A MODEL FOR OTOLITH DYNAMIC RESPONSE WITH A VISCOELASTIC GEL LAYER

Journal of Vestibular Researh, Vol. 1, pp 139-151, 1990/91 Printed in the USA. All rights reserved. 0957-4271/!:l1 :J>j.UU +.uu Copyright 1991 Pergamon Press pi A MODEL FOR OTOLITH DYNAMIC RESPONSE WITH A VISCOELASTIC GEL LAYER J. W. Grant and J. R. Cotton Engineering Siene and Mehanis Department,. Virginia Polytehni Institute and State University, Blaksburg, Virginia Reprint address. Dr. J. W Grant, Engineering Siene and Mehanis Department, Virginia Polytehni Institute and State University, Blaksburg, VA 24061-0219 o Abstrat - The otolith organs were modeled mathematially as a 3-element system onsisting of a visous endolymph fluid in ontat with a rigid otoonial layer that is attahed to the skull by a gel layer. The gel layer was onsidered to be a visoelasti solid, and was modeled as a simple Kelvin material. The governing differential equations of motion were derived and nondimensionalized, yielding 3 nondimensional parameters: nondimensional density, nondimensional visosity, and nondimensional elastiity. The equations were solved using finite differene tehniques on a digital omputer. By omparing the model's response with previous experimental researh, values for the nondimensional parameters were found. The results indiate that the inlusion of visous and elasti effets in the gel layer are neessary for the model to produe otoonial layer defletions that are onsistent with physiologi displaements. Future experimental data analysis and mathematial modeling effets should inlude visoelasti gel layer effets, as this is a major ontributor to system damping and response. o Keywords - otolith; visoelasti; gel layer; distributed parameter model; dynami response. Introdution A distributed parameter model was previously developed representing otoonial layer displaement in response to inertial motion of the skull and gravity (5,6). This work modeled the otoliths as a system of 3 homogenous isotropi materials: the endolymph fluid, the gel layer, and the otooniallayer. Eah material maintained a onstant height, the otooniallayer was treated as a rigid body, and edge effets were negleted. The simplified model is shown in Figure 1. This work modeled the gel layer as a linearly elasti material. The material was in simple shear, whih produes a linear displaement profile when deformed. This model produed displaements that greatly exeeded expeted results. For reasonable values of gel elastiity, the otoonial layer would displae several times its height. Suh large displaements would almost ertainly ause permanent damage to the material. Also, when the elastiity was inreased to provide smaller displaements, the otoonia! layer returned to its equilibrium position too rapidly and the system would osillate. The system has long been reognized to be overdamped (2) and one's sensation of motion is unidiretional not osillating. Therefore, the osillatory response was onsidered to be in error. More importantly, if the system elastiity was inreased dramatially it would not funtion as an overdamped aelerometer. In an effort to evaluate what needed to be hanged in the distributed parameter system model, a lumped parameter model was developed (7). This analysis indiated that more system damping was required. It was speulated that the gel layer material is visoelasti, having both elasti and visous harateristis. By making the gel visoelasti, the short time displaements should be redued, and the long-term effets would be strethed out RECEIVED 9 Marh 1990; REVISED 26 June 1990; ACCEPTED 4 July 1990. 139

140 J. W. Grant and J. R. Cotton y da Endolymph Fluid 2b Otoonia! Layer b o v, Figure 1. Simplified model of the otolith as used in referene 5. further in time. More importantly the elasti stiffness ould be inreased without system osillation, and the steady state.displaement in response to head tilt ould be redued. This is not a new idea and was first suggested by De Vries (2) in what was probably the first lumped parameter modeling effort of this system. The present work is a ontinuation of the earlier distributed parameter model to whih a visoelasti gel layer material has been added. The results indiate that this addition produes reasonable displaements and realisti dynami results. Governing Equations The basi formulation of this problem began as did the previous work of Grant et al. (5). The system was treated as 3 separate parts: the endolymphati fluid, the rigid otooniallayer of dense rystals, and the gel layer, previously modeled only as an elasti material, but in this work modeled as a visoelasti material. For eah of the 3 omponents, the free-body diagram was drawn (shown in Figure 2) and its governing equations were derived. Beause the length and width of the otoonia! membrane are severa! orders of magnitude larger than the height, the end or edge effets on the plate were negleted. The problem then approximated a plate with infinite extent on top of a gel layer, with visous fluid above. The gelatinous layer was onsidered as homogenous. The effets of the sensory ilia, whih are enapsulated in fluid-filled avities within the gel layer, and the striolar region were negleted. The ilia stiffness undoubtedly has an effet on the gel layer elasti deformation. Values for the ilia tuft stiffness have reently been measured and are available. However, nothing is available on the gel material deformation properties. Histologi studies indiate that the lower gel layer has less struture, whih would imply less stiffness. The ilia would tend to add stiffness in this region, whih may ompensate. In any event, not enough information is available to onstrut a meaningful model of the gel layer stiffness variation with depth. The striolar re-

