Topological Approach of 1D Solid Solutions for Singularity in Semicircular Canals

Author(s): Yves Gourinat, Arnaud Rolland, Thomas Hanchin, Marie-Stéphane Guillaumont Quentin Legois

The dynamic beam equations provide an analytical model and a generic solution for continuous systems. The constants for amplitudes and frequencies are given by the boundary conditions, which are here considered looped, in order to represent a closed element, by topological closure. This type of model is applied to the representation of a semicircular canal. This approach addresses the deformability of this anatomical element, both in Lagrangian solid parts (bone, membranes, intermediate materials) and in fluids (Eulerian pressure waves). The model is proposed for both physiological representation and pathology modeling. The latter is represented by a passive or retroactive singularity. An original analytical approach is thus developed for each segment, representing physiological modes – nodes and bellies - and possible disturbances. The damping entropy is also the subject of a special segmented treatment, to take account of fluid-solid interactions, providing a coherent model. The result is a simplified but robust model that both reproduces the vibroacoustic modes of the semicircular canal and anticipates the effects of a singular pathology, such as a third window or neuritis. Mathematically, this model opens the way to structural analytical models of the inner ear, and to possible dynamic couplings between equilibration and acoustics.