Mouse middle-ear forward and reverse acoustics

Abstract

The mouse is an important animal model for hearing science. However, our knowledge of the relationship between mouse middle-ear (ME) anatomy and function is limited. The ME not only transmits sound to the cochlea in the forward direction, it also transmits otoacoustic emissions generated in the cochlea to the ear canal (EC) in the reverse direction. Due to experimental limitations, a complete characterization of the mouse ME has not been possible. A fully coupled finite-element model of the mouse EC, ME, and cochlea was developed and calibrated against experimental measurements. Impedances of the EC, ME, and cochlea were calculated, alongside pressure transfer functions for the forward, reverse, and round-trip directions. The effects on sound transmission of anatomical changes such as removing the ME cavity, pars flaccida, and mallear orbicular apophysis were also calculated. Surprisingly, below 10 kHz, the ME cavity, eardrum, and stapes annular ligament were found to significantly affect the cochlear input impedance, which is a result of acoustic coupling through the round window. The orbicular apophysis increases the delay of the transmission line formed by the flexible malleus, incus, and stapes, and improves the forward sound-transmission characteristics in the frequency region of 7–30 kHz.

I. INTRODUCTION

The mouse has become an important experimental animal for hearing science, in part because of the availability of a number of genetic point mutations of the inner ear (e.g., Cheatham et al., 2014; Lee et al., 2016). To support hearing at frequencies up to 86 kHz (Heffner and Masterton, 1980) in the “forward direction” from the ear canal (EC) to the cochlea, the mouse middle ear (ME) provides a smoothly varying wideband frequency response without a significant number of isolated peaks or notches in the vibration spectrum (Saunders and Summers, 1982; Dong et al., 2013).

TABLE I.

List of abbreviations and definitions.

ME	Middle ear	MEC	Middle-ear cavity (tympanic cavity)
EC	Ear canal	TM	Tympanic membrane (eardrum)
OA	Orbicular apophysis (malleal processus brevis)	PF, PT	Pars flaccida, pars tensa
OW	Oval window	RW	Round window
SAL	Stapes annular ligament	IMJ	Incudomallear joint
ISJ	Incudostapedial joint	ST	Stapes
OAE	Otoacoustic emission
Open EC	The impedance boundary condition at the EC entrance is set to the characteristic impedance of air during reverse drive.
Closed EC	The EC entrance is occluded with a rigid surface during reverse drive.
Removed PF	The PF region of the TM is removed, and the MEC airspace is exposed to the EC.
Open MEC	The MEC airspace is removed, but the PF remains intact.
${\mathbf{p}}_U^F$	Pressure ($\mathbf{p}$) measured in the EC near the umbo (U), for forward drive (F)
$V_i^N$	Velocity ($V$) measured at a location indicated by subscript i (U = umbo, ST = stapes head, FP = stapes footplate) and normalized by ${\mathbf{p}}_U^F$
${\mathbf{p}}_{\mathit{MEC}}^F$	Pressure measured in the MEC, in front of the RW, for forward drive
${\mathbf{p}}_k^{i,j}$	The general form of pressure components, with superscript i indicating the drive direction (F = forward and R = reverse), superscript j indicating the component (i = incident, r = reflected, t = transmitted), and subscript k indicating the measurement location (EC = EC entrance, P = within the EC probe, PF = in the ME at the stapes footplate, C = in the cochlea at the OW, RW = in the cochlea at the RW)
${\overrightarrow{\mathbf{U}}}_k^{i,j}$	The general form of volume-velocity ($\overrightarrow{\mathbf{U}}$) components, with superscript i indicating the drive direction (F = forward and R = reverse), superscript j indicating the component (i = incident, r = reflected, t = transmitted), the vector direction indicating the direction of the volume velocity according to Figs. 1(c) and 1(d) ($\overrightarrow{\mathbf{U}}$ = EC to cochlea, $\overleftarrow{\mathbf{U}}$ =cochlea to EC), and subscript k indicating the measurement location (EC = EC entrance, P = within the EC probe, PF = in the ME at the stapes footplate, OW = in the cochlea at the OW, RW = in the cochlea at the RW)
${\overrightarrow{Z}}_{EC}$	EC input impedance at the probe–EC interface in the forward direction, equivalent to ${\mathbf{p}}_{EC}^{F,t}/{\overrightarrow{\mathbf{U}}}_{EC}^{F,t}$
${\overleftarrow{Z}}_{ST}$	ME input impedance at the stapes–cochlea interface in the reverse direction, equivalent to ${\mathbf{p}}_{ST}^{R,t}/{\overleftarrow{\mathbf{U}}}_{ST}^{R,t}$
$Z_{\mathit{MEC}}$	Equivalent lumped impedance representing the MEC airspace
${\overrightarrow{Z}}_C$	Cochlear input impedance at the stapes–cochlea interface in the forward direction, equivalent to ${\mathbf{p}}_C^{F,t}/{\overrightarrow{\mathbf{U}}}_C^{F,t}$
${\overrightarrow{\widehat{Z}}}_C$	Cochlear input impedance at the stapes–cochlea interface in the forward direction, based on the pressure difference between the cochlea and MEC, which is equivalent to ${(\mathbf{p}}_C^{F,t}-{\mathbf{p}}_{\mathit{MEC}}^F)/{\overrightarrow{\mathbf{U}}}_C^{F,t}$ or ${\overrightarrow{Z}}_C$ with an open MEC
${\overrightarrow{\widehat{Z}}}_{RW}$	Cochlear input impedance at the RW in the forward direction, based on the pressure difference between the cochlea and MEC, which is equivalent to ${(\mathbf{p}}_{RW}^{F,t}-{\mathbf{p}}_{\mathit{MEC}}^F)/{\overrightarrow{\mathbf{U}}}_{RW}^{F,t}$
${\overrightarrow{Z}}_{\mathit{diff}}$	Cochlear differential input impedance at the stapes–cochlea interface in the forward direction, equivalent to (${\mathbf{p}}_C^{F,t}-{\mathbf{p}}_{RW}^{F,t})/{\overrightarrow{\mathbf{U}}}_C^{F,t}$ or ${\overrightarrow{\widehat{Z}}}_C-{\overrightarrow{\widehat{Z}}}_{RW}$
${\mathit{GME}}^F$	ME pressure gain in the forward direction, equivalent to ${\mathbf{p}}_C^{F,t}/{\mathbf{p}}_{EC}^{F,t}$
${\mathit{GME}}^R$	ME pressure gain in the reverse direction, equivalent to ${\mathbf{p}}_{EC}^{R,t}/{\mathbf{p}}_{ST}^{R,t}$
${\mathit{GME}}^{RT}$	ME round-trip pressure gain, equivalent to ${\mathit{GME}}^F\times {\mathit{GME}}^R$
$\omega$	Angular frequency (2πf)
$E$	Young's modulus
$E_A^{PT}$	Young's modulus of the PT in direction A (r = radial, $\theta$= circumferential, $\phi$= transverse)
$G_A^{PT}$	Shear modulus of the PT along vector A (r$\phi$ = circumferential, r$\theta$= transverse, $\theta \phi$=radial)
$E^{PF}$	Young's modulus of the PF
$E_{\mathit{SAL}}$	Young's modulus of the SAL
$E_r^{PT*}$	Constant (i.e., independent of frequency) Young's modulus of the PT region of the TM in the radial direction for the sensitivity analyses (the PT components $E_{\theta }^{PT}$, and $E_{\phi }^{PT}$ are all scaled by the same ratio as for the frequency-dependent $E_r^{PT}$)
$E\left(f\right)$	Complex Young's modulus as a function of frequency
$E^s\left(f\right)$	Storage Young's modulus as a function of frequency
$E^l\left(f\right)$	Loss Young's modulus as a function of frequency
$\eta \left(f\right)$	Loss factor as a function of frequency
ρOSS	Density of the ossicles
$C_{\mathit{air}}$	Compliance of the enclosed MEC airspace

ME function is also relevant for otoacoustic emissions (OAEs). OAEs are low-amplitude sounds that are evoked or generated spontaneously in the cochlea, which travel in the “reverse direction” through the ME from the cochlea to the EC (Kemp, 1978). For evoked OAEs, the ME shapes both the evoking stimulus as it travels in the forward direction and the evoked OAE as it travels back to the EC in the reverse direction.

While ME sound-transmission transfer functions in the forward and reverse directions have been characterized in several animals, e.g., human (Puria, 2003) and gerbil (Dong and Olson, 2006), they have not been characterized in mouse. Saunders and Summers (1982) measured mouse ossicular motion at the umbo, the transverse portion of the malleus (the thin bony plate of the transversal lamina), and the head of the incus, from 1.5 to 35 kHz. Rosowski et al. (2003) measured the umbo velocity of mice in three different age groups, with an intact tympanic membrane (TM) and middle-ear cavity (MEC). To access the mouse ossicles, typically the pars-flaccida (PF) region of the TM and the MEC are opened, such that the ME anatomy is significantly altered. With these alterations, Dong et al. (2013) measured one-dimensional (1D) motions of the mouse ME ossicles in response to sound stimulation in the forward direction. Pressure transfer functions between the EC and cochlea have yet to be directly measured in either direction. To date, mouse ME transfer functions with an intact ear have not been measured, and the effects of opening the PF and MEC have not been fully evaluated.

The mouse ossicular chain is of the microtype category common to many small high-frequency-hearing animals (Fleischer, 1978). Besides the obvious differences in anatomical size, the mouse ME has two key features that distinguish it from the human ME. First, the mouse malleus has a bony anterior process that is fused to the tympanic annulus, and second, the malleus has a large anterior bony protuberance near the base of the manubrium called the orbicular apophysis (OA), which is illustrated in Fig. 1. It has been hypothesized that at low frequencies, the malleus–incus (MI) complex rotates about an anterior–posterior axis (axis 1) but that at high frequencies the malleus begins to rotate with respect to a different, perpendicular, axis (axis 2) that passes through the OA (Fleischer, 1978). There is controversy about whether axis 2 really exists, however, and the functional importance of the OA on ME sound transmission has yet to be established. While it is known that the MEC airspace can affect low-frequency hearing (Ravicz and Rosowski, 2012), the mouse ear contrasts with other small mammals in having a relatively small MEC volume. The effect of this on sound transmission through the mouse ME is not well understood.

FIG. 1.

(Color online) (a) Meshed geometries of the EC, ME, ME components, and MEC, reconstructed from micro-computed tomography (μCT) images of a left ear. The uncoiled tapered model of the mouse cochlea is coupled to the ME at the stapes. The cochlear model contains all of the organ of Corti details (not shown) from Motallebzadeh et al. (2018). The viscous-fluid domain surrounding the organ of Corti is shown as a dark-blue dense mesh. (b) The ME in a lateral-to-medial view after removing the EC and MEC. Shown are the malleus, anterior process of the malleus (APM), transversal lamina of the malleus, OA, superior ligament of the malleus (MSL), malleal ligament (ML), tensor-tympani tendon (TTT), incudomallear joint (IMJ), incus, incudostapedial joint (ISJ), stapes, pars tensa (PT) and PF regions of the TM, stapedius-muscle tendon (SMT), lateral–medial ligament of the incus (IL), and stapes annular ligament (SAL). (c) For forward drive, the stimulation pressure emitted by the probe, ${\mathbf{p}}_P^{F,i}$, is shown at the EC entrance to represent the pressure incident upon the probe–EC interface at x = 0. The probe refers to a general impedance boundary condition. In this study, to replicate open-canal measurements, the characteristic impedance of air was assigned to the probe–EC interface impedance, but the model could be adapted to account for the realistic mechanical properties of any probe. Due to the mismatch between the EC input impedance ${\overrightarrow{Z}}_{\mathit{EC}}$ and the characteristic impedance of the probe (equivalent to that of air in this study), a portion of the incident pressure is reflected back into the probe, ${\mathbf{p}}_P^{F,r}$, and the remainder, ${\mathbf{p}}_{EC}^{F,t}$, is transmitted into the EC. Another reflection happens at the stapes–cochlea interface at x = l due to the mismatch between the effective characteristic impedance of the ME and the cochlear input impedance ${\overrightarrow{Z}}_C$. The pressure incident upon the stapes–cochlea interface for forward drive is ${\mathbf{p}}_{ST}^{F,i}$, the pressure reflected back into the ME is ${\mathbf{p}}_{ST}^{F,r}$, and the pressure transmitted into the cochlea is ${\mathbf{p}}_C^{F,t}$. The inset lateral-to-medial view (rotated to show the ossicles from an anterior–posterior orientation) shows the round window (RW), scala tympani, and scala vestibuli. (d) For reverse drive, a stimulation pressure is applied at the RW (see inset). The resultant hydraulic pressure travels within the cochlea toward the stapes–cochlea interface at x′ = 0, where it produces an incident pressure, ${\mathbf{p}}_C^{R,i}$, and reflected pressure, ${\mathbf{p}}_C^{R,r}$. The pressure transmitted into the ME is ${\mathbf{p}}_{ST}^{R,t}$, which is comparable to what would be measured experimentally with a hydrophone inserted into the scala vestibuli near the stapes footplate. The probe refers to a general impedance boundary condition. At the probe–EC interface (x′ = l), the reverse incident pressure is ${\mathbf{p}}_{EC}^{R,i}$, the reflected pressure is ${\mathbf{p}}_{EC}^{R,r}$, and the net pressure at the EC entrance that would be picked up by the probe microphone in an equivalent experimental setup is ${\mathbf{p}}_P^{R,t}$.

