibule sound pressure is delayed by about 0.2 ms relative to the input. About half of this delay can be attributed to sound conduction down the ear canal, but a significant component is related to delays associated with the transformation and filtering of sound power by the middle ear (Gottlieb et al., 2018). The predicted time waveform of the stapes displacement is illustrated by the dotted line. The predicted peak stapes displacement (scaled on the right of the figure) is about 8 microns, and that peak is delayed by 0.4 ms relative to the input pulse. The additional 0.2 ms delay relative to P_V is related to the resistive cochlea input impedance at the entrance to the vestibule. Because of this resistance, the peak stapes velocity would coincide with the peak in vestibule sound pressure, but displacement is the integral of velocity, and its peak occurs when the vestibular pressure waveform first crosses zero from positive to negative. Of course, the details of these linear impulse predictions depend on our choice of external and middle-ear models.

We now introduce static nonlinearities into the extended O'connor and Puria circuit and continue our sinusoidal analyses. The nonlinear elements we introduce follow the example of Price and Kalb (1991) and Zagadou et al. (2016). They use the sound-induced stapes-displacement measurements made at 120 and 130 dB sound pressure level (SPL) by Guinan and Peake (1967) in cat to intuit an impedance of the annular ligament (the ligament that supports the stapes in the oval window of the inner ear) made up of a level-dependent stiffness and resistance. From the O'Connor Puria model description⁴ we define a linear acoustic stiffness K_AL and resistance R_AL associated with the annular ligament. These elements are then replaced by a nonlinear level-dependent stiffness K′_AL and resistance R′_AL that grow with large stapes displacements (X_S) and velocities (V_S), where

According to Eq. (3), when |X_S| equals 1 μm, K′_AL = 2 K_AL. According to Eq. (4), when |V_S| = 10 mm/s, R′_AL = 2 R_AL. With smaller values of |X_S| and |V_S|, K′_AL and R′_AL approximate their linear values, while larger |X_S| and |V_S| lead to larger K′_AL and R′_AL [Fig. 4(A)]. Since Eq. (3) defines K′_AL in terms of X_S, the level-dependent stiffness of Fig. 4(A) is independent of frequency when stapes displacement is held constant at all frequencies. However, Eq. (4) defines R′_AL in terms of V_S, and in the sinusoidal steady state, stapes velocity magnitude is the product of |X_S| and sound frequency (|V_S|= 2πf |X_S|). The frequency dependence of the relationship between velocity and displacement means that, with a constant stapes displacement magnitude and therefore constant K′_AL, a factor of 10 increase in frequency will produce a similar increase in the level-dependent part of R′_AL [Fig. 4(A)]. Our choice of a frequency dependent R′_AL and the magnitude of the nonlinear component of R′_AL has a significant effect on the level-dependent impedance of the stapes and annular ligament, where

and L_AL is the level-independent stapes inertance, and $j=−1$ is the imaginary number.

The frequency dependence of Z′_SAL, with K′_AL and R′_AL defined by Eqs. (3) and (4), and with different levels of stapes displacement is illustrated in Fig. 4(B). With |X_S| < 0.1 μm, Z′_SAL is essentially linear and the combination of the stiffness and resistance with the stapes inertance produces a moderately sharp resonance with a well-defined minimum in impedance magnitude and a π phase change in impedance phase angle near 3 kHz. At frequencies below the resonance, the impedance magnitude is inversely proportional to frequency and has an angle of −π/2, both of which are consistent with the stiffness of the annular ligament. At frequencies above the resonance, the impedance with low X_S has a magnitude proportional to frequency and a +π/2 phase angle consistent with domination by the stapes inertance.

At higher levels of X_S, the stiffness of the annular ligament increases and the low-frequency impedance grows in magnitude, but the angle of −π/2 suggests continued stiffness domination. In contrast, the regular increase in R′_AL with stimulus level and frequency [Fig. 4(A)] leads to a significant alteration in the frequency dependence and phase angle of the stapes-annular ligament impedance at mid and high frequencies. At frequencies above the resonance, the impedance magnitude still grows proportionally with frequency, but that growth becomes dominated by the frequency and level dependent resistance as stapes displacement increases, and the phase angle of the impedance at high-frequencies becomes more resistive (closer to 0) with larger X_S.

