How the Human Cochlea Moves: Biomechanical Modeling of a Wide, Layered Osseous Spiral Lamina

Abstract

Purpose

The human cochlear partition (CP) at the high-frequency region features a radially wide, layered osseous spiral lamina (OSL) and a soft-tissue bridge connecting it to the basilar membrane (BM). The OSL consists of two thin bony plates separated by a cavernous space that serves as a conduit for auditory nerve fibers. We used a finite element model with two fluid chambers, incorporating novel implementations of the CP features, to study the human cochlea. Model results were compared with experimental measurements of CP motion.

Methods

Model geometrical and material properties either came from the literature or were tuned to produce a frequency-place map for the passive human cochlea and measurements of the CP velocity normalized to the stapes velocity in human cadaver temporal bones. The best frequency (BF) for the experimental measurements’ seven specimens ranged from 9.5 to 14.4 kHz.

Results

The model motion results of the basal CP had similar trends to the experimentally measured results in both magnitude and phase. Sensitivity analysis studies changing material-property parameters of the nerve-fiber layer between the OSL plates produced small changes and showed negligible stress along a neutral axis compared to the outer OSL plates.

Conclusion

Our model, which incorporated human cochlear structures like the wide OSL with a layer sandwiched between the plates for auditory nerve fibers, successfully simulated CP motion, exhibiting trends that closely resembled experimental data. The relatively wide three-layered OSL structure’s neutral axis may serve as a stress-free conduit for the passage of auditory nerve fibers.

Introduction

Humans have a significantly larger skull than laboratory rodents such as mice and gerbils, and their cochlear volume and cochlear duct cross sections are also significantly larger [1]. It follows that the cochlear partition (CP) is wider in larger animals. Due to these factors, the anatomy of the basal CP in humans is larger than in laboratory animals [1]. A cross-sectional view of the human CP from histology is shown in Fig. 1A. From the medial end near the modiolus, two thin bony plates of the osseous spiral lamina (OSL) fan out laterally to the soft-tissue bridge. Auditory nerve fibers (ANFs) run from inner and outer hair cells between the bony plates of the OSL to the spiral ganglion cell bodies at the modiolus. On the lateral end, from the spiral ligament (SL), collagen fibers project medially through the basilar membrane (BM), through the bridge, and insert onto the lateral edges of the OSL plates [2]. Altogether, this OSL-bridge-BM structure forms a CP that separates the cochlear fluid volume into the ducts of scala vestibuli and media on one side of the partition and the scala tympani on the other. The limbus, with its roughly triangular profile, spans a significant portion of the OSL and extends into part of the bridge. Notably, the organ of Corti, which houses the sensory hair cells, is situated atop the BM and is located near the SL.

Fig. 1.

A Labeled histological section of the human cochlear partition towards the base. B Simplified graphical representation of the cochlear partition related to present model geometry. ANFs, auditory nerve fibers; BM, basilar membrane; OoC, organ of Corti; OSL, osseous spiral lamina; SL, spiral ligament; ST, scala tympani; SV, scala vestibuli; tymp. plate, tympanic plate; vest. plate, vestibular plate

The OSL has a vestibular plate and a tympanic plate with a complex space in between. Throughout this space, thin bony pillars connect the vestibular and tympanic plates in a randomly distributed manner [2–5]. Some bony projections from the plates are not connecting pillars; rather, they resemble cave-like “stalactites” or “stalagmites.” Between the two OSL plates, ANFs course through the bony cavernous structures radially from the organ of Corti sensory hair cell synapses, through the bridge, and towards Rosenthal’s canal and into the modiolus. The space between the two OSL plates is thus a composite of bone, soft tissue, and fluid. Motion measurements of the CP in human cadaveric specimens have been measured at the 2 kHz frequency [6] and 9- to 15-kHz place [7] with motion increasing from the modiolus to the lateral aspect of the OSL, reaching a peak and then decreasing approaching the SL. These measurements of the human OSL indicate that the OSL appears to have more motion [7, 8], in comparison to measurements in laboratory animals [9, 10].

In addition to measurements, there has been work towards understanding the OSL through modeling efforts. Allaire et al. [11] predicted that the human OSL would be mobile based on its cantilever structure and included it in an analytical model. Steele [12] considered the flexibility of the OSL to be important for high frequencies based on Rhode’s measurements in the 7-kHz region in the squirrel monkey [9]. Raufer et al. [7] created a single-beam model that was consistent with experimental measurements of the radial profile of the CP.

Several finite element models of the human cochlea have been made. Most models are in the form of a box, i.e., a tapered rectangular approximation of an uncoiled cochlea, or cochlear spiral, and only account for the motion of the BM with rigid constraints on the medial and lateral ends, and sometimes soft tissues of the organ of Corti as well (e.g., [13–15]). Most human computational models have not included the OSL because they assumed that the OSL was rigid and thus has no significant motion. A few modeling studies have included an OSL. Koike et al. [16] added a single-layered OSL with width varying from base to apex and did not explicitly constrain its motion with fixation. However, in this model, the Young’s modulus of the OSL was made to be exceedingly stiff and thus effectively fixed with the intention to make it move very little. Borkowski et al. [17] included a single-layered mobile OSL with varying width and Young’s modulus from base to apex, and favorably compared results to Stenfelt et al. [6]. Generally, computational models of the human cochlea have not considered the human structural anatomy, such as the complex multilayered structure of the OSL with internal soft-tissue space for nerve bundles, the soft-tissue bridge, or the placement of the organ of Corti and limbus relative to these structures.

The aim of this study was to model the human cochlea with a more realistic human CP anatomy. The human cochlea was modeled as a 3D fluid-filled cochlear duct, incorporating both anatomical findings and measurements of CP motion. The modeled CP, which emphasizes the mechanics of the layered OSL, is shown in Fig. 1B and compared on the same scale with histological imaging in Fig. 1A. Note that in the model, the space for ANFs between the OSL plates (Fig. 1A) is referred to as the “middle layer” of the OSL (Fig. 1B). Multiple sources for anatomical measurements and material properties were used, and thus the model is a composite of many ears, not representing any single specimen. This model was tuned to be consistent with the ½-octive shifted Greenwood cochlear frequency-place map [18] representing the passive cochlea. Model responses were compared to our OCT-derived measurements of the CP displacements from the basal high-frequency region [19] and with measurements of the CP motion using laser Doppler vibrometry (LDV) in fresh intact human cadaveric temporal bone specimens.