There exist a number of ME finite-element (FE) and lumped-parameter circuit models for different species, such as human (Gan et al., 2004; Khaleghi and Puria, 2017; Motallebzadeh et al., 2017b; O'Connor et al., 2017), chinchilla (Bowers and Rosowski, 2019), gerbil (Maftoon et al., 2015), monkey (Liang et al., 2018), and cat (Funnell and Laszlo, 1978; Shera and Zweig, 1992), and even for non-mammals such as birds and lizards (Livens et al., 2019; Muyshondt et al., 2019). For mouse, there was only a simple FE model, which lacked the EC, MEC, and cochlea (Gottlieb, 2017). In that model, the TM thickness was overestimated (due to insufficient image resolution), and the malleus and incus were fused (which was not discovered until after publication). For the present study, we expand upon and refine the Gottlieb (2017) effort by coupling together models of the mouse EC, ME, MEC, and cochlea (Fig. 1). Anatomically realistic FE models that have been tested against available experimental data offer a promising way of characterizing the mouse ME, both in experimentally altered and fully intact forms, and can provide an important supplement to experimental approaches, which are often limited in scope and challenging in such a small animal. Such models can also provide a powerful platform for studying and understanding normal responses and the effects of pathological conditions. A realistic FE model can also be used to evaluate how the ME affects OAEs (Motallebzadeh and Puria, 2017).

In this paper, an FE model of the mouse ear is developed and calibrated against available EC-sound-driven in vivo measurements of (1) ossicular motion with the PF region of the TM removed, such that the MEC airspace is directly exposed to the driving sound field from the EC (Dong et al., 2013), and (2) umbo motion with an intact TM and MEC (Rosowski et al., 2003). The model is then used to characterize the EC input impedance, cochlear input impedance, and reverse ME impedance at the stapes footplate, as well as the forward (drive pressure at the EC entrance), reverse (drive pressure at the round window), and round-trip ME pressure gains. Rotations, translations, and deformations of the ossicles are inferred from the three-dimensional (3D) motions and rotational fields of the model. Model predictions of the effects of anatomical features such as the PF, MEC, and OA on sound transmission are reported. The model is then used to perform sensitivity analyses showing how alterations to the (1) SAL stiffness, (2) TM stiffness, and (3) ossicular mass affect sound transmission. A list of abbreviations and definitions is provided in Table I. This study provides a platform for interpreting the effects of the mouse ME on experimental studies of cochlear function. For example, since the ME sits between the cochlea, where OAEs are generated, and the EC, where OAEs are measured, knowing how the mouse ME behaves for forward and reverse sound transmission will help to clarify the ways in which OAE measurements in mice can be affected by the ME.

TABLE II.

ME structures, parameters, and values.

Structure	Parameter	Value	Units	Reference
PF	Thickness	30	μm	Human, cat, gerbil, mouse (MacArthur et al., 2006; Decraemer and Funnell, 2008)
	E	10	MPa	Gerbil, human (Maftoon et al., 2015; O'Connor et al., 2017)
PT	Thickness	12	μm	Human, cat, gerbil, mouse (MacArthur et al., 2006; Decraemer and Funnell, 2008)
	Er	30–300 (from 1 to 60 kHz)	MPa	Human, gerbil (Zhang and Gan, 2013; Maftoon et al., 2015; O'Connor et al., 2017)
	Eφ	0.5 × Er	MPa	Human (Fay et al., 2006)
	Eθ	0.8 × Er	MPa	Human (Fay et al., 2006)
	Grφ, Gφθ, Grθ	0.3 × (Eθ, Eφ, Er)	MPa
Ossicles	E	14 × 103	MPa	Human, cat, gerbil (Funnell et al., 2013; Maftoon et al., 2015; O'Connor et al., 2017)
SAL	E	2	MPa	Human, gerbil (Maftoon et al., 2015; O'Connor et al., 2017)
Ligaments	E	40	MPa	Human (O'Connor et al., 2017)
IMJ and ISJ	E	20	MPa	Human (O'Connor et al., 2017)
Tendons	E	20	MPa	Human (O'Connor et al., 2017)

II. METHODS

A. Model development

1. Geometry

The interconnected anatomies of the EC, ME, and MEC in the model were reconstructed from μCT images with 5.56-μm resolution (obtained from a ZEISS Xradia 520 Versa x-ray microscope, Jena, Germany). The reconstructed structures include the EC, PT, and PF regions of the TM, malleus, incus, stapes, incudomallear joint (IMJ), incudostapedial joint (ISJ) suspensory attachments,¹ TTT, stapedius-muscle tendon (SMT), SAL, and MEC (Fig. 1). The ME and MEC are then coupled to a tapered-cylinder cochlear model. The cochlear model is a modified version of a previous model (Motallebzadeh et al., 2018) and includes the scala-vestibuli (including the scala media) and scala-tympani fluid-filled chambers. The cross-sectional areas of the scala vestibuli and scala tympani vary as functions of longitudinal position according to values obtained from reported μCT measurements (Rau et al., 2012).

2. Material properties

Since no experimental data on the material properties of the mouse ME have been reported, in this study a priori values (Qi et al., 2006; Motallebzadeh et al., 2017a) were adopted, and the initial ME material parameters were taken from a human FE model (O'Connor et al., 2017). The ME parameters (e.g., Young's moduli) were then adjusted such that the resulting model motions resembled the Dong et al. (2013) ossicular-chain measurements. Table II shows the values used in this model, along with references to values in the literature from other species for comparison. All ME tissues are represented as isotropic elastic materials except for the pars tensa (PT), which is represented as having orthotropic elastic properties (Salamati et al., 2011). In addition, the PT is modeled as a viscoelastic material (Luo et al., 2009; Zhang and Gan, 2010) with a frequency-dependent Young's modulus (Mase, 1970, Chap. 9) and constant structural damping (Fung, 2013, p. 288). Constant structural damping of 10% is applied to all tissues (Zhang and Gan, 2010). The Poisson's ratios for all soft tissues and ossicular components are 0.485 and 0.3, respectively. While the ME model has frequency-dependent viscoelastic components, it is a linear model that is independent of the input sound pressure level (SPL).

For the cochlear model, parameters from our previous study are used for the most part (Motallebzadeh et al., 2018), with only the gain factor α for the outer hair cells adjusted to represent low and high cochlear gains corresponding to input levels of 80 and 30 dB SPL, respectively. The cochlear fluid is modeled as water. The fluid region in the cochlea immediately surrounding the organ of Corti structures (the “cochlear near field”) is modeled as viscous fluid using linearized Navier–Stokes equations that produce viscous dissipation, but this dissipation is likely underestimated because the viscosity of the boundary layer surrounding the tectorial membrane has not been incorporated (Prodanovic et al., 2018; Sasmal and Grosh, 2019). Other dissipation sources are due to structural damping (10%) of the organ of Corti components in the model. The fluid in the “cochlear far field” and air inside the EC and MEC are modeled as inviscid fluid. The air in the EC and MEC and the fluid in the cochlea are modeled as interacting with the structures surrounding them. However, the beam elements forming the Y-shaped elements of the organ of Corti do not interact with the fluid surrounding them.

3. Mesh

To reduce the computational cost of the model, the fine mesh domains of the viscous boundary layers of the organ of Corti are modeled as structured meshes, whereas the rest of the model consists of free tetrahedral and triangular elements, similar to our previous cochlear model (Motallebzadeh et al., 2018). All elements have quadratic shape functions. The ME model consists of 128 356 tetrahedral elements (70 900, 12 937, 11 445, and 33 074 elements for the ossicles, ligaments, tendons, and joints, respectively), with 7258 triangular shell elements representing the TM (2190 and 5068 elements for the PF and PT regions, respectively). The EC and MEC are composed of 26 515 and 148 630 elements, respectively. Of these elements, the longest mesh edge of the densest components (e.g., the ossicles) is 0.05 mm. Since the elements are quadratic, this mesh resolution guarantees at least 10 nodes along the shortest wavelength (Ihrle et al., 2013) at 100 kHz. The cochlea is constructed of 446 387 elements (268 169 and 178 218 elements for the cochlear near-field viscous and far-field inviscid domains, respectively). The basilar membrane, reticular lamina, and RW consist of 4478, 12 198, and 73 shell elements, respectively. The Y-shaped cellular structures in the organ of Corti sandwiched between the basilar membrane and reticular lamina (consisting of outer hair cells, phalangeal processes, and Deiters's cells) are modeled with 5535 beam elements.

4. Boundary and loading conditions

The distal-end surfaces of the ligaments and tendons are clamped where they attach to the bony wall of the MEC, as is the TM annulus. The symmetric boundary condition of the cochlear model in the previous study (Motallebzadeh et al., 2018) has been removed, such that full cross-sections of the scala-vestibuli and scala-tympani fluid chambers are modeled. The EC, MEC, and cochlear walls are assumed to be rigid.

The input boundary forming the EC entrance is represented by an impedance boundary condition. For forward drive, it is assumed that the incident pressure is a planar wave propagating in an infinite air domain that reaches the EC-entrance surface. Due to the impedance mismatch at the EC entrance between the characteristic impedance of air (lateral to the EC entrance) and ${\overrightarrow{Z}}_{EC}$ (medial to the EC entrance), some portion of the incident wave is reflected back into the open far-field domain.

For reverse drive, a uniform pressure load is applied to the surface of the RW membrane, such as what might be provided by a piezoelectric driver in an experimental setup. Two different boundary conditions are assigned to the lateral entrance of the EC. To represent an open EC, a planar-wave radiation boundary condition is applied. This boundary condition is called “open EC” in this study. For the “closed EC” boundary condition (Pitaro et al., 2016), the lateral entrance to the EC is terminated by a rigid wall. A clinical or experimental probe would have a finite impedance that falls between these two boundary conditions (neglecting its damping). Thus, the EC boundary condition is depicted as a general virtual probe or ear tip in Figs. 1(c) and 1(d).

5. Computational requirements

The FE simulations were computed using comsol version 5.3a (COMSOL, 2017) on a Dell Tower Station computer with 40 cores (two 20-core Intel E5-2698 v4 Xeon processors) and 512 gigabytes of physical RAM, running Linux. Each simulation frequency took about 10 min to run. By running six parallel simulations with all physical RAM in use and an additional 400 gigabytes of swap space, the effective computation time could be reduced to 3.3 min per frequency.

B. Equations

1. Ossicular velocities

To compare the model against experimental results (Dong et al., 2013), the umbo-pressure-normalized velocity $V_i^N$ of a given location on the ossicular chain, represented by index $i$, is obtained (for intact and altered models) as

$$V_i^N=\frac{{\overrightarrow{v}}_i}{{\mathbf{p}}_u},$$ (1)

where ${\overrightarrow{v}}_i$ is the velocity of a given point on the ossicular chain projected onto the approximate direction of the laser Doppler vibrometer (LDV) in the experimental setup, and ${\mathbf{p}}_u$ is the pressure measured in the EC within 10 μm of the umbo [Fig. 1(c)]. In addition to projecting these velocities onto the direction of the LDV, the maximum motions based on the 3D velocities of the umbo and stapes are also calculated using the formulation described in Dobrev and Sim (2018).

