Relationships between semicircular duct radii with some implications for time constants.

The aims of this paper are (1) to provide accurate, statistically founded and "generalized" morphological parameters of labyrinth parts which can be used in recently developed hydrodynamical models of endolymph and cupula movement (cf. OMAN et al., 1987; MULLER & VERHAGEN, 1988a, b), (2) to discuss the implications of duct length and radius, cupular stiffness, ampullar and utricular geometry on the time constants of the semicircular duct system, (3) to discuss the mutual influence of the terms of the equation of motion on the transducing properties of this system ("velocity"- or "acceleration" transducer). Original, new measurements of the outer radii of the semicircular ducts (a, p, h) crus commune (c) and utriculus (u) of 225 labyrinths are given, based on scaled drawings or photographs in three literature sources: RETZIUS (1881, 1884), GRAY (1907) and BIERBAUM (1914) together with the statistical data provided by RAMPRASHAD et al. (1980, 1984, 1986). No survey of this extent has been published. The measurements of the averaged inner radii of the ducts measured by JONES & SPELLS (1963) (rJs) are statistically compared with the present measurements of the outer radii (raph = (ra+rp+rh)/3) giving a relationship of: rJs = 0.015 + 0.73. r aph . This provides a statistically founded measure for the thickness of the duct walls which has been used to correct measurements of outer duct radii to inner duct radii (correction factor 0.80). Relationships of these radii were investigated by statistical analysis (reduced major axis method). This analysis revealed the following results (radii given in mm): rp = -0.001 + 0.99. r a r c =-0.05 +2.14. r a r u =-0.066+2.77. r a r h = 0.003+0.90. r a r h = -0.02 + 1.06. r a (birds) r h = -0.09 + 0.90. r a (fishes) r h -0.001 +0.96. r a (without fishes). A strikingly linear relationship between duct radii has been found. This shows that there are strong allometric correlations within the labyrinth system. Although measurements of duct lengths and ductal volumes are (scarcely) reported in the literature, it is risky to use these data in calculations of time constants and endolymph movement. Ampullar and utricular volumes may also affect the mechanical properties of the labyrinth. It is argued that the sensitivity of the semicircular duct system is increased when the latter volumes can be kept small. By narrowing of the flow path towards a subcupular space the flow near the hair cells of the cristae is maximized. The question whether the semicircular duct system acts as a velocity- or an acceleration transducer is discussed. It is argued that it is unlikely that animals make significant movements below the natural frequency of the system (i.e. very slow movements). So, it is not needed to consider the long time constant (and the mechanical properties of the cupula) during movement. For movements faster than the natural frequency the magnitudes of the M.x term and the F.x term of the equation of motion mostly are identical. This means that both the inertial and frictional properties of the endolymph determine the mechanical behaviour of the system. Because the equation of motion with these terms may be integrated, the system acts technically as a velocity transducer, although this may give a misleading idea of the mechanical nature of the components of this transducer. For the 3-duct system the endolymph displacement in each duct is composed of six exponential terms each containing a time constant (three long, three short). The nature of these time constants is discussed considering a very simple 3-duct labyrinth. The time constants of the system are calculated using the values of duct radii, found from the regression analysis.