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. Extending known hydrodynamic calculations and making reasonable geometric and physical approximations, we derive an expression for the transcupular pressure $\Delta P_c$ exerted by a settling solid particle in canalithiasis. By tracking settling otoconia in a model two-dimensional geometry, the cupular 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 (almost twenty-fold in humans); 2) An average-sized otoconium requires approximately five seconds to settle through the wide ampulla, where $\Delta P_c$ is not amplified, which suggests a mechanism for the observed latency of BPPV; and 3) An average-sized otoconium can cause a volumentric cupular displacement on the order of 30 pL, corresponding to a step increase in angular velocity of $2^\circ$/s, approximately the threshold for sensation. Larger cupular displacement and nystagmus could result from larger and/or multiple otoconia.