A linearly tapered box model of the cochlea

The cochlea is a sensory organ of the hearing system responsible for converting sound-induced motion into electrochemical impulses for perception. Although the cochlea has a coiled structure, e.g., Viergever (1978), Loh (1983), Steele and Zais (1985), the fluid chambers are often modeled using a straight and uniform box model (Ranke, 1950; Lesser and Berkley, 1972; Siebert, 1974; Sondhi, 1978; de Boer, 1996; Vetesník and Nobili, 2006; Elliott et al., 2011; Elliott et al., 2013). Although the box shape is an overly simplified representation of the real cochlear geometry, the coupled¹ basilar membrane, BM, response to tones, especially the frequency-position tonotopy relation, can still be reproduced reasonably well (de Boer, 1996). The assumption that the box model is rectangular may appear to be physically unrealistic; however, de Boer (1991) has shown that similar results for the three-dimensional, 3D, fluid coupling are obtained if the cross section is assumed circular. This reflects the fact that the far-field, plane wave component of the fluid pressure depends mainly on the cross-sectional area of the chambers rather than their shape. If the uniform box model represents a first approximation to the cochlear geometry, for the calculation of the fluid coupling (Steele and Taber, 1981; de Boer, 1996; Elliott et al., 2011; Ni and Elliott, 2015), a sensible second approximation would thus appear to be a linearly tapered box model, as shown in Fig. 1.

Puria and Allen (1991) include exponential scalae tapering in a transmission line model of the cochlea and find it has a significant effect on the cochlear input impedance. Similarly, Shera and Zweig (1991) also consider the effects of tapering and point out that the width of the BM and the scalae area taper in opposite directions, enabling a slow variation of wavelength near the stapes, which provides an efficient energy transfer between the middle and inner ears at low frequencies. The differences between the linearly tapered box model and the transmission line models previously used are that the cross-sectional area varies in a linear way in this paper rather than exponentially from the base and the BM width varies linearly as well. Although not shown in this paper, the cochlear input impedance of the linearly tapered model increases with frequency and is greater than that calculated using a uniform model, which is consistent with previous prediction using the transmission line models. More recently, Kim et al. (2011) develop a finite element model of the cochlea for analyzing the mechanism of bone conduction hearing, in which the BM and cochlear partition width together with the height of the fluid chambers vary approximately linearly along the straight box model.

There are many methods to solve the cochlear box models, for example, the Wentzel–Kramers–Brillouin, WKB, method (Zweig, 1976; Steele and Taber, 1979a,b; Wang et al., 2016), the finite element method (Kim et al., 2011; Elliott et al., 2013; Ni and Elliott, 2015), and the finite difference method (Neely, 1985). Different methods have their own assumptions and limits, such as only one wave with slow variation traveling in one direction in the WKB method, or at least six elements within the shortest wavelength interested in the finite element/difference methods (Fahy and Gardonio, 2007). The method used in this paper is an extension of Elliott et al. (2011), which employs a matrix formulation for solving the fluid coupling and the cochlear response, as described below in Sec. III. One advantage of this approach is to provide a closed-form solution to the far-field fluid coupling, which depends on the cochlear geometry, that can be used in analyzing complicated cochlear structure at a low computational cost compared with the finite element solution (Ni et al., 2011), for example.

The fundamental mechanics of the passive, postmortem, cochlea can be thought as an interaction between the fluid inertia along the fluid chambers and the dynamic behavior of the cochlear partition that separates them (de Boer, 1996). The fluid pressure due to a moving element of the BM can be decomposed into two components (Steele and Taber, 1979a; Elliott et al., 2011). That due to the plane wave component of the pressure is called the far-field (or long-wavelength) component, and that due to the pressure variations close to the BM is called the near-field (or short-wavelength) component. The far-field component, which is associated with wavelengths that are large compared with the size of the fluid chambers, plays an important role in the cochlear traveling wave propagation. The near-field component is associated with short wavelength and affects the cochlear partition motion by introducing a fluid added mass on the BM (Neely, 1985; de La Rochefoucauld and Olson, 2007) and an inertial term to the local longitudinal coupling along the cochlea (Steele and Taber, 1981; Parthasarathi et al., 2000; Elliott et al., 2011). The far-field pressure mainly depends on the geometric features such as cross-sectional area and coiling of the cochlear fluid chambers (Manoussaki and Chadwick, 2000; Cai et al., 2005; Ni et al., 2011), whereas the near-field pressure is not substantially affected by the fluid chamber geometry and could be calculated from the local arrangement of the BM using various methods (Ni and Elliott, 2015).

