Mechanics of the Cupula: Effects of Its Thickness

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). lows this hypothesis. Such a method allows a rea,listic model of The median section of the ampullar dia strains and stresses in a structure of arbitrary phragm, in the plane of the semicircular ca shape to be built up, taking into account all nal, and the finite element mesh are presented the elementary knowledge, such as any behav in Figure 1. The shape results from a discus ior equation, peculiar boundary conditions, sion with P. Valli of the University of Pavia, and structural details of any scale. The limits Italy. The hairs of the sensory cells are located of this numerical approach are the inaccuracy in this plane. An appropriate mesh has been or lack of some data, and possibly the cost of constructed by hand. The cupular elements calculation. which are in contact with the crista ampullaris Due to the have a side which joins the crista and two sides dimensiona: 33Y.C ::o:-mal tc crista. The normaL sides of these crista) and taking into account the fact that elements stand for a distribution of hairs on cupula and crista share the same two geomet the crista ampullaris~ Two types of elements rical and mechanical planes of symmetry, a are used, isoparametric quadrangles (4 nodes, first approach will consider the two-dimen 8 degrees of liberty in plane deformation) and sional problem of a plane slice of the ampullar a small amount of triangles (3 nodes, 6 de diaphragm, with constant thickness, obtained grees of liberty in plane deformation). by cutting the ampullar diaphragm in the Cupula and crista are assumed homoge plane of the semicircular canal. neous and isotropic, following a linear elastic As we study only a plane slice of the am behavior. The mechanical characteristics and pullar diaphragm, the elasticity coefficient of dimensions are given in Table 1. the cupula can not be calculated directly, fur The value of Young's modulus for the cu~ ther hypotheses are necessary. But the defor- pula comes from reference (16). Young's mod- 230 E. Njeugna et al

No special strain is introduced at the junc tion cupula-crista ampullaris, which means that the stiffness of the hairs is supposed to be zero. For a given load on the ampullar dia phragm, the numerical model computes the strain and the deformations at each node of the mesh. We are especially interested in the (jeformations. The meaningful displacements of the nodes are those located on the two sides ~he at ;:he junction crista ampullaris. The global elasticity ',:oeffi dent of the cupula will be calculated from the deformations of the two sides of the cupula. The deformations at the junction cupula crista ampullaris lead to the distribution of the ciliar deflections.

The Global Cupular Elasticity Coefficient

The initial shape of the median slice of the ampullar diaphragm and the deformed shape are represented in Figure 2. In this case, the load is a transcupular pressure difference of Figure 1. The median slice of the ampullar diaphragm 2 2 2 in the plane of the semicircular canal (total height 1 N/m (0.5 N/m on one face, -0.5 N/m 1.5 mm). on the other). The nodes at the apex of the cu pula and those at the base of the crista are fixed. The displacements of the nodes are multiplied by a factor 100. The maximum dis ulus for the crista was chosen in such a way placement is loc'ated in the central zone of the that the deformation of the crista remains cupula. This has also been demonstrated ex very small compared with that of the cupula. perimentally by McLaren (3) who worked The value of Poisson's ratio is near that of an with the cupula of the frog. incompressible body. In a problem of this di The numerical value of the global elastic mension and with the elements we have cho ity coefficient K of the human cupula can be sen, the value 0.49 is a good compromise; the calculated from this result. Let So dy be the computer program does the tests of conver volume swept over by the median slice of the gence and of truncature. cupula during deformation. The assumption that for each parallel slice located at the dis tance y the volume swept over varies accord ing to a sinusoidal law is compatible with the boundary conditions of the problem. Thus V Table 1. Mechanical Characteristics and Dimensions of the Cupula and Crista will be given by the relation:

Cupula Crista Rc Y) 4Rc V = 2 So cos ('If-- dy = - So Young's modulus (N/m2) 513 61,300 1o 2 Rc 'If Poisson's ratio 0.49 0.4 Height (mm) 1 .1 0.6 where Rc is the radius of the cupula. Mechanics of the Cupula 231

Figure 2. The initial shape of the median slice of the ampullar diaphragm and the deformed shape. The stress 2 is a transcupular pressure difference of 1 N/m . The displacements are magnified by 100. The ciliar deflections are also represented. "

A transcupular pressure difference of 1 NI value obtained by in vivo experiments on an 2 2 m gives So = 0.1 mm • V is thus equal to imals, measuring the activity of the vestibular 7.33 10-2 mm3 and the value of K will be nerve (18). 4 1.36 1010 kg m- S-2. This value is close to the value determined in vitro by Grant for the cupula of the pigeon. Spatial Distribution of the The long time constant of the human hor Ciliar Deflection izontal semicircular canal is given by the fol lowing relation (5). The general hypothesis (infinitesimal de formations in linear elasticity) allow the appli OJ R T _ ~ pJ! cation of the principle of superposition. In 1 - q tior and comput(' the where ciliar CLC'~lections) Inret Kinas of loacis are ap plied separately on the ampullar diaphragm. a =" inner radius of the membranous canal The results for the three following cases are 1.58 lO-4m (17) presented: R = radius of curvature of the membra nous canal 3.17 10-3 m (17) 1. Both sides of the ampullar diaphragm are v = kinematic viscosity of the endolymph 2 8.52 10-7 m 2/s (11) loaded by a static pressure of 2 N/m • The nodes at the apex of the cupula and p = density of the endolymph at 37°C 1020 kg/m3 (11) those at the base of the crista are fixed. This simulates the effects of the hydro Thus Tl is 3.32 s for the human horizontal static pressure of the endolymph on the semicircular canal. This value is close to the cupula. 232 E. Njeugna et al