Gel Layer in Otolith Dynami Response Model 141 gion is small in proportion to the total surfae area and should have little effet on the dynamis. In Figure 3, the free body diagram of an element of the gel in shear an be seen. The fores ating on the element are the fores produed by shear stress on the top and bottom layers, 7 g + d7g and 7 g, the pressure on both sides of the element, Px and Px+dx, and the body fore on the element due to the omponent of gravity in the x diretion, gx. Using the momentum equation, Grant et al (5) derived the equation Y r Endolymph Fluid [1] representing gel motion (nomenlature definitions are given at the end). Previously, the shear stress 7 was modeled as purely elasti, resulting in deformations muh larger than expeted for realisti values of elastiity. Most gels, espeially bio-gels, are of a visoelasti nature. By treating the 'gel as a visoelasti material, the motion will be damped and the model should more losely resemble reality. The Kelvin-Voight model for visoelasti materials was used beause of its mathematial simpliity. In this model, the shear stress 0-6-r7..,.,...,.,r-r:r-r7TTTT7-r----... is onsidered to be a funtion of both disx plaement and veloity in a way analogous to -V 5 a spring and a dashpot linked in parallel. Hene, the shear stress is expressed as ao aw 7 g = G a + f.lg -a' Yg Yg [2] where G is the shear modulus, 0 is the displaement, and f.lg is the gel visosity. By substituting into the equation (2) for 0 the time integral of the veloity, ---I" w(y,,t) Figure 2. Free body diagram of the 3 omponents of the otolith model. By substituting this result into equation 1) tl-j gel layer governing: eauation of motion beomes [5] we get i1 aw aw 7 g = G - dt + f.lg -. o ayg ayg [3] [4] The otoonial layer's equation of motion began by examining the free body diagram of the layer as shown in Figure 2. The shear stress experiened at the underside of the otoonial layer, whih is aused by the gel, 7 g, is given by equation (4). Other fores ating on the

142 J. W. Grant and J. R. Cotton I... ax o------------------------------------------------.- o x Figure 3. Free body diagram of an element within the gel layer of the otolith. otooniallayer are the buoyant fore per unit volume Bx = Pj(a Vslat - gx)' the weight omponent in the x-diretion Wx = Pogx, and the fluid shear stress ating above 7j = -/J-j(aul ayj)lo' When the fores are summed and equated to the produt of mass and inertial aeleration, the plate governing equation of motion redued to au I it aw I aw I =/J-j- -0 - dt-/j-g-. ayj 0 0 ayg b ayg b [6] This is similar to the previous work of Grant et al (5) exept for the last term, whih represents the visous shear in the gel. The fluid governing equation is derived from the linearized version of the Navier Stokes equation. The linearized Navier Stokes equation assumes that there is no onvetive fluid motion, whih is a reasonable assumption for suh a highly visous, low Reynolds number flow. The Navier Stokes equation assumes a linearly visous fluid. When pressure, body fores, and visous fores are onsidered along with inertial motion of the fluid in the x-diretion, the Navier-Stokes equation redues to This analysis produes a set of 3 oupled partial differential equations similar to those derived earlier (5) exept for the inlusion of the visoelasti gel. The boundary onditions are that of no veloity and no displaement at the gel-skull interfae w(o, t) = 0, and no endolymph veloity at the upper wall of the hamber u (d, t) = where d is the height of the endolymph above the otooniallayer. Previous analysis (5) onsidered d to be an infinite distane. This boundary ondition and its effets are disussed later. In addition, the onsisteny of the interfae shear stress between the endolymph and the otooniallayer, as well as that between the otoonial layer and the gel, is built into the governing equations. N ondimensionalization [7] The equations were nondimensionalized in order to promote a numerial solution and

