A mathematical approach enabling the calculation of the total endolymph flow in the semicircular ducts

Mechanical aspects of the semicircular ducts in the vestibular system

The semicircular ducts (SCDs) of the vestibular system play an instrumental role in equilibration and rotation perception of vertebrates. The present paper is a review of quantitative approaches and shows how SCDs function. It consists of three parts. First, the biophysical mechanisms of an SCD system composed of three mutually connected ducts, allowing endolymph to flow from one duct into another one, are analysed. The flow is quantified by solving the continuity equations in conjunction with the equations of motion of the SCD hydrodynamics. This leads to mathematical expressions that are suitable for further analytical and numerical analysis. Second, analytical solutions are derived through four simplifying steps while keeping the essentials of the coupled system intact. Some examples of flow distributions for different rotations are given. Third, the focus is on the transducer function of the SCDs. The complex structure of the mechano-electrical transduction apparatus inside the ampullae is described, and the consequences for sensitivity and frequency response are evaluated. Furthermore, both the contributions of the different terms of the equations of motion and the influence of Brownian motion are analysed. Finally, size limitations, allometry and evolutionary aspects are taken into account.

Semicircular canal geometry, afferent sensitivity, and animal behavior

The geometry of the semicircular canals has been used in evolutionary studies to predict the behaviors of extinct animals. These predictions have relied on an assumption that the responses of the canals can be determined from their dimensions, and that an organism's behavior can be determined from these responses. However, the relationship between a canal's sensitivity and its size is not well known. An intraspecies comparison among canal responses in each of three species (cat, squirrel monkey, and pigeon) was undertaken to evaluate various models of canal function and determine how their dimensions may be related to afferent physiology. All models predicted the responses of the cat afferents, but the models performed less well for squirrel monkey and pigeon. Possible causes for this discrepancy include incorrectly assuming that afferent responses accurately represent canal function or errors in current biophysical models of the canals. These findings leave open the question as to how reliably canal anatomy can be used to estimate afferent responses and how closely afferent responses are related to behavior. Other labyrinthine features, such as orientation of the horizontal canal, which is reliably held near earth-horizontal across many species, may be better to use when extrapolating the posture and related behavior of extinct animals from labyrinthine morphology.

A mathematical model for top-shelf vertigo: the role of sedimenting otoconia in BPPV

Benign paroxysmal positional vertigo (BPPV) is a mechanical disorder of the vestibular system in which calcite particles called otoconia interfere with the mechanical functioning of the fluid-filled semicircular canals normally used to sense rotation. Using hydrodynamic models, we examine the two mechanisms proposed by the medical community for BPPV: cupulolithiasis, in which otoconia attach directly to the cupula (a sensory membrane), and canalithiasis, in which otoconia settle through the canals and exert a fluid pressure across the cupula. We utilize known hydrodynamic calculations and make reasonable geometric and physical approximations to derive an expression for the transcupular pressure DeltaPc exerted by a settling solid particle in canalithiasis. By tracking settling otoconia in a two-dimensional model geometry, the cupular volume displacement and associated eye response (nystagmus) can be calculated quantitatively. Several important features emerge: (1) a pressure amplification occurs as otoconia enter a narrowing duct; (2) an average-sized otoconium requires approximately 5 s to settle through the wide ampulla, where DeltaPc is not amplified, which suggests a mechanism for the observed latency of BPPV; and (3) an average-sized otoconium beginning below the center of the cupula can cause a volumetric cupular displacement on the order of 30 pL, with nystagmus of order 2 degrees/s, which is approximately the threshold for sensation. Larger cupular volume displacement and nystagmus could result from larger and/or multiple otoconia.

Optimization of the mechanical performance of a two-duct semicircular duct system--part 2: excitation of endolymph movements

The endolymph flow inside the semicircular ducts is analytically investigated by considering a system of two hydrodynamically interconnected ducts. Rotation of this system adds an amount of motion (momentum) to parts of it. This results in an endolymph flow in generally all vestibular parts. The "external impulses" are the impulses which emerge by rotation of exclusively a particular vestibular part. The real impulses can be calculated from a set of equations which contain the external impulses. Analytical expressions are derived for the initial velocities in the ducts and for the maximum endolymph displacements. These formulae contain the external impulses and the ratios of: (1) the radii of crus commune and ducts (gamma), (2) the lengths of crus commune and ducts (lambda). It was proven that an interconnected system composed of two ducts, and also a system composed of two such semicircular duct systems, behaves as a pure rotation transducer (like a single duct does), also when it is rotated excentrically. Duct systems with polygonal and circular geometries were used to evaluate whether an optimal value of lambda would exist (gamma was already considered elsewhere). Optimum values of lambda in a range of about 0.10-0.52 were found. This rather wide range of values agrees with values from measurements. Optimization of an interconnected duct system appeared to be equal to optimization of a system composed of separate ducts.

Optimization of the mechanical performance of a two-duct semicircular duct system--part 1: dynamics and duct dimensions

The classical representation of the semicircular duct system consists of three separate duct circuits. The ducts are, however, in reality, hydrodynamically interconnected. Muller & Verhagen (1988a,b) derived equations for the mechanical behaviour of an interconnected system with three ducts (anterior, posterior and horizontal). An analytical solution of these equations would, however, be too complex to provide surveyable formulae. A system of two interconnected ducts avoids this complexity whilst keeping the essentials of the coupling of ducts intact. The solution of the equation of motion leads to expressions for time constants and maximal endolymph excursions which are functions of morphological parameters, viz. the ratios of radii (gamma) and lengths (lambda) of the common vestibular part (crus commune or utriculus) and the ducts. The system possesses two short time constants which are shown to have similar values. The maximum endolymph displacements in the two ducts after a steplike stimulus are the products of the respective initial velocities and combinations of time constants. The initial velocities depend strongly on the position of the labyrinth with respect to the excitating rotation vector. Measured data of gamma and lambda are compared with the theoretical results. For gamma, excellent agreement was found. lambda is treated elsewhere.

Optimization of the mechanical performance of a two-duct semicircular duct system--part 3: the positioning of the ducts in the head

In the majority of vertebrates, the horizontal duct of the vestibular system lies approximately in the yawing plane of the head. The positioning of the vertical ducts, however, is not in the pitch- and roll planes but the vertical ducts generally lie under an angle of about 30-45 degrees relative to the medial plane. Using the equations for a hydrodynamically interconnected two-duct system, optimal positions of the vertical and horizontal ducts in different vertebrate groups can be derived. It was stated that the mean response of the vertical ducts should be optimized. This leads to a symmetrical positioning of the vertical ducts with respect to the medial plane. In all observed vertebrate groups, a solution of mu =(pi-alpha)/2 is found (mu is the angle of the vertical ducts relative to the medial plane, alpha is the angle between the vertical duct planes). For alpha=90 degrees, this provides an equal sensitivity for pitch- and roll- movements. For alpha>90 degrees, a larger sensitivity for pitch movements is obtained, at the expense of a lower sensitivity for roll movements. It is argued that the angle alpha between the vertical ducts may vary from 90 to 120 degrees. In most vertebrates, the centre of mass is stabilized by e.g. fins, tri- or quadrupedal stability, a crawling body or upside-down resting positions (e.g. bats). Birds are generally biped, so in walking they are also rather sensitive to roll. These features are related to labyrinth positioning in the head.

Three-dimensional reconstruction of the membranous vestibular labyrinth in the toadfish, Opsanus tau

Membranous vestibular labyrinths from the oyster toadfish, Opsanus tau, were fixed, dissected from the animal, stained, and embedded in rectangular blocks of clear histological resin. Photomicrographs of complete embedded labyrinths were taken from six orthogonal directions and used to construct three-dimensional (3D) geometrical models of the semicircular canals, ampullae, utricular vestibule and common crus. Membraneous ducts and ampullae were modeled using a set of cross-sectional elliptical curves laced together to generate curved tubular models of each structure. The ensemble of these curved tubes was used to generate a complete 3D reconstruction of the outside surface of the membranous labyrinth. When viewed from six orthogonal directions, reconstructions closely matched the embedded tissue. Dimensions of the reconstruction and histological sections were compared to measurements of fresh tissue taken from the same animals prior to fixation and used to correct the reconstructions for tissue shrinkage. Results provide estimates of the endolymphatic volumes, local cross-sectional areas and elliptical eccentricities as well as 3D orientations of the geometric canal planes relative to the skull. Ten micrometer histological sections of the material were also prepared to measure wall thickness in various regions of the labyrinth.

Comparative review of the human bony labyrinth

The bony labyrinth inside the petrous part of the temporal bone houses the organs of hearing and balance. Being functionally linked with sensory control of body movements and located in a part of the basicranium that is closely associated with the brain, this structure is of great interest in the study of human evolutionary history. However, few aspects of its morphology have been studied in nonhuman primates. This review compares the bony labyrinth of humans with that of the great apes and 37 other primate species based on data newly acquired with computed tomography combined with previous descriptions. With body mass taken into account, consistent differences are found between the size of the semicircular canals in humans, the great apes, and other primates. In particular, the arcs of the anterior and posterior canals are larger in humans than in the African apes. The functional implications of semicircular canal dimensions for registering angular head motion are evaluated in relation to locomotor behavior. Biophysical models, comparative studies, and some neurophysiological experiments all support a link between semicircular canal size and agility, or more specifically the frequency contents of natural head movements, but the evidence on the exact nature of this link is ambiguous. It is concluded that any link between the characteristic dimensions of the human canals and locomotion will be more complex than a simple association with the broad categories of quadrupedal vs. bipedal behavior. The functionally important planar orientations of the semicircular canals are similar in humans and the African apes as well as in many other species. In contrast, other aspects of the human labyrinth differ markedly in shape, following a pattern that seems to reflect the characteristic architecture of the human basicranium. Indeed, it is found that labyrinthine and basicranial shape are interspecifically correlated in the sample, and in most respects the human morphology is consistent with the general trend among primate species. Differences in brain growth and development are proposed as the predominant factor underlying both the unique shape of the human labyrinth as well as the interspecific labyrintho-basicranial correlations.

A new quantitative model of total endolymph flow in the system of semicircular ducts

1. A new concept of endolymph flow in the vertebrate vestibular system is presented. This approach describes quantitatively the flow in the entire system of three semicircular ducts interconnected by the utriculus and the crus commune. This approach is quite distinct from the classical theory in which the labyrinth is generally conceived to consist of three separate duct circuits. 2. The present approach shows the following set of distinct differences to the classical view: (a) In a labyrinth composed of three ducts perpendicular to each other the flow is non-zero in the other ducts when the labyrinth is rotated in the plane of a particular duct. (b) In a labyrinth with two equal ducts and with the duct planes under approximately 73 degrees the flow in one duct is zero when the rotation takes place in the plane of the other duct. Previous measurements of duct angles reflect this value surprisingly well. An obtuse or sharp angle between duct planes can lead to better performance of a particular labyrinth because the "external impulses" in the different ducts may amplify or compensate each other. (c) The behaviour of the flow in the entire labyrinth is a non-linear function of direction or rotation (cf. points (d), (e]. (d) Six time constants for the entire labyrinth can be distinguished (three long, three short); the flow in a particular duct is composed of six terms with these time constants. The composition of this flow and thus the relative importance of the terms depends on the positioning of the labyrinth with respect to the rotation vector. (e) The time constants also depend, for different labyrinths, on a shared influence of the dimensions of the ducts and the elastic properties of all three cupulae. (f) The forces in a particular duct depend also on the amount of motion the fluid will acquire in the other ducts. (g) The sensitivity of a particular duct depends also on the dimensions of the other parts in the vestibular system. 3. Equations for a system consisting of two ducts and for the classical single duct system are also given. Both systems are special cases of the three-duct system. The single duct equations are equivalent with equations given by Oman (1980) and Oman et al. (1987) which include the contribution of a wide utriculus. 4. The present theory of endolymph flow is mainly supported by the outcome of previously performed experiments concerning time constants and rotation of human subjects in different planes.(ABSTRACT TRUNCATED AT 400 WORDS)