Mechanical model of an arched basilar membrane in the gerbil cochlea

Accepted Manuscript Mechanical Model of an Arched Basilar Membrane in the Gerbil Cochlea Wei Xuan Chan, Seong Hyuk Lee, Namkeun Kim, Choongsoo S. Shin, Yong-Jin Yoon PII:

S0378-5955(16)30323-9

DOI:

10.1016/j.heares.2016.12.003

Reference:

HEARES 7285

To appear in:

Hearing Research

Received Date: 23 July 2016 Revised Date:

2 December 2016

Accepted Date: 8 December 2016

Please cite this article as: Chan, W.X., Lee , S.H., Kim, N., Shin, C.S., Yoon , Y.-J., Mechanical Model of an Arched Basilar Membrane in the Gerbil Cochlea, Hearing Research (2017), doi: 10.1016/ j.heares.2016.12.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Mechanical Model of an Arched Basilar Membrane in the Gerbil Cochlea

Wei Xuan Chana , Seong Hyuk Leeb,∗, Namkeun Kimc , Choongsoo S. Shind , Yong-Jin Yoona,∗ a School

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of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, S 639798, Singapore b School of Mechanical Engineering, Chunh-Ang University, 221 Heuksuk-dong, Dongjak-gu, Seoul 156-056, Korea c Division of Mechanical System Engineering, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Republic of Korea d Department of Mechanical Engineering, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, South Korea, 121-742

Abstract

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The frequency selectivity of a gerbil cochlea, unlike other mammals, does not depend on varying thickness and width of its basilar membrane from the basal to the apical end. We model the gerbil arched basilar membrane focusing on the radial tension, embedded fiber thickness, and the membrane arch, which replace the functionality of the variation in thickness and width. The model is verified with the previous gerbil cochlea model which estimated the equivalent basilar membrane thickness and shown to be more accurate than the flat sandwiched basilar membrane model. The simple sinusoidal-shaped bending mode assumption in previous models is found to be valid in the present model with < 12% error. Parametric study on the present model shows that fiber thickness contribution to the membrane stiffness is close to the 3rd order, higher than the 1st order estimation of previous models. We found that the effective Young’s modulus of the fiber bundle is at least 6 orders higher than the shear modulus of the soft-cells and the membrane radial bending stiffness is more sensitive to the membrane arch and the shear modulus of the soft-cells near the apical end. Keywords: gerbil, cochlea, basilar membrane, arch fiber, radial tension

1. Introduction The mammalian cochlea is a sensitive transducer. It maps the input acoustic energy according to frequency onto the basilar membrane (BM) as vibrations which excite the inner hair cells, sending electrical signal to the brain. ∗ corresponding

authors Email addresses: [email protected] (Seong Hyuk Lee), [email protected] (Yong-Jin Yoon)