Gel Layer in Otolith Dynami Response Model 143 ahieve some nondimensional parameters that haraterize the problem. We began nondimensionalization by introduing the dimensionless variables for spae YJ and Yg, time r, and veloities ii, ii, iv, and fs. These are defined as and _ u _ v _ w u= - v= - w= - V' V' V' [8] [9] - fs Vs = V' [10] where V is some harateristi veloity of the problem. By introduing these nondimensionalized variables, the governing equations beome -= au + (1 - R) [afs] -- - g at at x and [11] [12] [13] where the nondimensional parameters R, E, and M, represent, in order, nondimensionalized density, elastiity, and visosity, and are defined as and R = PJ, [14] Po M= I1-g. [16] I1-J Solution and Results for a Step Change in Skull Veloity The governing equations were solved for a step hange in skull veloity to establish the dynami response of the system. The entire system was onsidered suh that when time is started, t = 0, the skull will be given a step hange in veloity of - VS' The fluid endolymph and gel material will begin at rest, at time zero, and would experiene the same step hange in skull veloity, sine they are in a losed volume within the skull. Due to inertia, the otooniallayer, whih also begins at rest at time zero, will be aelerated to the skull veloity, produing a relative displaement of the otooniallayer. Sine, however, the veloities in the governing equations are measured with respet to the skull, and the fluid ompartment is a losed apsule of equal density material (exept for the otooniallayer with its dense alium arbonate rystals), the effets are equivalent to the otoonial layer being given a step hange in veloity of magnitude (1..:;: R) Vs, but opposite in diretion, to that of the skull, while the skull and surrounding gel and fluid remain at rest. The solution to the governing equations was found using finite differene analogues for the differential terms and oding the results into Fortran for solution by digital omputer. Solution involved using impliit analogues for numeri stability, along with the method of imaginary points, whih involves shear stress mathing for the plate boundary between the fluid endolymph and the gel layer. The integral terms were approximated using a summation proess with 3 terms based on Simpson's rule, whih ontained the new veloities at the next Slep lorwari ill time. The resulting differene equations were solved using a linear algorithm. The displaements were then updated using the new veloities at the new time. Errors due to the disretization of the integration terms [15] by summation were heked by hanging time step sizes and were well into the 5th signifiant figure even for extremely long times. The model was reated suh that the effets of fluid visosity, gel visosity, and gel elastiity ould be independently removed.

144 The parameter M an be taken to zero, thereby removing the effets of solid visosity. The parameter E an also be made zero, whih removes the effets of gel elastiity. Finally, a parameter K was introdued in front of every term aused by the fluid visosity. Hene, when K is unity, we have a ase where the fluid is visous, while if we set K to zero, the fluid visosity is removed and we have a "no fluid" model. The parameter K is essentially an on/off svvith for fluid '!lsslt:'. B' independently setting these 3 parameters, we an rereate earlier work, whih negleted one or more of these harateristis, and ompare results. By removing both gel elastiity and fluid visosity, we have the lassial flow problem, whih has been solved analytially by Couette. The results of the model are onsistent with this. With the gel modeled as a visous fluid and the effets of the endolymph fluid visosity removed, as time grows, t -+ 00, the displaement also grows indefinitely, 0 -+ 00. As the elastiity is slowly added, the displaements begin to beome finite, whih is expeted. Also by removing the gel visosity, we an rereate the earlier results (5) seen by the elasti-only gel material. Eah of these steps J. W. Grant and J. R. Cotton onfirmed the orret funtioning of the model solutions. With the model's validity onfirmed, the effets of eah of the nondimensional parameters E, R, and M were studied by varying one while holding the others onstant at probable values. The parameter that is best estimated is R, whih is the ratio of the gel density (equal to fluid density) to the density of the otoonial membrane. The alium arbonate rystals have an approximate density value of 2.7 g/ em 3 (1), while both the gelatinous membrane that makes up the gel layer and the matrix for the otoonial membrane and the endolymph fluid have been measured at 1.0 g/m 3 (12). Hene, the value for R must be between 0.35 and 1.00. More signifiantly, the work of Steinhausen (14) puts the speifi weight of the otooniallayer at between 1.32 and 1.39. This translates to an R value of between 0.76 and 0.72. In order to investigate the effets on the system, values of R were run from 0.25 to 0.9. Plate displaement versus time is shown for different values of R in Figure 4. Notie that as R inreases (dereasing otoonial layer density), displaement dereases; however, the time needed to reah the maximum displaement remains nearly 3.2e-02 Plate Displaement vs. Time (E=0.2, M=S.O) --------------- R = 0.25 ==.... <.I e.3 Q =. =.. e Q 2.4e-02 1.6e 02 8.0e-03 _---------------- R = 0.5 _---------------- R = 0.75 0.0e+00 o 0.2 0.4 Dimensionless Time 0.6 0.8 Figure 4. Dimensionless displaement of the otoonial layer versus dimensionless time for various values of R (E =0.2, M =5.0).