The areas of the upper fluid chamber, which include the scala media, SM, and scala vestibuli, SV, and the lower fluid chamber, which includes the scala tympani, ST, vary along the length of the cochlea, x, as A₁(x) and A₂(x), respectively. The cochlear fluid is assumed to be incompressible and inviscid since the effects of fluid viscosity and compressibility are generally considered to be not as important as the fluid inertia for the cochlear macromechanics (Viergever, 1980; Neely, 1985), although viscosity is shown to play a key role in cochlear power flow (Wang et al., 2016). The following derivations are based on these assumptions but are not limit to inviscid fluid. The far-field component of the pressure can be determined by combining the conservation of mass and momentum equations. Assuming that the fluid longitudinal velocity in a single chamber, here, the upper fluid chamber is chosen for illustration, averaged across its cross-sectional area is $u ¯ ( x )$⁠, and that the radial BM velocity, averaged across the width of the chamber is $v ¯ ( x )$⁠, which is positive toward the SV (upward). The continuity equation for the upper chamber can then be written as

where A(x) and W(x) are the chamber cross-sectional area and width, respectively. This is equivalent to the equation used by Peterson and Bogert (1950).

The momentum equation can also be written in terms of the complex pressure, $p ¯ ( x )$⁠, averaged across the cross-sectional area, as

where ρ is the density of the cochlear fluid, ω is angular frequency, and $i = − 1$⁠.

Substituting $u ¯ ( x )$ in Eq. (2) into Eq. (1) gives an expression for $p ¯ ( x )$ in terms of $v ¯ ( x )$ as

which is an incompressible form of Webster's horn equation, as described by Fletcher and Rossing (2008), for example. Shera et al. (2004) derive a similar expression, Eq. (A1) in Shera et al. (2004), Appendix A, for the far-field, long-wavelength, fluid coupling in a tapered cochlear model with a constant BM width.

Geometrical features, such as the cross-sectional area of the fluid chambers, are crucial for quantitative modeling of the cochlea. Thorne et al. (1999) derived cochlear fluid space dimensions for different species from reconstructions of 3D magnetic resonance microscopy, including areas of the SV, SM, and ST, which are used in this study for constructing a tapered model of the cochlea. The mouse cochlea geometry, however, is taken from a more recent work by Rau et al. (2012) to cover a larger number of spatial positions.

In calculating the coupled response of the cochlea, we focus on the passive case, whose normalized distribution is similar to those measured at high sound pressure levels, SPLs, excitations, e.g., over 80 dB SPL, because of the simplicity of the micromechanical model in this case. The structure of the paper is that Sec. II reviews the geometric variation of the fluid chambers along the length of different cochleae. In Sec. III, expressions for the far-field and near-field fluid coupling of tapered fluid chambers are derived. In Sec. IV, the fluid coupling in the tapered and uniform model of the human cochlea is calculated, as well as the fluid added mass distribution in the cochleae of five different species. The uniform model here takes mean geometrical values over the length of the linearly tapered model. In Sec. V, the coupled response of the cochlea is calculated using the linearly tapered and uniform models of the human and mouse cochleae and compared with experimental measurements.

There are several sources of information about the variation in the physical dimensions of the fluid chambers along the length of the cochlea in different species, among which the database of Salt's laboratory² provides the most comprehensive set. The measured geometrical variations in the fluid chambers and cochlear partition in a number of species are used here to calculate the cross-sectional area, the BM width, and derived variables such as the effective area.

