The Role of 3-Canal Biomechanics in Angular Motion Transduction by the Human Vestibular Labyrinth

Abstract

The present work examines the role of the complex geometry of the human vestibular membranous labyrinth in the process of angular motion transduction by the semicircular canals. A morphologically descriptive mathematical model was constructed to address the biomechanical origins of temporal signal processing and directional coding in determining the inputs to the brain. The geometrical model was developed based on shrinkage-corrected temporal bone sections using a segmentation/data-fitting procedure. Endolymph fluid dynamics within the 3-canal labyrinth was modeled using an asymptotic form of the Navier–Stokes equations and solved to estimate endolymph and cupulae volume displacements. The geometrical model was manipulated to study the role of major morphological features on directional and temporal coding. Anatomical results show that the bony osseous canals provide reasonable estimates of the orientation of the delicate membranous canals—the two differed by only 3.48 ± 1.89°. Biomechanical results show that the maximal response directions are distinct from the anatomical canal planes, but can be closely approximated by fitting a flat plane to the centerline of the canal of interest and weighting each location along the centerline with the inverse of the cross-sectional area squared. Vector cross-products of these maximal response directions, in turn, determine the null planes and prime directions that transmit the direction of angular motion to the brain as three independent directional channels associated with the nerve bundles. Finally, parameter studies indicate that changes in canal cross-sectional area and shape only moderately affect canal temporal and directional coding, while three-canal orientation is critical to directional coding.

Appendix

The fluid filled labyrinth was modeled using the approach of Damiano and Rabbitt14 adjusted to the three-dimensional geometry of the human labyrinth. The endolymph was modeled as a Newtonian fluid undergoing low Reynolds number, low Stokes number, small displacement laminar flow. Inertial forcing due to acceleration of the head is introduced using a Galilean transformation. A slender body asymptotic expansion14 or, alternatively a control volume approach,34 reduces the Navier–Stokes equations to an ordinary differential equation acting along the curved centerline of each duct segment. Following the notation of Rabbitt31, for each short segment n of the labyrinth (n = HC, PC, AC, CC, UA, or UP), the volume displacement of the endolymph (Q) during head movements are represented by:

$$ m_n \frac{d^2Q_n}{dt^2}+c_n \frac{dQ_n}{dt}+k_n Q_n =P_n (l_n)-P_n (0)+f_n.$$

(1)

Parameters m _n , c _n and k _n are proportional to the equivalent mass, damping and stiffness of the endolymph (or cupula depending on location in the canal, as will be discussed later), respectively, and can be estimated from endolymph density (ρ), viscosity (μ), shear stress (γ), and cross-sectional area function A(s) according to the following expressions:

$$ m_n =\int_0^{l_n} {\frac{\rho }{A(s)}ds}, $$

(2)

$$ c_n =\int_0^{l_n } {\frac{\mu \lambda }{A(s)^2}ds}, $$

(3)

$$ k_n =\int_0^{l_n } {\frac{\gamma \lambda }{A(s)^2}ds}. $$

(4)

The independent variable s _n defines a curved coordinate running along the centerline of the canal segment. Integration limits are along the length (l _n ) of the duct segment (n). The term λ is a dimensionless number that relates the viscous drag to the flow rate based on the cross-sectional shape and frequency of excitation.14, 30 For low Reynolds number, low Stokes number flow, the velocity distribution in an elliptical cross-section is${u=\frac{\Updelta P}{2\mu \Updelta \ell }\frac{\left({ab} \right)^2}{\left({a^2+b^2} \right)}\left({1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} \right)}$, where (x,y) are cross-sectional coordinates with origin at the center, (a,b) are the major and minor radii, and ${\Updelta P/\Updelta \ell}$ is the pressure gradient. This results in a viscous drag factor ${\uplambda =8\uppi \frac{2-\upvarepsilon ^2}{2\left({1-\upvarepsilon ^2} \right)^{3/2}}},$ where the elliptical eccentricity ${\varepsilon =\sqrt {1-\left(a/b \right)^2}}$. The parameter λ reduces to 8π for a circular cross-section and is easily extended to higher frequencies (Stokes number >1) by making λ a frequency-dependent complex-valued parameter following Damiano.14

The right hand side of Eq. (1) includes a pressure gradient, ${\Updelta P=P_{n}(l_{n})-P_{n}(0),}$ relating the pressures on either end of the labyrinth segment, and inertial forcing f _n , due to angular acceleration of the head. The inertial forcing was calculated as

$$f_n =\int_0^{l_n } {\rho (\ddot {\vec {\Upomega }}\times \vec {R}(s))} \cdot d\vec {s}, $$

(5)

where ${\vec {R}}$ is a vector extending from the stereotactic head-fixed coordinate system origin to the centerline of the segment of interest. Angular acceleration (${\ddot {\vec {\Upomega }})}$ is presented as a vector resolved in the head-fixed coordinate frame (Fig. A.1). It is calculated from angular acceleration relative to the ground-fixed inertial frame (${\ddot {\vec {\Upomega}}_{\rm inertial})}$ by applying the time dependent orthonormal rotation matrix, N, relating the head-fixed frame to the ground-fixed frame using:

$$ \ddot {\vec {\Upomega }}=N\ddot {\vec {\Upomega }}_{\rm inertial}. $$

(6)

Figure A.1.