Gel Layer in Otolith Dynami Response Model the same. Other than this, there is no effet on the urves. R only affets the amplitude. This is expeted, sine the displaement is driven by the differene in densities. As R approahes one, the density of the plate equals that of the fluid, and no perturbation of the otolith is possible. The value of E, whih represents nondimensionalized elastiity, ould be varied greatly. Values approahing zero were studied as well as values that drove the system so elasti that it osillated. Sine the system is generally aepted to be overdamped, values of E that ause osillation shall onstitute the upper limit of the investigation. For short times, there was little differene between ases of zero elastiity and values on the order of magnitude of 10-2, while osillation beame obvious between E values of 1 and 10. Grant et al (5) used a ombination of measured and estimated values for parameters that make up E and arrived at a value of E around 0.028. However, many of the values used are quite unstudied and require extrapolation from known values of other biomaterials. All in all, we seriously onsidered values of E ranging from 0.002, whih gave the same response as E = 0.0, to 10, whih would ause the sys- 145 tern to osillate notieably. Establishing a loser range for the value of E will be onsidered in the Disussion setion. Plate displaement versus time is shown in Figure 5 for various E values from 0 to 5. Finally the third parameter, M, whih is the ratio of the gel visosity to fluid visosity, was investigated. With inreasing M, the gel visosity inreases, produing a derease in all veloity magnitudes. This has two effets. First, the higher M values ause a smaller maximum displaement. Seondly, the higher the M values, the more slowly the plate displaement returns to zero after a displaement, in essene, inreasing system damping. Plate displaement versus time for various values of M are shown in Figure 6. To establish the proper value of the fluid layer height required some refletion during the solution proess. It was originally assumed by Grant et al (5) that the order of magnitude differene between the otolith height and the fluid height would suggest an infinite fluid layer, while modeling the system. However, the present analysis showed signifiant flow near this upper boundary. Sine the upper boundary was held still, this shear traveled bak down through the fluid,.... e Q "i.2 e 1.0e-02 7.Se-03 S.Oe-03 2.Se 03 <= Z Plate Displaement vs. Time (M=S., R=0.7S) E = 0.2 ========:::::::::::: 0.0 E = 1.0 E = 2.0 E = S.C O.Oe+OO o 0.2 0.4 0.6 0.8 Noodimensional Time Figure 5. Dimensionless displaement of the otoonial layer versus dimensionless time for various values of E (R =0.75, M =5.0).

146 J. W. Grant and J. R. Cotton 4.0e 02 Plate Displaement vs. Time (E=O.2, R=0.7S) ---M=1... " E " ::i -;. 3.0e 02 2.0e 02 :a 1.0e 02 o Z O.Oe+OO --------- M = 2. _ --M=5. _ M = 10. ------------------ M = 20. o 0.2 0.4 0.6 0.8 Nondimensional Time Figure 6. Dimensionless displaement of otoonial layer versus dimensionless time for various values of M (E =0.2, R =0.75). slightly impeding plate motion. Therefore, the height of the fluid layer in the model was proportional to its value in vivo of approximately 1 mm. Another interesting result was observed when the veloity profiles of the gel layer were examined. Figure 7 shows these profiles for different times in the present solution of a step hange in skull veloity. At the earliest times, the profiles display an approximately exponential urve, indiating higher visous shear stress near the otooniallayer. After the otoonial layer reahes its maximum defletion, the veloity profile shallows to a straight line. This would imply that the gel layer is in simple shear after the otooniallayer reahes maximum displaement. This behavior indiates that the most important ontribution of the visoelasti gel is in this initial fast displaement phase where the otoonial layer reahes maximum displaement. Disussion The dimensionless displaement and time must be onverted bak to real length and time so that they may be ompared to physially observed results. By inverting the results of nondimensionalization, we get p O b 2 ] _ t = -- t. [ jj.f Approximate human values for the above parameters are: Po = 1.35 g/m 3 (14), b = 15.jJ.m (8), jj.f = 0.85 m Pa s (l3). The harateristi veloity V is taken to be 1 mis, a kind of average maximum for normal human movement. Using these parameters, the dimensional time and displaement beome and (j = (3.57 x 10-4 )8 meters t = (3.57 x 10-4 ) i seonds. These onversions result in an otooniallayer displaement of 3.5 jj.m in 0.10 ms, when R = 0.75, E = 0.2, and M = 5.0. The values of R, E, and Mused here are good estimates of the physiologi values, for reasons whih are disussed later in this setion. This displaement

