1979 - Steele - Permeability of Fluid Flow Through Hair Cell Cilia

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Permeability of fluid flow through hair cell ciliaGlenn H. Frommer a) and Charles R. Steele

Stanford University, Stanford, California 94305(Received 24 August 1976; revised 4 December 1978)

Models (--•400X) were constructed of the one row of inner hair cell cilia bundles and the three row arrayof "w" shaped outer hair cell cilia bundles ound in the mammalian cochlea. These were placed betweenparallel plates simulating the under surface of the tectorial membrane and the upper surface of the

reticular lamina. The pressure equired to produce a given volume of flow through the obstacles wasmeasured. For low flow rate, the behavior is linear, i.e., the ratio of flow rate to pressure drop(permeability) s independent f flow rate. For higher flow rates (Reynold's number A_• 0.08) the behavioris nonlinear, characterized by an increase n the effective permeability to a maximum at A--• 10 and anasymmetry n the flow through the OHC cilia bundles. nterpretation in terms of cochlear function isbeyond he scope of the present work. A very tentative estimate places he onset of nonlinearity at around80 dB SPL, so the flow around the cilia bundles n the cochlea s linear for normal intensity levels

PACS numbers: 43.63.Bq, 43.63.Ds, 43.63.Kz

INTRODUCTION

Whatever the exact cochlear mechanism may be, itis likely that the fluid layer between the reticularlamina and the tectorial membrane plays a significantrole. Lim (1972) finds no attachment of the inner haircilia with he ectorial membrane. Although oss i974)does find some evidence of such an attachment, it isquite light in comparison with the stiff attachment of theouter hair cell ciliaø The conclusion is that the outerhair cell cilia take all or most of the shear stress act-

ing between the tectorial and reticular membranes,leaving the inner hair cell cilia subject primarily to theforce arising from the motion of the fluid. Ross (1974)does conclude that the tectorial membrane adheres to

the reticular membrane except for a fluid pocketaround the cilia from each outer hair cell, but her spe-cific evidence for the absence of continuous fluid path-ways from the spiral sulcus to the scala media seems

lackingø

Zwicker 1972) has observed strong dc fluid flow \emitting from the subtectorial membrane region underpure tone acoustic stimulation of the pig cochlea. Ex-perimental models (Helle, 1974; Zwicker, 1974)whichcontain a "tectorial membrane" without any attachmentto the "organ of Corti" show that a sharpened dc flowdoes occur when the stimulation is of sufficient strength.

In the present study, attention is restricted to thedetermination of the properties of the flow about theinner and outer hair cell cilia, with the assumption thata continuous luid layer does exist. Billone (1972) has

considered the flow through the array of cilia on asingle hair cello However, Flock (1977) gives com-pelling evidence of a substance which binds the individualcilia together into a cohesive unit on a single hair cell.The fluid flow is then around, rather than hrough, eachcilia bundle. Scale models (~ 400 x) of individual "V"'sand "W"'s as well as three row arrays and a single rowof slats were placed between two parallel flat plates,and the resistence of flow measuredø Fluids rangingfrom water to those with 100x the viscosity of waterwere used. Although in principle this experiment isrelatively simple, it was found that inconsistent resultscould be caused by many factors including leaks, sur-

_

a)Present ddress: Biology nstitute, Oderise, Denmark.

face imperfections, and external sound vibration. Theresults reported herein are from a model and measure-ment procedure which proved to be repeatable to a highdegreeø

The fluid in the cochlea is subject to oscillatoryforces. Because of the thinness of the layer under con-sideration, the flow around the cilia has a very shorttransient timeø For flow in a pipe of radius a which isinitiated by the sudden application of a pressure, thetime to reach the steady-state flow is given by (Batche-lot, 1967, po 195) he quantity .75 a•'/Vo For waterin a tube of radius 2 /xm, this gives the transient timeof 3/XSo Thus, in the normal frequency range, thefluid surrounding the cilia will follow the fluctuationsin the driving pressure in a quasisteady state mannerwithout any significant time delayø Therefore, ourmeasurements are on only the steady-state flow.

I. EXPERIMENT

The models for the three-row array of outer haircell "cilia bundles" and the single-row array of inner

ß

ß

•..

• .• • •:...: .:.• ,.•"" ......1. o o'fi mrg-

ment of cilia array. Average distance between centers ofis q = 2.13 ram; the average gap between w's is d= 0.35 ram;the free space between rows is 1 mm. The surface of the ciliahas been painted in an attempt to show the orientation moreclearly. (b) Baseplate of test section; length c= 101 mm withb = 38.1 mm. Cilia affixed to baseplate with a layer of epoxy

whose thickness is small in comparison to cilia height.

759 J. Acoust. Soc. Am. 65(3), Mar. 1979 0001-4966/79/030759-06500.80 (D 1979 Acoustical ociety of America 759

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FIG. 2. Model of inner hair cell cilia bundles. (a) Enlarge-ment of cilia array; average distance between centers of bun-dles is q = 2.75 ram; average gap between bundles is d = 0.25mm. (b) Baseplate of test section. The dimensions are thesame as in Fig. l(a).

5O

0.13•_ a .78 m

0.89

i-o.o

h=1.60 mm

FIG. 3. Model of outer hair cell cilia bundle (in Fig. 1) cutfrom brass stock. Inner hair cell cilia bundle (in Fig. 2) is

2.50x1.60x0.25 mm.

FLEXIBLE

• ii ii i

INNERBOX

O-RING•

FLOW

rEST SECTION

OUTERBOX

PUMP

END PLATE

VIEW PORT

FIG. 4. Exploded view of the experimental apparatus.

hair cell "cilia bundles" are shown n Figs. 1 and 2,respectively. The arrangement is similar to that foundin the mammalian cochlea; however, a precise scalingof dimensions and angle of orientation, such as shownby Lim (1972), has not been made. Each "cilia bundle,"whose dimensions are given in Fig. 3, is milled frommetal stock and then epoxied to the jig plate formingthe test section shown in Figs. 1 and 2. In earlier workwith bent pieces of wire simulating the cilia bundles,it was found that the presence of a small notch at thevertex, making the "W" rather than the "V" shape, didhave a quantitative effect on the flow.

To produce a measurable flow through the "cilia,"the apparatus shown in an exploded view in Fig. 4 andin cross section in Fig. 5 was constructed. The de-vice consists of two boxes constructed of half-inch

aluminum jig plate with milled surfaces. The test sec-

tion (Figs. i and 2) is inverted and then bolted to theunderside of the inner box, which is then bolted to thebottom of the outer box. A viewing port consisting ofa glass plate located in the bottom of the outer boxallows examination of the model during testing. Theendplates of the outer box are placed flush against theends of the inner box and are then bolted in place. Em-ployment of O-rings in these endplates and grease onadjacent surfaces throughout the device assures a fluid-proof seal between the two boxes. Completion of thisassembly procedure results in the formation of twofluid tight chambers lengthwise in the apparatus joinedsolely by the channel in which the model is situated

(Fig. 5).

760 J. Acoust. Soc. Am., Vol. 65, No. 3, March 1979 G.H. Frommer nd C. R. Steele' Fluid low through air cell cilia 760




VERNIEREIGHTAUGECLAMP,• RASS

,/- MENISCUS

I•J •-FLUID

.oxLIA"IN TEST SECTION

FIG. 5. Height gauge and a cross section of the apparatus.The pump shown in Fig. 4 maintains the head e, which causesflow from right to left through the test section.

Since the "W"'s are at right angles between twoparallel plates and completely fill the space betweenthem, the test section is actually a Hele-Shaw cell(Happel and Brenner, 1973). However, Hele-Shaw flowis typically associated with bodies whose width is muchgreater than the space between the parallel plates. Inthe present case, the width and height of the body (thecilia bundle), are about the same, while the gaps be-tween bodies are small in comparison with the height.

Concurrent measurement of the volume flowrate Qand pressure drop Ap, is fairly simple. A fluid flowbetween the two chambers is initiated and maintainedby a Masterflex variable-speed tubing pump visible inFig. 4. Gilmont flowmeters are used to measure thisvolume flow from 0.008 to 2.3 ml/s to within an ac-curacy of ñ 2%. The height difference between thelevels of the fluid in these chambers is determined bya probe attached to a vernier height gauge, visible inFig. 5. The probe consists of three bars, one hori-zontal support and two vertical bars. The probe iscalibrated by setting the tips at the same height,(ñ 0. 002 cm) relative to the undisturbed fluid level. Inoperation, the probe is lowered until one tip touches thefluid surface, where a visible miniscus is then formed.

