Biomechanics Principles and Applications - Donald R. Peterson & Joseph D. Bronzino

expansion of the cell over a narrow frequency band, which is related to the place for the cell along the cochlea. Khanna and coworkers [1989] observe a similar tonic displacement of the whole organ of Corti.

17.5 Active Models

De Boer [1991], Geisler [1993], and Hubbard [1993] discuss models in which the electromotility of the outer hair cells feeds energy into the basilar membrane. The partition stiffness K is expanded from Equation 17.3 into a transfer function, containing a number of parameters and delay times. These are classed as phenomenological models, for which the physiological basis of the parameters is not of primary concern. The displacement gain may be defined as the ratio of ciliary shearing displacement to cell expansion. For these models, the gain used is larger by orders of magnitude than the maximum found in laboratory measurements of isolated hair cells.

Another approach [Steele and colleagues (1993)], which is physiologically based, appears promising. The outer hair cells are inclined in the propagation direction. Thus the shearing of the cilia at the distance x causes a force from the hair cells acting on the basilar membrane at the distance x + x. This “feed-forward” law can be expressed in terms of the pressure as

where p, the total pressure acting on the basilar membrane, consists of the effective pressure acting on the basilar membrane from the hair cells pohc and the pressure from the fluid pf. The coefficient α is the force gain supplied by the outer hair cells. With this law, Equation 17.11 is replaced by:

from which the local wave number n must be computed numerically. Only two new parameters are needed, the gain α and the spacing x. With physiologically reasonable gain α = 0.18 and spacing x = 20 μm, the result is an increase of the response of the basilar membrane for higher frequencies by a factor of 102 in a narrow sector, apical to the passive peak. The simple feed-forward addition in Equation 17.17 enhances a narrow band of wave lengths without a closed control loop. At this time, it appears that much of the elaborate structure of the organ of Corti is for the purpose of such a “feed-forward.” This approach is also found to work well for the 1-D model by Geisler and Sang [1995]. The results from the 3-D model for the effect of adding some feed-forward are shown in Figure 17.3. One defect of the current feed-forward results is that the shift of the maximum response point is about one octave, as seen in Figure 17.3, rather than one-half octave consistently shown in normal cochlear measurements of mechanical and neural response.

The significant nonlinear effect is the saturation of the active process at high amplitudes. This can be computed by letting the gain α be a function of the amplitude.

In the normal, active cochlea, it was first observed by Khanna and colleagues [1989] and subsequently by Gummer and colleagues [1996], that the tectorial membrane (TM in Figure 17.1) has a substantially higher amplitude than the basilar membrane. Presumably, the electromotile expansion of the outer hair cells encounters less resistance from the tectorial membrane than the basilar membrane. This shows the importance of getting the correct stiffness and geometry into a model.

17.6 Fluid Streaming

Bek´esy´ [1960] and many others have observed significant fluid streaming in the actual cochlea and in experimental models. Particularly for the high frequencies, it is tempting to seek a component of steady streaming as the significant mechanical stimulation of the inner hair cells [Lighthill, 1992]. Passive models indicated that such streaming occurs only at high sound intensity. Among the many open questions is whether or not the enhancement of amplitude provided by the feed-forward of energy by the outer hair

cells and the mechanical nonlinearity at low amplitudes of displacement provided by the ciliary gating can trigger significant streaming. It is clear from Figure 17.1 that the motion and corresponding pressure in the fluid of the inner sulcus is the primary source of excitation for the inner hair cells. The DC pressure associated with DC streaming would be an important effect.

17.7 Clinical Possibilities

A better understanding of the cochlear mechanisms would be of clinical value. Auditory pathology related to the inner ear is discussed by Pickles [1988] and Gulick and colleagues [1989]. The spontaneous and stimulated emissions from the cochlea raise the possibility of diagnosing local inner ear problems, which is being pursued at many centers around the world. The capability for more accurate, physically realistic modeling of the cochlea should assist in this process. A significant step is provided by Zweig and Shera [1995], who find that a random distribution of irregularities in the properties along the cochlea explains much of the emissions.

A patient with a completely nonfunctioning cochlea is referred to as having “nerve deafness.” In fact, there is evidence that in many cases the nerves may be intact, while the receptor cells and organ of Corti are defective. For such patients, a goal is to restore hearing with cochlear electrode implants to stimulate the nerve endings directly. Significant progress has been made. However, despite electrode stimulation of nerves at the correct place along the cochlea for a high frequency, the perception of high frequency has not been achieved. So, although substantial advance in cochlear physiology has been made in the recent past, several such waves of progress may be needed to adequately understand the functioning of the cochlea.