2. EC input impedance

The EC input impedance ${\overrightarrow{Z}}_{EC}$ is calculated in the forward direction at the probe–EC interface [at $x=0$ in Fig. 1(c)]. The continuity of pressure at this interface dictates that

$${\mathbf{p}}_P^{F,i}+{\mathbf{p}}_P^{F,r}={\mathbf{p}}_{EC}^{F,t},$$ (2)

where ${\mathbf{p}}_P^{F,i}$, ${\mathbf{p}}_P^{F,r}$, and ${\mathbf{p}}_{EC}^{F,t}$ are the incident and reflected pressure components at $x=0^{-}$ and the transmitted pressure at $x=0^{+}$, respectively. The continuity of the volume velocity² at $x=0$ is represented by

$${\overrightarrow{\mathbf{U}}}_P^{F,i}+{\overleftarrow{\mathbf{U}}}_P^{F,r}={\overrightarrow{\mathbf{U}}}_{EC}^{F,t},$$ (3)

where ${\overrightarrow{\mathbf{U}}}_P^{F,i}$, ${\overleftarrow{\mathbf{U}}}_P^{F,r}$, and ${\overrightarrow{\mathbf{U}}}_{EC}^{F,t}$ are the incident and reflected volume-velocity components at $x=0^{-}$ and the transmitted volume velocity at $x=0^{+}$, respectively. The volume velocities are calculated as $\iint \overrightarrow{\mathrm{u}}·\widehat{n}\hspace{0.17em}dA,$ where $\overrightarrow{\mathrm{u}}$ and $\widehat{n}$ are the velocity and unit-normal vectors at the plane of interest for surface element $dA$. ${\overrightarrow{Z}}_{EC}$ is calculated as

$${\overrightarrow{Z}}_{EC}=\frac{{\mathbf{p}}_{EC}^{F,t}}{{\overrightarrow{\mathbf{U}}}_{EC}^{F,t}}.$$ (4)

3. Cochlear input impedance

The continuity of pressure at the interface between the stapes footplate and cochlear fluid [at $x=l$ in Fig. 1(c)] dictates that

$${\mathbf{p}}_{ST}^{F,i}+{\mathbf{p}}_{ST}^{F,r}={\mathbf{p}}_C^{F,t},$$ (5)

where ${\mathbf{p}}_{ST}^{F,i}$, ${\mathbf{p}}_{ST}^{F,r}$, and ${\mathbf{p}}_C^{F,t}$ are the incident and reflected pressure components at $x=l^{-}$ and the transmitted pressure at $x=l^{+}$, respectively. The continuity of the volume velocity at $x=l$ is represented by

$${\overrightarrow{\mathbf{U}}}_{ST}^{F,i}+{\overleftarrow{\mathbf{U}}}_{ST}^{F,r}={\overrightarrow{\mathbf{U}}}_C^{F,t},$$ (6)

where ${\overrightarrow{\mathbf{U}}}_{ST}^{F,i}$, ${\overleftarrow{\mathbf{U}}}_{ST}^{F,r}$, and ${\overrightarrow{\mathbf{U}}}_C^{F,t}$ are the incident and reflected volume-velocity components at $x=l^{-}$ and the transmitted volume velocity at $x=l^{+}$, respectively. The standard definition of the cochlear input impedance ${\overrightarrow{Z}}_C$ in the forward direction is

$${\overrightarrow{Z}}_C=\frac{{\mathbf{p}}_C^{F,t}}{{\overrightarrow{\mathbf{U}}}_C^{F,t}}.$$ (7)

4. Reverse ME input impedance at the stapes footplate

The ME input impedance as viewed from the cochlea side of the stapes footplate, ${\overleftarrow{Z}}_{ST}$, is calculated for the reverse direction, as shown in Fig. 1(d). The continuity of pressure at the interface between the stapes footplate and cochlear fluid [at $x\prime =0$ in Fig. 1(d)] dictates that

$${\mathbf{p}}_{ST}^{R,t}={\mathbf{p}}_C^{R,i}+{\mathbf{p}}_C^{R,r},$$ (8)

where ${\mathbf{p}}_C^{R,i}$, ${\mathbf{p}}_C^{R,r}$, and ${\mathbf{p}}_{ST}^{R,t}$ are the incident and reflected pressures at $x\prime =0^{-}$ and the transmitted pressure at $x\prime =0^{+}$, respectively. The continuity of the volume velocity at $x\prime =0$ is represented by

$${\overleftarrow{\mathbf{U}}}_{ST}^{R,t}={\overleftarrow{\mathbf{U}}}_C^{R,i}+{\overrightarrow{\mathbf{U}}}_C^{R,r},$$ (9)

where ${\overleftarrow{\mathbf{U}}}_C^{R,i}$, ${\overrightarrow{\mathbf{U}}}_C^{R,r}$, and ${\overleftarrow{\mathbf{U}}}_{ST}^{R,t}$ are the incident and reflected volume-velocity components of the cochlear fluid at $x\prime =0^{-}$ and the transmitted volume velocity of the stapes footplate at $x\prime =0^{+}$, respectively. ${\overleftarrow{Z}}_{ST}$ is calculated as

$${\overleftarrow{Z}}_{ST}=\frac{{\mathbf{p}}_{ST}^{R,t}}{{\overleftarrow{\mathbf{U}}}_{ST}^{R,t}}.$$ (10)

The pressure continuity at the probe-EC interface in the reverse direction results in ${\mathbf{p}}_P^{R,t}={\mathbf{p}}_{EC}^{R,i}+{\mathbf{p}}_{EC}^{R,r}$, where ${\mathbf{p}}_{\mathit{P}}^{\mathit{R},\mathit{i}}$, ${\mathbf{p}}_{EC}^{\mathit{R},\mathit{r}}$, and ${\mathbf{p}}_{EC}^{\mathit{R},\mathit{t}}$ are the incident and reflected pressures at $x\prime =l^{-}$, and the transmitted pressure at $x\prime =l^{+}$, respectively.

5. Middle-ear pressure gains

The pressure gain through the ME in the forward direction [Fig. 1(c)], ${\mathit{GME}}^F$, is calculated as

$${\mathit{GME}}^F=\frac{{\mathbf{p}}_C^{F,t}}{{\mathbf{p}}_{EC}^{F,t}}.$$ (11)

This transfer function normalizes the pressure reaching the cochlea (${\mathbf{p}}_C^{F,t}$) by the forward drive pressure in the EC (${\mathbf{p}}_{EC}^{F,t}$), similar to what would be done experimentally.

The pressure gain in the reverse direction [Fig. 1(d)], ${\mathit{GME}}^R$, is calculated as

$${\mathit{GME}}^R=\frac{{\mathbf{p}}_P^{R,t}}{{\mathbf{p}}_{ST}^{R,t}}.$$ (12)

Here, the pressure transmitted to the stapes from the cochlea, ${\mathbf{p}}_{ST}^{R,t}$, results from a drive stimulus of a constant pressure applied at the surface of the RW membrane. Due to the continuity conditions, the combination of the incident and reflected pressure components on one side of the probe–EC or stapes–cochlea interface is equal to the transmitted pressure on the other side, namely ${\mathbf{p}}_P^{R,t}$ or ${\mathbf{p}}_{ST}^{R,t}$, respectively.

The round-trip ME pressure gain, ${\mathit{GME}}^{RT}$, is simply the product of the forward and reverse pressure gains:

$${\mathit{GME}}^{RT}={\mathit{GME}}^F\times {\mathit{GME}}^R.$$ (13)

III. RESULTS

A. Calibrating and verifying the model against ossicular-velocity measurements

The model was calibrated against measured vibration data (Dong et al., 2013) under forward stimulation with the PF removed (thus exposing the MEC to the EC), so as to match the Dong et al. experimental conditions (Fig. 2). The velocities of the umbo (black lines), incus (blue lines), and stapes head (red lines) from the model (solid lines) are projected along the approximate direction of the measurement LDV beam and are normalized by the pressure measured in the vicinity of the umbo inside the EC. The locations indicated on the inset image of the model (black, red, and blue circles) were selected to be close to the experimental measurement locations. The magnitudes of the experimental results [Fig. 2(a), dashed lines] show that the pressure-driven vibrations of the ME locations are offset from one another and are more or less uniform across a wide range of frequencies from 15 to 60 kHz, with the umbo (black) being approximately 1 (mm/s)/Pa and stapes (red) being approximately 0.2 (mm/s)/Pa. The motion of the incus (blue) is somewhat less uniform but is generally about 20 dB or more below the umbo motion. The model responses and measurements exhibit overall consistency with each other in terms of their smoothness across frequencies and offsets with respect to one another. A notable exception is the significantly lower vibration magnitudes of the model below 6 kHz that deviate from the experimental lines by up to 40 dB for the malleus and even more for the stapes and incus. It is expected that the pressure gradient across the TM would be reduced when the PF is removed, but in the measurements this pressure gradient appears to be smaller than in the model, resulting in a higher transfer function (see Sec. IV). Given the expected variability of ME measurements across individual ears (e.g., Rosowski et al., 2003), the general trends of the model in terms of ossicular vibrations and wideband ME filtering appear to be consistent with the reported measurements.

FIG. 2.

(Color online) Measured vibrations of three points on the ossicular chain (at the umbo, incus, and stapes head) from a single mouse ear (dashed lines) compared against corresponding model results (solid lines) for both the magnitude (a) and phase (b). The orientation of the experimental specimen and measurement locations are indicated in the inset model image, with the measurement locations shown as dots in colors that match the corresponding plotted lines. To be consistent with the experimental setup, the PF region of the TM has been removed for the plotted model calculations, such that the MEC airspace is connected to the EC. The velocities are all normalized by the averaged acoustic pressure near the lateral surface of the umbo.

The measurements show largely in-phase motion of the ossicular chain below 6 kHz [Fig. 2(b), dashed lines]. The stapes and incus (dashed blue and red lines) remain largely in-phase through to higher frequencies as well, dropping by slightly more than 2 cycles over the 2–60 kHz range, unlike the umbo, which decreases by only about 1 cycle of phase over that range. The model shows in-phase ossicular motion up to 6 kHz as well. From around 6 to around 45 kHz, the stapes phase (solid red line) decreases more than the incus and umbo (solid blue and black lines), with the incus and umbo staying in-phase up to the resonance frequency around 20 kHz. Similar to the measurements, the decreases in phase of the umbo and stapes are nearly 1 and 2 cycles, respectively, over the 2–60 kHz range. The incus phase drops by more than a cycle between 50 and 60 kHz, but the phase response is dependent on the measurement location on the incus (not shown).

B. The effects of removing the pars flaccida and opening the middle-ear cavity

The effects of removing the PF or opening the wall of the MEC are examined in Fig. 3 through comparisons between intact and modified versions of the model. The plotted velocities, normalized by ${\mathbf{p}}_U^F$, represent the 3D-resultant maxima of the orthogonal vector components at the umbo and at the center of the stapes footplate. The individual velocity components in the x, y, and z directions in the intact model are presented in supplemental Fig. 1³ for the umbo and stapes. Removing the PF means that the EC and MEC airspaces are connected. Opening the MEC means that the MEC airspace has been removed from the model, so as to simulate a large opening of the MEC wall, with the PF still intact.

FIG. 3.

(Color online) The magnitude (a) and phase (b) of the 3D-velocity maxima at the umbo (black) and stapes footplate (magenta), normalized by the umbo pressure, for the fully intact model (solid lines), a model with the PF removed but an intact MEC (dashed lines), and a model with the MEC removed (“open MEC”) but an intact PF (dotted lines). The intact model represents the physiological condition, the removed-PF case represents the Dong et al. (2013) experimental setup, and the open-MEC case represents a common experimental setup convenient for laser-Doppler vibrometry.