The pattern of displacement-dependent impedance change is highly sensitive to our choice of a velocity-dependent resistance, as well as the choice of the magnitude of the nonlinear component of the resistance. Figure 4(C) illustrates the level-dependent impedance of the stapes and annular ligament when we reduce the nonlinear component of the R′_AL by a factor of 10, such that obvious nonlinear growth in R′_AL (when the linear and nonlinear components of R′_AL are equal) is delayed until |V_S| ∼100 mm/s. This order of magnitude reduction in the level-dependent component of R′_AL has two visible effects on the level dependence of Z′_SAL: (1) Not unexpectedly, the slower growth of the resistance allows a more prominent linear inertance component (as defined by a similar high-frequency impedance magnitude and a high-frequency impedance phase angle closer to +π/2 in Fig. 4(C), over an extended displacement range, and (2) related to this increased influence of the level-independent stapes-inertance component, there is a clear shift in the resonance-defined frequency of the impedance minimum as K′_ALincreases along with X_S.

This second effect is relevant to one of the controversies in model predictions of hearing loss due to high-level impulsive sounds. The model of Price and Kalb (1991) predicts the hearing loss in the middle frequencies does not grow monotonically with impulse level, specifically an increase in impulsive stimulus level can result in a decrease in the predicted hearing loss at some frequency-level combinations. As the minima of the predicted impedances shift in frequency in Fig. 4(C) we see non-monotonic growth in impedance magnitude. Specifically, near 5 kHz an increase in |X_S| from 0.1 to 2 μm produces a factor of 2 reduction in |Z_AL|. Whether such a non-monotonicity in impedance produces a non-monotonicity in the predicted hearing loss depends on the details of the different model structures, as well as the choice of nonlinear elements.

The non-monotonic behavior observed in Fig. 4(C) is directly related to the choice of K′_AL and R′_AL; such behavior is not observed in Fig. 4(B). Zagadou et al. (2016) point out a similar dependence of the non-monotonic behavior in the Price and Kalb (1991) model on the choice of non-linear elements. The elimination of both the shift in resonance and the non-monotonicity in the growth of the impedance of the annual ligament by an increased nonlinear component of R′_AL [Fig. 4(B)] is consistent with the approach taken by Zagadou et al. (2016) to reduce non-monotonic growth of the predicted hearing loss. At this time, there is little data available to test whether the nonlinear resistance described in Eq. (4) and Fig. 4(A) (and used throughout the rest of our analysis) is the best descriptor of level-dependent impedance of the annular ligament.

Figures 5(A) and 5(B) show the effect of the static nonlinearities described by Eqs. (3) and (4) on the transfer functions describing Middle-Ear Pressure Gain and Stapes Displacement vs Ear Canal Sound Pressure.⁵ Not surprisingly, the level-dependent increase in the model's annular ligament impedance produces decreases in sound transmission through the middle ear. A fixed stapes motion of 1 μm produces a small but noticeable reduction in middle-ear transfer function magnitudes relative to those observed when X_S is fixed at 0.1 μm, while fixed stapes motions of 10 and 31 μm are associated with magnitude reductions of factors near 10 and 30 in the predicted transfer functions. The level-dependent changes in the phase angle of the transfer functions are relatively small (± 0.5 radian), and appear largest in the middle frequency ranges.

The decreases observed in MEE [Fig. 5(C)] are somewhat larger than the reductions in the magnitudes of the pressure and displacement transfer functions, with a reduction of a factor of 20 and 100 with stapes motions of 10 and 30 μm. The larger level-related decreases in MEE compared to sound transfer may be related to an increase in sound power absorption by the increased R′_AL [Fig. 4(A)] but could also result from an increase in shunted energy caused by the increased impedance of the annular ligament. Nonetheless, Figs. 5(A)–5(C) all point out that a general effect of the nonlinearity is to reduce the output of the middle-ear relative to its input. The variations in the shape of the transfer functions and MEE with level are consistent with the presence of non-monotonicities in growth at specific frequencies.