Methods

Model Development

Geometry

A tapered-box cochlear model with a partition separating the cochlear duct into two fluid chambers, one representing the scala vestibuli and scala media combined without Reissner’s membrane separating them, and the other representing scala tympani, was developed (Fig. 2A). A two-chambered box structure is a common simplification for cochlear models that approximates the response of the spiral cochlea while simplifying mathematical analysis [20, 21]. The partition, shown in Fig. 2B, included an OSL, a “BM-bridge” region (combination of BM and bridge labeled in Figs. 1B and 2B), limbus, and organ of Corti. Since the collagen fibers of the BM continue through the soft-tissue bridge region, both the bridge and BM are collectively labeled as “BM bridge” in the present model. A helicotrema region was included at the apical end of the model by tapering off the partition to the modiolar side of the cochlea, leaving a triangular fluid connection between the two fluid chambers. Details about the geometrical dimensions of each model component follow below.

Fig. 2.

Model geometry and measurements with a cross-section of the cochlear partition at the base and boundary conditions. A Full-length box model geometry. The model’s 1.8 mm location from the base is labeled. B Scaled-up view of the basal-most partition with radial boundary conditions. OoC, organ of Corti; OSL, osseous spiral lamina; RWM, round window membrane; SM, scala media; ST, scala tympani; SV, scala vestibuli; tymp. plate, tympanic plate; vest. plate, vestibular plate

Scalar cross-sectional areas at the base and apex were reported by Thorne et al. [22] and Wysocki [23]; however, since the model excludes the vestibule of the inner ear to instead form a geometrically simpler cochlear box, we needed to make accommodations for the windows. Ultimately the widths of both scalae were 4 mm throughout the length. The heights of both scalae at the base were 3 mm, and at the apex, 0.15 mm for scala vestibuli and 0.45 mm for scala tympani.

The model partition was given a longitudinal length of 32 mm [24]. The radial widths of the OSL and BM-bridge as a function of longitudinal location were taken from histological measurements [2]. The BM and bridge had approximately equal widths along the length of the cochlea, both increasing from approximately 0.1 mm at the base to 0.5 mm at the apex. The OSL, on the other hand, has the opposite relation. The width decreased from 1.1 mm at the base to 0.3 mm at the apex. The total CP width (i.e., the sum of the OSL, bridge, and BM) remained nearly constant at about 1.3 mm along the length of the cochlea.

The transverse height (i.e., thickness) of the total OSL (including the two plates and space for ANFs between the plates) remained constant along the length of the cochlea. We looked at histological reconstructions to measure OSL plate thicknesses and total OSL thickness. On average, each plate had a thickness of approximately 9 μm but was variable within a range of 6–16 μm in both radial and longitudinal directions. The model used individual plate thicknesses of 9 μm and a middle layer thickness (representing where the ANFs run) of 22 μm, totaling 40 μm in height (Fig. 2).

The BM-bridge thickness varied from base to apex in the model, which, along with the width, sets the stiffness gradient that produced the cochlear place-frequency map. Even though Bhatt et al. [25] measured the human BM thickness, such histological measurements could have inaccuracies. Alterations in dimensions of cochlear tissue occur by way of fixation, dehydration, or mechanical distortion from the cutting process, and such alterations differ across different types of tissue [26–28]. Thus, the BM-bridge thickness as a function of longitudinal position was the one geometrical parameter that was tuned to experimental data (see “Model Tuning to Experimental Data” section). A piecewise linear function was ultimately used to describe the thickness (Fig. 3). The BM-bridge thickness was set to 4 μm for the basal-most 8 mm, then decreased linearly to 0.45 μm at the helicotrema. Our thickness estimate is based on the assumption that the thickness of the bridge is similar to that of the BM; however, the collagen fibers running through the BM bridge likely dominate the mechanics. Average thickness data at four longitudinal locations from Bhatt et al. [25] is also plotted in Fig. 3, showing that our estimate is within a factor of 2–3 of the experimentally measured thickness range.

Fig. 3.

BM thickness in the model as a function of longitudinal place from base to apex

Oval window area (3.64 mm²) and round window area (1.92 mm²) were calculated from an average of measurements from several literature sources [29–33]. Round window membrane thickness was uniform and approximated as 70 μm [34].

Material Properties

The majority of material properties for the human cochlear structures have not been measured. Starting with parameters previously used by other modelers, the present model relied on sensitivity analysis to select key parameters that were then adjusted to produce results closest to the experimental data (see section “Model Tuning to Experimental Data” for more details on parameter sensitivity analysis). A summary of the material parameters is provided in Table 1. Tuned parameters are denoted with an asterisk.

Table 1.

Material parameters used for the finite element model 

	Poisson’s ratio	Density (kg/m3)	Young’s modulus (MPa)	Shear modulus (MPa)	Damping
OSL bone	0.3	2,000	10,000*	–	0.1
OSL inner layer	0.485	1,100	3000*	–	0.1
Basilar membrane	0.485 (RT), 0 (RL), 0 (TL)	1,100	100 (R)*, 0.1 (L), 0.1 (T)	33.3 (R), 0.033 (L), 0.033 (T)	0.1
Organ of Corti	0.485	1,100	10*	–	0.1
Limbus	0.485	1,100	10*	–	0.1
Round window membrane	0.485	1,100	1.4	–	0.8

L, longitudinal; R, radial; T, transverse

Young’s modulus values varied among the modeled tissues: 10 GPa for bony OSL plates, 3 GPa for the cavernous OSL middle layer, 10 MPa for the organ of Corti and limbus, and 1.4 MPa for the round window membrane.