In the intact model, the umbo-pressure-normalized umbo velocity $V_U^N$ (solid black lines) increases toward 11 kHz (referred to as the ME resonance in this study) with a slope of approximately +10 dB/octave, reaches a prominent peak, and then rolls off with a slope of –10 dB/octave at higher frequencies. The low-frequency slope is slightly steeper than the +6 dB/octave from Rosowski et al. (2003). At frequencies above the peak, the measured slope appears to be age-dependent (Rosowski et al., 2003). The model (Fig. 1) is representative of the middle-aged (12–14 months) and old-aged (18–24 months) animals, but not the young (1.5–3 months) animals. Thus, the model peak and high-frequency slope are consistent with middle-aged and older mice reported in Rosowski et al. The umbo-pressure-normalized stapes-footplate velocity $V_{FP}^N$ (magenta solid line) is lower than $V_U^N$, but their difference decreases more or less uniformly from 27 to 23 dB between 1 and 10 kHz and down to 12 dB at 100 kHz. $V_{FP}^N$ has a peak magnitude of 0.2 (mm/s)/Pa near the broad resonance in the 11–13 kHz range. The phase response of $V_U^N$ for the intact model [Fig. 3(b), solid black] drops by less than 1 cycle across the full frequency range; however, the $V_{FP}^N$ phase for the intact model (solid magenta) drops precipitously above 60 kHz, leading to a total drop of over 3 cycles across the full frequency range.

Removing the PF lowers the magnitude of $V_U^N$ (dashed black line) and $V_{FP}^N$ (dashed magenta line) by up to 30 and 43 dB, respectively, at frequencies below 5 kHz and generates a broad ME resonance between 6 and 20 kHz [Fig. 3(a), dashed black and magenta lines]. A few additional peaks in both $V_U^N$ and $V_{FP}^N$ are also visible at frequencies above 60 kHz. The phase responses of both $V_U^N$ and $V_{FP}^N$ [Fig. 3(b), dashed black and magenta lines] deviate from the corresponding intact responses above 20 kHz by as much as 1 cycle by 100 kHz. In the experimental data, however, removing the PF resulted in about 1 cycle of phase deviation between the umbo and footplate at frequencies below 5 kHz (Dong et al., 2013) (Fig. 3).

With the open MEC and intact PF (dotted black and magenta lines), the resonances of both $V_U^N$ and $V_{FP}^N$ broaden and shift to 7–9 kHz, with a boost of about 10 dB at frequencies below 10 kHz. There is no significant effect due to the MEC for frequencies above 20 kHz. The phase responses of $V_U^N$ and $V_{FP}^N$ [Fig. 3(b), dotted black and magenta lines] differ from the corresponding intact responses by less than 0.25 cycles in the 5–20 kHz range but otherwise remain nearly identical.

C. Ear-canal, stapes, and cochlear impedances

An interesting phenomenon happens when the MEC is open (i.e., removed): The cochlear impedance at low frequencies drops significantly (Fig. 4). Figure 4(a) shows the forward-looking EC input-impedance magnitude |${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}|$ (in GΩ), as well as the reverse-looking ME input-impedance magnitude |${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}|$ and forward-looking cochlear input-impedance magnitude ${\overrightarrow{|\mathit{Z}}}_{\mathit{C}}|$ (both in TΩ). With an open MEC [Fig. 4(a), dotted red line], the magnitude of ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ is essentially flat between 1 and 2 TΩ over the 1–100 kHz range, except for the sharp peak and notch between 70 and 80 kHz, which occur due to the formation of standing waves inside the cochlea (Mm. 1). With an intact MEC (solid red line), the low-frequency long-wavelength MEC pressures exerted at the RW membrane are coupled through the cochlea and add to the internal cochlear pressure near the oval window due to the stapes motion, thus resulting in a higher ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ magnitude below 5 kHz (e.g., 5.2 TΩ at 1 kHz). Since the acoustic impedance of the air in the MEC decreases as the frequency increases, the effect of the MEC on ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ vanishes at higher frequencies. The phases of ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ for the open- and intact-MEC models [Fig. 4(b)], respectively, start at –0.17 and –0.22 cycles (stiffness-dominant), reach 0 cycles at 3.5 and 5 kHz, and then both merge near 10 kHz. From there, they remain almost flat near 0 cycles up to 60 kHz, (viscosity-dominant) and then, with the exception of a sharp notch at 76 kHz (due to a standing-wave mode within the cochlea), converge smoothly to 0.25 cycles over the 60–100 kHz range (mass-dominant response). The activity of the outer hair cells in the cochlea, as represented by gain factor α, introduces subtle changes to ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$, especially at frequencies >30 kHz, including sharp peaks up to 8 dB relative to a cochlear model without active amplification (supplemental Fig. 2).³

FIG. 4.

(Color online) The magnitude (a) and phase (b) of the model EC input impedance, ${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$ (solid and dotted blue lines), and characteristic impedance of the air at the EC entrance, Z₀ (dashed blue lines), all in GΩ, as well as the reverse ME input impedance at the stapes footplate (for the open-EC case), ${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$ (solid green lines), and the forward cochlear input impedance, ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ (solid and dotted red lines), all in TΩ. The EC and cochlear calculations are shown for intact (solid lines) and open-MEC cases (dotted lines).

Mm. 1.

Download video file ^{(1.4MB, mp4)}

DOI: 10.1121/10.0004218.1

Animations of the standing-wave mode of the cochlear fluid pressure occurring at 76 kHz. This is a file of type “m4v” (1.3 MB).

While |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| is fairly constant, the reverse stapes impedance |${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$| varies above and below this relatively constant value. Below 11 kHz, |${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}|$ [Fig. 3(a), green line] is above |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$|, starting at 25 TΩ at 1 kHz. The magnitude continues to mostly decrease as the frequency rises, reaching the lowest value of 0.3 TΩ at 48 kHz, and then increases to around 1 TΩ at 100 kHz. The phase of ${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$ [Fig. 3(b)] stays in the stiffness-dominant region (close to –0.25 cycles) up to 50 kHz. Above 80 kHz, the phase becomes greater than 0 cycles and leans toward mass dominance as the phase increases toward 0.25 cycles. Since ${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$ is measured in the reverse direction, it is affected by the EC termination condition (either open or closed). As shown in supplemental Fig. 3,³ the effects of changing the EC termination are less than 5 dB and appear in the 10–20 and 60–70 kHz ranges, corresponding to the first and third standing-wave modes in the EC. Opening the MEC affects the ${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$ magnitude and phase by less than 3 dB and 0.08 cycles, respectively, throughout the frequency range (supplemental Fig. 3).³

|${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}|$ is typically lower than |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| by about 4 orders of magnitude and hovers around the 0.26 GΩ characteristic impedance, Z₀, of the EC (dashed blue line), calculated as

$${\mathit{Z}}_0=\frac{\rho c}{A},$$ (14)

where $\rho$ and $c$ are the density of air and speed of sound in air, respectively, and A is the area of the EC entrance.

Below 10 kHz, |${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$| for the intact model is inversely proportional to frequency [Fig. 4(a), blue line]. The local minimum happens slightly below the umbo resonance for the intact model (Fig. 3, solid black line). The patterns of peaks at 15 and 64 kHz and valleys at 40 and 86 kHz happen as a result of standing waves inside the EC. The frequencies of these resonances depend on the place in the EC chosen as the reference measuring point. The ${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$ phase [Fig. 4(b), blue line] alternates rather sharply between –0.25 and 0.25 cycles at the standing-wave frequencies. ${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$ for the model with an open MEC (dotted blue line) deviates from the intact model at frequencies below 20 kHz, and its magnitude is lower by 11 dB at frequencies below 7 kHz. Its first minimum and maximum shift toward lower frequencies by 3 and 1 kHz, respectively. A significant phase deviation of over 0.25 cycles occurs in the 4–15 kHz range.

D. Middle-ear pressure transfer functions

Figure 5 shows the ideal and actual ME pressure gains of the mouse model. The ideal forward ME gain is the ratio of the TM area to the oval-window area (30 dB) multiplied by the ratio of the malleus length to the incus length (the lever ratio of 7 dB). Calculated from the model anatomy, the ideal gain is therefore 37 dB for forward transmission [Fig. 5(a), dotted red line] and −37 dB for reverse transmission [Fig. 5(c), dotted blue line]. The reported forward (${\mathit{GME}}^F$) and reverse (${\mathit{GME}}^R$) pressure gains in the model are calculated similarly to what has been done experimentally in other animals (Olson, 2001): with a microphone at the EC entrance (${\mathbf{p}}_{EC}^{F,t}$) and a hydrophone in the scala vestibuli of the cochlea (${\mathbf{p}}_C^{F,t}$). These measured and calculated values are typically lower than the ideal values due to losses in the ME.

FIG. 5.

(Color online) The respective magnitudes and phases of the calculated ME pressure-gain transfer functions for the forward direction, ${\mathit{GME}}^F$ [(a) and (b)], reverse direction, ${\mathit{GME}}^R$ [(c) and (d)], and round trip, ${\mathit{GME}}^{RT}$ [(e) and (f)]. ${\mathit{GME}}^F$ is calculated for two cases: one with respect to the average input pressure at the EC entrance (${\mathbf{p}}_{EC}^{F,t}$; solid red lines) and one with respect the umbo pressure (${\mathbf{p}}_U^F$; solid green lines). For the ${\mathit{GME}}^R$ and ${\mathit{GME}}^{RT}$ calculations, two boundary conditions at the EC entrance are considered: one in which the lateral side of the EC matches the characteristic impedance of the EC (open EC, which approximately represents the physiological condition) and one in which the EC opening is terminated with a rigid surface (closed EC). Horizontal dotted lines in red [(a) and (b)], blue [(c) and (d)], and black [(e) and (f)] indicate the ideal ME pressure gains of 37, –37, and 0 dB for forward, reverse, and round-trip transmission, calculated based on anatomical features of the model.

The gain in the forward direction is calculated for two input terminals: ${\mathbf{p}}_{EC}^{F,t}$ (i.e., ${\mathit{GME}}_{EC}^F={\mathbf{p}}_C^{F,t}/{\mathbf{p}}_{EC}^{F,t}$) and the pressure at the umbo ${\mathbf{p}}_U^F$ (i.e., ${\mathit{GME}}_U^F={\mathbf{p}}_C^{F,t}/{\mathbf{p}}_U^F$). One should note that the uniformity of the pressure distribution on the TM surface does not hold at higher frequencies, such that ${\mathbf{p}}_U^F$ is not representative of the total pressure driving the whole TM.

The magnitudes of ${\mathit{GME}}_{EC}^F$ (red line) and ${\mathit{GME}}_U^F$ (green line) are nearly identical ( –1 and 2.5 dB at 1 kHz, respectively) and flat up to 4 kHz [Fig. 5(a)]. |${\mathit{GME}}_{EC}^F$| increases to a peak of 34 dB (3 dB lower than the ideal ratio) at 10 kHz. Two other sharp peaks appear at 39 and 86 kHz, corresponding to the anti-node standing-wave patterns seen in Fig. 4(a) (solid blue line). After the 86 kHz peak, |${\mathit{GME}}_{EC}^F$| sharply decreases to 2.5 dB at 100 kHz. |${\mathit{GME}}_U^F$| has a flatter dominant peak of 34 dB in the 11–13 kHz range and gradually decreases toward 2.5 dB at 100 kHz without any of the prominent peaks that are visible for |${\mathit{GME}}_{EC}^F$|. The phase responses of both cases remain rather flat around 0 cycles (with less than half a cycle of variation) up to 10 kHz. Above 10 kHz, the ${\mathit{GME}}_{EC}^F$ phase decreases faster than that of ${\mathit{GME}}_U^F$, reaching –2.5 cycles at 100 kHz, as compared to –1.5 cycles for ${\mathit{GME}}_U^F$ [Fig. 5(b)]. Unless otherwise stated, for the rest of this study, ${\mathit{GME}}^F$ refers to ${\mathit{GME}}_{EC}^F$, the case with ${\mathbf{p}}_{EC}^{F,t}$ as the input terminal.

For reverse drive, two boundary conditions at the EC entrance are considered: (1) open EC, in which the EC-entrance boundary is assigned a load matching the characteristic impedance of air, $\rho c/A$, which mimics an infinite tube, and (2) closed EC, for which the EC-entrance boundary is assumed to be a rigid and ideal reflective surface. The realistic impedance of a measurement probe would lie somewhere between these two conditions, plus the addition of some absorbance. The magnitude of ${\mathit{GME}}^R$ with an open EC [Fig. 5(c), solid blue line] starts at −86 dB and increases to around −47 dB at 13 kHz (10 dB below the ideal reverse gain). It remains between −55 and −45 dB from 13 to 64 kHz and then finally decreases to −68 dB at 100 kHz. The magnitude of ${\mathit{GME}}^R$ with a closed (i.e., occluded) EC (dashed blue line) stays close to −67 dB up to 4 kHz, after which it starts to increase to form a peak of −36 dB in the 13–15 kHz range. It forms another rather sharp peak of −22 dB at 64 kHz, corresponding to the node pressure seen in Fig. 4 (solid blue line). Similar to the open-EC case, the ultimate gain reaches −68 dB at 100 kHz. The phase responses of the open- and occluded-EC cases show similar patterns [Fig. 6(d)]. Unless otherwise stated, for the rest of this study, ${\mathit{GME}}^R$ refers to the case with an open EC.