Preprint submitted to Elsevier

December 12, 2016

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The performance of such a system depends on its frequency selectivity where sharper frequency tuning (higher frequency selectivity) signifies that the cochlea is able to discern two signals with closer frequency from one another. The frequency selectivity of a mammalian cochlea predicates on the varying properties of its basilar membrane from the basal to the apical end. These properties are typically the width and thickness in most mammalian cochlea including cat, chinchilla, and guinea pig (Lim and Steele, 2002; Yoon et al., 2011). A recent frequency tuning comparison of mammalian cochlea across species in Ruggero and Temchin’s work (Ruggero and Temchin, 2005) indicated that the frequency tuning sharpness of mammalian cochlea are similar among most mammals including the gerbil. Such similar frequency tuning sharpness normally suggests that the gerbil cochlea has a similarly large variation of BM width and thickness. However, in the gerbil cochlea, the width of the BM remains relatively constant from the base to the apex (100% increased (Naidu and Mountain, 2007; Edge et al., 1998)) compared to other mammals such as cat and chinchilla (300% increased (Yoon et al., 2011; Dallos, 1970; Cabezudo, 1978)) and the thickness of gerbil cochlea BM increases slightly (Edge et al., 1998; Naidu and Mountain, 2007) from the basal to the apical end, compared to other mammals such as cat (Cabezudo, 1978), chinchilla (Eldredge et al., 1981), and guinea pig (Wada et al., 1998) which decrease along their cochlea. Without the variation of BM width and thickness to change its radial bending stiffness (y-axis), the BM in gerbil cochlea relies on the fiber bundle thickness as well as its arch toward the scala tympani from base to apex to achieve comparatively sharp frequency selectivity (Chan and Yoon, 2015). There are various attempts to simplify the unique structural mechanism of gerbil BM which is flat on the side of the scala vestibuli and has an arch toward the scala tympani (Richter et al., 1998; Kapuria et al., 2011)(see Figure 1). Yoon et al. (2007b) used the flat BM model by inserting estimated effective BM thickness into their mechanical gerbil cochlea model. Although the mechanical gerbil cochlea model formulated with this uniform fiber density model agrees with experimental measurements (Yoon et al., 2007b), the effective thickness in the gerbil model which decreases from basal to apical end of the cochlea do not represent experimental measurements of the BM thickness of gerbil cochlea (Edge et al., 1998; Naidu and Mountain, 2007). Moreover, the simulated gerbil cochlea using this model (Yoon et al., 2012) deviates from experimental input impedance (Decraemer et al., 2007). Liu and White (2008) as well as Naidu and Mountain (2007) modeled the BM as a composite with fiber bundles across the width of the BM. Liu and White (2008) modeled the fiber bundle as an array of rods at the center of the BM thickness and Naidu and Mountain (2007) modeled the gerbil BM as a sandwich plate with the fiber bundles as two arrays of rods at the top and bottom of the BM. Although these composite models use the experimentally measure parameters, a parametric study comparing these models with an experimentally measured BM behavior shows a mismatch in the acoustic traveling wave number (Chan and Yoon, 2015). In our previous study (Chan and Yoon, 2015), we have identified the radial bending stiffness, as one of the important factors affecting the acoustic travel2

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ing wave in the gerbil cochlea and shown that previous gerbil BM models (Yoon et al., 2007b; Naidu and Mountain, 2007) are unable to account for significant changes in the radial bending stiffness along the gerbil cochlea. In this work, an arched basilar membrane model is formulated. Although the structural stiffness of composites with complex curve can be simulated with Finite Element Analysis (FEM) (Vafaeesefat and Khanahmadlu, 2011; Jang and Kim, 2012), this approach is less ideal when the deformation of the entire membrane is unknown. The structural model of a complex curve composite is analytically simplified, verified for accuracy against the radial bending stiffness variation of Yoon et. al.’s mechanical gerbil cochlea (Yoon et al., 2007b,a) and compared to Naidu and Mountain’s sandwich BM model. With the verified model, the stiffness variation with arch shape and fiber bundle thickness are analyzed. The results show that the shearing of the soft-cells and the arch shape of the BM enable significant variation of the radial bending stiffness with lesser dependency on the BM width and thickness than the fiber bundle thickness variation along the cochlea.

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2. Model of an Arched Basilar Membrane

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Figure 1: (a)Illustration of the pectinate zone of present arched basilar membrane model with denotation on parameters of the basilar membrane and (b) a section, ∆x∆y, of the arched plate showing forces on the section. M and Q are the external moments and shearing force in the respective axes. δα is the change in angle of the cross-section plane in the respective axes and q is the z-direction pressure on the BM section.

Distance from base,x(mm) BM Area,A(µm2 ) BM height,h(µm) BM Width,b(µm) Fiber thickness,t(µm) Fiber width,dwidth (µm) Fiber spacing,dspacing (µm)

3.61 3998 35 168 1.02 1.13 1.6

6.86 7005 49.7 192 0.57 0.8 1.62

11.24 8225 55.3 207.3 0.28 0.35 1.67

Table 1: Properties of Gerbil arched basilar membrane (Naidu and Mountain, 2007; Edge et al., 1998; Schweitzer et al., 1996). The basilar membrane width is the measurement of the pectinate zone.

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The gerbil cochlea BM in this work is modeled as a homogeneous, soft ground substances (soft-cells) sandwiched by fiber bundles lining the top and bottom of the membrane as shown in Figure 1(a). We used parameters extracted from measurements of gerbil’s BM in Edge’s work (Edge et al., 1998) as well as the fiber bundle geometries from Naidu et al.’s measurements(Naidu and Mountain, 2007) as listed in Table 1.