The middle layer between the OSL plates, representing nerve fibers and bony pillars, was treated as having a composite Young’s modulus. Although not explicitly stated in Bom Braga et al. [4], we estimated that the bony pillars could constitute 30% of the volume of the middle layer portion of the OSL. The OSL middle layer was therefore given a composite Young’s modulus value using the rule of mixtures for composite materials:

$$E_{\text{middle}}={\mathit{vf}}_{\text{bone}}{E}_{\text{bone}}+{\mathit{vf}}_{\text{nerve}}{E}_{\text{nerve}}$$ 1

where E_bone is the Young’s modulus of bone given above (10 GPa), vf_bone is the volume fraction of bone at 0.3, E_nerve is the Young’s modulus of nerve tissue taken as 10 MPa, and vf_nerve is the volume fraction of nerve tissue at 0.7. The composite middle layer Young’s modulus is ~ 3 GPa. Note that because the bone Young’s modulus is much greater than that of the nerve tissue, Eq. (1) is approximately reduced to vf_bone*E_bone.

To investigate the contribution of the stiffness of the middle layer, we performed a sensitivity analysis by modifying the baseline model's middle layer material. For the purposes of this sensitivity study, the baseline model is referred to as the “composite middle layer” model. Other variations of the model were investigated: in one model, the middle layer was given the same material as the bony plates (i.e., Young’s modulus, Poisson’s ratio, and density were made that of bone), effectively making the three OSL layers into a single thick bony plate. This model instance will be referred to here as the “bony middle layer” model. In another model, the Young’s modulus of the middle layer was reduced to 270 MPa, or 2.7% of the baseline value, which is in the range of biological soft tissues [35]. This instance will be referred to as the “soft-tissue middle layer” model.1

The BM bridge was treated as an orthotropic material, where the radial direction was stiffest since collagen fibers within the BM bridge are primarily oriented radially. There is no data on how collagen fiber volume fraction changes from the base to the apex in the human CP. The Young’s modulus value of the radial fiber direction was chosen to be 100 MPa. This value did not change from base to apex. The model relied on the BM thickness and width as a function of longitudinal position to form the stiffness gradient in the model that produced the place-frequency map shown in Fig. 4. The longitudinal and transverse direction values (0.1 MPa) were lower than the radial direction by three orders of magnitude, consistent with a very soft ground substance that reduces longitudinal coupling [36].

Fig. 4.

Tuned-model frequency-place map vs. shifted-Greenwood cochlear frequency map [46]. The Greenwood map was shifted basally by a half octave to mimic a passive cochlea

While we do not know the Young’s modulus of the BM-bridge for certain, we know that the BM bridge is one of the stiffest soft tissues in the partition due to the presence of collagen fibers [37–39], which are typically in the range of 1–20 GPa [40, 41]. The overall stiffness of the BM bridge depends on the volume fraction of collagen fibers. Although we did not specify volume fraction in this model, we can calculate volume fraction from our chosen material property values using the rule of mixtures for composite materials:

$${\mathit{vf}}_{\text{fiber}}=\frac{E_{\text{composite}}-E_{\text{matrix}}}{E_{\text{fiber}}-E_{\text{matrix}}}$$ 2

Here, vf_fiber is the fiber volume fraction (ratio) of collagen, and E is Young’s modulus. Given E_composite of 100 MPa, E_matrix of 0.1 MPa (that of the BM-bridge transverse and longitudinal directions), and E_fiber within the range of values for collagen, the volume fraction is estimated to be 0.5–10%. This is close to the range of volume fractions from base to apex calculated by Kapuria et al. [42] for guinea pig (1–8%) and gerbil (0.8–16%).

Since the BM bridge was orthotropic, the shear modulus was also specified in the model. The relation between shear modulus and the other mechanical properties for an orthotropic material is complex and not well understood; therefore, for simplicity, each individual orthogonal direction was given a shear modulus value equal to one-third of the respective Young’s modulus value, as has been done in other modeling studies (e.g., [43]).

Structural damping represents frictional effects in materials and is distinct from viscous losses in fluids. For all model components, with the exception of the round window membrane, structural damping was set to 0.1 (e.g., [43]). Damping of the round window membrane was set to 0.8. The higher damping at the round window was chosen to compensate for the radiation impedance of the middle ear cavity.

The density of bone was set to 2000 kg/m³, and the density of all soft tissue components was set to 1100 kg/m³, similar to other auditory models [44, 45]. Poisson’s ratio of bone (the OSL plates) was set to 0.3, a typical value used in modeling bone (e.g., [46, 47]), and all soft tissue components were assumed close to incompressible at 0.485 (e.g., [43]). Cochlear fluid was modeled as water for simplicity.

Physics and Boundary Conditions

Three aspects of physics were combined in this model: thermoacoustics for the fluid domains (i.e., scala vestibuli and scala tympani); solid mechanics for the two OSL plates, middle ANF layer, limbus, and organ of Corti; and shell mechanics for the BM bridge and round window membrane. Thermoacoustics is used for its inclusion of fluid viscosity. Shell mechanics is a computationally efficient approximation to model structures like the BM-bridge and round window membrane that are transversely thin in comparison to other dimensions.

The partition was given a fixed boundary condition on the medial side of the OSL. On the lateral end of the BM, the applied boundary was given a pinned condition corresponding to fixed translation and free rotation (Fig. 2B).

The normal vectors of the BM-bridge shell project towards the scala tympani so that the top of the BM bridge is level with the top of the OSL vestibular plate (see Fig. 1B).

Input velocity stimulus at the oval window was constant across frequencies at 4 μm/s. All model results, as with experimental results, are reported as velocity relative to the stapes input velocity. The model is linear, and measurements were made at levels where linearity is maintained (less than 110 dB SPL).

Finite Element Model Details

The model was created using COMSOL Multiphysics (www.comsol.com) version 6.1. The final mesh was composed of 469,919 elements consisting of 296,946 tetrahedra, 31,925 pyramids, 57,158 prisms, and 83,890 hexahedra. There were 3,962,263 degrees of freedom in total. The computer used was a Dell desktop running a Linux operating system with two 20-core Intel Xeon Processors, 40 cores in total, and 512 GB of RAM. With this hardware, the simulation time was approximately 30 min per frequency. The frequencies used in the model were between 0.2 and 14 kHz at one-third octave intervals, with additional frequencies added after 10 kHz. In total, the model was solved at 18 frequencies.