FIG. 6.

(Color online) Sensitivity analyses showing the effects of varying the Young's moduli of the SAL (E_SAL; left column) and TM (E_TM; middle column) and varying the ossicular mass density (ρ_OSS; right column) on ${\overrightarrow{Z}}_C$ (top two rows), ${\overleftarrow{Z}}_{ST}$, (middle two rows), and ${\overrightarrow{Z}}_{EC}$ (bottom two rows). The black dotted lines indicate the effects of opening the MEC of the baseline model (solid black lines), whereas the colored dashed and solid lines indicate the results of, respectively, decreasing or increasing a parameter with respect to the baseline value.

The magnitudes of the ME round-trip pressure gains ${\mathit{GME}}^{RT}$ for the open-EC (solid black line) and closed-EC (dashed black) cases are similar to the corresponding ${\mathit{GME}}^R$ magnitudes [Fig. 5(e)]. |${\mathit{GME}}^{RT}$| for the open-EC case increases to around −20 dB at 10 kHz and remains rather flat (±6 dB) toward 37 kHz. In the 37–45 kHz range, it forms small peaks of –7 and –9 dB [corresponding to sharp |${\mathit{GME}}_{EC}^F$| peaks in Fig. 5(a)], after which it decreases to −65 dB at 100 kHz. |${\mathit{GME}}^{RT}$| for the closed-EC case (dashed black line) is higher than for the open-EC case overall. Not surprisingly, each ${\mathit{GME}}^{RT}$ phase exhibits a phase change of around 5.5 cycles from 5 to 100 kHz [Fig. 6(f)], due to the summation of the forward and reverse phases. This corresponds to a round-trip ME delay of approximately 61 μs, with about half of the total delay provided by the transit time in each direction.

E. Sensitivity analyses of ear-canal, stapes, and cochlear impedances

Figure 6 shows the results of sensitivity analyses in which the stiffness of the SAL (E_SAL; left column) and TM (E_TM; middle column), as well as the density of the ossicular chain (ρ_OSS; right column) are varied to test their effects on the cochlear impedance, ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$, (top two rows), stapes impedance, ${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$, (two middle rows), and EC impedance, ${\overrightarrow{\mathit{Z}}}_{EC}$, (bottom two rows). These simulations are performed for the active cochlea with an 80 dB SPL input (using an α value of 0.2) as reported by Motallebzadeh et al. (2018). Earlier it was shown that the MEC can affect ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ (Fig. 4, red lines). The sensitivity analyses show that the mechanical characteristics of the ossicular chain can affect the MEC pressure, which in turn can influence the cochlear impedance in surprising ways through the RW membrane. To demonstrate this new finding that ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ can be a function of the ME parameters when the mouse MEC is closed, we also present the responses of the baseline open-MEC case (dotted black lines).

1. Sensitivity to changing the stiffness of the SAL (E_SAL)

Changing E_SAL has a proportional effect on |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| below 4 kHz [Fig. 6(a)]. As stated above, however, an unexpected result is that |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| for the closed-MEC case is generally higher than for the open-MEC case in this frequency range (black dotted line). Increasing E_SAL by 8 times (solid red lines) raises |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| to 17 TΩ at 1 kHz, which is higher than the baseline (solid black lines) by a factor of 3.3. Scaling E_SAL by a factor of $\frac{1}{8}$ (dashed red lines) reduces |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| to 3.8 TΩ at 1 kHz, which is lower than the baseline by a factor of 1.4. The effects of changing E_SAL almost vanish above 6 kHz or so, above which the ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ model calculations merge and become similar to the open-MEC case, with slight deviations above 60 kHz. The phase responses [Fig. 6(b)] show that higher E_SAL values keep ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ in the stiffness-dominant region longer (i.e., phase below 0), and, as is also evident in Fig. 4(b), the open-MEC case is more viscosity-dominated at frequencies below 4 kHz (i.e., the phase is closer to 0) than any of the intact models.

As expected, E_SAL has a proportional effect on |${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$|, with a stiffer SAL producing a higher impedance magnitude out to 50 kHz [Fig. 6(c)]. For example, |Z_ST| for the stiffest case (8×, solid red line) is higher than the most-compliant case ($\frac{1}{8}$×, dashed red line) by more than an order of magnitude around the ME resonance (11 kHz). Higher E_SAL values extend the stiffness-dominant regions where the phase remains close to –0.25 cycles [Fig. 6(d)]. For the stiffest case (solid red line), the phase remains negative for all plotted frequencies.

The responses of ${\overleftarrow{Z}}_{EC}$ [Figs. 7(e) and 7(f)] are all nearly independent of the SAL stiffness.

FIG. 7.

(Color online) The sensitivity of ${\mathit{GME}}^{RT}$ to increases or decreases in E_SAL [(a) and (b)], E_TM [(c) and (d)], and ρ_OSS [(e) and (f)]. See the caption to Fig. 6 for line descriptions.

2. Sensitivity to changing the stiffness of the TM (E_TM)

Figures 6(g) and 6(h) show that the stiffness of the TM (E_TM) also influences ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ at low frequencies in a bizarre and unintuitive manner and in opposition to the effects of E_SAL. A less-stiff (more-compliant) TM increases |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| [Fig. 6(g), dashed lines]. For instance, the most-compliant TM (dashed red line) can result in a |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| value of 17 TΩ at the lowest frequency of 1 kHz, which is 3.3, 4, and 9 times higher than the respective values for the baseline (solid black), stiffest-TM (i.e., 8×; solid red), and open-MEC (dotted black) cases. Consistent with the effects on |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$|, the phase of ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ below 10 kHz drops closer to –0.25 cycles when the TM becomes more compliant (dashed lines), suggesting that a less-stiff TM somehow causes ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ to appear more stiff.

${\overleftarrow{Z}}_{ST}$ is not affected by the increases or decreases in TM stiffness below 11 kHz, the umbo resonance, whereas at higher frequencies, E_TM has a moderate effect on ${\overleftarrow{Z}}_{ST}$ [Figs. 6(i) and 6(j)]. This is because at lower frequencies, the stiffness of the SAL dominates ${\overleftarrow{Z}}_{ST}$, whereas at higher frequencies the TM stiffness dominates, such that changes in E_TM cause changes to the TM modes that are reflected in ${\overleftarrow{Z}}_{ST}$. The phase responses for all cases start at –0.25 cycles at low frequencies and converge to +0.25 cycles at 100 kHz, with E_TM-dependent variations in the intervening frequencies [Fig. 6(j)].

As the medial termination of the EC, it is not surprising that the TM stiffness affects |${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$|, especially at frequencies below 50 kHz [Fig. 6(k)]. Below 10 kHz, |${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$| is proportional to frequency, with the highest E_TM value (8×, solid red line) producing |${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$| values as much as 8 times higher than for the most-compliant TM [$\frac{1}{8}$×, dashed red line; Fig. 6(k)]. Increasing E_TM by 8 times extends up to 12 kHz the stiffness-dominant region where the phase remains close to –0.25 cycles [Fig. 6(l), solid red line]. The first and second standing-wave modes at 10 and 40 kHz are affected by the TM stiffness, but the responses above 50 kHz are less sensitive to E_TM.

3. Sensitivity to changing the mass density of the ossicles (ρ_OSS)

Unlike the stiffnesses of the SAL and TM, changing the mass density of the ossicles, ρ_OSS, has virtually no effect on ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$ for all frequencies [Figs. 6(m) and 6(n)], as ossicular density does not affect the impedance of the MEC airspace.

Increasing or decreasing ρ_OSS by a factor of 2 has a minor effect on |${\overleftarrow{Z}}_{ST}$| below 30 kHz [Fig. 6(o)]. The effects become moderately significant above 40 kHz. The phase of ${\overleftarrow{Z}}_{ST}$ shows a downward shift of the ME resonance toward 10 kHz when ρ_OSS is doubled, which shifts the stiffness-to-mass transition toward lower frequencies. Halving the mass causes the phase to cross 0 cycles at a higher frequency [Fig. 6(p)].

|${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$| is significantly affected by ρ_OSS in the 8–20 kHz range [Figs. 6(q) and 6(r)], likely due to changes in the ME resonance. Above about 30 kHz, EC acoustics dominate such that any effects of ρ_OSS on ${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$ are negligible.

F. Sensitivity analyses of the middle-ear round trip pressure gain (${\mathit{GME}}^{\mathit{RT}}$)

Across individual animals of the same species, there are anatomical variations that can affect sound transmission. Figure 7 shows the sensitivity of three key ME parameter variations on the ME round-trip pressure gain. The corresponding changes to just the forward pressure gain are reported in supplemental Fig. 4.³

1. Sensitivity to changing the stiffness of the SAL (E_SAL)

The magnitude of ${\mathit{GME}}^{RT}$ systematically and predictably decreases as E_SAL increases [Fig. 7(a)], with a maximum decrease from the baseline of 20 dB for an 8-times increase in the baseline value (solid red line) near the 11-kHz ME resonance. Decreasing E_SAL by a factor of 8 results in up to a 6 dB increase in |${\mathit{GME}}^{RT}$| at 11 kHz (dashed red line). The ${\mathit{GME}}^{RT}$ phase is altered by less than ±0.4 cycles when altering E_SAL by a factor of 8 [Fig. 7(b)].

2. Sensitivity to changing the stiffness of the TM (E_TM)

E_TM has a profound effect on |${\mathit{GME}}^{RT}$| in the 10–60 kHz range [Fig. 7(c)]. In general, a stiffer TM (solid lines) has better performance at frequencies above the 11-kHz ME resonance, in that the forward [supplemental Fig. 4(c)]³ and round-trip pressure gains [Fig. 7(c)] are higher, but this is at the expense of reduced gains at the lower frequencies. For instance, the stiffest TM (solid red line) results in approximately flat gains in the 20–60 kHz region that are up to 24 dB above the baseline (solid black line), which reaches as high as 8 dB below the ideal |${\mathit{GME}}^{RT}$| of 0 dB at 40 kHz. Below 10 kHz, the gain for the stiffest TM is reduced by up to 25 dB relative to the baseline. The most-compliant TM (dashed lines) has its gain reduced by as much as 38 dB near 30 kHz [Fig. 7(c), dashed red line]. The phase of ${\mathit{GME}}^{RT}$ varies by less than a cycle up to 70 kHz [Fig. 7(d)], above which it varies by as much as ±1.25 cycles relative to the baseline phase.

3. Sensitivity to changing the mass density of the ossicles (ρ_OSS)

The magnitude of ${\mathit{GME}}^{RT}$ is affected by ρ_OSS mostly in the 5–20 kHz region and above 60 kHz [Fig. 7(e)]. For the densest ossicles (solid red line), the gain surrounding the 11-kHz ME resonance increases by as much as 15 dB relative to the baseline (solid black line). At frequencies above 60 kHz, the densest ossicles reduce the gain by as much as 30 dB at 90 kHz. The phase responses remain within 1 cycle around the baseline up to 80 kHz, above which the lower-density cases decline by a lesser amount (dashed lines).

This systematic study of key ME parameters might prove to be useful in future studies for characterizing the effects of normal vs pathological ME variations.

G. Effects of the OA on sound transmission

While in Sec. III F the effects on the ME pressure gain of altering the overall ossicular mass are reported, in this section the more-localized effects of the OA mass are investigated. The mass of the OA comprises about 12% of the total malleus mass, and is located far from the anterior–posterior anatomical axis of the malleus. Figure 8 showcases the effects of the OA on the mouse ME ossicular-velocity transfer functions [(a) and (b)] and forward ME pressure gain [(c) and (d)]. In general, removing the OA (dotted lines) causes a moderate reduction in transmission through the ME between 7 and 30 kHz (red lines). The OA has very little effect above 40 kHz, indicating that the mass it adds to the ossicular chain does not improve or compromise high-frequency hearing. Although removing the OA increases the umbo velocity by up to 6.6 dB between 11 and 40 kHz (a) (black lines), the stapes velocity decreases by up to 5 dB between 7 and 30 kHz (magenta lines).