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2.1. Basilar membrane fiber bundle

Rn R

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Ma Na εa




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Figure 2: Illustration of a section, ∆θ, of the arched fiber. R and Rn are the radius of the centroid axis and the neutral axis respectively. The external moment per length, Ma , and tension per length, Na , causes a strain, ǫa , at the centroid axis and change in angle, ∆α, of the cross-section plane.

The arched fiber bundle is approximated to a rectangular cross-sectional beam of thickness t, width dwidth , with an arch profile w0 (see Figure 1(b)) and a radius of curvature, R. With the following assumptions;

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1. The cross-sectional plane cut through the center of curvature and remains in a plane after bending. 2. The radius of curvature is much larger than the thickness, R >> t. 3. The deflection of the arched fiber, wp , is small.

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The strain along the with of the BM, ǫyy , is, ǫyy

=

δα δθ

=

R

=

γ

=

δα Rn ( − 1) δθ r δ 2 wa R − 2 δy γ 3/2 γ δ 2 w0 δy 2

s

1+(

4

δw0 2 ) δy

(1) (2) (3)

(4)

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units (in S.I ) N N · m−1 Pa Pa m 1 m3 m3 m 1

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Definition Moment per sectional length Force per sectional length Young’s modulus Shear modulus Radius of curvature Curvature correction Second moment of area per length Equivalent second moment of area per length Deflection Strain

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Variables M N E G R Am I η w ǫ

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where Rn is the radius of the neutral axis, R is the radius of curvature at the centroid axis, wa is the deflection of the arched fiber due to bending and δα/δθ is the change in angle of the cross-sectional plane along a small angular section ∆θ of the arch (see Figure 2).

Table 2: Definition and units of variables.

The moment, Ma , and tension, Na , per sectional length along the length of the BM in the arched fiber are, (5)

Na

=

(6)

=

Am

=

=

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= −E

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γη δα (ǫa + ) R δθ δα E[RAm ǫa + (RAm − t)] δθ dwidth Ef dspacing Z 1 dz t r R + t/2 ln( ) R − t/2 R2 (Rm Am − t) γ

Ma

η

=

(7) (8) (9) (10)

where ǫa is the strain at the centroid axis (r = R), η is the equivalent second moment of area per length of the arched fiber, Ef is the Young’s modulus of the fiber bundle and E is the equivalent Young’s modulus due to the fiber bundle spacing, dspacing (see Figure 1(a)). The total deflection of the arch fiber, wp , can be expressed as, δwp δy

=

δwa δw0 + ǫa δy δy

(11)

For the straight fiber, the moment and tension per sectional length along

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Ms Ns

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the length of the BM are, δ 4 wp δy 4 = Etǫs

(12)

= EI

(13)

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where I is the second moment of area per length and ǫs is the strain of the straight fiber at the centroid. 2.2. Soft-cells of basilar membrane

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To formulate the bending equation for the soft-cells in the BM, we simplify the problem with the following assumptions; 1. The cross-sectional plane (x-z-axis plane) remains in a plane after bending. 2. The Young’s modulus of the soft-cells is much smaller than that of the fiber bundle, Ef >> Eg (Young’s modulus of soft-cells in human is in the order of 104 Pa (Ahn and Kim, 2009)). 3. The deflection of the soft-cells, wp , is small. The deflection of the soft-cells can be separated into bending and shearing deflections, (14)

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wp (y) = wb (y) + ws (y)

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where ws and wb are the deflections of the soft-cells due to shearing and bending, respectively. We equate the strain of the soft-cells with that of the top and arched fiber bundles,

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ǫs − ǫa

Ns − ǫa Et

δ 2 wb δy 2 δ 2 wb = w0 2 δy

= −|w0 |

(15) (16)

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Using the continuity of traction at the straight fiber, G(ǫyz ) G(

δws ) δy Gws

δNs δy δNs = δy = Ns + Cs

=

(17) (18) (19)

where ǫyz is the engineering shear strain, G is the shear modulus, and Cs is the tension due to residual shearing stress. As the Young’s modulus of the soft-cells is much smaller than that of the fiber bundles, the change in total tension in the y-axis direction can be expressed

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by, py

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Nay

=

δ [Ns + Nay ] δy δw0 δMa γNa + δy δy

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(20)

(21)