OCT Measurement Methods

CP motion in response to sound presented to the ear canal was measured while visualizing structures with OCT vibrometry through the intact round window membrane on a fresh cadaveric human specimen (16 h postmortem). Multiple radial location measurements near the scala tympani surface of the CP were made from an approximately transverse direction. Cross-sectional imaging and vibrometry measurements of the CP were made using a spectral-domain OCT system with a 900-nm center wavelength and A line-scan camera frame rate of 46 kHz (GAN620C1, Thorlabs, Germany). The imaging axial resolution was 2.23 µm (in water) and the lateral resolution approximately 8 µm, using a 36 mm, 0.055NA, × 2 objective lens. Custom-built LabVIEW-based software (VibOCT versions 2.1.5) recorded images and made vibration measurements. SyncAv (version 0.47) generated pure-tone sequences from 0.1 to 21 kHz, ranging from ~ 80 to 110 dB in the ear canal, where linearity was confirmed. The measurement noise floor was 0.3 nm or better above 2 kHz, and an SNR criterion of 6 dB was used for all measurements reported. Stapes velocity was also measured with laser Doppler vibrometry (CLV-3D, Polytec, Germany). OCT displacement measurements of the CP were converted to velocity V_CP within the CP and referenced to the stapes velocity V_st to calculate the reported V_CP/V_st ratio.

LDV Measurement Methods

The details of the LDV measurements are described in greater detail in Raufer et al. (2019). Briefly, they measured the stapes velocity (V_st), scala vestibuli pressure (P_SV), and, similar to the OCT measurements, obtained CP velocity V_CP at radial locations including the OSL-bridge and BM-bridge junctions. One difference between the OCT and LDV measurements of the CP is that for the OCT measurements, the round window membrane was kept intact, while in Raufer et al. [7] it was removed for access to the CP surface. They reported V_CP/P_SV ((mm/s)/Pa) from six temporal bones. For comparison to our model calculations of V_CP/V_st (dimensionless), we calculated V_CP/V_st for the same six temporal bone specimens. Measurement locations for the six specimens using LDV (specimens 12–17) and 1 specimen using OCT (specimen 44) were all different. Their BF ranged from 9.5 to 14.4 kHz. For ease of comparison across temporal bones, we normalized the responses by their respective BF such that the results are plotted on a normalized frequency axis f/BF.

Model Tuning to Experimental Data

The parameters chosen for model tuning were those that were not well known in the literature or those that produced significant changes in the model response during our initial single-parameter sensitivity studies. The tuned parameters were BM-bridge thickness as a function of length, and Young’s moduli of the three OSL layers, BM-bridge, organ of Corti, and limbus.

To quantitatively assess how well the model performed with parameter variations, we calculated the difference between the model and respective experimental measurements: the frequency-place map, frequency response normalized to oval window input, and radial profile of motion relative to stapes input. By randomizing unknown material properties within physiological limits, we converged on model parameters that produced the closest match to the measurements, i.e., the minimization of differences between model and data for all measurement types simultaneously. Minimization criteria were partly quantitative in determining the difference between model and data response values at each frequency, and partly qualitative in looking at the difference in response shapes. The best set of parameters used in the model satisfied the minimization criteria, but is not necessarily unique.

A hallmark of hearing is the tonotopic organization of sound, which is found even in the passive cochlea. High frequencies have their best response at the base, and low frequencies in the apex, and this was succinctly described by Greenwood (1990) [18] for multiple species, including humans. The Greenwood place-frequency map developed from psychoacoustic measurements corresponds to a living cochlea, which, when shifted by about ½-octave to lower frequencies, represents passive response seen in cadaveric specimens [48]. The model was tuned to this shifted Greenwood map.

Both the normalized-frequency (f/BF) response and radial (spatial) profile of motion were compared between the model data and experimental vibrometry measurements. Two key anatomical locations, the OSL-bridge junction and the BM-bridge junction (these junctions are labeled in Fig. 2B), were used to compare frequency responses between experimental and model results. These junction points will be referred to as the OSL-bridge and BM-bridge, respectively, in the discussion of results going forward. Points along the scala tympani surface of the CP were used to compare radial motion responses.

Model and Measurement Reference Frames

The reference frames used for model and experimental measurements with OCT and LDV differed, and this section describes how we reconciled those reference frames, which is particularly important for phase calculations. The phase of the measurements has traditionally been defined such that a positive input at the stapes corresponds to a downward motion of the CP at least on the basal end and relatively low frequencies [49]. However, the model axes have a right-hand coordinate system such that an inward displacement of the stapes in the longitudinal x direction corresponds to negative z-direction transverse displacement (Fig. 2A). Thus, for an inward stapes motion, the phase of the model was chosen to start at + 1/2 cycles at low frequencies.2

Both stapes and CP measurements were measured with LDV. For this setup, an inward stapes displacement corresponded to a CP displacement that differed by + 1/2 cycles.

The CP displacement from specimen 44 was measured using OCT; however, the stapes velocity measured for reference was performed using LDV. The OCT displacement measurement was converted to velocity by multiplication with $j\omega$, and the ratio (V_CP/V_st) was calculated to obtain the CP velocity referenced to stapes velocity. Because the sensitivity of the LDV and OCT had opposite phases, the measured ratio (V_CP/V_st) was adjusted by + 1/2 cycles.

All phases were unwrapped and then adjusted by integer multiples of cycles so that they lined up at the low frequencies (f/BF < 0.2). The phases for specimens 12, 14, and 17 were further decreased by a cycle for f/BF above ~ 0.3 so that their phases lined up with the other measurements near f/BF ~ 1. With these phase adjustments, all phases could then be compared with the same reference frame.

Results

Baseline Model Tuning

As detailed in the “Methods” section, the baseline model was constructed from multiple sources for the geometrical representations of the anatomy and material properties. As such, the resulting finite element model results do not represent measurements from any single specimen but rather a composite representing multiple ears.

Cochlear Map

As shown in Fig. 4, the FE model captured the tonotopic organization of sound for the passive cochlea. Inputting tone frequencies to the model produced a peak response longitudinally within approximately ± 1 mm of the locations expected from the Greenwood place-frequency map corresponding to the passive cochlea. The Greenwood map is a fundamental characteristic of cochlear physiology, and Fig. 4 demonstrates that model anatomical and material parameters globally capture its essence.