FIG. 8.

(Color online) The effects of removing the OA are shown for the 3D-velocity maxima [(a) and (b)] of the umbo (black lines) and stapes (red lines), each normalized by the umbo pressure, and for ${\mathit{GME}}^F$ [(c) and (d)], with the intact- and removed-OA cases depicted in the inset of panel (d). The solid and dotted lines indicate the intact- and removed-OA models, respectively [(a)–(d)].

The stapes moves mostly in phase with the umbo for frequencies up to about 5 kHz. However, at higher frequencies the stapes lags behind the umbo by nearly 1 cycle at 40 kHz and 1.5 cycles by 80 kHz [Fig. 8(b)]. Much of this phase lag is along the malleus, with little added lag across the IMJ and some added lag across the ISJ (see supplemental Fig. 5³ for plots of the changes in the phase components from the umbo to the stapes in the x, y, and z directions). The phase of the umbo velocity does not alter significantly when the OA is removed [Fig. 8(b), black lines]. However, the phase of the stapes velocity shows an increase in up to 0.15 cycles between 10 and 40 kHz, resulting in less steepness of the phase and therefore less delay when the OA is removed [Fig. 8(b), magenta dotted line].

While the stapes velocity represents the local motion of a single point at the center of the footplate, the cochlear pressure is a result of the overall 3D motion of the stapes footplate. Removing the OA decreases |${\mathit{GME}}^F$| by up to 5 dB between 4 and 30 kHz, suggesting that the OA provides a moderate advantage for mid-frequency hearing, but not for the upper half of the mouse hearing range. Similar to the stapes velocity, the phase of ${\mathit{GME}}^F$ [Fig. 8(d)] is shifted somewhat higher (up to 0.14 cycles), such that removing the OA produces less delay between 10 and 40 kHz.

The OA has been hypothesized to have its greatest influence on the rotational axes of the ossicular chain. To analyze this, the rotational field of the ossicles was calculated with and without the OA. According to the Helmholtz decomposition theorem (Mase and Mase, 1999, Chap. 6), a displacement field can be decomposed into translational and rotational components. The rotational components of displacement vector u are calculated by

$$\nabla \times \mathbf{u}=\left(\frac{\partial u_z}{\partial y}-\frac{\partial u_y}{\partial z}\right)\mathit{i}+\left(\frac{\partial u_x}{\partial z}-\frac{\partial u_z}{\partial x}\right)\mathit{j}+\left(\frac{\partial u_y}{\partial x}-\frac{\partial u_x}{\partial x}\right)\mathit{k},$$ (15)

where $\nabla \times$ is the curl operator and u is the displacement vector with components $u_x$, $u_y$, and $u_z$, and $\mathit{i}$, $\mathit{j}$, and $\mathit{k}$ are unit vectors pointing in the Cartesian $x$, $y$, and $z$ directions. The direction of the resulting curl vector indicates the axis of rotation, and the magnitude of the curl vector is twice the speed of rotation.

Figure 9 uses vector fields to summarize the rotational motions of the mouse ossicles (not to be confused with translational displacements) for the intact OA (blue arrows) and removed OA (red arrows) at six different frequencies. The length of each arrow indicates the local magnitude of rotation, and the direction of each arrow indicates the local axis of rotation, according to the right-hand rule. In this representation, the axis 1 and axis 2 rotational axes from Fleischer (1978) are largely aligned with the respective anterior–posterior y and dorsal–ventral z axes (Fig. 9, 1-kHz panel). Thus, if the calculated rotational vectors were to align closely with the y axis at a given frequency (shown in subpanels), then that would confirm the existence of axis 1 in the model at that frequency.

FIG. 9.

(Color online) Vector fields summarizing the rotations of the mouse ossicles with the OA intact (blue arrows) and with the OA removed (red arrows), as calculated from the curl of the displacement field at six frequencies. The length of each arrow indicates the magnitude of rotation, and the direction of each arrow indicates the axis of rotation (according to the right-hand rule). Note that the scale factor for the vector lengths is different at each frequency, ranging from 2e3x at 10 kHz to 5e5x at 80 kHz. In the 1-kHz panel, the anterior–posterior y axis is closely aligned with axis 1 (green), the conventional “anatomical” rotation axis. The dorsal–ventral z axis is closely aligned with axis 2 (magenta), as proposed by Fleischer (1978), which passes through the OA.

For rigid bodies, all rotational vectors should have the same magnitude and direction for each mode of vibration. However, Fig. 9 shows that the magnitudes and directions of the rotational vectors are spatially varying, especially within the malleus, and in a manner that tends to increase with frequency, which is inconsistent with rigid-body motion but is consistent with ossicular bending. In the intact-OA model, the blue arrows near axis 1 remain aligned primarily with axis 1 for all frequencies. Along the manubrium and within the OA, the blue arrows are aligned primarily with axis 2 for higher frequencies (at and above 60 kHz). The presence of both axes of rotation in the malleus indicates that the malleus is bending. The changes in phase along the malleus (supplemental Fig. 5)³ are also consistent with bending of the malleus.

Removing the OA (Fig. 9, red vs blue arrows) has negligible effects on the rotational field at lower frequencies (e.g., at 30 kHz and below). At higher frequencies (e.g., 60 kHz and above), the rotational field near axis 1 is more closely aligned to axis 1 with the intact OA than without it. At 80 kHz, the red arrows are less aligned with axis 2 than the blue arrows representing the intact model, which indicates that removing the OA reduces motion around axis 2.

While the intact OA improves the magnitude of sound transmission to the stapes by up to about 5.5 dB in the 10–30 kHz range [Figs. 8(a) and 8(c)], the lack of major differences between the blue and red arrows indicates that this improvement is not due to a switch to some lower-inertia rotational axis. Rather, the improved transmission for the intact-OA case is associated with a change in the phase delay through the ossicles [Fig. 8(b) and supplemental Fig. 5],³ which is primarily caused by greater bending of the malleus (Mm. 2 and Mm. 3). This greater bending of the malleus due to the presence of the OA is therefore what enhances ME transmission in the 10–30 kHz range. Because there are minimal observed changes in the rotational field, this enhancement is likely due to an increase in translation rather than rotation (not shown).

Mm. 2.

Download video file ^{(9.6MB, mp4)}

DOI: 10.1121/10.0004218.2

Animations of the intact mouse ossicles at six frequencies, shown with the lateral–medial axis perpendicular to the frame. The color maps and legends indicate the displacement (in μm) in response to a 1-Pa sound stimulation at the EC entrance. The dashed-green and magenta lines (a) indicate axes 1 and 2, respectively. This is a file of type “m4v” (10.6 MB).

Mm. 3.

Download video file ^{(7.9MB, mp4)}

DOI: 10.1121/10.0004218.3a

Animations of the intact mouse ossicles at six frequencies (spread between two files), shown with axis 1 (upper rows) and axis 2 (lower rows) perpendicular to the frame. The blue points in the upper and lower rows, respectively, indicate the perpendicular viewing axes axis 1 and axis 2. The color maps and legends indicate the displacement (in μm) in response to a 1-Pa sound stimulation at the EC entrance. The green closed curves are presented to track the motion of the non-fixed axes. These results are spread between two files of type “m4v” (a: 8.4 MB, b: 8.5 MB).

Download video file ^{(8MB, mp4)}

IV. DISCUSSION

A. Summary of key findings

In this study, we developed a coupled EC–ME–cochlea FE model for the mouse (Fig. 1), derived model parameters through an iterative process, and calibrated the resulting model against available in vivo experimental ossicular-velocity measurements (Fig. 2). We used the model to make several predictions about the mouse ME, including input impedances (${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$, ${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$, and ${\overleftarrow{\mathit{Z}}}_{\mathit{ST}}$; Fig. 4) and ME pressure gains (${\mathit{GME}}^F$, ${\mathit{GME}}^R$, and ${\mathit{GME}}^{RT}$; Fig. 5). We then performed sensitivity analyses showing how changes to the stiffness of the SAL and TM and mass of the ossicles affect these aspects of the ME. Opening the MEC increases the stapes response below 12 kHz. The bony OA slightly improves the stapes response in the 7–40 kHz mid-frequency region. Removal of the PF decreases the stapes response below about 10 kHz. A significant and surprising finding of this study is that the MEC and tympano-ossicular chain can have an effect on the cochlear input impedance, especially at frequencies below 10 kHz, which is around the lower extent of the mouse hearing range. In this frequency range, the MEC pressure on the RW membrane influences the pressure at the oval window. From the 3D ossicular-chain motions over a wide frequency range of 1 to 80 kHz (e.g., Mm. 1 and 2), vector fields were calculated for which the phase of the ossicular velocity (supplemental Fig. 5),³ and magnitude and direction of the ossicular rotational components (Fig. 9) all suggest that the malleus bends. This bending, which has generally been neglected, increases transmission bandwidth at the expense of increasing transmission delay.

B. The middle-ear cavity affects the cochlear input impedance

The effects of the MEC on ME sound transmission have been known for some time (Guinan and Peake, 1967; Shera and Zweig, 1992; Keefe and Levi, 1996; Motallebzadeh et al., 2017b). This study newly reveals that the MEC can also affect cochlear impedance and thus acoustical drive to the inner ear. Interestingly, the results imply that ME parameters can alter the cochlear response via the action of the MEC pressure upon the oval window and RW membrane.

To a first-order approximation, a closed MEC airspace with hard walls can be considered an ideal lumped-impedance element defined as Z_MEC = 1/($j\omega C_{\mathit{air}})$, where j is $\sqrt{-1}$ and $\omega$ is the angular frequency. $C_{\mathit{air}}$, the compliance of the enclosed volume of air in the MEC (Vol) is calculated by C_air =$\hspace{0.17em}\mathit{Vol}$/($\rho c^2)$, where $\rho$ and c are the respective density of air and speed of sound in air (Motallebzadeh et al., 2017a). These relationships indicate that the impedance of the MEC is inversely proportional to frequency and the MEC volume. This means that the magnitude of $Z_{\mathit{MEC}}$ increases for low MEC volumes, but can become negligible at high-enough frequencies. Ravicz and Rosowski (2012) showed that $Z_{\mathit{MEC}}$ for a small animal such as gerbil can contribute up to 70% of the acoustic ME impedance at frequencies below 1 kHz. However, Keefe and Levi (1996) predicted that the MEC can also affect the ME impedance at high frequencies, and Motallebzadeh et al. (2017b) showed that this effect is due to the occurrence of standing waves inside the MEC chambers, which cannot be accounted for using a lumped-impedance representation of the MEC. One of the subtle features of the MEC standing-wave modes can be seen in Fig. 5 as the sharp peak at 43 kHz for both ${\mathit{GME}}_{EC}^F$ (red line) and ${\mathit{GME}}_U^F$ (green line).

Shera and Zweig (1992) were the first to develop equations that phenomenologically describe the ME and its interaction with the “three windows,” namely the TM, oval window, and RW, that interface with the MEC. In their formulation, the MEC pressure was assumed to be uniform throughout the MEC volume due to their treatment of the MEC as a single lumped element. The implications of a non-uniform pressure distribution in the MEC, with resulting pressure differences at the oval window and RW, and the effects of the three windows on the MEC compliance were not addressed in that study but are presently being tackled using the FE-modeling approach employed in this study.