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Further Information

The following are workshop proceedings that document many of the developments. De Boer E., and Viergever M.A., Eds. 1983. Mechanics of Hearing. Nijhoff, The Hague.

Allen J.B., Hall J.L., Hubbard A., Neely S.T., and Tubis A., Eds. 1985. Peripheral Auditory Mechanisms. Springer, Berlin.

Wilson J.P. and Kemp D.T., Eds. 1988. Cochlear Mechanisms: Structure, Function, and Models. Plenum, New York.

Dallos P., Geisler C.D., Matthews J.W., Ruggero M.A., and Steele C.R., Eds. 1990. The Mechanics and Biophysics of Hearing. Springer, Berlin.

Duifhuis H., Horst J.W., van Kijk P., and van Netten S.M., Eds. 1993. Biophysics of Hair Cell Sensory Systems. World Scientific, Singapore.

Lewis E.R., Long G.R., Lyon R.F., Narins P.M., Steele C.R., and Hecht-Poinar E., Eds. 1997. Diversity in Auditory Mechanics. World Scientific, Singapore.

FIGURE 18.1 Schematic of the otolith organ: (a) Top view showing the peripheral region with differential area d A where the model is developed, and the striolar area that makes an arch through the center of the otolith. (b) Cross section showing the layered structure where dx is the width of the differential area d A shown in the top view.

In the equation of motion for the endolymph fluid, the force τfdA acts on the fluid at the fluid–otoconial layer interface (Figure 18.2). This shear stress τf is responsible for driving the fluid flow. The linear NavierStokes equation for an incompressible fluid are used to describe this endolymph flow. Expressions for the pressure gradient, the flow velocity of the fluid measured with respect to an inertial reference frame, and the force due to gravity (body force) are substituted into the Navier-Stokes equation for flow in the x-direction yielding

with boundary and initial conditions: u(0, t) = v(t); u(∞, t) = 0; and u(yf, 0) = 0.

The gel layer is treated as a Kelvin–Voight viscoelastic material where the gel shear is acting in parallel. This viscoelastic material stress has both an elastic and viscous component acting in parallel. This viscoelastic material model was substituted into the momentum equation and the resulting gel layer equation of

with boundary and initial conditions: w (b, t) = v(t); w (0, t) = 0; w (yg, 0) = 0; and δg(yg, 0) = 0. The elastic term in the equation is written in terms of the integral of velocity with respect to time, instead of displacement, so the equation is in terms of a single dependent variable, the velocity.

FIGURE 18.2 The free-body diagrams of each layer of the otolith with the forces that act on each layer. The interfaces are coupled by shear stresses of equal magnitudes that act in opposite directions at each surface. The τg shear stress acts between the gel–otoconial layer and the τf acts between the fluid–otoconial layer. The forces acting at these interfaces are the product of shear stress τ and area dA. The Bx and Wx forces are the components of the buoyant and weight forces acting in the plane of the otoconial layer, respectively. See the nomenclature table for definitions of other variables.

(a) Endolymph fluid layer: The spatial coordinate in the vertical direction is yf and the velocity of the endolymph fluid is u(yf, t), a function of the fluid depth yf and time t. (b) Otoconial layer: The otoconial layer with thickness b and at a height b above the gel layer vertical coordinate origin. The velocity of the otoconial layer is v(t), which is a function of time only. (c) Gel layer: Gel layer of thickness b, vertical coordinate yg, and horizontal coordinate δg the gel deflection. The gel deflection is a function of both yg and time t. The velocity of the gel is w (yg, t), a function of yg and t.

The otoconial layer equation was developed using Newton’s second law of motion, equating the forces that act on the otoconial layer: fluid shear, gel shear, buoyant, and weight to the product of mass and inertial acceleration. The resulting otoconial layer equation is

(18.3)

with the following initial conditions v(0) = 0. The otoconial layer displacement can be calculated by integrating the velocity of this layer with respect to time

Displacement of the otoconial layer is the variable of importance, it is proportional to acceleration, and is the variable transduced into a neural signal.

18.4 Nondimensionalization of the Otolith Motion Equations

The equations of motion are then nondimensionalized to reduce the number of physical and dimensional parameters, and to combine them into some useful nondimensional numbers. The following nondimensional variables, which are indicated by overbars, are introduced into the motion equations:

Several nondimensional parameters occur naturally as a part of the nondimensionalization process; these parameters are

These parameters represent the following: R is the density ratio and represents the system nondimentional mass, ε is a nondimensional elastic parameter and represents the system elasticity, M is the viscosity ratio and represents the system damping, and g¯x is the nondimensional gravity.

The governing equations of motion in nondimensional form are then as follows: