ANNALS OF BIOMEDICAL ENGINEERING 5, 1--11

(1977)

The Effect of the Utricle on Fluid Flow in the Semicircular Canals WILLIAM: C. VAN BUSKIRK Biomedical Engineering Program, Tulane University, New Orleans, Louisiana 70118

Received January 16, 1976 The fluid mechanics of the semicircular canals is examined theoretically. This analysis eonfirms earlier studies that indicate that the semicircular canal can be modeled ~ a heavily damped, second-order system. Included for the first time, however, is an accurate accounting of the effect of the utriele on fluid flow in the canals. Using available physiological data, quantitative predictions of canal response to angular acceleration are made. INTRODUCTION T h e basic mechanical response of the semicircular canals to angular acceleration is qualitatively well understood. An angular acceleration of the head causes the b o n y canals and the m e m b r a n o u s structures attached to t h e m to rotate in a similar manner. B u t the endolymph, which is within the m e m b r a n o u s system, tends to lag behind the motion of the head because of its inertia. Due to the small caliber of the m e m b r a n o u s canal, this relative motion of the e n d o l y m p h within the duet is retarded b y a relatively large viscous force. T h e eupula is displaced and exerts a restoring force on the endolymph. Q u a n t i t a t i v e l y this basic mechanical response is not so well understood. While a m a t h e m a t i c a l model exists, the values for the coefficients in the governing equation are still disputed. I n no w a y is the confusion more evident t h a n in the discussions in the literature concerning the effects of the utriele on fluid flow in the semicircular canals. T h e purpose of this p a p e r is to p u t the fluid Ineehanies of the semicircular canals on a solid footing and to clarify the role played b y the utriele. REVIEW OF THE LITERATURE T h e m o s t popular model for the response of the semicircular canals to angular acceleration is due to Wilhelm Steinhausen (1933). He proposed t h a t the canals respond to angular acceleration in the same m a n n e r as would a heavily d a m p e d torsion pendulum. T h e "torsion p e n d u l u m equation" he proposed m a y be written in the f o r m J ~ + b~ + k~ = --Jo~

(1)

where }, ~, ~ represent the m e a n angular displacement, m e a n angular velocity, and m e a n angular acceleration respectively of the e n d o l y m p h with respect to the

Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in a n y form reserved.

ISSN 0090-6964

WILLIAM C. VAN BUSKIRK skull. The term a is the inertial angular acceleration of the skull in the plane of the canal. The coefficient J is an inertia term defined by the mass and distribution of the endolymph. The damping coefficient b denotes the ratio of torque resulting from viscous forces to the mean angular velocity of the endolymph. The stiffness of the cupula is represented by k. The transfer function from a to ~ is readily found from Eq. (1) : } -

--J (8)

-

(2)

Js 2 q- bs + k

Since the system is heavily overdamped, a convenient and sufficiently accurate form of Eq. (2) is } o~

-- T1T2 (8) ~

(3)

(T18 -~ 1)(T2s q- 1)

where TI = "long" time constant ~ b/l~ T2 = "short" time constant ~ J / b and where T2