The same procedure is undertaken for the second bar.The pressure drop through the channel is equal to thepressure due to the height differential between the twolevels of fluid. So we have

aP = pge , (1)

where p is the fluid density, and g is the gravitationalconstant. Correct measurement of e is insured by theuse of a long settling time to allow the fluid levels toreach equilibrium, and a pump that automatically com-pensates for change in torque or system pressure.

II. RESULTS

The laminar (Poiseuille) flow through the test sectionin Fig, 5 without cilia is well-known (Happel and

Brenner, 1973; Batchelor, 1970). The pressure dropis

(aP).o = R,,•.. U, (2)where U is the average velocity of fluid through the testsection, and the "resistance" is

R,a•r• = 12/• c/h ' , (3)

in which/x •s the viscosity, and c and h are the len•hand height of the test section. With the "cilia" in thetest section the pressure drop is increased;

a•=nU . (4)

It is convenient to use the dimensionless flowrate (Rey-nolds number)

A , (5)

and the dimensionless pressure drop

p = (phz/12/x c)(aP) ot,• ß (6)Then Eq. (4) is

=A (n . (7)The experimental results for flow through the OHC ar-ray (Fig. 1) are shown n Fig. 6. The resistance Rdoes depend on the rate as well as the direction of flow,but for low flowrate A < 0.08 is constant. The transition

to.the linear behavior is more clearly seen on the log-arithmic scales in Fig. 7 of the dimensionless permea-ability (R/R.a•) ' =Alp. The results or flow throughthe IHC array (Fig. 2) are also shown in Fig. 7.Rather unexpected was the increase in permeability,for both the OHC and IHC arrays, with the flowratesA> 0.08. For A > 10 the permeability decreases withincreasing flow rate, as would be expected.

Water was used to obtain the experimental points forA> 1, while Dow Corning 200 fluid, with a viscosity of10 cs, was used for the points A < 1. The continuity of

[ FLOW INTO

/ FLOW TOWARD

241 TIPS-' -.4/ -

/ /LINEAREHAVIOUR',6[--/ -12 -

-

•02 4 16 88 i0 12 14

FIG. 6. Experimental results for relation of dimensionlesspressure drop p and flowrate A for flow through OHC ciliabundles.

761 J. Acoust. Soc. Am., Vol. 65, No. 3, March 1979 G.H. Frommer nd C. R. Steele: Fluid low through air cell cilia 761




1.0 _ ]1• [ i i i i i[[ [ i [ [ i [i

FLOWTHROUGH IHC) __- SLAT ARRAY% •...---• I

- ,ow cu;%

FLOW TOWARD TIP --

-

-

i , i i i i i i i i i i i i i i i i i i i i

0.5

0.050.1 1 10

A

FIG. 7. Permeability of test section with cilia bundles'. Flowis linear and reversible for flowrate of A < 0.08.

the curves validates the procedure. Each experimentalpoint shown in Fig. 6 is the result from one of fourindependent tests at about that value of A. The scatteras well as the estimated error in measurement are

within 4%. Additional confirmation of the procedurecame from tests with only the base plate without cilia,which gave results within 4% of the theoretical valueEq. (2).

The effects of walls and cilia can be separated by con-.sidering the two as independent resistances in series

•P = (R.a•. + R ci,ia)U ß (8)Thus the permeability of the cilia alone is given by

- 1) . (9)As seen in Fig. 8, this factor has the linear values forA <0.08'

(P/A1)z=0.14,HC (10).67, IHC

then increases ninefold for the OHC and threefold for

the IHC array as the flowrate increases to A = 10.