OSL and BM Frequency Responses

Figure 5 plots the magnitude and phase of the CP motion measured on the scala tympani surface at the OSL-bridge junction (A and B) and at the BM-bridge junction (C and D) with respect to normalized frequency. The experimentally measured and model motion ratio is the velocity ratio (V_CP/V_st), and the frequency is normalized with respect to the best frequency BF (f/BF). The magnitude ratios are dimensionless, while the phases are in cycles. The model results were taken at 1.8 mm from the base, corresponding to approximately the 11 kHz BF location based on the Greenwood place-frequency map [18].

Fig. 5.

The left column shows OSL-bridge responses while the right column shows BM-bridge responses (collectively V_CP) normalized by stapes velocity V_st. The top panel shows V_CP/V_st magnitudes (dimensionless), while the bottom panel shows phase (cycles). Comparisons of V_CP/V_st to model calculations, with OCT measurement from one ear #44 (dashed lines), and for the six other specimens using LDV (thin colored lines). Because the measurement locations for the model and measurements differed, their BF ranged from 9.5 to 14.4 kHz. For ease of comparison, we normalized the responses by their respective BF such that the results are plotted on a normalized frequency axis f/BF that ranged from 0.018 to 1.3

For f/BF below ~ 0.8, the BM-bridge and OSL-bridge magnitude curves of the model and measurements both decrease with a slope of about − 6 dB/oct, which is expected for a stiffness-dominated response. Around BF (f/BF ~ 0.8 to 1.2 range), the measured BM-bridge data is variable, with some ears not very well tuned, while others, such as specimen 14 (red line in Fig. 5), are reasonably tuned with a steep roll off in magnitude above BF. The response from the model is more sharply tuned than even this tuned measurement. The OSL-bridge measurements are similarly not well tuned, except specimen 14, which demonstrated some degree of tuning. The model results for OSL bridge were flatter near BF than BM-bridge; however, like BM-bridge, the roll-off above BF was steeper than for specimen 14.

For the model, both BM and OSL phases were similar and decreased approximately from + 0.35 cycles at f/BF = 0.02 to − 0.2 cycles at f/BF = 0.9 (linearly when viewed on a linear frequency axis), above which the phases decreased faster and approximately quadratically. The linear phase response is consistent with the long-wavelength region, and the quadratic phase response is consistent with the dispersive short-wavelength region of the traveling wave [12, 50]. While variable, the phases of the measurements also show similar trends. There were phase unwrapping issues for f/BF between 0.3 and 0.6 for specimens 12, 14, and 17, which we corrected to ensure consistency with other measurements near BF.

These comparisons suggest that fitting to any single specimen is unlikely to be feasible, as the model anatomy itself does not accurately represent any one specimen. Nevertheless, the model results have overall similar trends with measurements from the seven temporal bones.

Model Wavenumber Frequency Relationship

For the model, we also had the ability to compute the longitudinal spatial response of the CP $V_{\text{CP}}\left(f_0,,,x\right)$ for a given input tone frequency $f_0$. From the phase ${\varphi }_{V_{\text{CP}}}\left(f_0,,,x\right)$, we calculated the real part of the wavenumber $\mathcal{R}e\left(k,,,f_0\right)=-d{\varphi }_{V_{\text{CP}}}/dx$ at each frequency $f_0$, to determine the wavenumber frequency relationship (WFR) of the model at the basal BF location of 1.8 mm. The results (not shown) indicated that the WFR increased linearly with frequency to 0.2 cycles/mm at f/BF ~ 0.45 (5 kHz) and grew more steeply to 0.8 cycles/mm at about f/BF ~ 1.1 (12 kHz). The linear increase in WFR is consistent with the long-wave region of the traveling wave, while the steep increase in WFR is consistent with the dispersive short-wave region of the traveling wave [12, 50].

We do not have measurements of the human CP as a function of longitudinal location. However, since the phase of the model and measurements were similar (Fig. 5B, D), it can be expected that WFR across the measurements, while variable, will be similar to the model results. Future studies should measure the WFR by making frequency response measurements at nearby locations (e.g., [51]).

OSL and BM Radial Profile

Another approach to analyzing CP mechanics is to look at the radial profile of transverse motion for a fixed frequency. Fig. 6 shows the model CP transverse motion at three frequencies normalized to stapes input motion plotted against radial position. The radial positions ranged from near the medial end of the CP, near the modiolus, extending approximately 1.3 mm radially towards the spiral ligament. The LDV and OCT measurements were similarly plotted from approximately 0.4–1.3 mm radial locations. Measurements from the more medial locations were not possible due to limited surgical access to the CP. The three plots are for three different normalized frequencies f/BF: (A, B) at BF (f/BF = 1), (C, D) about ½-octave below BF (f/BF = 0.73), and (E, F) and far below BF (f/BF = 0.36). The LDV data were smoothed using the MATLAB smooth function with the Savitzky-Golay method, with a window length of 7.

Fig. 6.

Comparison of the model and experimental data for transverse motion referenced to the input motion of the stapes, plotted against radial place. A, B Both experimental and model data lines correspond to their respective best frequencies (f/BF = 1); C, D Model and measured responses at normalized frequency of 0.73. E, F Model and measured responses at a normalized frequency of 0.36. Note the change in y-axis range for the magnitude ratios. The solid black lines are from the model, the dashed line is from OCT measurements, while the thin colored lines are from LDV measurements

At BF (f/BF = 1), for both model and experimental measurements, the relative CP motion increased in magnitude with radial position from the medial end to the OSL-bridge junction (Fig. 6A). Then further laterally, past the limbus, the motion increased rapidly, reached a peak ratio-metric gain of 40–45 near the BM-bridge junction and then decreased. For the model, the ratio between the BM peak and OSL-bridge motion was about 5.4 (15 dB). For the measurements, this ratio varied across specimens from about a factor of 2 (6 dB, specimen 15) and up to 9 (19 dB, specimen 44).