While the effect of nonlinearity is better predicted by time-domain analyses, we can still use the model above to estimate the effects of the non-linearity on impulse transfer. Specifically, we can use the predicted alterations in the stapes-displacement transfer function [Fig. 5(B)] to compute the impulsive stapes motion, normalized by the peak of a diffuse-field sound pressure impulse. [Again, we use the Friedlander impulse described in Fig. 3(D) as input.] Figure 5(D) compares the predicted stapes displacement in response to the stimulus impulse where the linear displacement transfer function is replaced by transfer functions computed with different large stapes displacements pictured in Fig. 5(B).

Figure 5(D) suggests that the circuit's nonlinearity not only reduces the amplitude of the transmitted impulse relative to its input, but also reduces its width as the time to the positive to negative zero crossing is decreased with higher-level transfer characteristics. This reduction in width may be related to the larger decrease in low-frequency displacement transfer compared to the decrease at higher frequencies: In Fig. 5(B), |X_S/P_T| at 0.5 kHz is decreased by a factor of 20 as average stapes motion increases from 0.1 to 31 μm, while the same change in average stapes motion is associated with factor of 6 decrease in |X_S/P_T| at 5 kHz. The observed decreases in the relative height and width of the stapes impulse with increasing level are both consistent with a reduction in the relative transmission of sound power with increased stimulus levels.

A lesson from Fig. 4, and the current controversy over how to model blast-induced hearing loss in humans (Price et al., 2017; Zagadou et al., 2017) is the need to test models of the middle-ear response to high-level stimuli with improved measurements made in live humans or cadaveric human ears. Until recently, the evidence for such comparisons has been rather sparse, and primarily consisted of measurements of hearing loss after exposure of humans to impulsive noise (e.g., Johnson, 1994; Remenschneider et al., 2014) that were supplemented with animal studies of impulse-induced hearing loss (e.g., Patterson et al., 1993) and nonlinear middle-ear responses (e.g., Guinan and Peake, 1967). Here, we compare the predictions of the nonlinear model described above with new measurements of level-dependent middle-ear responses made in human temporal bones using high-level tones.

The comparison dataset we use is a subset of new measurements of stapes and umbo motions in response to moderate and high-intensity tones (Cheng et al., 2017; Cheng et al., 2018). (The data we show are part of a larger study that is being prepared for publication.) This dataset has several advantages over other datasets in the literature (e.g., Greene et al., 2017, 2018). The measurements were gathered using long duration (on the order of one second) pure tones of selected frequencies that covered a broad frequency range, from 0.2 to 18 kHz. The long-duration tones are consistent with sinusoidal analysis, which matches well with the use of a static-nonlinearity in our model. On the negative side, the dependence on a single broad-range sound source limited the stimulation levels achieved to about 500 Pa peak (∼145 dB SPL).

Figure 6(A) illustrates a series of stapes-velocity growth with stimulus-level functions recorded from a single temporal bone. The techniques are similar to those used in other studies (Gan et al., 2004; Chien et al., 2007). The measurements were made with a sound source sealed in the ear canal, with stimulus sound pressure measured near the lateral surface of the TM, and a measurement of the sound-induced displacement of a single point on the stapes or umbo using a laser-Doppler vibrometer (LDV).⁶ The upper panel of Fig. 6(A) illustrates the growth of stapes displacement magnitude with increasing ear-canal sound pressure at selected frequencies. Both the sound level (in Pa) and the displacement (in μm) are plotted on log scales. The lower panel illustrates the phase angle of the displacement normalized by the phase angle of the ear canal sound pressure plotted against stimulus level on a log scale. The results indicate there is a stimulus range from 0.1 to 10 Pa, where the stapes displacement magnitude generally grows linearly (with a log-log slope of 1) and the phase between the stimulus and the displacement is constant. At stimulus levels near 30 Pa (but as low as 5 Pa at 14 kHz), we first start to see deviations from linear growth, where the displacement level at the lowest visible alteration in growth rate varies between 0.03 μm (at 14 kHz) and 0.3 μm (at 0.5 kHz). We also start to see alterations in the phase in this level range. The nonlinear growth of displacement magnitude can be compressive (with log-log slopes <1, e.g., at most frequencies at the highest stimulus levels), or expansive (with log-log slopes >1, e.g., at most frequencies with stimulus levels between 10 and 100 Pa). While many of the stimulus-level related changes in phase slowly accumulate, there are instances of rapid changes in phase with level. These changes may reflect alterations in the 3D motion of the stapes that complicate the single point motion measurements (Decraemer et al., 2007), or they can result from level-dependent shifts in the resonant features of the middle ear (Rosowski et al., 1984).