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where Nay is the y-axis component of the arched fiber tension and py is the y-axis component of the external pressure. The pressure function along the cross-section of the BM(Equation 20) can be expressed in terms of the BM total deflection by inserting the mechanical properties of the soft-cells and fiber bundle as well as the appropriate boundary conditions (at the ends of the BM width). 2.3. Arched basilar membrane model In the present model for the arched BM, the boundary conditions at the shelves are taken to be hinged and the total deflection of a sinusoidal shape is assumed, with the width ranging from −b/2 to b/2. This shape can be considered the first mode and higher order modes will be discussed later, wp (y) =

Wp cos(

πy ) b

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The profile of the arch is approximated to a polynomial function;

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w0 (y) = c0 + c2 y2 + c4 y4

w0 |y=b/2

= h/10

(23) (24)

Ns + Nay

= Ne

(25)

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where the coefficients c0 , c2 and c4 are found by substituting the BM area A, width b, and height h into Equation 23. It is also approximated that there is no motion in y-direction, indicating py = 0. Equation 20 becomes,

where Ne is the external tension in the BM. The profile shape w0 and the deflection wp are referenced to the rest state of the BM with the presence of the resting radial tension (Naidu and Mountain, 2007). Therefore, Ne does not include the radial tension and is taken to be 0. From Equation 19, using the hinged boundary condition where Ms = 0 and ws = 0 at the shelves, Cs

=

0

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The solution to ws is found by discretizing along the y-axis (500 elements) and solving the simultaneous equation matrix. The matrix is filled via automatic differentiation with differential order reduced by central finite difference, thus,

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Wp (Pxx n4 + Pxy n2 + Pyy + Nyy ) δ 2 wp δy 2 2 δ Mxx δx2 δ 2 Mxy 2 δxδy δ 2 Myy δy 2 δMa δMs + − |w0 |Ns δy δy

Wp Nyy

= Nr

Wp Pxx

=

Wp Pxy Wp Pyy Myy

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there is no need to evaluate the coefficients of different orders of differentiation of ws manually (Chan, 2016). The pressure on the BM (see Figure 1) can be expressed with the coefficients of the wave number n (Chan and Yoon, 2015) as follows;

=

=

=

(27)

(28) (29) (30) (31) (32)

where Nr is the radial tension estimated from Naidu et al.’s work (Naidu and Mountain, 2007), = 1.51e−580x

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Nr (x)

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where x is the position along the BM from the basal end and all units are in SI units. Integrating the pressure along the y-axis of the BM, its inertia and the fluid pressure would yield the eikonal equation (formulation in Lim’s work (Lim, 2000)), R qdy + FM = Ftf (34) Wp

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where FM is the pressure integral due to the vibrating mass of the BM and Ftf is the total fluid pressure integral at the scalar vestibuli and scalar tympani. 3. Results and Discussion There are limited measurement data on the Young’s modulus of the fiber bundle and the estimations of fiber bundle Young’s modulus in different works differ significantly. Yoon et. al. (Yoon et al., 2007b) estimated the Young’s modulus of the fiber bundle to be in the order of 1 GPa, while Naidu et al. (Naidu and Mountain, 2007) predicted it to be around 14 MPa. ToR remove the error in such estimation, we compare the BM integral of pressure, Pyy dy, variation along the gerbil cochlea after scaling with Yoon et al.’s model (see Figure 3). The adapted arched BM model matches Yoon et al.’s gerbil cochlear 8

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Nr

Nr

Naidu et. al

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Yoon et. al’s model Naidu et. al’s model Present model Adapted model

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Scaled pressure integral, (N/ m)

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0.004 0.006 0.008 Distance from base (m)

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Figure 3: Plot of scaled pressure integral for comparison of Yoon et al.’s gerbil model, Naidu et. al.’s flatRsandwiched basilar membrane model, present BM model without inclusion R of radial tension( λPyy dy), and the adapted model with the inclusion of radial tension ( λ(Pyy + Nyy )dy). The scaling factors are λ = 1, 0.03, 52 and 47 respectively. The right panel illustrates Yoon et al.’s, Naidu et. al.’s, the present, and the adapted BM models. The thick lines indicate the fiber bundles. Yoon et al.’s model uses an equivalent BM thickness with correction for fiber bundles contribution instead of a composite. Naidu’s flat sandwich model is similar to present model except that both the top and bottom surface are flat. The present and adapted models have an arch toward the scalar tympani and the adapted model considers the radial tension.