Since the MEC airspace has three compliant windows in addition to its bony walls, its impedance is affected by any ME parameters that can change the pressure regime of that airspace. Figure 10 compares different methods of determining the cochlear impedance. The ${\overrightarrow{\mathit{Z}}}_C$ calculations for the intact-MEC (solid black) and open-MEC (${\overrightarrow{\mathit{Z}}}_C^O$, dashed black) models are functions of the cochlear pressure ${\mathbf{p}}_C^{F,t}$ (the pressure transmitted by the stapes into the scala vestibuli) and the volume velocity ${\overrightarrow{\mathbf{U}}}_C^{F,t}$ at the oval window. Subtracting the MEC air pressure measured in front of the RW (${\mathbf{p}}_{\mathit{MEC}}^F$) from the cochlear fluid pressure next to the oval window (${\mathbf{p}}_C^{F,t}$) produces a new measure of cochlear input impedance (${\overrightarrow{\widehat{\mathit{Z}}}}_C$, green line),

$${\overrightarrow{\widehat{\mathit{Z}}}}_C={(\mathbf{p}}_C^{F,t}-{\mathbf{p}}_{\mathit{MEC}}^F)/{\overrightarrow{\mathbf{U}}}_C^{F,t},$$ (16)

which closely replicates ${\overrightarrow{Z}}_C$ for the open-MEC model (dotted black line). This indicates that when the MEC air pressure ${\mathbf{p}}_{\mathit{MEC}}^F$ acts on the RW, it affects the fluid pressure measured inside the cochlea at the oval window, by up to 9 dB at 1 kHz. Similarly, the cochlear impedance can also be calculated at the RW (${\overrightarrow{\widehat{\mathit{Z}}}}_{RW}$, blue line) as

$${\overrightarrow{\widehat{\mathit{Z}}}}_{RW}={(\mathbf{p}}_{RW}^F-{\mathbf{p}}_{\mathit{MEC}}^F)/{\overrightarrow{\mathbf{U}}}_{RW}^{F,t},$$ (17)

where ${\mathbf{p}}_{RW}^F$ and ${\overrightarrow{\mathbf{U}}}_{RW}^{F,t}$ are the respective cochlear fluid pressure and volume velocity adjacent to the RW. Another measure of the cochlear impedance is the lumped “cochlear input impedance” defined by Shera and Zweig (1992), which is also termed the “differential impedance” ${\overrightarrow{\mathit{Z}}}_{\mathit{diff}}$, by Nakajima et al. (2009) and Frear et al. (2018). This impedance is calculated by subtracting the cochlear fluid pressures adjacent to the oval and round windows and dividing that pressure difference by the volume velocity of the oval window (or stapes footplate), as follows:

$${\overrightarrow{\mathit{Z}}}_{\mathit{diff}}={(\mathbf{p}}_C^{F,t}-{\mathbf{p}}_{RW}^F)/{\overrightarrow{\mathbf{U}}}_C^{F,t}.$$ (18)

Assuming the cochlear fluid is incompressible, this measure is equivalent to the difference between the oval-window and RW impedances (i.e., ${\overrightarrow{\widehat{\mathit{Z}}}}_C-{\overrightarrow{\widehat{\mathit{Z}}}}_{RW}$; Fig. 10, red lines). The magnitude of ${\overrightarrow{\mathit{Z}}}_{\mathit{diff}}$ is lower than that of ${\overrightarrow{Z}}_C$ for the intact-MEC (solid black lines) and open-MEC (dotted black lines) cases by up to 17 and 8 dB, respectively, and ${\overrightarrow{\mathit{Z}}}_{\mathit{diff}}$ converges upon the other cochlear-impedance formulations depicted in Fig. 10 above 6 kHz for the magnitude and above 12 kHz for the phase.

FIG. 10.

(Color online) Comparisons among cochlear input impedances at the oval window (${\overrightarrow{\mathrm{Z}}}_C,\hspace{0.17em}{\overrightarrow{\mathrm{Z}}}_C^O$, and ${\overrightarrow{\widehat{\mathrm{Z}}}}_C$), the cochlear input impedance at the RW (${\overrightarrow{\widehat{\mathrm{Z}}}}_{RW}$), and the cochlear differential input impedance (${\overrightarrow{\mathrm{Z}}}_{\mathit{diff}}$) as defined by Shera and Zweig (1992) and Frear et al. (2018). The input impedances at the oval window are calculated in terms of the cochlear fluid pressure adjacent to the stapes footplate (${\mathbf{p}}_C^{F,t}$) for the intact (${\overrightarrow{\mathrm{Z}}}_C$; solid black lines) and open-MEC (${\overrightarrow{\mathrm{Z}}}_C^O$; dotted black lines) cases and in terms of the difference between ${\mathbf{p}}_C^{F,t}$ and ${\mathbf{p}}_{\mathit{MEC}}^F$, the pressure in the MEC airspace next to the RW (${\overrightarrow{\widehat{\mathrm{Z}}}}_C$; solid green lines). ${\overrightarrow{\widehat{\mathrm{Z}}}}_{RW}$ is calculated only for the intact model, in terms of the difference between ${\mathbf{p}}_{RW}^F$, the pressure in the cochlea next to the RW, and ${\mathbf{p}}_{\mathit{MEC}}^F$ (solid blue lines). ${\overrightarrow{\mathrm{Z}}}_{\mathit{diff}}$ is calculated in terms of the difference between ${\mathbf{p}}_C^{F,t}$ and ${\mathbf{p}}_{RW}^F$ and is equivalent to ${\overrightarrow{\widehat{\mathrm{Z}}}}_C-{\overrightarrow{\widehat{\mathrm{Z}}}}_{RW}$ under the assumption of incompressible cochlear fluid. The volume velocities of the stapes footplate and RW membrane (${\overrightarrow{\mathbf{U}}}_C^{F,t}$ and ${\overrightarrow{\mathbf{U}}}_{RW}^{F,t}$, respectively) are calculated over the corresponding surfaces.

In general, at low frequencies, a small-volume and thus stiffer MEC results in reduced sound transmission through the ME, a higher ${\overrightarrow{\mathit{Z}}}_C$ magnitude, and a reduced pressure difference across the cochlear partition. These effects can possibly contribute to the very poor low-frequency hearing in mice below 2 kHz (Heffner and Masterton, 1980). Such effects do not occur to the same extent in larger animals like humans and cats, because their MEC volumes are much larger.

As for the effects of the MEC on ME function, in this study we show that opening the MEC, and thus decoupling its impedance from the ME, reduces the stiffness acting medially on the TM, consistent with previous observations (Rosowski et al., 2006). This increases both $V_U^N$ and $V_{FP}^N$ by up to 15 dB at frequencies below 5 kHz, and shifts the ME resonance downward from 11 kHz to 7–9 kHz (Fig. 3, dotted black line). The effects of the MEC become negligible above 20 kHz because the lumped impedance of the closed MEC volume is inversely proportional to frequency, and because standing-wave modes at higher frequencies are negligible, as discussed above. The EC input impedance ${\overrightarrow{\mathit{Z}}}_{EC}$ is also affected by the MEC at frequencies below 20 kHz (Fig. 4). For instance, an open MEC shifts the first minimum and maximum of $\left|{\overrightarrow{\mathit{Z}}}_{EC}\right|$ toward lower frequencies by 1–3 kHz, with a decrease in over 8 dB at frequencies below 7 kHz.

C. The middle-ear cavity could mediate low-frequency vestibular sensation

While the cochlea of the inner ear detects sounds, the otoconial vestibular organs detect gravity and accelerations. Of these, it has been established that the sacculus and utriculus detect linear accelerations. Evidence suggests that the sacculus retains some acoustic sensitivity for low-frequency sounds in cats (McCue and Guinan, 1994) and mice (Jones et al., 2010). Additional input pressure passing through the RW and into the inner ear below 2–3 kHz may also stimulate the sacculus in mice. We hypothesize that this may be facilitated by the relatively small MEC that then produces an alternate path for sound pressure to enter the inner ear through the RW, past the oval window, and ultimately into the vestibule where the sacculus resides.

D. PF effects

In the Dong et al. (2013) experiments, the PF region of the TM was removed, such that the MEC was exposed to the open field. The TM is driven by the pressure gradient across its lateral and medial surfaces, and not just the absolute pressure on its lateral side (e.g., O'Connor and Puria, 2006). By removing the PF, at lower frequencies a largely uniform pressure is presented on the two sides of the TM, since the hole where the PF used to be shunts the pressure gradient across the TM (supplemental Fig. 6).³ This is especially the case at lower frequencies because the wavelengths are relatively long (e.g., at 1 kHz the wavelength is around 340 mm, as compared to an EC length of around 2 mm). In comparison to the intact model, opening the PF decreases the umbo velocity at frequencies below about 6 kHz (Fig. 3). These model calculations are consistent with measurements by Dong et al. (2013), who concluded that “the only significant change in umbo motion was at frequencies below 5 kHz,” and these results of opening the PF are consistent with similar changes observed in gerbils (Dong et al., 2012). The model has a peak near 18 kHz that is not present in the published measurements from a single mouse (Fig. 2). By increasing the structural damping of all components above 10%, this peak could be reduced. However, doing so would further decrease the response at higher frequencies and cause greater discrepancies. Because the published ossicular-chain velocity magnitudes and phases are only available from a single mouse with a different and unknown anatomy, we have not attempted to replicate the fine details of those responses when selecting parameter values for the model.

E. Wideband middle-ear transmission and tympanic-membrane viscoelasticity

Does the ME or the cochlea determine the 86-kHz upper hearing limit of mouse (Heffner and Masterton, 1980)? It has been previously argued that the ME is not the limiting factor in auditory responses and that the limit is instead determined by the auditory nerve fibers near the high-frequency end of the cochlear map (Ruggero and Temchin, 2002). The present stapes-velocity calculations [Fig. 3(a)] predict that there is no sharp decrease in ME transmission up to at least 100 kHz, which is consistent with the claim that mouse hearing is not limited by the ME. This response characteristic is made possible due to the viscoelastic properties of the PT region of the TM, for which the stiffness increases with frequency. For the very simple case of a one-degree-of-freedom mass-spring-dashpot system, a rather sharp resonance (depending on the damping value) occurs at the frequency where the response switches from being stiffness dominant to being mass dominant. On the other hand, if the spring stiffness is no longer constant, but instead varies with frequency, then the result is a broader resonance spread over a wider frequency range. For the case of the frequency-dependent stiffness of the PT, the overlap between the resonances at neighboring frequencies produces a smooth transition from frequency to frequency and prevents a subsequent steep decay of sound transmission to the cochlea.

In this study, the frequency-dependent Young's modulus of the PT begins with a value of 30 MPa at 1 kHz and then exponentially increases to 300 MPa at 100 kHz for the radial direction. Models that have a constant Young's modulus (supplemental Fig. 7)³ generally produce either good low-frequency responses but poor high-frequency responses (e.g., 30 or 95 MPa, the latter being the geometrical mean of 30 and 300), or good high-frequency responses but poor low-frequency responses (e.g., 300 MPa). O'Connor et al. (2017) showed that low or high TM stiffness, respectively, yields more gain at low or high frequencies. Fay et al. (2006) explored the transmission efficiencies of constant-stiffness radial and circumferential collagen fibers in the TM at low and high frequencies, but frequency-dependent variations in the stiffness of those fibers were not explored. Presently, the circumferential and transverse Young's moduli are defined as linear functions of the radial Young's modulus Fay et al. (2006), such that they also vary with frequency. Defining the PT as an orthotropic material with a frequency-dependent Young's modulus (as compared with constant values) produced the closest resemblance to the Dong et al. (2013) experimental data over the 1–60 kHz frequency range.

F. Effects of the OA on sound transmission, ossicular deformations, and rotational axes

Many rodents, shrews, and bats have ossicular-chain configurations that fall into the microtype category, which features a bony protuberance at the base of the manubrium called the OA [Fig. 1(b)]. Fleischer (1978) hypothesized that this mass reduces the resonant frequency of the ossicular chain to improve low-frequency transmission through the ossicles.

The classical view of mammalian ME mechanics is that the malleus and incus rotate together about an anatomical axis (i.e., the axes that passes through the anterior process of the malleus (APM) and short process of the incus) that effectively forms a class-1 lever (axis 1, Fig. 9, 1-kHz panel). However, the OA mass is placed far from this anatomical axis, which increases the rotational moment of inertia around the axis. Since torque increases with angular acceleration, a lower angular moment of inertia in turn increases sound transmission to the cochlea at higher frequencies. To explain this paradox, Fleischer proposed a secondary rotational axis (axis 2, Fig. 9, 1-kHz panel) that passes through the OA and comes into play at higher frequencies to reduce the rotational moment of inertia due to the OA mass as the malleus rotates with respect to this axis.

The existence of axis 2 has been tested experimentally. Saunders and Summers (1982) concluded that the difference in the velocity responses of the umbo and transversal-lamina portion of the malleus at high frequencies (above 20 kHz) is consistent with “the shift in the rotational axes” proposed by Fleischer. Dong et al. (2013) concluded that axis 1 is the primary rotational axis for the 1–60 kHz frequency range tested. However, motion of the incus and malleus above 30 kHz is the result of the summation of rotation about axis 1 and rotation about a second axis that appears to be slightly different from Fleischer's second axis (i.e., axis 2). Fleischer's second axis passes through the OA but is perpendicular to axis 1. In Dong et al., the second axis is said to go through the OA and the center of mass of the MI complex, which forms an acute angle with axis 1 (Dong et al., 2013) (Fig. 9). Lavender et al. (2011) calculated the “principal rotational axes” and showed that they are almost aligned with the two axes that Fleischer proposed.

