Journal of Vestibular Research, Vol. 2, pp. 227-234, 1992 0957-4271/92 $5.00 + .00 Printed in the USA. All rights reserved. Copyright © 1992 Pergamon Press Ltd. MECHANICS OF THE CUPULA: EFFECTS OF ITS THICKNESS E. Njeugna, * J.-L. Eichhorn, t C. Kopp,t and P. Harlicot§ *Centre Universitaire de Douala, Douala, Cameroun, tlnstitut de Mecanique des Fluides, URA CNRS 854 tEcole Nationale Superieure de Physique, Universite L. Pasteur, Strasbourg, France, and ~Laboratoire de Genie Civil, Universite R. Schuman, Strasbourg, France f:Jeprint address. Jean-Louis Eichhorn, Institut de Mecanique des Fluides, URA CNRS 854, Universite L. Pasteur, Strasbourg, France D Abstract - Mechanical aspects of the ampullar di­ The cupula deforms under the effects of aphragm, that is the crista ampuliaris and the cu­ the transcupular pressure difference when the pula, related to its thickness, are studied by a head is moved, but also when the endolymph numerical method. Numerical methods are able to pressure varies. This last pressure variation go beyond the limits of analytical approaches and deforms the cupula directly through the pres­ are the only methods able to take into account this sure variation on its two faces and indirectly thickness. A finite elements method is applied to the through the deformation of the ampulla at median plane slice of the ampullar diaphragm. One assumes that the cupula sticks firmly without sUp­ which the cupula sticks firmly (1). ping, to the ampullar wall and to the crista ampul­ These deformations of the cupula are im­ laris. The computation takes into account the portant for two purposes: (i) the global elastic pressures on the liquid interfaces and the deforma­ behavior of the cupula is generally described tions of the ampulla. So the volume swept over by by its elasticity coefficient which is the ratio the cupula during quasi-static deformations can be between the transcupular pressure difference evaluated and the global elasticity coefficient of the and the volume swept over by the cupula dur­ human cupula can be calculated. The related value ing its deformation, (ii) at the junction be­ of the long time constant of the semicircular canal tween crista and cupula, the deformations is close to the value obtained when measuring, in shear the cilia of the sensory cells and are at vivo, the activity on the vestibular nerve in animals. the origin of the afferent neurological signal. The thick cupula model clearly shows two different spatial distributions of strain on the hairs of the sen­ Our purpose is to gain a good evaluation sory cells, leading to a discrimination between the of the elasticity coefficient of the cupula and vestibular inflating pressure and the transcupular a better insight into the mechanical part of the pressure difference. This result matches recent neu­ mechano-neural transduction. rophysioiogicaE data and brings a new insight. ill' the mechanics of the vestibtdar ~mguiar acceieromeki amI its Keywords - cupula; mecltmnics; fhnHe ehemell11:: The 10rc( cupula" mechano-neura~ tran.sduction. is balanced by the hydrostatic pres­ sure on the two sides of the cupula, because in normal conditions its density is very close Introduction. to the endolymph density. Moreover, our pur­ pose is to analyse quasi-static deformations. The mechanical study of the cupula is an as­ Then inertial forces are also negligible. Quasi­ pect of the modelisation of the semicircular static deformations can be considered if the canals, that is the system cupula-endolymph. frequencies are much lower than the frequency RECEIVED 5 December 1991; REVISED 22 May 1992; ACCEPTED 22 June 1992. 227 228 E. Njeugna et al corresponding to the short time constant of pula as a thin plate, an analytical approach is the semicircular canal (T2 = 7.3 10-3 s) that possible; if we want to take into account the is 20 Hz. So the quasi-static approximation is thickness of the cupula, only a numerical ap­ correct when the frequencies are lower than proach is suitable. To emphasize the specific 2 Hz. To neglect the short time constant is, interest of taking into account the thickness of within these limits, a justified and usual ap­ the cupula, we will first give the results of the proximation in modelisation of the semicircu­ thin plate model. lar canal. Thus only the forces acting on the surface of the cupula can deform it. The shape of a The Thin Plate l.Vlodel 'normal the ~n ~his a given tinle depend way study the mechanics of the cu­ on its contour, that is the boundary conditions. pula is to consider it as a two dimensional :hin The ideas about the boundary conditions, plate. Though the assumption of a thin plate which were very controversial in the past, are is not realistic, the radius and the thickness now clarified, especially with regards to the being quite equivalent, important