More specifically, the peak occurred directly at the BM-bridge junction in the model (also where the foot of the inner pillar cells would have been), while for the experimental measurements, the peak occurred more laterally at the BM. This subtle difference in peak radial locations can possibly be attributed to the absence of relatively stiff pillar cells and the tunnel of Corti in the model, which were likely present in the measured temporal bone. The tunnel of Corti can stiffen the BM, causing the peak to occur more around the outer pillar [9]. Far below the BF (f/BF ~ 0.36), the radial profiles of the magnitudes between model and experiment were mostly similar, with a peak gain of about 6 (Fig. 6E).

At about ½-oct below BF (f/BF ~ 0.73), at the OSL portion of the CP, the model and measurements were similar. But in the BM-bridge region, the magnitude peak of the model was only half that of the experiment (Fig. 6C). Fig 6 and Raufer et al. [7] showed that there is significant variability in the radial profile of the measurements, and these differences could just be due to variability across ears.

The measured experimental phases were nearly flat with respect to radial position region for all frequencies tested, although in the measurements there was a slight “kink” in the phase near the OSL-bridge junction (Fig. 6B, 6D, and 6 F), but also in the magnitudes (Fig. 6A, 6 C, and 6E). The model phase responses were also nearly flat and close to the measured data for the lower frequencies (f/BF ~ 0.73 and 0.36), but with differences up to about 1/4-cycle at BF (Fig. 6B). Near BF (f/BF ~ 1), the model shows a kink in the phase near the OSL-bridge junction similar to the experimental data.

Near BF (f/BF ~ 1), a decrease in phase of ¼-cycle is observed between the OSL-bridge and BM-bridge junctions of our OCT measurements (specimen 44). Similar phase changes were also present in the radial motion measurements with LDV when the ratio between BM peak and OSL-bridge motion was high (Fig. 6). Overall, the radial response of the composite FE model is consistent with measurements across several specimens.

The Layered OSL

A stated goal of this study was to explore the role of the OSL in the overall CP mechanics, and in particular, the role of the “middle layer” where the auditory neurons traverse between the thin bony vestibular and tympanic plates. Towards this end, all three layers were modeled as solid elements. To understand the effect of varying the material properties of the middle ANF layer, we computed the model with three different middle layer Young’s modulus values that varied from the lowest to highest by a factor of 30: (1) With baseline value of 3 GPa representing a somewhat cavernous middle layer labeled “composite middle layer”; (2) A “soft-tissue middle layer” with a Young’s modulus of 0.27 GPa representing a reduction, compared to (1), of the bony elements that connect the plates; and for comparison (3) A “bony middle layer” with a Young’s modulus of 10 GPa that effectively models the OSL as a single-material beam rather than a three-layered beam. This last scenario (3) is not realistic as it does not provide a pathway for the neurons, but serves as a comparison study for OSL mechanics.

Figure 7 shows the frequency responses of the OSL-bridge (left column) and BM-bridge (right column) for all three middle layer Young’s modulus variations, both magnitude (A, C) and phase (B, D). As shown in Fig. 7A, there is little difference in the response of the BM-bridge motion, but not surprisingly, there were differences of up to 10 dB (factor of 3) in OSL motion near BF due to Young’s modulus variation of the middle OSL layer. The radial profiles of motion at BF for the three model variations are shown in Fig. 8. Consistent with Fig. 7, greater differences in motion at OSL-bridge motion (factor of 9/5 = 1.8, or 5.1 dB) were observed than at BM-bridge location (factor of 44/38 = 1.2, or 1.3 dB). There is only a slight decrease in phase for the soft-tissue middle layer (Fig. 8B). Frequency-place maps were similar for the three models (not shown). These results indicate that a greater OSL motion does not necessarily result in a proportional increase in BM motion (that is, the BM/OSL ratio is not generally affected by absolute OSL motion). It could be surmised that the OSL does not really function to act as a lever that might increase BM motion, nor necessarily improve the sensitivity of hearing.

Fig. 7.

The frequency responses for the transverse displacement of the OSL-bridge (left) and BM-bridge (right) referenced to input displacement on a dB scale. The OSL middle layer Young’s modulus was varied for the three different model variations: bony middle layer model (10 GPa) similar to bone of upper and lower plates (red); the composite middle layer model (3 GPa, baseline) representing a composite of bone and soft tissue (black); and a low Young’s modulus middle layer model (0.27 GPa) approximately representing a more cavernous bony layer and more soft tissue at 3% of the baseline composite value (cyan)

Fig. 8.

The transverse motion referenced to input displacement plotted against different radial locations. The OSL middle layer Young’s modulus was varied as described in Fig. 7. The longitudinal place was 1.8 mm with a f/BF = 1 (11 kHz)

Stress in the OSL Layers

The result that the Young’s modulus of the middle layer could be changed by a factor of about 30 without significantly affecting the motion of the BM, while also affecting the OSL response near BF by a factor of 3 (10 dB) was surprising. We thus investigated the effect of the middle layer further by looking at the stress within it. In the OSL, there are several different types of directionally dependent axial stresses.

The von Mises stress (σ_v) is a measure that gives an overview of multiaxial stress in a structure as a scalar value. The von Mises stress in the baseline model is plotted at 1.8 mm from the base (BF ~ 11 kHz) in Fig. 9. The phase of the motion cycle plotted for each frequency was chosen to correspond to the maximum stress, and color encodes the stress at that phase. It turned out that the maximum von Mises stress was at the maximum deflection of the OSL-bridge junction. The model also shows that stress is confined to the OSL plates while the stress in the OSL middle layer, limbus, and OoC was relatively insignificant. The composite, bony, and soft-tissue middle layer models all had similar profiles where maximum stress occurred in the OSL plates, but the radial location of high stress depended on the stimulation frequency (Fig. 9).

Fig. 9.

