Size limitations in semicircular duct systems

The present article discusses mechanical requirements and limitations which are applicable to the construction of the system of semicircular ducts, especially to its size. The simplified case of a single, uniform duct system has been considered which can be described by a second order equation of motion. The principal functional quantities for this rotation-sensor are: (1) response speed; (2) sensitivity; and (3) regular flow. The response speed of a single, uniform semicircular duct is characterized by the short time constant (T-2) which is dependent on the duct radius (r). Its estimated range is from 0.04 ms in the smallest to 140 ms in the largest known labyrinth. The sensitivity is characterized by the maximal endolymph displacement after a step stimulus (x(max)). Its estimated range is from 0.0016 mu m to 5.97 mm (6.56 decades!), assuming an input angular velocity of omega = 1 rad s(-1). The Reynolds number is a measure for an undisturbed laminar flow. Its estimated range varies from 7.38.10(-4) to 45.1 for omega = 1 rad s(-1). The above data follow from graphs in which, for a single uniform duct, circuit radius (R) is plotted against duct radius (r) for labyrinths of 233 species belonging to different vertebrate-groups. A relation R = 38.9.r(1.60) was determined. The smallest labyrinth was found in a carp larva (Cyprinus), the largest in a whale shark (Rhincodon). Large whales possess labyrinths of average mammalian size. It is revealed that semicircular duct size is bound by requirements concerning regular flow and by a too low response speed for large labyrinths, and by a too low sensitivity for small labyrinths. Other important quantities are mechanical amplification factors which are a consequence of more complex vestibular constructions than a single uniform duct circuit. Allometric relationships are interpreted as compromises between the quantities mentioned. A hypothesis for the relatively large semicircular duct sizes of fishes, especially Elasmobranchii, compared with mammals and birds is presented. (C) 1999 Academic Press.