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model in term of variation along the cochlear length while Naidu et al.’s sandwich BM model does not vary significantly. This result is in agreement with the parametric study in our previous study (Chan and Yoon, 2015) where the insufficient variation of radial bending stiffness is shown. In Figure 3, the present model excluding the radial tension is also shown and the data is much closer to that of Yoon et. al.’s simulation. Naidu and Mountain (2007) calculated the radial tension with measurements of the point-load deflection in the gerbil BM and a sandwich model without shearing. Therefore, the results might deviate from the actual radial tension values due to the presence of shearing in the soft-cells. One of the significant assumptions in the present model is that only the first harmonics is considered for the deflection. In reality, the true deflection, wp′ , is, wp′ (y) =

X m

Wm cos(

mπy ) b

(35)

where Wm is the amplitude associated with the mode, m. A first mode BM deflection profile does not necessarily produce a BM pressure profile with only the first mode (see Figure 4) due to the BM arch. However, the fluid pressure solution profile of the linearized Navier-Strokes equation with a 9

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1.0

wp /(wp|y=0)

Pyy /(Pyy |y=0)

0.8

Difference

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Ratio

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−0.2 0.00000

0.00002 0.00004 0.00006 0.00008 Distance from midpoint of BM y-axis cross-section (m)

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Absolute ratio of integral of Pyy to its quadratic integral

Figure 4: Comparison between ratio of wp /(wp |y=0 ) and Pyy /(Pyy |y=0 ) for the present model (G/E = 0) at 6.86 mm from base. The difference is also plotted. Both the wp and Pyy simulated here used only the first mode approximation.

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Figure 5: Absolute ratio of pressure integral to its quadratic integral for the present model (G/E = 0) at 6.86 mm from base. The boundary condition, wp |y=b/2 = 0, limits the modes such that only odd modes have amplitude and are calculated in this plot.

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quadratic integral

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single-wavelength (2b), sinusoidal-shaped fluid displacement is also sinusoidally shaped with the same wavelength (Yoon et al., 2007b). Therefore, the BM pressure and deflection profiles must match so that the fluid and BM displacements are equal at their boundary and the quadratic integral (or quadratic mean) of these different modes must be in the same order, sZ

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Figure 5 shows the ratio of the integral of pressure to its quadratic integral in different modes. The ratio of the first mode is more than 2.5 times higher than that of other modes. Although this indicates that the first mode assumption has a maximum error of 40%, the actual error is much smaller. This is because the quadratic integral of the first mode is much larger than that of other modes as the difference between the profiles of wp and Pyy is relatively small (see Figure 4). Considering the quadratic integral of higher modes is only about 30% of that of the first mode (see Figure 4), the calculated pressure integral shows a good estimation with approximately 12% error when the single mode assumption was used. 3.1. Simplification of arched basilar membrane model

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Because the present model is relatively complex, we attempt to make possible simplifications by emphasizing the significance of the fiber bundle radius of curvature, the gradient of the arch profile, and the shearing stiffness on the radial bending stiffness variation. The equivalent second moment of area per length, η, determines the effect of the radius of curvature on the bending moment (see Equation 5). When we only look at the change in the radius of curvature across the width of the BM, we can approximate the radius of the arched membrane (to value at y = 0) as a constant for regions away from the basal and apical ends of the BM (6.86 mm from base). The radius needs to be discretized and calculated nearer to the basal and apical ends due to the large changes near the shelves. At 11.24 mm from base, the radius at the shelves is twice the value to that at y = 0 and the difference is up to 10 times at the basal end (3.61 mm from base) (see Figure 6(a)). However, the value of η (second moment of area per length) is a function which depends on both the radius of curvature and the arch profile gradient (see Equation 10). It decreases to more than 15% from the center to the shelves of the BM (see Figure 6(b)). The change in equivalent second moment of area per length is therefore too significant to make any constant approximation for the calculation of the moment of arched fiber. It is also common (Yoon et al., 2007b; Naidu and Mountain, 2007) to estimate the BM as a flat beam (without arch profile in Figure 7(a)) and assumed that, δw0 δy

<< 1

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(a)

0.95 0.90

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Ratio of second moment of area per length η to value at y=0

1.00

0.75 0.70

3.61 mm from base 6.86 mm from base 11.24 mm from base

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0.65 0.00000 0.00005 0.00010 −0.00010 −0.00005 Distance from midpoint of BM y-axis cross-section (m)

(b)

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Figure 6: Plot of (a) radius of curvature and (b) equivalent second moment of area per length of the arched membrane at x= 3.61 mm, 6.86 mm and 11.24 mm from the basal end of the BM.