One parameter for the far-field fluid coupling is the effective area as described in detail in Sec. III, which is calculated based on the upper and lower fluid chamber areas, A₁(x) and A₂(x), respectively. Since the fluid coupling is not sensitive to the shape of the fluid chamber cross-sectional area (de Boer, 1991), it is mathematically convenient to assume that the cross section is square. So that the equivalent fluid chamber width and height, W_e and H_e, respectively, are equal to square root of the effective area, A_e(x), as

where the effective area is written in the form of the harmonic mean as (Peterson and Bogert, 1950; Zwislocki, 1953)

Variations of the upper and lower fluid chamber areas and the calculated effective area in different species are shown in Fig. 2 with respect to a non-dimensional longitudinal position, i.e., longitudinal position, x, over the cochlear length, L. The variation of the lower fluid chamber area is seen to be steeper than that of the upper fluid chamber among all the selected species, especially at the basal end. All measured geometrical data, to our knowledge, does not cover the entire length of the cochleae, which implies extrapolation is necessary for modeling purposes.

Figure 3 shows the variation of the fluid chamber equivalent height, H_e, and its linearized approximation using the least squares method. It is interesting to see that although the area variations in the two fluid chambers are different among the different species, the square root of the effective area can be reasonably well approximated by a linear fit. The linear fit overestimates the true values by about 11% near the base where the anatomical geometry is known to be more complicated and many other factors such as aqueducts, round window dynamics, etc., may play a more significant role, so that we assume that this overestimate of the effective area can be ignored when we compare model predictions with measurements later. There are other approximations, such as exponential, e.g., Puria and Allen (1991), or polynomial functions, which may provide a better fit to the measured data. In this paper, however, a linearly tapered model is used as motivated by the results shown in Fig. 3 and also as the simplest development of a uniform box model. It should also be noted that this model is reasonably consistent with an exponential tapering, proving the tapering is not too great since

provided $( α x / 2 ) 2 ≪ 1$⁠.

Values for those linearized fluid chamber equivalent height are listed in Table I, in which a constraint has been imposed that the fluid chamber equivalent width [equal to the equivalent height, as defined in Eq. (4)] should be no less than the BM width. The correlation coefficient, r, for the least squares fit, as also shown in Table I, is calculated using corr command in matlab^TM (R2015b, The Mathworks, Natick, MA).

The BM is assumed to take up the middle part of the cochlear partition that separates the two fluid chambers, as shown in Fig. 1. The variation of the BM width, B(x), from the base to the apex was measured by Fernández (1952) for the guinea pig and human, and measured or assumed by others for other species, as shown in Table I and Fig. 3. Previous observations show that the variation of the BM width is approximately linear along the length of the cochleae and can be given by

where B_A and B_B stand for the BM width at the apex and base, respectively, and L is the length of the cochleae.

The mechanism underlying the cochlear frequency-position mapping characteristic is believed to be similar in most mammals (Robles and Ruggero, 2001). There is, however, a wide range of physical dimensions of the cochlea in different mammals, resulting in difference in perceptible hearing frequency range. Greenwood (1990) develops a function that maps the characteristic frequency, CF, onto locations along the length of the cochlea, as

where x stands for distance away from the base, f_h is a constant controlling the high-frequency limit, α is a constant that controls the slope of the frequency map, L is again the length of the cochlea, and f_l is a constant controlling low-frequency limit. Here, we ignore the factor f_l in Eq. (9), and simplify the CF function to

where f_B is the CF at the base, equal to f_h10^α, and l is the characteristic length, equal to x/ln 10^αx/L.

Figure 4 shows the spatial variation of CF for the five species, based on parameters listed in Table I. It is interesting to see that although different species have different CF distributions along their length, they are almost parallel with each other if they are plotted against position normalized by the length of the cochlea. The ratio of the cochlear length, L, to the characteristic decay length, l, is thus approximately constant and has a value of about 5, as shown in Table I among different species, and so the frequency ranges of the cochleae in different species are always about 7 octaves, as shown in Fig. 4.

The fluid coupling and the BM velocity can be analyzed using a single longitudinal variable, x, along the cochleae (Steele and Taber, 1979b; de Boer, 1984; Elliott et al., 2011), which reduce the 3D model to a uni-dimensional while still accounting for its radial, y, and vertical, z, characteristics. The averaged BM radial velocity, $v ¯ ( x )$⁠, in Eq. (3), can thus be defined as

where ψ(y) is the BM velocity, v(x), radial profile and generally assumed as half-sinusoidal and normalized over the BM width, B, as $ψ ( y ) = 2 W / B sin ( π y / B )$ (Steele and Taber, 1979b).