Gel Layer in Otolith Dynami Response Model 147 Veloity Profile (short time) 0.8..:: -a Qi 0.6 C;.;; = Q,I 0.4 5 : = Z 0.2 All times in dimensionless units xlo"5 o+----------------------------------+ o 0.04 0.08 0.12 0.16 0.2 0.24 0.28 Nondimensional Veloity (a) Veloity Profile (long time) \.0 +----r...,-,-_: --'--,----'--- +---- +..:: =- Q,I "C 0.6 C; e 0t;. Q,I E : = Z 0.8 t :0.5 0.4 0.2 All times in dimensionless units xl0"3 o -----r---------------r------r----------+ o 0.04 0.08 0.12 0.16 0.2 0.24 0.28 Nondimensional Veloity (b) Figure 7. Gel layer veloity profiles for short and long times (M =5.0, E =0.2, R =0.75).

148 and time reflet the nature of the model to produe realisti values. We an get a reasonable value for the nondimensional elastiity E by observing the steady-state ase of an otolith under a onstant gravitational aeleration. Here the governing equation for the plate (equation 6) loses all transient terms, and we are left with the gravitational aeleration term equaling the elasti term aa =-E-. ayg [17J We note from earlier observations that for long time solutions, whih this ertainly is, the gel is in simple, linear, shear. Hene ab b p -=ay LlY [18] g where b p is the defletion of the otoonial membrane and LlY g = 1, as the gel height is b. Hene, we an solve for E, E = (l - R) x. op [19] We an now plae E between 0.3 and 0.08 when we take R to be between 0.5 and 0.75 and the physiologi displaement due to one g to be 2 to 4 JLm (3). The most likely value of E is 0.2. By lumped parameter modeling along with linial and physiologial observation, we know that the relative displaement response of the otolith to a step hange in veloity an be approximated as an inverse exponential limb in a very short time followed by an inverse exponential deay of very long time. Mathematially, the system an be desribed by the lumped parameter, linear system model solution, where Dmax is the maximum otoonial layer displaement. Tt and 'Fs are the long and short time onstants respetively (6). This J. W. Grant and J. R. Cotton mathes the quik displaement of the plate followed by a slow elasti return, seen both physiologially and in the present model. Values of the long and short time onstants whih should be used in this approximate model will be disussed in the following paragraphs along with their impliations in the present model. The short time onstant derived from afferent nerve reordings, originating in the otolith, range from 100 to 16 ms (3,4,9,10). Values trom oular LOrsion are even longer (11,2). Lumped parameter modeling indiates that this short time onstant should not exeed 2 ms (7). The present analysis indiates that this time onstant is in the viinity of 0.02 ms. This is muh shorter than that predited or measured in earlier work. This value is obtained when R = 0.75, E = 0.2, and M = 5.0. Sine the values of Rand E are fixed by previous arguments, the range of M is reasonable, and physiologi defletions are mathed, the 0.02 ms short time onstant is an exellent estimate from the present work. The earlier, muh longer measured values of the short time onstant inlude ell transdution and nerve transmission effets. The present model muh more losely represents the true end organ response. It should be reognized that the exponential rise of displaement with time is only an approximation to the solution of the partial differential equations. Observed values of the human long time onstant average around 10 s (11,15). These were obtained by pereption and oular torsion. These measured values have been shown, from later work, to originate in the entral nervous system rather than refleting true end organ mehanial system effets. When Mis given a value between 5 and 15, the value of the long time onstant resulting from the present work is 0.1 s. In order to see if a long time onstant of 10 seonds ould be reahed, we attempted raising M to unreasonably high numbers. This resulted in two negative results. First, the system beame so visous that the displaements were too small for hair ell detetion. Seond, the model began to show numerial instabilities in both time and spae