Figure 6(B) contains stapes displacement level functions computed from the model. Similarities between the measurements and the model include: (1) linear growth at low levels, and the presence of nonlinear growth at higher levels. (2) The range of displacements where the nonlinearity is first apparent is similar in the model and the data (between 0.2 and 1 μm). There are also differences: (1) The model only predicts compressive growth at higher levels, and there is no indication of expansive growth. (2) There are no step changes in the models predicted level-dependence of the phase responses. (3) The sound pressures required to produce low-level stapes displacements differ significantly at the highest frequency.

The temporal bone dataset also contains measurements of umbo displacement made concurrently in the same temporal bone specimen, which differ significantly from what was observed at the stapes. Figure 6(C) shows measurable nonlinearities in the growth of umbo displacement magnitude in the temporal bone, especially at the three lower frequencies; this growth was initially expansive at 0.5 and 1 kHz, and compressive at 2 kHz. At higher frequencies, the magnitude of the displacement appears to grow linearly with stimulus level. Despite the more linear growth of umbo displacement magnitude, there are significant level dependent phase angle variations at all frequencies, with larger and more apparent phase steps in response to frequencies of 4 kHz and above. Again, these phase steps may represent alterations in the 3D motion of the umbo that complicate our single point measurements. Figure 6(D) demonstrates that the model we use exhibits little nonlinear growth in the umbo velocity with stimulus level, with small changes in the displacement phase angle relative to the stimulus phase being the most apparent indication of level-dependent growth.

Our data-based model with simple structure provides useful predictions of sound-power collection and conduction by the external and middle ear, where a similar analysis for the chinchilla ear predicted the hearing loss in chinchillas exposed to moderate level impulse noise of varied bandwidth (Patterson et al., 1993). This analysis separates the fraction of sound power reflected at the TM and the fraction absorbed within the middle ear. Both processes limit the sound power that reaches the inner ear. While these analyses are dependent on the choice of data used in the calculations, multiple datasets lead to similar conclusion [Figs. 2(A) and 2(B)]. One limitation of this approach is its dependence on the quantification of time-averaged sound power such that it is not conducive to predictions of temporally varying events. A second limitation of this approach is the effects of structural changes, e.g., TM perforations, on power transfer through the ear are not easily predicted, and can only be described after a complete set of measurements of the middle-ear input and output variables with the structural variation is in place.

Structure-based linear circuit models of the ear are better at defining temporal variations in model inputs and circuit variables [e.g., Fig. 3(D)]. as well as the effect of alterations in circuit parameters. These models also depend on the choice of target data as well as the choice of model structure and element values. The middle-ear is complex enough that any one middle ear output can be mimicked with multiple circuit structures and parameter values; however, fitting such models to multiple mechanical and acoustical measurements [e.g., Figs. 3(A) and 3(B)] can increase the degree of fit between model and ear form and function. If the relationship between model and ear form is relatively strong, controlled changes in model element values can be used to predict the effect of pathological changes on ear function, e.g., Rosowski and Merchant (1995). A major limitation of such models is that the middle ear does not respond linearly with high levels of sound stimulation (Guinan and Peake, 1967; Greene et al., 2017; Cheng et al., 2017), and such linear models would not predict level-dependent variations in sound transfer through the middle ear.