In the far-field limit, the pressure is uniform over the BM so that the modal pressure in the upper fluid chamber, p₁(x), can be given as

Substituting Eqs. (11) and (12) into Eq. (3) gives the far-field fluid coupling in the upper chamber, as

In general, the far-field fluid coupling in the lower fluid chamber is similarly related to the modal BM velocity as

The integral of the right-hand side of Eqs. (13) and (14) with respect to x is thus equal to both of the expressions

where $x ′$ is dummy integration variable. The pressure gradients in the two chambers are related by

We can relate the far-field component of the modal pressure difference, p_F(x) = p₁(x) − p₂(x), to the modal BM velocity, v(x), via the effective area A_e(x) in the expression as

An analytic solution to the pressure difference can be obtained for excitation of a single BM element, having a velocity of v₀ from x₀ − Δ to x₀ and zero elsewhere. The boundary condition that p_F(x) is zero at x = L and the fact that Δ is small compared with L can then be used to integrate Eq. (17) twice to give

where $x ′$ is again dummy integration variable. It is computationally convenient to divide a continuous system into a number of discrete elements, which may be taken as an accurate representation of the continuous system if there are at least six elements within the shortest wavelength present, which is a condition commonly used in finite element analysis (Fahy and Gardonio, 2007). If the areas of the fluid chambers in the cochlear models are divided up into N discrete sections, as for the BM, the integrals in Eqs. (18) and (19) can be approximated by summations to give the pressure at the nth element as

where n₀ = x₀/Δ.

It is significant that the near-field pressure is greatest in the immediate vicinity of the vibrating BM and is small on the further surfaces of the fluid chamber. The near-field pressure is thus relatively independent of the shape of the fluid chambers. Elliott et al. (2011) show that there are not many variations over the range B/W = 0.11 to B/W = 0.99, as displayed in the tapered model, which suggests that the shape of these near-field components is relatively independent of B/W.

Apart from the physical mass per unit area of the BM, fluid added loading, due to the fluid inertia, will provide some additional mass. In a one-dimensional, 1D, model, in which only the far-field component is considered, of the cochlea, there is an effective additional thickness of the BM due to the fluid loading that can be calculated using the wavenumber analysis of fluid coupling (Steele and Taber, 1979b; Elliott et al., 2011). When taking Eq. (4) into account, this effective thickness of the BM can be given as

where C(x) is the distance from one side of the cochlear partition to the corresponding edge of the BM, as shown in Fig. 1.

The total BM mass per unit area is thus given by

where T_t is the sum of the physical thickness due to the organ of Corti, OC, T_OC, and the BM, T_BM, and T_f is the effective thickness due to the fluid loading.

The physical thickness of the organ of Corti and the BM can be given by

where A_OC is the average area of the organ of Corti and A_BM is the average area of the BM.

The fluid added mass, due to the near-field component, on the BM along its length is calculated using Eq. (22), as shown in Fig. 5, for five different species. Since the sum of the physical thickness due to the organ of Corti and the BM, T_t, is significantly less than the effective thickness due to the fluid loading, T_f, the variation of T_t along the cochlea is not shown here and an average value is used in the models.

The distribution of the modal pressure differences with both far- and near-field components, due to excitation by a single element of the BM at 5 mm, 15 mm, and 25 mm away from the base are calculated for both the uniform and tapered box models for the human cochlea, as shown in Fig. 6. The far- and near-field components have also been calculated separately for the tapered and uniform cases, in which Eqs. (20) and (21) are used for the far-field component and the wavenumber approach (Steele and Taber, 1979b) is used for the near-field component. In the uniform case, the BM width, B, and fluid chamber width, W, are assumed to be the mean value of the linear fits, B_l and W_l, which are equal to 0.32 mm and 0.9 mm, respectively. It can be seen from Fig. 6 that close to the basal region, when a single element of the BM is driven, at 5 mm, for example, the pressure difference of the uniform model is greater than that of the tapered case implying higher fluid impedance in the uniform model, which leads to a smaller BM motion. This is not surprising since the effective area of the tapered model is greater than that of the uniform model in the basal region, as seen in Fig. 3, in which the effective area is equal to the square of the equivalent height. The curvature in the pressure difference distribution for x greater than x₀ is due to the reduction of the effective area with distance. The near-field component is seen to be insensitive to the tapering, as the difference between the two models is hardly visible.