Gel Layer in Otolith Dynami Response Model that beame signifiant at values of M above 200. At M = 200, the long time onstant was 1.0 s. This seond negative result reflets a hange in the basi paraboli nature of the governing equations. By extrapolating the data from these earlier points, it is estimated that the long time onstant would approah 10 seonds as M was around 2000 to 3000. Thus, the present analysis supports a long time onstant from 0.1 s to 1.0 seonds, with weight toward the 0.1 s low end. The most signifiant result of the previous disussion is the apparent physial impossibility of the otolith system ontaining enough damping to support a 10 s long time onstant. The value of 0.1 s, indiated by the present work, would plae the system's lower orner frequeny at approximately 10 rls (1.6 Hz). This value is in agreement with afferent nerve reordings (3,4). This high value of the lower orner frequeny establishes the otolith system as an overdamped aeleration transduer over the frequeny range from D.C. to 1 Hz, the frequeny range where gravity and aeleration is most important and useful in the entral nervous system. In summary, reasonable values of Rare lose to 0.75. E has been braketed between 0.3 and 0.08, with values around 0.2 being more relevant. Values of M between 5 and 15 fit expeted short time response and produe a long time response with an approximate 0.1 s time onstant. Experimental data from otolith afferent nerve reordings were reently analyzed assuming a visoelasti gel layer (3). In this analysis, the gel layer was modeled as an infinite number of visoelasti relaxation elements arranged in series. This di5tribi.ll.e model was integrated between 2 rate onstants, approximately 3 deades apart, and was used to analyze the nerve dynami data from the utrile of the hinhilla. The utrile was given speifi sinusoidal inertial motion as a stimulus. The model proved to be an almost exat math to the dynami data. This type of visoelasti gel model will result in spreading the attenuation (smearing over a broader range than expeted by linear system 149 analysis) of the dynami response near the lower orner frequeny. A linear lumped parameter system ould not represent this type of response. These results are an additional indiation that a visoelasti gel layer is ative in the dynamis of otolith response. The present work inluded only a single relaxation element and will also result in some smearing of the dynami response near the lower orner frequeny. Physiologists and physial sientists have utilized frational derivatives and frational powers of the Laplae transform variable to haraterize this type of behavior where only time is onsidered as an independent variable. The frational dynamis used to desribe vestibular response have mostly been of an ad ho nature with little physial basis. The present work desribes this type of dynami response with two independent variables: onedimensional spae and time. Linear partial differential equations with one time and one spatial independent variable produe transfer funtions-whih have frational dynamis. The system of equations developed in this work, when Laplae transformed, ontain terms with frational powers of the transform variable and a hyperboli otangent term whose argument has frational s terms. The time transfer funtion is definitely not linear. The ability of the model developed here to desribe otolith dynamis remains to be seen; however, it is based on physial priniples. Conlusions This model provides good results in many lea.;.,,os: notably, he agnitude of the defletions of the otoonial layer are orret. Also, the model is able to test various hypotheses onerning variations in both gel and fluid visosity, as well as gel elastiity. The greatest suess of this work is the ability of the model to have inreased elastiity ompared to earlier purely elasti models, and maintain reasonable displaements. The higher elastiity values orrespond to expeted values of E and limit otoonial layer