While it is clear that the middle-ear does not respond linearly to high-level sound pressure stimuli, the dataset describing this response is small. Until recently, other than demonstrations that structures within the middle ear responded nonlinearly to large static stresses and strains (Dirckx and Decraemer, 1991; Liang et al., 2016), the best evidence for the magnitude and form of nonlinearities in sound transduction, and the sound-pressures where they first appear, came from a few measurements of sound-induced stapes displacement made in the cat middle ear with high-level tonal stimuli at three frequencies (Guinan and Peake, 1967). These limited observations were the basis for the nonlinearity introduced in the Price and Kalb (1991) and Zagadou et al. (2016) models. With limited data on the form and growth of the nonlinear response, multiple forms of the nonlinearity were used in these models, and as Zagadou et al. (2016) and comparisons of Figs. 4(B) and 4(C) point out, the selected form of the nonlinearity can significantly affect model predictions. Variations in the form and location of different nonlinear elements within the model can affect the level where nonlinear behavior is first predicted, the frequency dependence of the nonlinear behavior, and the presence or absence of nonlinear behavior in different mechanical or acoustic measurements. For example, to produce a significant nonlinearity in the predicted umbo motion in the nonlinear model we present, we may need to include a non-linear element within the TM or incudo-mallear joint.

As we investigate how high-intensity sound is transferred through the middle-ear, a topic relevant to the two published models of middle-ear nonlinearities that predict the hearing loss induced by intense impulsive sounds and blasts (Price and Kalb, 1991; Zagadou et al., 2016), it is worthwhile to compare those earlier models with our efforts.

1. As is done here, those studies are based, in part, on computer simulations of electrical analogs that describe how sound is transferred from the environment to the inner ear. Our analog model and that of Zagadou et al. (2016) are based on relatively modern data of human middle-ear function that rely heavily on acousto-mechanical measurements made in cadaveric human ears. The Price and Kalb analog model preceded much of the data used in this report and Zagadou et al. (2016) and is based on measurements and models of sound transmission in live cats and guinea pigs. While the forms of the electrical analogs are similar, the differences in base data lead to significant differences in specific element values.

2. The nonlinearity introduced in all three models is similar, and is based on limited data describing a non-linearity in the annular ligament that supports the stapes (Kobrak, 1948; Guinan and Peake, 1967).

3. There are significant differences in how the different models deal with the transformation of sound from the environment to the entrance of the ear. Both the Price and Kalb and the Zagadou models used measurements and models of the ear canal and pinna and head diffraction to impart directionality, and allow the introduction of protective devices, although some of the assumptions regarding head size and diffractive effects are quite different in the two models (Price and Kalb, 1991; Price et al., 2017; Zagadou et al., 2016, 2017). Our simpler model does not deal with such issues and assumes a non-directional diffuse-field stimulus and simple two-tube model of the external ear.

4. The Price and Kalb (1991) model uses estimates of the motion of basilar membrane as an indicator of the output of the model ear. The Zagadou et al. (2016) model and our approach uses estimates of the sound power that reaches the inner ear with different stimuli.

5. The output of both the Price and Kalb (1991) and the Zagadou et al. (2016) models have been used to predict measurements of hearing loss in human subjects. The intent of our analysis is more mechanistic and such test are beyond the scope of this study.

6. In line with their use to predict hearing loss in live humans, both Price and Kalb (1991) and Zagdou et al. (2016) make assumptions concerning how the middle-ear muscles affect sound transfer through the middle ear, though these assumptions differ greatly between the two (Price et al., 2017; Zagadou et al., 2016, 2017). The contribution of the middle-ear muscles in reducing sound transmission through the middle ear is also outside the scope of our mechanistic analysis.