However, such assumption is invalid because the change in height of the BM is in the order of 1 near the shelves (see Figure 7(b)). We can also observe that change in gradient profile of the arch is almost similar along the cochlea from the basal to apical end (see Figure 7(b)) while the height of the BM increases (see Figure 7(a)). In this case, strain on the arched fiber bundle can be approximated with the profile gradient, δw0 /δy, expressed in terms of the distance from the center of the BM width (see Equation 11). We also simulated and calculated the pressure integral (without radial ten-

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Arched membrane profile, w0 (m)

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(a)

3.61 mm from base 6.86 mm from base 11.24 mm from base

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δw0/δy

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−1.0

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0.00000 0.00005 0.00010 −0.00010 −0.00005 Distance from midpoint of BM y-axis cross-section (m)

(b)

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Figure 7: (a) Arch profile w0 and (b) changes in arch profile along the y-axis of the BM at x= 3.61 mm, 6.86 mm and 11.24 mm from the basal end of the BM.

sion) for a finite value of the ratio of Young’s modulus to shear modulus, E/G (see Figure 8). The pressure integral variation along the BM approaches the simulation results of Yoon et. al.’s result as G tends to 0. In the present model, the shear modulus is approximate to 0. However, any approximation in the order of E/G ≥ 107 is quite accurate with less than an order difference. Near to the basal end of the BM, the variation of bending stiffness is relatively small with a large shear modulus (E/G ≤ 105 ). Significant changes in the variation occur when the ratio of Young’s modulus to shear modulus, E/G, is between 105 to 106 . The bending stiffness near the apical end of the BM is much more sensitive

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Yoon et. al’s model E/G=102 E/G=103 E/G=104 E/G=105 E/G=106 E/G=107 E/G=108 E/G=109

105

104

103

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0.004 0.006 0.008 Distance from base (m)

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Scaled pressure integral, (N/m)

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R Figure 8: Plot of scaled pressure integral, λPyy dy, (without radial tension) with different ratio of E/G. The scaling factors for E/G = 102 , 103 , 104 , 105 , 106 , 107 , 108 , 109 are λ = 0.020, 0.079, 0.59, 5.2, 27, 48, 52, 52 respectively.

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to the value of shear modulus of the soft-cells in the BM with a larger variation of pressure integral due to different shear modulus value. We are able to observe an optimum-like value (E/G ≈ 104 near the basal end and E/G ≈ 105 near the apical end, see Figure 8) where the variation of pressure integral along the BM is minimal. This is due to two factors which affect the pressure integral; the moment due to tension in the fiber, Ns and Na (Equation 32) and the large gradient near the shelves of the BM where the pressure, Pyy can be negative (see Figure 2) and thus, deviates more significantly from the first mode assumption of the deflection, wp . 3.2. Parametric study of the arched basilar membrane

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We study the effects of different parameters of the arched BM model on the resultant pressure integral with the present model (G = 0) as well as when E/G = 107 (an adequate approximation for a non-zero value of the shear modulus). The height at y = b/2 is varied in order to change the shape of the arch without changing the BM area, width and height (see Figure 9). When G is close to 0, the change in arch shape has insignificant effect (< 5%) on the calculated pressure integral. When G increases,the region near the basal end remains insensitive to the change in arch shape. However, the pressure integral near the apical end decreases > 50% when the height of the BM at the shelves decreases and the arch becomes more curved. With this decrease, the variation of pressure integral along the BM is in better agreements with Yoon et. al.’s model (see Figure 3). This variation of pressure integral with shape is also considered when choosing an appropriate E/G value (107 ). We also examine the variation of pressure integral with fiber bundle thickness (see Figure 10) and compare to the pressure integral of a pure shearing flat beam