2. A displacement of 0.2 mm is imposed on ticularly sensitive to variations of the transcu the nodes at the apex of the cupula. The pular pressure difference. nodes at the base of the crista ampullaris These results from a mechanical model are fixed. There is no pressure imposed on have to be compared with results from phys the sides of the ampullar diaphragm. This iology and anatomy. Honrubia (19) showed simulates the deformations of the ampul that there are two types of sensory cells differ lar diaphragm due to the deformation of ently distributed on the crista. These two the ampullar wall when the inflating pres types of sensory cells are respectively con sure increases. nected to two types of neural fibres, leading 3. The stress exerted is a transcupular pres- to separate zones of the vestibular nuclei. Fi- sure jifference for the NlL1 a (that is:ast conduc- cularion of K). The nodes at the apex of Lion) respond to var:ations of the transcupular the cupula and those at the base of the pressure difference. The thin fibres have a con crista are fixed. stant activity. So the spatial distribution of the stresses Even without any stimulation, the stresses acting on the hairs of the sensory cells, lo of type 1 and 2 exist in the ampullar dia cated on the crista ampullaris, allows a differ phragm and are undissociable, but the numer entiation in the measurement of the vestibular ical simulation allows them to dissociate. inflating pressure and of the transcupular When the semicircular canal is stimulated, a pressure difference. stress of type 3 is added. This matches known facts about the mac The coefficients needed to actually build ula and the cochlea. The geometry of the the linear combination for the loads of type 1 macula and the spatial distribution of its sen and 2 are unknown, due to the lack of infor sory cells allows the two components of the mation about the elasticity of the ampulla. linear acceleration to be measured. In the co In Figure 3A the distributions of the ciliar chlea, there is not only a spatial discrimina deflections for stresses of type 1 and 2 are rep tion of the frequencies but also the distinction resented. The values for the loads are arbi between sensory cells sensitive to the system trary. For type 1, the value is weak; for type related signal and the sensory cells involved in 2, the value is high. The difference in order of the system regulation. magnitude for the computed deflections has We can also assume that the resting firing no significance. The valuable result is that in rate on the vestibular nerve probably corre the two cases the distribution is antisymmet sponds to the inflating pressure. Variations of ric and the deflections are maximum for the firing rate measured on the vestibular nerve hairs located on the side of the crista. For when the endolymph pressure varies (20) con each combination of loads of type 1 and 2 the firm this point of view. distribution will be antisymmetric. The distribution obtained when the stress is a transcupular pressure is represented in Conclusion Figure 3B. This distribution is symmetric and the deflections are maximum for the hairs lo cated at the top of the crista ampullaris. Taking into account the thickness and a re For a vestibular stimulation, the distribution alistic shape of the cupula considerably im of the ciliar deflection is the addition of a sym proves the determination of the value of the metric and an antisymmetric distribution. global human cupular elasticity coefficient. These results are of qualitative interest; they The long time constant of the lateral semicir show that the sensory cells located on the cular canal, derived from this value, is near sides of the crista ampullaris are more sensi the values obtained by in vivo experiments on tive to variations of the inflating pressure, and animals measuring the activity of the vestib those located at the top of the crista are par- ular nerve. Mechanics of the Cupula 233

Distribution of ciliar deflections (0)

30 4e-2 A 20 2e-2 10 s tress type 2 stress type 1 0 Oe+O 0

-10 -2e-2 -20

-30 "i--r---r-,-,.--.....-...,.--r---r-,,....-,.--.....--.--.,..-..,.-.--.... -4e-2 -120 -90 -60 -30 0 30 60 90 120 Position of the cilia (0)

Distribution of ciliar deflection (0)

6e-l

5e-l

4e-l

3e-l

2e-l

le-1

Oe+O ------...... ,...------(': -120 -90 -60 -30 o 60 9" 120 Position of the cilia (0)

Figure 3. (A) Distributions of ciliar deflections for stresses of type 1 and 2. (8) Distribution of eiliar deflection for a transcupular pressure difference. The position of the cilia is given by the angle between the symmetry axis and the radius of the circle centered at the pOint of convergence of the elements of the crista upper part.

Even without any stimulation (rotational inflating pressure. This deflection increases or caloric), the hairs of the sensory cells lo when the inflating pressure increases. The cu cated on the sides of the crista are perma pula thus appears to have two functions: the nently subjected to a deflection due to the measurement of the classical, stimulation- 234 E. Njeugna et al related, transcupular pressure difference and the elastic forces exerted by the hairs of the the measurement of the inflating pressure sensory cells. needed for the parametric control of the me Despite the fact that this study of the me chanics of the semicircular canal. chanics of the cupula is limited to the median A further study of the cupular mechanics slice and the fact that the most simple model needs more reliable data on: (i) the three was chosen (neglecting subcupular space, ac dimensional geometry of the cupula and the tive role of the cilia, and so on), the qualita crista ampullaris, (ii) the rheological proper tive results are general, only the numerical ties of the ampullar wall and of the crista, (iii) values can change.

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