Recent studies are beginning to provide the data necessary for better descriptions of the external- and middle-ear's response to high-level sound stimuli, including the data of Greene et al. (2017) and Greene et al. (2018), and the preliminary data in this report. The data from these studies is complementary, but their combination still falls short of a complete description of nonlinear sound conduction through the ear. The tonal measurements of inner-ear sound pressure and stapes displacement by Greene et al. (2017) cover a sound intensity range of as high as 170 dB SPL (∼20 kPa) but are restricted to frequencies ≤2 kHz. These data suggest the nonlinear response in this low-frequency range eventually leads to a saturation in stapes motion at a level of ∼100 peak-to-peak μm or 50 μm peak (consistent with our definition of displacement magnitude), which can be used to refine the definition of nonlinear model elements [e.g., Eqs. (3) and (4)], as well as their interaction with linear elements within the various circuit models. [For example, Price (1974) assumes a cat-based maximum stapes displacement of 5 μm peak.] The tonal laser-Doppler measurements of Cheng et al. (2017) included in Figs. 6(A) and 6(B), do not cover as wide an intensity range as Greene et al. (2017), but they do cover a broader frequency range with finer level resolution (5 dB steps), include both the stapes and the umbo, and shed light on the frequency dependence of both the stimulus and response levels where the nonlinearity is first observed. The observations of nonlinear growth in umbo as well as stapes motion is a potentially significant finding, as is the presence of expansive growth (greater than a slope of 1) at certain frequencies and levels. Further measurements of these sorts are required before we can describe the response of the middle-ear to high level sound with any degree of certitude.

An issue related to cochlear sound-pressure measurements vs laser-measurements of displacement in one dimension is the complication of how the umbo and stapes move in 3D, and how accurately that is captured by 1D motion measurements. Sound-pressure measurements in the vestibule are thought to be unaffected by these directional complications: Decraemer et al. (2007) describe a relatively simple frequency dependence of sound driven P_V, while V_S measured nearly simultaneously did exhibit phase and magnitude changes that occurred over narrow frequencies. For the most part, however, they found P_V and the piston-like translational component of V_S to be well correlated. Also, while both P_V and V_S would be reasonable indicators of an annular ligament nonlinearity, neither would necessarily be able to separate out the contribution of a more peripheral nonlinear process.

The data in these new studies can provide information on the lowest displacement and the lowest sound pressure where noticeable nonlinear events first occur, as well as descriptions of the rate of growth and possible saturation of the nonlinear response with increasing level. Comparisons between model predictions and data (Fig. 6) suggest the model we presented here has several obvious failings including an inability to produce any signs of the expansive nonlinearity seen at selected levels and frequencies in the temporal bone data, and failure to reproduce the observations of nonlinearity in the umbo motion. The presence of the later are indicative of the need for either other nonlinear mechanisms within the middle ear, or a stronger coupling of the nonlinear stapes response to the umbo than the present model allows.

At a bigger level, any model that argues it is true to physiology has to break down when high-level stimuli produce damage to the ear (e.g., Greene et al., 2018). An example of such a breakdown is Fig. 7, which illustrates temporal bone measurements of the growth of stapes velocity with increasing tonal stimulus levels at four frequencies. The measurements are made under two conditions: with an intact TM and after a posterior-inferior perforation that covers 25% of the TM surface (Cheng et al., 2018). One of the more physically-based models of perforation available today (Voss et al., 2001) predicts perforations effectively reduce the driving pressure to TM motion, which should result in a simple rightward shift of the stapes-growth function along the sound-pressure axis. Such a shift would also be consistent with the perforation having no direct effect on the nonlinear mechanisms that produce the level-dependent growth of stapes motion. The measurements shown in Fig. 7 are not consistent with this view; yes, the linear parts of the growth function are shifted to the right on the sound-pressure axis, but the shape of the nonlinear part of the growth curve is altered and the displacement magnitude where the nonlinear growth is first observed changes after perforation.

If it were demonstrated that the nonlinear mechanism itself is not affected (or is affected in a characterizable manner) by specific pathologies, a well-tested model of external- and middle-ear transduction of high-level sounds may provide useful predictions of how pathology, blast-induced or otherwise, will reduce the sound power that reaches the inner ear by way of ossicular conduction. More generally, a partial TM perforation or partial ossicular interruption might reduce the sound energy that reaches the inner ear from a blast stimulus in predictable ways (e.g., Voss et al., 2001; Röösli et al., 2012; Sim et al., 2013; Farahmand et al., 2016). If so, model-based estimates of the sound-power transferred to the inner ear from a blast, before and after such a pathology will be useful predictors of the protective effect of such pathology. For example, some have speculated that tympanostomy tubes, which produce controlled perforations of the TM, could help reduce the effects of impulsive sounds on hearing. A potential flaw in that suggestion is that tympanostomy tubes primarily reduce sound stimuli at frequencies <0.5 kHz (Voss et al., 2001), and power models, e.g., Fig. 2(D), predict these frequencies are not the major contributors to the sound power delivered to the ear.

