Fluid-particle dynamics in canalithiasis

Abstract

The semicircular canals (SCCs; located in the inner ear) are the primary sensors for angular motion. Angular head movements induce a fluid flow in the SCCs. This flow is detected by afferent hair cells inside the SCCs. Canalithiasis is a condition where small particles disturb this flow, which leads to benign paroxysmal positional vertigo (top-shelf vertigo). The present work investigates the interaction between the fluid flow and the particles on the basis of an idealized analytical model. Numerical solutions of the full model and a thorough analytical study of the linearized model reveal the principal mechanisms of canalithiasis. We propose a set of dimensionless numbers to characterize canalithiasis and derive explicit expressions connecting these dimensionless numbers directly to the typical clinical symptoms.

Figure 1

Nystagmus velocity N (dots) and head velocity (solid line) of a canalithiasis patient during and after a head manoeuvre. The time T_P and the magnitude N_max of the maximum nystagmus velocity are only estimates due to incomplete data between t=6 and 10 s. After its peak (t>10 s), the positional nystagmus decays approximately like exp (−t/4.2 s) (dot-dashed line).

Figure 2

The human vestibular system with schematic of a single SCC. (Reproduced from an original illustration by M. Pinker.)

Figure 3

(a) Cupular displacement V(t) in an SCC without particles during and after (b) a head manoeuvre from α=0° to 120° (ϵ=0.09752, σ_c≈0.00610). Solid line, α(t); dashed line, $\dot{\alpha }(t)$.

Figure 4

Numerical simulation of the (a) cupula displacement V(t) (solid line, with particles; dashed line, without particles) and (b) particle position x_p(t) and velocity ${\dot{x}}_{\text{p}}(t)$ (solid line, x_p/R; dashed line, x_p/R; dotted line, −α) for an SCC with seven particles (n_p=7) of radius a_p=14 μm (ϵ=0.09752, Fr²=0.79052, ξ=217.69, Χ=0.02089). For comparison, (a) shows again the cupula displacement of a healthy SCC from figure 3a.

Figure 5

Schematics of the SCC with particles (a_p=11 μm, n_p=10; ϵ=0.09752, Fr²=0.79052, ξ=352.61, Χ=0.02345) during a head manoeuvre. (a) Per-rotatory phase: (i) initial state, cupula relaxed (α=0°, $\dot{\alpha }$=0° s⁻¹), (ii) maximum head velocity, cupula fully deflected (α=60°, $\dot{\alpha }$=87.5° s⁻¹) and (iii) end of head manoeuvre, cupula overshoot (α=120°, $\dot{\alpha }$=0° s⁻¹). (b) Post-rotatory phase: (i) end of latency period, t=T_L (α=120°, $\dot{\alpha }$=0° s⁻¹), (ii) peak nystagmus, t=T_P (α=120°, $\dot{\alpha }$=0° s⁻¹) and (iii) return to relaxed state, t→∞ (α=120°, $\dot{\alpha }$=0° s⁻¹) (the arrow shows the absolute particle velocity ${\dot{x}}_{\text{p}}+R\dot{\alpha }$; the cupula is drawn separately below the canal as a bulged membrane).

Figure 6

Influence of the particle size a_p and the particle number n_p on the characteristic values of the positional nystagmus (curves for n_p=1, 2, …, 50); the dotted line in (c) indicates the maximum cupula displacement in the per-rotatory phase.

Figure 7

Vanishing onset latency. (a) Latency T_L as a function of the particle size a_p and number n_p and (b) cupula displacement V(t) for a configuration without onset latency (a_p=20 μm, n_p=10).

Figure 8

(a) Time to peak T_P and (b) cupula displacement V_max as a function of a_p if the total particle mass M_p is kept constant ($n_{\text{p}}\propto a_{\text{p}}^{-3}$). The dotted line in (a) indicates the proportionality of T_P to $1/a_{\text{p}}^2$, and the dotted line in (b) indicates the maximum cupula displacement in the per-rotatory phase.

Figure 9

Particle trajectories for n_p=5 and (a) a_p=5 μm, (b) 15 μm and (c) 25 μm (dot indicates the location of the particles at the end of the head manoeuvre).

Figure 10

Axial particle velocity ${\dot{x}}_{\text{p}}(t)$ for n_p=5 and (a) a_p=5 μm, (b) 15 μm and (c) 25 μm (solid line, one-dimensional particle model; dashed line, two-dimensional particle model; note the different ranges for (a)).

Figure 11

Cupula displacement V(t) for n_p=5 and (a) a_p=5 μm, (b) 15 μm and (c) 25 μm (solid line, one-dimensional particle model; dashed line, two-dimensional particle model).

Figure 12

(a) Eigenvalues $\sigma ={\sigma }_{\text{r}}+\text{i}{\sigma }_{\text{i}}$ for 10 particles with radius a_p=10 μm. (b) Eigenvalues σ_s (slow particle mode) and σ_c (cupula mode) as a function of the particle radius a_p (n_p=10). The dashed lines show the estimates ϵ/16 and 1/(Fr²ξ).

Figure 13

(a) Ratio θ between the average fluid velocity $\overline{u}$ and the particle velocity $-\sigma {\hat{x}}_{\text{p}}$ of the cupula and slow particle modes as a function of the particle radius a_p (n_p=10). (b) The same data on a logarithmic scale together with the approximate expression θ_u.

Figure 14

Estimate for the difference between time to peak T_P and onset latency T_L as a function of the particle size according to the relationship (5.11) (ϵ=0.09752, Fr²=0.79052).

Figure 15

Two possible cases of transition from co-flow to counter-flow. (a) For A₁>0 the fluid velocity $\overline{u}$ changes the sign at $\tilde{t}=T_{\text{p}}$ when the cupula displacement V reaches an extremum. (b) For A₁<0 the particle velocity $\partial {\tilde{x}}_{\text{p}}/\partial \tilde{t}$ changes the sign whereas the fluid velocity and the cupula displacement decay monotonically. Solid line, $\overline{u}$; dashed line, $\partial {\tilde{x}}_{\text{p}}/\partial \tilde{t}$; dot-dashed line, V.

Figure 16

Value of 1−θ₀/θ_u as a function of the particle size a_p and number n_p (values indicated at the contour lines).