Biomechanics Principles and Applications - Schneck and Bronzino

(b)

(c)

FIGURE 19.3 (continued) (b) Long time response for various M values. Note the small undershoot for M = 1, no conditions of an overdamped system were put on the model. As the system is highly overdamped this set of nondimensional parameters (M = 1, R = 0.75, and ε = 0.2) would not be an acceptable solution. (c) Short time response for various values of ε. Note that this elastic parameter does not have much effect on the otoconial layer maximum displacement for a step change in velocity solution. This is not true for a step change in acceleration, see part (f).

Starting with the nondimensional fluid and gel layer equations, taking the Laplace transform with respect to time and using the initial conditions give two ordinary differential equations. These equations can then be solved, using the boundary conditions. Taking the Laplace transform of the otoconial layer motion equation, combining with the two differential equation solutions, and integrating otoconial layer velocity to get deflection produces the transfer function for displacement reacceleration:

(d)

(e)

FIGURE 19.3 (continued) (d) Effect of the mass or density parameter R on response. R changes the maximum otoconial layer displacement. (e) Response to a step change in acceleration for various values of the damping parameter M. Note the overshoot for M = 1.

(f)

FIGURE 19.3 (continued) (f) Acceleration step change time response for various values of the elastic parameter ε. Note that this parameter affects the final displacement of the system for this type of constant stimulus (compare with part (c)).

where the overbars denoting nondimensional variables have been dropped, s is the Laplace transform variable, and a general acceleration term A is defined as:

19.5 Otolith Frequency Response

This transfer function can now be studied in the frequency domain. It should be noted that these are linear partial differential equations and that the process of frequency domain analysis is appropriate. The range of values of ε = 0.01–0.2, M = 5–20, and R = 0.75 have been established by Grant and Cotton [1991] in a numerical finite difference solution of the governing equations. With these values established, the frequency response can be completed.

In order to construct a magnitudeand phase-vs.-frequency plot of the transfer function, the nondimensional time will be converted back to real time for use on the frequency axis. For the conversion to real time the following physical variables will be used: ρo = 1.35 g/cm3, b = 15 µm, µf = 0.85 mPa·s. The general frequency response is shown in Fig. 19.4. The flat response from DC up to the first corner frequency establishes this system as an accelerometer. These are the range-of-motion frequencies encountered in normal motion environments where this transducer is expected to function.

The range of flat response can be easily controlled with the two parameters ε and M. It is interesting to note that both the elastic term and the system damping are controlled by the gel layer, and thus an animal can easily control the system response by changing the parameters of this saccharide gel layer. The cross-linking of saccharide gels is extremely variable, yielding vastly different elastic and viscous properties of the resulting structure.

The otoconial layer transfer function can be compared to recent data from single-fiber neural recording. The only discrepancy between the experimental data and theoretical model is a low-frequency phase lead and accompanying amplitude reduction. This has been observed in most experimental single-fiber recordings and has been attributed to the hair cell.

FIGURE 19.4 General performance of the octoconial layer transfer function shown for various values of the nondimensional elastic parameter E and M. For these evaluations the other parameter was held constant and R = 0.75. The value of M = 0.1 shows an underdamped response which is entirely feasible with the model formulation, since no restriction was incorporated which limits the response to that of an overdamped system.

19.6 Semicircular Canal Distributed Parameter Model

The membranous SCC duct is modeled as a section of a rigid torus filled with an incompressible newtonian fluid. The governing equations of motion for the fluid are developed from the Navier–Stokes equations. Refer to the nomenclature section for definition of all variables, and Fig. 19.5 for a crosssection of the SCC and utricle.

everyday movement. Environments such as an aircraft flight can produce frequencies outside the linear range for these transducers.

Rabbit and Damino [1992] have modeled the flow of endolymph in the ampulla and its interaction with a cupula. This model indicates that the cupula in the mechanical system appears to add a highfrequency gain enhancement as well as phase lead over previous mechanical models. This is consistent with measurements of vestibular nerve recordings of gain and phase. Prior to this work this gain and phase enhancement were thought to be of hair cell origin.

Defining Terms

Endolymph: Fluid similar to intercellular fluid (high in potassium) which fills the membranous labyrinth, canals, utricle, and saccule.

Kelvin–Voight viscoelastic material: The simplest of solid materials which have both elastic and viscous responses of deformation. The viscous and elastic responses appear to act in parallel.

Otolith: Linear accelerometers of the vestibular system whose primary transduced signal is the sum of linear acceleration and gravity in the frequency range from DC (static) up to the maximum experienced by an animal.

Semicircular canals: Angular motion sensors of the vestibular system whose primary transduced signal is angular velocity in the frequency range of normal animal motion.

References

Grant JW, Cotton JR. 1991. A model for otolith dynamic response with a viscoelastic gel layer. J Vestibular Res 1:139.

Grant JW, Best WA, Lonegro R. 1984. Governing equations of motion for the otolith organs and their response to a step change in velocity of the skull. J Biomech Eng 106:203.

Grant JW, Huang CC, Cotton JR. 1994. Theoretical mechanical frequency response of the otolith organs.

J Vestibular Res 4(2):137.

Lewis ER, Leverens EL, Bialek WS. 1985. The Vertebrate Inner Ear, Boca Raton, FL, CRC Press.

Rabbit RD, Damino ER. 1992. A hydroelastic model of macromachanics in the endolymphatic vestibular canal. J Fluid Mech 238:337.

Van Buskirk WC. 1977. The effects of the utricle on flow in the semicircular canals. Ann Biomed Eng 5:1. Van Buskirk WC, Grant JW. 1973. Biomechanics of the Semicircular Canals, pp. 53–54, New York, Bio-

mechanics Symposium of American Society of Mechanical Engineers.

Van Buskirk WC, Grant JW. 1987. Vestibular mechanics. In R. Skalak, S. Chien (Eds.), Handbook of Bioengineering, pp. 31.1–31.17, New York, McGraw-Hill.

Van Buskirk WC, Watts RG, Liu YU. 1976. Fluid mechanics of the semicircular canal. J Fluid Mech 78:87.

Nomenclature

Otolith Variables

x = coordinate direction in the plane of the otoconial layer

yg = coordinate direction normal to the plane of the otolith with origin at the gel base yf = coordinate direction normal to the plane of the otolith with origin at the fluid base t = time

u(yf,t) = velocity of the endolymph fluid measured with respect to the skull v(t) = velocity of the otoconial layer measured with respect to the skull

w(yg,t) = velocity of the gel layer measured with respect to the skull δg(yg,t) = displacement of the gel layer measured with respect to the skull

δo = displacement of the otoconial layer measured with respect to the skull

Vs = skull velocity in the x direction measured with respect to an inertial reference frame V = a characteristic velocity of the skull in the problem (magnitude of a step change)

Semicircular Canal Variables

u(r,t) = velocity of endolymph fluid measured with respect to the canal wall r = radial coordinate of canal duct

a = inside radius of the canal duct

R = radius of curvature of semicircular canal ρ = density of endolymph fluid

v = endolymph kinematic viscosity

ω = angular velocity of the canal wall measured with respect to an inertial frame

α = angular acceleration of the canal wall measured with respect to an inertial frame K = pressure-volume modulus of the cupula = ∆p/∆V

∆p = differential pressure across the cupula

∆V = volumetric displacement of endolymph fluid

β = angle subtended by the canal in radians (β = π for a true semicircular canal)

λn = roots of Jo(x) = 0, where Jo is Bessel’s function of order 0 (λ1 = 2.405, λ2 = 5.520)