The von Mises stress in Pa for the baseline model at 1.8 mm from the base for f/BF = 1 (f = 11 kHz), f/BF = 0.73 (f = 8 kHz), and f/BF = 0.36 (f = 4 kHz). Note, each frequency has a separate corresponding scale. Color denotes the maximum stress experienced at each point in the structure at the phase of displacement corresponding to the maximum downward deflection of the OSL tip. Deformation is scaled for visibility (× 36,000 at 11 kHz, × 75,000 at 8 kHz, × 171,000 at 4 kHz). The undeformed graphical representation of the partition cross section at 1.8 mm is shown at the bottom of the figure. The majority of the nonzero stress is situated in the OSL plates

While the von Mises stress gives an overall measure of the different stresses, another way to look at stress in an OSL is through the second Piola–Kirchhoff (second PK) stress tensor. The second PK stress has normal (σ_x,y,z) and shear (τ_xy,xy,yz) stress components and helps us to understand the various stress components of the different layers of the OSL model for the CP. Fig 10 shows the three highest stresses of the baseline model, namely normal radial stress σ_y, normal longitudinal stress σ_x, and shear stress τ_xy in the radial-longitudinal plane of the OSL. The phase of the deflection cycle plotted for each frequency was chosen such that it corresponds to the maximum positive stress. Note that, unlike the von Mises stress, the phase corresponding to maximum 2nd PK stress components in most cases does not correspond to the phase at maximum deflection. The model shows that all stress components, regardless of frequency, were localized to the OSL plates while the stresses in the OSL middle layer, limbus, and OoC were relatively insignificant. The opposite polarity of stress between the upper and lower plates can be appreciated with the second PK components, denoted by color, where warm reddish colors represent positive normal tension, while cold bluish colors represent negative normal compression. The middle layer of the OSL remains stress-neutral in the three OSL models at all frequencies. These modeling results suggest that the upper and lower bony plates of the OSL bear most of the CP motion-related stress, leaving the middle layer stress-free for the passage of the auditory nerve fibers from hair cell synapses to their spiral ganglion cell bodies.

Fig. 10.

The second Piola–Kirchhoff stress in Pa for the baseline model at 1.8 mm from the base for f/BF = 1 (f = 11 kHz), f/BF = 0.73 (f = 8 kHz), and f/BF = 0.36 (f = 4 kHz). Each frequency and component have a separate corresponding scale. Color denotes the maximum stress experienced at each point in the structure at a phase of displacement corresponding to greatest magnitude of positive stress in the deformation cycle. Deformation is scaled for visibility (× 36,000 at 11 kHz, × 75,000 at 8 kHz, × 171,000 at 4 kHz). The majority of the nonzero stress is situated in the OSL plates

Discussion

The goal of this study was to understand the motion of the human CP through a finite element model, which incorporated the anatomy of the human CP, specifically the multilayered OSL joined to the BM with a non-osseous soft-tissue bridge. The study investigated the mechanics of the OSL, encompassing two thin outer bony plates (9 μm each) and its thicker cavernous middle layer (22 μm) where the ANFs pass through. The model was built using a combination of measurements from literature, histology, and OCT imaging of the human CP and, as such, does not represent any individual ear but is rather a composite from many observations. As is the case for many finite element modeling studies, determining material properties of anatomical structures was a challenge that required parameter tuning. We chose a set of parameters such that the model results were generally consistent with the cochlear frequency-place map for humans from Greenwood (1990) [18], but ½-octave shifted, approximately corresponding to a passive postmortem cochlea. At the basal high-frequency location where experimental measurements were available, the response of the baseline model had consistent trends with the experimental measurements. Similar trends in frequency response between the model and experiments were seen in the motion at the BM-bridge and OSL-bridge junctions at the scala tympani surface. Similar trends between model and experimental measurements of motion were also seen with respect to various radial locations at a few frequencies.

While model calculations and experimental measurements showed similar trends, there are noteworthy differences. The frequency response in the model was more sharply peaked at BF than for the OCT measurements of both the OSL (Fig. 5A) and BM (Fig. 5C). Comparison with LDV-based measurements also reveals this. However, there is significant variability in the frequency response in those measurements with some ears that are more sharply tuned (e.g., Specimen 14) than others. This suggested the possibility that there is more damping in the physical structure than incorporated into the model. This could be because the model did not incorporate the microstructures and fluid spaces of the organ of Corti. For example, the model did not contain a tectorial membrane, and thus did not include a sub-tectorial fluid gap.

To see if further improvements could be made to model results due to damping, we performed a parametric study of the effect of dynamic fluid viscosity and structural damping. We found that changes in viscosity by a factor of 2 did not significantly affect the response (not shown). Changes in structural damping from 0 to 0.3 decreased the BM response by about 3 dB for each 0.1 increment in structural damping. Increased damping increased the slope of the BM response above the BF.

Prodanovic et al. [52] found that increasing damping increased the slope above the BF, which was similar to our finding. They also reported an apical shift in BF and lower Q (greater bandwidth), which we did not observe. There are some important differences between the two models. Prodanovic et al. used Rayleigh damping, included damping due to the TM, and included sub-tectorial gap damping using Couette flow between the TM and RL plates. Our model did not have these structures. We were focused on modeling the OSL, and thus, other present structures in the model were given a simple representation. These modeling differences likely account for the differences between the results reported in the two studies. Future versions of our model will aim to investigate the mechanics of the sub-tectorial region, incorporating greater complexity of the anatomy.

Structure and Function of the Layered OSL

The middle layer of the OSL provides a passage for ANFs. The nerves travel medially from the hair cell synaptic terminals in the OoC, through the bridge, through the habenula perforata, then through the middle OSL layer, and into Rosenthal’s canal [24]. Within the middle layer, there are bony pillars that connect the two plates, but their columnar structure keeps the plates separated as visualized by Bom Braga et al. [4]. The cavernous non-bony space serves as a network of passages for the nerve fibers to go through. Deflection of the plates, for example, due to pathologies in the plates, could possibly traumatize the nerves by introducing stress in the auditory nerve fibers, and sensitivity can increase via mechanical stimulation [53]. This is the first study that shows, through a finite element modeling study, that the OSL construction is possibly optimized for minimal mechanical stimulation of nerve fibers.

Changing the Young’s modulus of the middle layer of the OSL from its lowest to highest value by nearly 30 times such that the entire OSL can be approximated entirely as solid bone at one extreme (the bony model variation of this study), or a layered OSL with a soft-tissue middle layer at the other extreme (the soft-tissue model variation), resulted in 10 dB differences in response of the OSL-bridge junction motion (Fig. 7C). Differences in the OSL middle layer material properties had little effect on BM motion (Figs. 7 and 8). This means that, while the general layered structure of the OSL may be anatomically important, the material (bony or soft-tissue) in the middle layer of the OSL had minimal impact on overall BM motion. We surmised from this that the mobile OSL does not result in an improvement in sensitivity of hearing, at least in this passive configuration.