Middle-ear models will be less able to predict differences in the the protective value of large conductive hearing losses. For example, in total ossicular path disruption the conduction of airborne sound to the inner ear by other pathways, e.g., whole-body conduction (Zwislocki, 1957) or sound pressure differences at the cochlear windows (Peake et al., 1992) can become dominant. These other mechanisms play a significant role in sound conduction to the ear when the air-conduction mechanism is reduced by 45 to 65 dB, and their contributions will limit the protective effect of such extreme pathology to 45 to 60 dB.

The sound-power collection and transfer metrics we introduced in Figs. 1 and 2 have been used to make predictions of hearing threshold (Rosowski et al., 1986; Rosowski, 1991), and predict the effect of impulse noise on hearing (Patterson et al., 1993; Zagadou et al., 2016). Zagadou et al. (2016) integrates the sound power that reaches the inner ear from an impulse [e.g., effectively computing the area under the curve that describes power at the OW in Fig. 3(D)] and has demonstrated that the integrated power can help predict the hearing loss produced in a series of controlled human experiments (Johnson, 1994). It is likely then, that a nonlinear model that accurately predicts the integrated power that reaches the inner ear can be a useful tool to assess damage risk criteria associated with blast-like sounds. Nonlinear models have also been used to compute some integrated displacement of the basilar membrane (Price and Kalb, 1991), and this predictor has also been correlated with animal and human studies of hearing loss (Price, 2007).

A significant area of research is the design and testing of different hearing protective devised for use by those exposed to blasts (Zagadou et al., 2016). The specification and design of such devices would be greatly helped by a nonlinear model that predicts how blasts that contain different frequencies and levels are conducted to the inner ear. Such a model would define the level of protection needed to maintain normal communication and hearing health. One potentially useful prediction from our analysis would be that most of the impulsive sound energy at frequencies less than 500 Hz is already filtered out by the normal action of the ear [Fig. 2(D)], and that this filtering is enhanced by the action of the middle-ear nonlinearity [Fig. 5(C)]. Therefore, efforts should be directed at increasing protection from the middle- and higher-frequency sounds.

The results of this and earlier analyses (Fig. 2) point out that only a small fraction, generally much less than 1%, of the sound power in a blast that could be collected by the human external and middle ear is absorbed within the inner ear. Much of the sound power at frequencies below 2 kHz is reflected at the TM at the entrance to the middle ear, and much of the sound power that enters the middle ear at frequencies above 3 kHz is absorbed within the middle ear before it can be passed on to the inner ear.

The introduction of a level-dependent stiffness and resistance into models of the annular ligament around the stapes of otherwise linear middle-ear models can significantly reduce the predicted transfer of high-level sound power into the inner ear (Fig. 5). The frequencies and levels where these reductions occur depend greatly on the character and size of the introduced nonlinearities. Nonlinear mechanisms that primarily act by way of increases in stiffness of the annular ligament with little or reduced increase in its resistance can lead to variations in the frequency of the stapes-annular-ligament resonance that may explain some predictions of nonmonotonic growth of hearing-loss with increases in stimulus intensity.

Measurements comparing the nonlinear growth of stapes and umbo displacement with high-intensity sounds in human cadavers can help constrain the nonlinearities introduced in such models. New data from cadaveric ears suggest that multiple nonlinear mechanisms may exist in the ear, and observations from these studies of complex growth patterns in umbo and stapes displacement with level (with expansion and compression) support the existence of complex level dependencies in the growth of hearing loss with level.

We thank our co-workers in the Eaton-Peabody Laboratory, past and present, for helping shape these analyses. Our colleagues at Worcester Polytechnic Institute's Center for Holographic Studies and Laser micro-mechaTronics (especially Cosme Furlong, Ph.D.) also made significant contributions. The work was partially supported by grants from the NIDCD of the NIH: Grant Nos. R01DC000194 (J.J.R.) and R01DC016079 (J.T.C.).