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R Ratio R of Pyy to respective value of Pyy when w0 |y=b/2 = −h × 0.1

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w0|y=b/2 = 0

w0|y=b/2 = -h × 0.05

0.96

w0|y=b/2 = -h × 0.075

0.004

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0.005

0.006 0.007 0.008 0.009 Distance from base (m)

0.010

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0.012

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0.011

0.012

(a)

1.0

R Ratio R of Pyy to respective value of Pyy when w0 |y=b/2 = −h × 0.1

0.9

0.7

D

0.8

w0|y=b/2 = 0

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0.6

w0|y=b/2 = -h × 0.05 w0|y=b/2 = -h × 0.075

0.5

0.004

EP

0.003

w0|y=b/2 = -h × 0.1 0.005

0.006 0.007 0.008 0.009 Distance from base (m)

(b)

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Figure 9: Ratio of pressure integral of different height at the shelves to that of w0 |y=b/2 = −h/10 for (a) G = 0 and (b) E/G = 107 .

which follows the equation as shown below; Wp Pyy I

= 2EI =

t3 12

δ 4 wp δy 4

(38) (39)

where the soft-cells only act as a link between the top and bottom flat fiber bundles and the stiffness is just twice that of a single fiber bundle. For both cases (G = 0 and E/G = 107 ), the pressure integral increases in roughly 2–3 orders in the fiber thickness. When the fiber bundles are thick 15

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10−5

10−7

SC

Pressure integral, (N/m)

10−6

Flat, pure shearing sandwich beam 3.61 mm from base 6.86 mm from base 11.24 mm from base

10−8

M AN U

10−9 0.0000000

0.0000005

0.0000010 0.0000015 Fiber thickness (m)

0.0000020

(a)

10−7

D

10−6

TE

Pressure integral, (N/m)

10−5

Flat, pure shearing sandwich beam 3.61 mm from base 6.86 mm from base 11.24 mm from base

EP

10−8

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0.0000000

0.0000005

0.0000010 0.0000015 Fiber thickness (m)

0.0000020

(b)

Figure 10: Variation of pressure integral with fiber bundle thickness for (a) G = 0 and (b) G/E = 10−7 . The dotted line estimates the pressure integral for a pure shearing, flat sandwich beam which include effects of respective fiber spacing.

(>0.5 µm), both the present model and BM model when E/G = 107 have close values of pressure integral. However, at lower values of fiber bundle thickness, t, there is a significant increase in pressure integral (increase in stiffness) of the BM with a lower ratio of Young’s modulus to shear modulus (E/G). This also explains the sensitivity of the pressure integral near the basal end (where the fiber bundles are thinner) to the shear modulus of the soft cells (Figure 8).

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4. Conclusions

Acknowledgements

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The arch shape enables the radial bending stiffness to vary with the fiber bundle thickness in higher order of dependency. The relationship between the fiber bundle thickness and the radial bending stiffness is in the 2–3 order in an arched membrane compared to previous flat model estimation with linear relationship. This allows the gerbil cochlea to achieve comparative frequency tuning despite having a basilar membrane with relatively constant width and height (equivalent BM thickness). The contribution of arch shape in the radial bending stiffness was found to be less significant than that of the change in fiber bundle thickness. The radial stiffness decreased slightly with a sharper arch (larger peak to edge height) especially in the pure shearing model (present model). We also found that the radial bending stiffness of the gerbil BM was more sensitive to changes in the arch shape, fiber thickness, and shear modulus of its BM soft-cells near the apical end of the BM. This paper models the change in radial bending stiffness of the gerbil arched BM more accurately compared to previous, flat, no-shearing BM models. However, further experimental measurements on the properties of the BM fiber bundle are required to develop a gerbil cochlear model which is fully derived from physical formulations and experimentally measured parameters.

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References

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This work was funded by School of Mechanical and Aerospace Engineering, Nanyang Technological University and MOE AcRF Tier 1 (M4010494; RG35/10).

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The arched gerbil basilar membrane with shearing deformation is modeled. The model radial bending stiffness variation is consistent with the gerbil cochlea. Radial bending stiffness varies in second to third order with fiber thickness. Shearing deformation consideration is crucial to modelling the gerbil cochlea.

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