150 defletions under stati gravity onditions. The model also shows that the same value of E an maintain reasonable defletions under both steady-state onditions and dynami onditions. Already this model offers some insight into the mehanial properties, and methods for modeling similar polysaaride gel materials found in the inner ear. Most notably, these results should be relevant to researh on :nater";al :-espoi'.se if'. t!"1e <"-11)111a of the semiirular anals and the tetorial membrane in the ohlea. In future analysis and modeling in the anals and the ohlea, visoelasti gel material should be inluded. The distributed parameter model, developed in this work, indiates that linear lumped parameter modeling is inadequate to desribe the physial behavior measured and observed in the otoliths. The distributed model, whih is haraterized here as a set of partial differential equations, along with the inlusion of visoelasti gel material, hanges the dynami response in a manner that is onsistent with observed and experimental effets. Aknowledgment-The help of W.A. Best for tehnial assistane and the Engineering Siene and Mehanis Department for omputing time is gratefully aknowledged. v(t) J. W. Grant and J. R. Cotton = time. = veloity of the endolymph fluid measured with respet to the skull. = veloity of the otoonial plate measured with respet to the skull. w (y g' t) = veloity of the gel layer measured with respet to the skull. o(y g, t) = displaement of gel or otoonial layer measured with respet to the SkUll. = omponent of skull veloity in the x-diretion measured with respet to inertial referene frame. Po = density of otoonial layer. Pj = density of the fluid and gel material. Tg = gel shear stress in the x-diretion /1-g = visosity of the gel material. /1-j = visosity of the endolymph fluid. G = shear modulus of the gel material. b = gel layer thikness and otoonial layer thikness (assumed equal). V = harateristi veloity in the problem. gx "= omponent of gravity in the plane of the otolith. Nondimensional variables Nomenlature Dimensional variables Y Yg Yj x = spatial oordinate normal to the plane of the otolith, with origin at the skull base. = spatial oordinate normal to the plane of the otolith, with origin at the skull base (Yg = y). = spatial oordinate normal to the plane of the otolith with origin at the top of otoonial layer (Y/ = Y - 2b). = spatial oordinate tangent to the plane of the otolith..y =.Yg = YJ t i1 = D = Y b Yg b Y/ b.j!:l t P o b 2 u V v V

Gel Layer in Otolith Dynami Response Model 151 w W = V Vs R = Pj Po V Nondimensional parameters b 2 po E gx = = G2"Po /-tjv g>; /-tj /-tg = ---..!!:.L 8 M VPob 2 /-tj b 2 REFERENCES 1. Carlstrom D, Enstrom H, Hjorth S. Eletron miro- organ's apparatus. Seond Symposium on the Role sope and X-ray diffration studies of statoonia. of the Vestibular Organ in Spae Exploration, NASA Laryngosope. 1953; 63: 1052-7. SP-115. 1966;47-53. 2. DeVries HL. Mehanis of the labyrinth organs. 9. Lowenstein 0, Saunders RD. Otolith-ontrolled re- Ata Otolaryngol. 1950;38:262-73. sponse from the first order neurons of the labyrinth 3. Goldberg JM, Desmadryl G, Baird RA, Fernandez of the bullfrog to hanges in linear aeleration. Pro C. The vestibular nerve of the hinhilla; 4: disharge R So Lond. 1975;BI91:475-505. properties of utriular afferents. J Neurolphysiol. 10. Lowenstein 0, Roberts TDM. The equilibrium fun- 1990 [in press]. tion of the otolith organs of the thornbak ray. 4. Goldberg JM, Fernandez C. Physiology of peripheral 1950; 110:392-415. neurons innervating otolith organs of the squirrel 11. Meiry JL. A revised dynami otolith model. Aeromonkey; 3: response dynamis. J Neurolphysiol. spae Med. 1968;39:606-8. 1976;39:996-1008. 12. Money KE et al. Physial properties of fluids and 5. Grant JW, Best WA, LoNigro R. Governing equa- strutures of vestibular apparatus of the pigeon. Am tions of motion for the otolith organs and their re- J Physol. 1971;220:140-7. sponse to a step hange in veloity of the skull. J 13. Steer R W J r. The influene of angular and linear a- Biomehanial Engineering. 1984; 106: 302-8. eleration and thermal stimulation on the human 6. Grant JW, Best WA. Mehanis of the otolith semiirular anal [S. D. Thesis], Massahusetts Inorgan-dynami response. Ann Biomed Eng. 1986; stitute of Tehnology, 1967. 14:241-56. 14. Steinhausen W. Conerning the fores released by the 7. Grant JW, Best WA. Otolith organ mehanis: otoliths. Pflugers Arh f d ges Physiol. 1934-5;235: lumped parameter model and dynami response. 538-544. Avia Spae Environ Med. 1987;58:970-6. 15. Young LR et al. Oular torsion on earth and in 8. Igarashi M. Dimensional study of the vestibular end weightlessness. Ann NY Aad Si. 1981;374:80-92.