conclusions interface between cupula and ampullar wall result from this study. and to the interface between cupula and crista In the different mechanical lumped param­ ampullaris. Since the work of Steinhaus en (2), eter models of the semicircular canal (2,4-6, the cupula has been considered to have, dur­ 8-12), the global elastic behavior of the cupula ing physiological stimulations, a movement of is characterized by one single real (2,5,6,8-12) deflection articulated on the crista ampullaris. or complex parameter. This real parameter in­ In this case, one has a relative displacement troduces only an elastic return term in the sys­ between the apex of the cupula and the am­ tem equation. A complex parameter contributes pullar wall. But in vivo experiments on ani­ also to the friction term; this contribution is mals (3) proved that the cupula is deformed weak compared to the friction of the endo­ like a diaphragm and sticks firmly to the am­ lymph in the canal (4) and will be neglected in pullar wall during physiological stimulations. the following. This parameter, the cupular elas­ Several mechanical models of the semicircu­ ticity coefficient K, is the ratio between the lar canal when taken into account of this kind transcupular pressure difference and the vol­ of cupular deformation have been proposed ume swept over by the cupula during its (4-6). deformation: The mechanical actions applied on the cu­ pula are: (i) the hydrostatic pressure exerted K = b-.p/V. by the endolymph on the two sides of the cu­ pula even in the absence of any stimulation, The main problem is to determine the ac­ (ii) a differential transcupular pressure due to tual value of K for the human cupula and its rotatory or caloric stimulus, (iii) the forces ex­ variations related to pathologies. If we con­ erted by the ampullar wall at the interface be­ sider the cupula as a thin plate, an analytical tween cupula and ampullar wall. These forces approach leads to the following results (13). can vary in relation to the deformation of the K depends on the stiffness k' of the hairs of ampulla subjected to the inflating pressure, the sensory cells and on the inflating pressure that is the pressure difference between endo­ of the vestibule PI. K increases if k' or PI in­ lymph and perilymph, and (iv) the forces (dis­ crease. K is controlled through the efferent tribution of moments) exerted at the interface nervous system because k' and PI are. between cupula and crista ampullaris by the These results simply allow the active regu­ hairs of the sensory cells. These hairs have a lation mechanisms at the interface between variable stiffness under efferent control (7). crista and cupula to be included in the bound­ There are two possibilities for the study of ary conditions on the cupula, and the func­ the cupular mechanics. If we consider the cu- tional conclusions match the observations on Mechanics of the Cupula 229 patients with Menieres disease (1). However, mations of the junction crista-cupula of the this thin plate model gives for K a higher median slice will be, at least qualitatively, value than that derived from physiological identical to those of the other slices. data, and gives no information about the dis­ To focus our attention on the effects of tribution of the deflection angles of the sen­ shape and thickness of cupula, the other phe­ sory cells' hairs. nomena will not be included in our model. Moreover, the effects of the rigidity of the cilia have already been studied with the thin The Thick Cupula plate model. To take into account the geometrical and mechanical complexity of the ampullar dia­ Computation of Cupular Deformations phragm, that is the crista ampullaris and the with a Finite Element Method cupula, a numerical method such as a finite elements method should be used. In our modelisation, the cupula and the Finite element methods are peculiar nu­ crista ampullaris are considered as elastic bod­ merical methods of approximation (14), used ies. The limitations of this hypothesis are to solve problems described by partial deriv­ creeping for very low frequencies as well as ative equations or by integraJ equations given the viscoelastic properties of the cupula for in their variational form. high frequencies. But there is no data about In solid mechanics, with this method one creeping, and the viscoelastic properties are can compute internal displacements, strains negligible for quasi-static deformations. and stresses for each element for a given set So we consider small elastic strains of the of external loads (volume and surface forces). median slice of the ampullar diaphragm with These methods are efficient and many finite the hypothesis of plane strain. The symmetry element analysis codes exist. We used the code of the general three-dimensional problem al­ CAD SAP (15).