Instead of the dimensionless form Eq. (10) which ismost convenient for the overall properties of the testsection, it is helpful to calculate the average propertyof flow through a single gap between the obstacles.Since the height of the gaps is greater than the width,

h/d={4'6'HC.4, IHC (11)the flow in the vicinity of the gaps should be nearly twodimensional. The solution for slow, viscuous flow

through an infinite slot is discussed by Morse and F esh-bach (1953), who make an error of a factor of 8. Thecorrect result for the average velocity { through a slotof width d due to a pressure drop Ap is

•= (•d/32iz)/xp (2D Slot Flow) . (12)

For our experiment we write

•'= Uq/d = (k•d/32tz) zxP, (13)

where k is a factor which gives the deviation from Eq.(12) due to the actual three-dimensional geometry. Thisfactor can be computed from the previous dimensionlessparameters with the formula

k = (32qh2/12•rcd•')(p/A 1) • x (No. of rows) . (14)

The experimental results Eq. (10) give the values

0.16,HCA<0.08) (15)= 0.56, HCThe finite height of the test section, the finite thicknessof the obstacles, and the interference between slotscause the decrease in k from unity. The interferencebetween the three rows of OHC cilia bundles causes thefurther decrease. Some substantiation comes from cal-

culations for the interference effect on flow through arow of parallel, infinite cylinders (Miyagi, 1958;Billone, 1972) and a square array of cylinders (Happeland Brenner, 1973). However, the cilia bundle config-uration, particularly the OHC, is such that no two-di-mensional flow calculation is justified.

In the nonlinear range (A > 0.08), the difference inpermeability for flow toward the "tips" of the W-shapedOHC cilia bundles and in the opposite direction, towardthe "cups," is evident in Fig. 9. In this earlier work,bent wires were used to simulate the cilia bundles.

Alumina particles were injected during a steady-stateflow for flow visualization. For the same pressure(p-4), hardly any particles find their way through whenthe flow is toward he "tips" [Fig. 9(a)], in contrast othe situation or flow toward he "cups" Fig. 9(b)]. Thedisturbance to the flow persists for many body lengthsdownstream, as seen in Figs. 9(b) and 9(c). As pointedout by Acrivos (personal communication), if the velocityfield is uniform both upstream and downstream, thenthe maximum value of permeability will be that of thelinear behavior for slow flow. As shown in Fig. 7, themeasured permeability exceeds the linear value at inter-mediate flow rates. The explanation from Fig. 9 is thatthe test section is not sufficiently long for the down-stream velocity to achieve a nearly uniform distribution.

Ill. ESTIMATES oR FLOW NTENSITY N COCHLEAWe now turn to the question of relating the flow pa-

rameter A to intensity level in the cochlea. One esti-mate of the physiological values can be obtained fromdata given by Wilson and Johnstone 1972). At a point onthe guinea pig basilar membrane with a best frequencyof 20 kHz, the rms displacement is 0. i izm at 100dB SPL. In his experimental model with a "tectorialmembrane," Zwicker (1974) found the induced low offluid from the subtectorial membrane region to have

5.0i

(-•-'-1)• .1.0-

0.5

.

0.1

I I I [ I I [ I I I I I I l[ I I I I If[ I

FLOW THROUGH (IHC)

.• ' 3ROWOHC)LINEAR--• ARRAY/.•/'• "- NTOUP

_/'///• FLOW///y - TOWARDIP

_•/Y • LINEAR__.... '"• •' BEHAVIORI I I I I I I I I i i ill i i i i i i ill

o.1 1.oA

FIG. 8. Effective permeability of cilia bundles alone.

762 J. Acoust. Soc. Am., Vol. 65, No. 3, March 1979 G.H. Frommer and C. R. Steele: Fluid flow through hair cell cilia 762




{a)

(b)

a velocity 10 times that of the basilar membrane.Thus, for a fluid layer of h-4 t•m and the viscosity anddensity of water in the actual cochlea, one calculates

A =pUh/• --0. ? (100 dB SrL), (16)

which indicates that the threshold of the nonlinearity inFigs. 6-8 is around 80 dB SPL, the intensity at whichneural and electric field nonlinearities become pro-nounced. Some models for the consequence of the fluidbehavior are tentatively explored in Frommer (1977).If, however, the pressure at which the flow nonlinearityfirst occurs is calculated, one obtains for the IHC ciliabundles (Fig. 7) with the cochlear dimensions c =0.1mm, h = 4 t•, the result

(AP)cilia 12g c/5ph =4 kNm 2 . (17)

Since 0 dB SPL is at 20 t•Nm ø•', the threshold pressurefor the flow nonlinearity is at some 160 dB SPL. Thereis of course a pressure increase from the eardrum toendolymph and undoubtedly a further increase to theinner sulcus. But such an increase to bring the value

given by Eq. (17) into the physiological range seems un-likely. At the moment we cannot say which of the esti-mates represents reality: Eq. (16), which indicates thatthe flow nonlinearity is in the physiological range, orEq. (17), which indicates the contrary.