The mechanics of beam theory gives insight into this result. The central portion of a symmetric cantilever beam forms a neutral plane with zero stress compared to the outer portions with opposite compression and tension phases that bear the load. Our result of the overall motion being very similar, whether the middle solid layer has high or low Young’s modulus, is consistent with this theory. When stress is applied to a beam, there is compression on one side, and expansion on the other with a neutral transition plane in between, parallel to the cross section of the beam and perpendicular to the plane of bending (illustrated in the stress results for the radial component in Fig. 10). The middle layer of the OSL contains the neutral plane which runs from the base to the apex. In reality, the partition cross section is not entirely symmetric due to the limbus situated adjacent to the vestibular plate; nevertheless, the neutral plane is still visualized to be between the OSL plates.

Because there is little stress along the middle neutral plane, the material used to construct it will have little consequence in comparison to the material properties of the upper and lower plates. This is very similar to the design of an I-beam in construction materials, where the middle portion of the beam is thinned to save material cost, or lower weight without sacrificing structural integrity. This type of design would optimally allow the auditory nerve fibers to pass through the OSL middle layer and into Rosenthal’s canal while minimizing stress to the nerve fibers. In short, what we see is localization of compressive and tensile stress to the OSL plates with neutral zero stress in the middle OSL transition layer where the auditory nerve fibers reside, and thus potential avoidance of nerve stimulation. However, just as in an I-beam construction, the middle I part joining the top and bottom plates is critical. Without it, the beam is weak and does not work as designed. Additionally, if there were no solid component between the upper and lower plates, and consequently, the bony cavernous structures between the OSL plates were absent, then there would only be fluid and nerve between the plates. This could potentially cause fluid to be squeezed in and out, thereby hypothetically increasing auditory nerve sensitivity, as described in Perez-Flores et al. [53]. In this study’s tapered box model with Cartesian coordinates, we have made the OSL without curvature for convenience and computational economy. In the very base of the cochlea, the OSL is seldom so straight as shown in Figs. 1 and 2. In our model, the OSL is symmetric, and therefore, the neutral plane is in the middle. But in the very base of the human cochlea, in the hook region, the OSL is a curved structure; thus, that neutral plane would shift towards the inner radius of the bend. We did not explore the curvature of the OSL, but we suspect that differences in Young’s modulus between the outer bony plates and the middle layer will still result in stresses being localized near the plates.

The motion of the OSL plates is primarily in the transverse direction with a longitudinally propagating traveling wave. This traveling wave was shown to result in high stress in the longitudinal direction of the bony OSL-bridge junction, especially at BF (Fig. 10, middle column showing σ_x has the highest value).

The OSL plates are described as being porous, meaning it has many tiny holes [2, 4]. We did not explicitly explore porosity in our model because of the increase in computational cost and the assumption that it may not be a major factor in understanding OSL mechanics. Given that our model was tuned with an equivalent porosity of around 50% [4] we would predict that we would need a higher Young’s modulus of bone by a factor of two with porosity implemented to achieve the same model tuning.

If the plates were to get thinner and break, then OSL motion could increase and possibly make the auditory nerve more sensitive, and thus stimulate it [53]. We are not aware of pathologies that would produce such a change in the OSL bone density, though it is possible that after trauma (e.g., a temporal bone fracture), a break in the OSL could occur.

Why Is the Human OSL so Wide?

One can wonder why the channel for auditory nerve fibers, or equivalently, the width of the OSL, in humans is relatively wider compared to laboratory animals in the basal high-frequency regions. We know that there is a correlation between skull size and inner ear volume, at least in primates [54] and across placental mammals [1]. Additionally, humans have a significantly larger skull mass than most laboratory animals, like mice and gerbils. Consequently, the cochlear volume in humans is much larger, and the corresponding cross-sectional area of their cochlear duct is also significantly larger than that of smaller animals (e.g., Puria and Allen [55]). Cochlear ducts are divided into two major fluid chambers by the CP, with larger animals having wider CPs for approximately the same frequency region. The OSL in larger mammals like humans makes up a larger fraction of the cochlear partition because the basilar membrane (BM) width is generally quite similar for a given frequency region [7, 56].

Why is the organ of Corti (OoC) typically closer to the stria vascularis than the modiolus in larger mammals, resulting in a larger OSL (e.g., Fig. 1)? This proximity may facilitate transcellular recycling of ions, especially potassium ions (K⁺) during outer hair cell (OHC) transduction. During mechano-transduction, K+ ions flow from endolymph into the OHC, moving through supporting cells and gap junctions back to the stria vascularis for pumping into endolymph, maintaining ionic homeostasis. The shorter ionic pathway near the stria facilitates faster recycling of high K+ to maintain endolymph homeostasis. However, this comes at the expense of a longer OSL. A three-layer structure allows peripheral axons from sensory inner hair cells to pass through the habenula perforata and between the OSL plates to reach spiral ganglion cell bodies.

Conclusions

At the middle layer of the human OSL, the cavernous structure with bony pillars may be important for creating a non-traumatizing normal passage for the auditory nerve fibers. The nerves pass through the middle layer along a neutral plane free from mechanical stress, while the stiff vestibular and tympanic plates of the OSL take on the stress-bearing load due to surrounding fluid pressures.

Author Contribution

Andrew Tubelli: methodology, validation, and writing—original draft preparation. Paul Secchia: OCT investigation, data curation, and review and editing. Stefan Raufer: LVD investigation. Heidi H. Nakajima: methodology, review and editing, and funding acquisition. Sunil Puria: methodology, validation, writing—original draft preparation, and supervision.

Funding

National Institute on Deafness and Other Communication Disorders (R01 DC013303).

Data Availability

Data and Finite Element model (in COMSOL ver 6.1) for this study are available on the Harvard Dataverse site https://doi.org/10.7910/DVN/NTUHYI, or from the corresponding author on request.

Declarations

Competing Interests

The authors declare no competing interests.