The coupled behavior of the linear cochlear dynamics can be calculated using a representation of the pressure distribution, driven by the BM movement, and the BM admittance in response to the imposed excitation (Neely, 1985; Elliott et al., 2011). The fluid-structure coupling defined here represents a weak coupling between the fluid and structure, the BM, since each domain is separately calculated and then combined to give the coupled response, as illustrated below. This type of discrete expression is generally used for uniform and symmetric box models of the cochlea, using the finite difference method (Steele and Taber, 1979b; de Boer, 1981; Neely, 1981; de Boer and Viergever, 1982).

If the stapes velocity is assumed to be zero, the vector of pressures, p, due to the vector of BM velocities, v, can be written as (Neely, 1981; Elliott et al., 2011)

where Z_FC is a matrix of the fluid coupling impedances and its columns can be obtained by calculating a position-shifted sequence of the pressure distributions, as shown in Fig. 6, normalized by the velocities of each element.

Similarly, the vector of BM velocities can be written as

where v_s is a vector that accounts for the stapes velocity and Y_BM is the BM admittances matrix. Y_BM will be a diagonal matrix, if the BM reacts only locally. The vector of BM velocities can be given by substituting Eq. (26) into Eq. (27) as

The boundary conditions of the tapered and uniform models in this paper are: (1) rigid wall on the external sides of the fluid chambers, (2) the BM is simply supported at two ends, (3) pressure difference at the helicotrema is zero to account for the fact that the upper and lower chambers exchange fluid there, and (4) excitation is the stapes velocity, v_s.

An advantage of this matrix form is that complicated geometries only need to be analyzed once to determine the elements of Z_FC and then the coupled responses can be calculated simply using Eq. (28), for a variety of models, with known or assumed BM dynamics.

The passive BM can be approximated by a series of isolated single-degree-of-freedom systems, whose mechanics at a given position and frequency is given by

where $v$ is the model BM velocity, p is the model pressure difference, and r(x), s(x), and m(x) are damping, stiffness, and mass of the BM, per unit area, respectively. The mass, m(x), includes the physical mass due to the organ of Corti and the BM and the fluid added mass as given by Eq. (23). The stiffness is chosen to give the correct natural frequency at each position (meet the frequency-position map of the cochlea) as

where ω_n(x) is the angular form of the CF, given by Eq. (10), and thus equal to 2πf_Be^–x/l. Damping is defined by the quality factor, Q, which is assumed to be constant along the cochlea, so is given by

Figure 7 shows a comparison of the calculated BM velocity along the cochlea for the linearly tapered and two uniform models of the human cochlea at three different excitation frequencies. The two uniform models have parameters that are either those in the middle of the tapered box model, i.e., average parameters, or those at base of the tapered model, as used in the comparisons of Puria and Allen (1991) and Shera and Zweig (1991). It can be seen that the amplitude distributions between the models are similar, except for the magnitude and position of the BM velocity peak change from one model to another. Since the BM admittance of all the models is defined to be identical, it is probably the difference in the fluid added mass, as shown in Fig. 5, that causes the shift in the position of the peak BM velocity. The fluid added mass of the linearly tapered model is small at the base (high-frequency region) and then increases until slightly greater than that of the uniform models at the apex (low-frequency region). Since the input impedance of the tapered model is greater than that of the uniform models (Puria and Allen, 1991; Shera and Zweig, 1991), then more power is supplied at the stapes of the tapered model for a constant stapes velocity than for the uniform box model, and this leads to a greater peak BM velocity. Another effect at low frequencies, however, is that the ratio between the BM width and the fluid chamber width, B/W, in the linearly tapered model becomes much larger than in the uniform model with averaged parameters, which leads to a decrease of the peak BM velocity.