It is clear from either estimate that the flow around

the cilia bundles is linear (i. e., A < 0.08) throughoutthe normal physiological range, from threshold to 70dB SPL. For this our experimental results Eq. (15)give the important property of the flow. The parameterk is dimensionless, and will remain constant for anysimilar geometric configuration, i.e., for the actualcilia in the apical region of a large cochlea as well as in

the basal region of a small high frequency cochlea.From k the forces acting against individual hair cellbundles due to a pressure difference between scalamedia and inner sulcus can be calculated. The overall

permeability given by Eq. (10) provides a necessary in-gredient in models for the interaction of tectorial mem-brane, organ of Corti, subtectorial membrane fluid,and basilar membrane.

ACKNOWLEDGMENT

This work was supported by a grant (5R01NS12086)from the National Institute of Neurological and Commu-nicative Disorder and Stroke. We thank the reviewers

for several suggestions which have been followed in therevised manuscript. We also thank M. Van Dyke andA. Acrivos for helpful discussion.

FIG. 9. Visualization of flow from right to left about ciliabundles. Alumina particles were injected during steady-stateflow. For the same pressure, the flow toward the tips of •.hecilia (a) seems blocked in comparison with the reversed flow(b). Asymmetry is apparent in flow about isolated bundles (c).

Batchelor, G. K., (1970). An Introduction to Fluid Dynamics(Cambridge U. P., Cambridge, England).

Billone, M. (1972). "Mechanical Stimulation of Cochlear HairCells," Ph.D. dissertation, Northwestern University.

Flock, A. (1977). "Physiological Properties of Sensory Hairsin the Ear," in t•sycholbhysics nd t•hysiology of Hearing,edited by E. F. Evans and J.P. Wilson (Academic, London).

Frommer, G. H. (1977). "Fluid Motion in the ReticularLamina," Ph.D. dissertation, Stanford University.

Happel, J., and Brenner, H. (1973). Low Reynolds Number

763 J. Acoust. oc. Am., Vol. 65, No. 3, March 979 G.H. Frommer ndC. R. Steele: luid low hrough air cell cilia 763




HydrodynamicsNoordhoff, roningen).Helle, R. (1974). "Enlarged Hydromechanical Cochlea Model

with Basilar Membrane and Teetorial Membrane," in Factsand Models in Hearing, edited by E. Zwieker and E. Terhardt(Springer, Berlin), pp. 77-85.

Lim, D. J. (1979.). "Fine Morphology of the Tectorial Mem-brane," Arch. Otolaryngol. 96, 199-215.

Miyagi, T. (1958). "Viscous Flow at Low Reynolds NumbersPast an Infinite Row of Equal Circular Cylinders," J. Phys.Soe. Jpn. 13, 493-496.

Morse, P.M., and Fesbaeh, H. (1953). Methods of TheoreticalPhysics (McGraw-Hill, New York), Vol. II.

Ross, M.D. (1974). "The Tectorial Membrane of the Rat,"Am. J. Anat. 139, 449-482.

Wilson, J.P., and Johnstone, J. R. (1972). "Capacitive ProbeMeasures of Basilar Membrane Vibration, "inHearing Theory(IPO, Eindhoven, Holland), pp. 172-181.

Zwicker, E. (1972), "Investigation of the Inner Ear of theDomestic Pig and the Squirrel Monkey with Special Regard tothe Hydromechanics of the Cochlear Duct," inHearingTheory(IPO, Eindhoven, Holland), pp. 182-185.

Zwicker, E. (1974). "A 'Second Filter' Established within theScala Media," in Facts and Models in Hearing, edited by E.Zwicker and E. Terhardt (Springer, Berlin).

764 J. Acoust. Soc. Am., Vol. 65, No. 3, March 1979 G.H. Frommer and C. R. Steele: Fluid flow through hair cell cilia 764

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