Fleischer predicted that the transversal lamina “could twist” because it is very thin (like a thin web of I-shaped beams with thicker flanges at the dorsal and ventral portions), but he did not predict the contribution of such deformation on rotational axes. However, Saunders and Summers (1982), Lavender et al. (2011), and Dong et al. (2013) assumed that the ossicles are rigid. Also, in those studies, the slippage in the ossicular-chain joints has been neglected, and it was assumed that the short process of the incus (suspended by the incus posterior ligament) is as rigid as the APM, which is fused to the MEC wall. In the present model, phase changes throughout the ossicles (supplemental Fig. 5),³ as well as animations of their motions (Mm. 2 and 3), show clear signs of deformations and bending of the malleus. Ossicular bending has also been shown in gerbil (de La Rochefoucauld et al., 2010), chinchilla (Ramier et al., 2018), and human (Nakajima et al., 2005).

By calculating the curl of the displacement field, the spatial rotations of the mouse ossicular chain were determined for the first time (Fig. 9). Plotting this field clearly shows that bending of the malleus results in more than one rotational axis being active at the same time, which is inconsistent with rigid-body motion. Rotation primarily with respect to axis 1 continues up through 10 kHz (cf. the alignment of the blue arrows with axis 1). Although the dorsal region of the MI complex continues to rotate with respect to axis 1, a pronounced secondary rotational axis along axis 2 occurs in the ventral region of the malleus around the umbo and OA for frequencies above 10 kHz (cf. the alignment of the blue arrows with axis 2). In summary, the observed ossicular rotations cannot be explained by rigid-body motion, and the flexibility (bending) of the malleus gives rise to multiple spatial vibration modes at each frequency (Fig. 9 and supplemental Fig. 5).³ Thus, Fleischer's hypothesis of ossicular rotations shifting from one axis to another axis is not supported in this study, insofar as both axes could be active simultaneously due to bending. However, his prediction that “the orbicular apophysis regulates the natural frequency of the MI complex around two axes of vibration” is consistent with the findings of this study (cf. blue and red arrows in the dorsal portion of the MI complex around the IMJ region at 60 and 80 kHz).

It has been proposed by Puria and Allen (1991) that the cat eardrum and ossicles form a transmission line that improves the bandwidth of the ME, but at the expense of delaying the passage of sound through it. In their formulation, the ossicular transmission line is formed by a series of mass and stiffness units, in which the malleus mass and a shunt stiffness due to the flexibility of the IMJ are coupled to the incus and a shunt stiffness due to the flexibility of the ISJ, which are then coupled to the stapes mass. Mason (2013) proposed that rodent ossicles may also form a transmission line, which is consistent with the calculated phase delay from the umbo to the stapes [Fig. 8(b) and supplemental Fig. 5].³ The intact-OA model has a steeper phase decrease for the stapes velocity [Fig. 8(b)] and ${\mathit{GME}}^F$ [Fig. 8(d)] in the 10–40 kHz range, resulting in more acoustic delay than the model without the OA. The intact OA also provides a slight boost in transmission over the 7–30 kHz range [Figs. 8(a) and 8(c)]. However, a key difference in the way transmission-line theory applies to the ossicular chains of cat vs mouse is that the flexibility of the cat ossicular chain is due to the joints, while the flexibility of the mouse ossicular chain is due to bending of the malleus.

The differences in the stapes velocity and ME pressure gain due to the removal of the OA are relatively small (below 6 dB), and this possibly explains why mice deficient in the Msx1 homeobox gene that lack an OA were reported to have hearing similar to wild-type mice (Zhang et al., 2003). Zhang et al. (2003) assessed hearing using auditory evoked potentials measured using tones at 4, 8, and 16 kHz in four wild-type controls and four Msx1 mutants. The current results suggest that they would not have measured a change at 4 kHz but might have measured a small change at 8 and 16 kHz. One possibility as to why they were unable to detect a significant difference between the wild-type and OA-knock-out mice groups could be that the number of animals in each group was insufficient to discern the effects of OA removal when the influence of many other inter-animal differences would have been occurring at the same time. In contrast, the differences predicted by our FE models are much easier to detect because everything about the models being compared remains exactly the same except for the removal of the OA [Figs. 8(a) and 8(c)].

G. Probe insertion depth and ear-canal standing waves

In an experimental setup, the insertion depth of the EC measurement probe affects the ${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$ response (Stinson et al., 1982; Voss and Allen, 1994), especially at high frequencies. Figure 4(a) shows that ${\overrightarrow{\mathit{Z}}}_{\mathit{EC}}$ exhibits sharp peaks and notches corresponding to resonance and anti-resonance nodes inside the EC, which depend on the EC length and are due to the presence of standing waves. A sensitivity analysis shows that $\left|{\overrightarrow{Z}}_{EC}\right|$ is significantly affected by E_TM but is less affected by ρ_OSS and only slightly affected by E_SAL [Figs. 6(e), 6(k), and 6(q)]. This suggests that EC-based measurements such as tympanometry should be mostly dominated by the impedance of the TM (Feldman, 1974; Motallebzadeh et al., 2017b).

H. Cochlear input impedance and standing waves

The passive version of the mouse cochlear model, with an open MEC, has an input-impedance magnitude |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| that is almost flat between 1 and 2 TΩ up to 100 kHz, excluding a sharp peak and notch between 70 and 80 kHz (Fig. 4, dotted red line). This is comparable to the 2.3 TΩ for the cochlear input impedance determined from Hemilä et al. (1995) and the 1.5 TΩ reported by Wang et al. (2016). The sharp peak and notch between 70 and 80 kHz are present in all |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| calculations [Figs. 4(a), 6(a), and 9(a)]. For a fluid-filled tube of length L, closed on both ends, with wave speed c, the standing-wave modes in the form of nodes (i.e., points of zero pressure) at one end and antinodes (i.e., points of maximum pressure) at other end occur at frequencies $f_n$, calculated by

$$f_n=\frac{nc}{2L}\hspace{1em}\left(n=1,\hspace{0.17em}3,\hspace{0.17em}5,\hspace{0.17em}\dots \right).$$ (19)

If the mouse cochlea were an ideal tube with two rigid terminations at the oval window and RW, then for the combined length of the scala vestibuli and scala tympani of 11.6 mm (2 × 5.8 mm), the first standing-wave mode would occur at 63 kHz. For the model used in this study, with its compliant oval and round windows, the first standing-wave mode occurs at 76 kHz (Mm. 1), which corresponds to the sharp notch in the cochlear input impedance |${\overrightarrow{\mathit{Z}}}_{\mathit{C}}$| between 70 and 80 kHz [Figs. 4(a), 6(a), and 9(a)].

I. Ideal middle-ear gains

It is well established that one advantage of the tympano-ossicular ME is that it acts as an impedance-matching device between the low-density air in the EC and the high-density cochlear fluid. The ideal ${\mathit{GME}}^F$ is calculated to be 37 dB for the mouse model of Fig. 1, as the product of the TM/footplate area ratio and the malleus/incus length ratio. This ideal assumes a fixed rotational axis for the MI complex along the lines of axis 1 (Fig. 9, 1-kHz panel), with the TM and oval window undergoing piston-like motions. The actual ${\mathit{GME}}^F$ model response reaches a maximum of 34 dB at 10 kHz [Fig. 5(a), solid red line], which is 3 dB lower than this ideal gain (dotted red line). Differences between the ideal and measured maximum forward gains range from 3 to 8 dB (6 dB on average) for other species studied, such as cat, chinchilla, guinea pig, and human (Puria et al., 1997). Further deviations from the ideal gain above and below the ME-resonance frequency can be attributed to force losses due to flexible attachments and joints (Guinan and Peake, 1967; Goode, 1994; Puria et al., 1997). Standing waves inside the EC give rise to two sharp |${\mathit{GME}}^F$| peaks at 39 and 78 kHz [Fig. 5(a), solid red line].

The ideal ${\mathit{GME}}^R$ is the reciprocal of the ideal ${\mathit{GME}}^F$ and therefore has a magnitude of −37 dB. Similar to ${\mathit{GME}}^F$, the actual ${\mathit{GME}}^R$ model response for the open-EC case [Fig. 5(c), solid blue line] is shifted below the ideal reverse gain by 9 dB near 10–13 kHz, around the ME resonance, indicating similar losses through the ME in the reverse direction. However, for frequencies above the ME resonance, the |${\mathit{GME}}^F$| slope of −6 dB/octave [Fig. 5(a), solid red line] is very different from the rather flat |${\mathit{GME}}^R$| slope in the 10–60 kHz range that is followed by a steep drop exceeding −12 dB/octave [Fig. 5(c), solid blue line]. This indicates that forward and reverse transmission losses in the mouse ME are not symmetrical, which had already been shown to be the case for the human ME (Puria, 2003). The phases of ${\mathit{GME}}^F$ and ${\mathit{GME}}^R$ are similar, with few notable differences.

The round-trip ME pressure gain ${\mathit{GME}}^{RT}$ stays below the 0-dB ideal and drops rapidly above 60 kHz with a slope of –16 dB/octave [Fig. 5(e)]. Attenuations below and above the 11-kHz ME resonance can be as much as −88 dB at 1 kHz and −65 dB at 100 kHz for the open-EC case.

J. Limitations of this study and future directions

Due to the significant amount of time required to develop FE models, this study is limited to the anatomy of a single ear. While there is general overall agreement between the model response and measurements of ossicular motion from an individual mouse ear from a different specimen, there are also some differences, particularly at the lowest and highest frequencies tested (Fig. 2). Perhaps the main reason why we were not able to further improve the fit between our model and the measurements is because the two are from different specimens. Ideally, the same specimen that provides the μCT-scanned anatomy for the model should also be used to obtain physiological measurements for model calibration and verification. Comparing the responses of this model to others developed using different μCT-based anatomies would allow functional sensitivity analyses to be performed based on the variability of anatomical features. The results and conclusions of this study can be extrapolated to other mouse anatomies when typical variations in the ME anatomy are considered.

In the absence of any known material properties for the mouse ME, the initial material-property values were taken from a human ME model (O'Connor et al., 2017) and then refined to calibrate the model to the Dong et al. (2013) vibration measurements of the mouse ossicles. This was an iterative process in which the ME parameters were systematically varied to visually optimize closeness to the ossicular-motion data over a broad range of frequencies spanning 1–60 kHz. This constituted the most challenging and time-consuming part of the model-development process. All but one of the final material properties of the mouse ME structures reported in Table II are independent of frequency. The exception to this is the Young's modulus of the PT region of the TM. Generally speaking, biological tissues are inherently frequency-dependent materials. The mouse and other animals with microtype MEs have an especially high upper frequency limit of hearing, such that frequency-dependent stiffness properties may be required (Charlebois et al., 2013; Motallebzadeh et al., 2013). In the present ME model, the frequency dependence of the Young's modulus of the PT region of the TM improves the fit to the ossicular-chain vibration measurements, but some discrepancies remain. For one, the mechanics of the ossicular joints in the current model are oversimplified. In the future, the synovial-fluid mechanics of the joints should be considered (Soleimani et al., 2020). Another area for improvement would be to incorporate frequency-dependent Young's moduli for the remaining ME structures (e.g., Cheng and Gan, 2008; Motallebzadeh et al., 2015).

It has been shown that the two extreme cases of lateral EC termination (i.e., open and closed) affect ${\mathit{GME}}^R$ [and thus ${\mathit{GME}}^{RT}$; Figs. 5(c)–5(f)]. This emphasizes the importance of the termination impedance of the probe used to make experimental measurements (Puria et al., 1997; Dong and Olson, 2006). Incorporating realistic probe impedances into the model would help to improve future modeling interpretations of measurements, especially with regard to OAE data.

The reported sensitivity analyses are limited to just a few material properties. Additional manipulations, such as those related to pathology, ossicular fusion, and dislocations, should be simulated in the model and compared to experimentally measured conductive-hearing-loss measurements (Nakajima et al., 2005; Prieve et al., 2013) as further tests of the present model formulation.