Figure 8 shows the BM frequency response calculated at about 12 mm from the base for the human cochlea in comparison with the BM response measured in a human cadaver by Gundersen et al. (1978) and Stenfelt et al. (2003). Results are plotted against a non-dimensional variable, f/BF(x₀) (Zweig, 1976; Shera, 2007), where f is the driving frequencies and BF(x₀) denotes the best frequency at which the BM motion is maximum for the level at which the measurement was performed. It can be seen that the human tapered model prediction is similar to those from experiments, although the amplitude has a sharper drop compared with Stenfelt et al. (2003). The uniform model predicts an even faster decrease with a peak that is about 10 dB less than the tapered model around the best frequency. The other difference between the two models is that amplitude of the tapered model shows a nearly flat distribution similar to the slow decrease found in the experiments but missing in the uniform model. The roll-off of both the tapered and uniform model is less than that from experimental measurements.

Figure 9 shows another comparison between the mouse models and a recent in vivo measurement (Lee et al., 2015) at the apex of a mouse cochlea when excited at 80 dB SPL. The tapered model of the mouse cochlea shows a better match in amplitude compared with the human case, although the measured phase shows a greater lag. It is interesting to see that the tapering in the mouse model does not play a significant effect, as the uniform model does not manifest as much difference, less than 3 dB, as the human case. We suspect that this less discrepancy of amplitude between the tapered and uniform models is due to the fact that the mouse cochlea has the least variation among the selected species, as seen in Figs. 2 and 3.

As an improvement and extension to the previous work, a modification of the widely used uniform box model is developed here with a linear variation of fluid chamber cross-sectional areas and the BM width, which both play an important role in the BM passive response. Discrete formulations for the far-field fluid coupling are derived, which are suitable for both uniform and tapered cochlea box models. Although a linear approximation is not accurate enough to represent the entire geometrical features in the cochleae, especially at the basal region where the anatomy is more complicated, it is found that this approximation is not bad when it reproduces the passive cochlear response measured at a certain distance away from the base. It is also interesting to see that the selected species all show an approximately linear variation when the square root of the effective area is considered.

The BM velocity calculated using the linearly tapered model is greater than that using the uniform models and the difference in BM velocity peak is frequency dependent, e.g., the difference is about 20 dB at 5000 Hz but about 5 dB at 500 Hz. This is because the ratio of the BM width to the fluid chamber width, B/W, increases with position in the tapered model, which leads to a decrease in the BM response. The tapered model has a smaller effective area and greater B/W at the apical end but still provides higher BM response than the uniform model with average parameters, which suggests that the tapering helps to maintain efficiency in the transfer of acoustic energy to the cochlea at low frequencies, as suggested by Shera and Zweig (1991). The other difference between the tapered and uniform models is that the best position, where the BM response is maximal, is different even though these models have the same BM admittance. Close to the base, the tapered model has smaller fluid added mass, which shifts its best position toward the apex. Whereas close to the apical region, this fluid added mass of the linearly tapered model becomes greater than that of the uniform models, thus, shifts its best position toward the base.

The dissipative effect is accounted for here by a damping term in the BM admittance only, but this does not seem to be sufficient for an accurate phase response. The mouse models show a slower phase drop compared with the experiment after the best frequency. The phase response could be better if a greater Q factor was used in the micromechanical model, Q = 5, for example, but this leads to a worse fit for the amplitude. This is contradictory since one would expect that a broad and smooth amplitude would need a small Q factor, high damping, whereas a fast phase drop would need a large Q factor, low damping. We suspect that a combination of fluid viscosity (Wang et al., 2016) and BM damping may improve model prediction.

An important advantage of this approach is that it provides a general way to calculate the fluid coupling even when the fluid chambers are non-uniform. By combining the fluid coupling with the BM dynamics, the coupled response of the cochlea can be readily calculated with low computational efforts.

The authors declare no existing or potential conflict of interest. G.N. is supported by grants from Engineering and Physical Sciences Research Council, UK (EPSRC) (Engineering nonlinearity, Grant No. EP/K003836/1) and Medical Research Council, UK (MRC) (Interaction between sensory and supporting cells in the organ of Corti: Basis for sensitivity and frequency selectivity of mammalian cochlea, Grant No. MR/N004299/1).