Model labyrinth geometry. The human membranous labyrinth has 3 natural bifurcation points (1–3) where the six labyrinthine segments join. The three-dimensional motion of each segment was specified by time-dependent angular acceleration in the ground-fixed intertial frame and resolved into the moving head-fixed system. This introduces a Galilean transformation and an inertial force that appears in the Navier–Stokes equations. Poiseuille flow and slender body approximations were assumed to further reduce the equations to a set of coupled ordinary equations (Damiano and Rabbitt14). Equations for six segments were coupled together at the bifurcations by conservation of mass and pressure continuity.

Equations (1)–(6) were employed to compute the endolymph fluid displacements and resultant pressure gradients in a single canal or other labyrinth component. Comparable equations have been used to describe the dynamic responses of a single the HC14, 30 and a 3-canal labyrinth.31, 35 In the single-canal models the pressure gradient, ΔP, corresponded to transcupular pressure. In the current work, pressure gradients and volume displacements apply to six individual duct segments, the HC, AC, PC, CC, UA, and UP. Equations for the segments were coupled together to form a coupled 3-canal model. This was accomplished by identifying three bifurcation points (Fig. A.1) where the labyrinth segments naturally connect to one another and applying conservation of fluid volume and pressure continuity to create a matrix equation governing whole labyrinth endolymph (^e) fluid mechanics:

$$ {\mathbf M}^e\frac{d^2\vec {Q}^e }{dt^2}+{{\mathbf C}^e}^e\frac{d\vec {Q}^e }{dt}+{{\mathbf K}^e}^e\vec {Q}^e =\vec {F}^e-\Delta \vec {P}. $$

(7)

The mass, ${{\mathbf M}^e,}$ damping ${{\mathbf C}^e,}$ and stiffness ${{\mathbf K}^e}$ are matrices are

$$ {\mathbf M}^e=\left[ {\begin{array}{lll} {\mathbf M}_{\rm HC} & m_{\rm UA} & -m_{\rm UP} \\ m_{\rm UA} & M_{\rm AC} & m_{\rm CC} \\ -m_{\rm UP}& m_{\rm CC} & M_{\rm PC} \\ \end{array} } \right], $$

(8)

$$ {\mathbf C}^e=\left[ {{\begin{array}{lll} C_{\rm HC}& c_{\rm UA} & -c_{\rm UP} \\ c_{\rm UA} & C_{\rm AC} & c_{\rm CC} \\ -c_{\rm UP} & c_{\rm CC} & C_{\rm PC} \\ \end{array} }} \right], $$

(9)

and

$$ {\mathbf K}^e=\left[ {{\begin{array}{lll} K_{\rm HC} & k_{\rm UA} & -k_{\rm UP} \\ k_{\rm UA} & K_{\rm AC} & k_{\rm CC} \\ -k_{\rm UP} & k_{\rm CC} & K_{\rm PC} \\ \end{array} }} \right]. $$

(10)

The diagonal elements are the addition of the segments forming a closed loop around the respective canal; e.g., ${M_{\rm HC} =m_{\rm HC} +m_{\rm UP} +m_{\rm UA}}$, ${M_{\rm PC} =m_{\rm PC} +m_{\rm CC} +m_{\rm UP} }$ and, ${M_{\rm AC}=m_{\rm AC} +m_{\rm CC} +m_{\rm UA}}$ . In the present work we have selected the local coordinate systems such that ${m_{\rm HC},m_{\rm UP},m_{\rm UA},m_{\rm PC} ,m_{\rm CC} }$ and m _AC are all positive. This simplifies the sign convention and clarifies the sign of each element in the matrix (but differs from previous work31, 36). Each element in the above matrices was calculated from Eqs. (2)–(4). The pressure across the three cupulae is

$$\Updelta \vec {P}=\left[ {{\begin{array}{l} \Updelta P_{\rm HC} \\ \Updelta P_{\rm AC} \\ \Updelta P_{\rm PC} \\ \end{array} }} \right] $$

(11)

and inertial forcing becomes

$$ \vec {F}^e=\left[ {{\begin{array}{l} f_{\rm HC} +f_{\rm UA} +f_{\rm UP} \\ f_{\rm AC} +f_{\rm CC} +f_{\rm UA} \\ f_{\rm PC} +f_{\rm CC} -f_{\rm UP} \\ \end{array} }} \right]. $$

(12)

The vector ${\vec {Q}^e }$ contains the volume displacements of the endolymph at the HC, AC, and PC cupulae:

$$ \vec {Q}^e =\left[ {{\begin{array}{l} Q_{\rm HC}^e \\ Q_{\rm AC}^e \\ Q_{\rm PC}^e \\ \end{array} }} \right] $$

(13)

The preceding equations provide a method of relating endolymph volume displacements to prescribed head angular accelerations. In order to characterize the effect of these fluid displacements on cupula movement, it is necessary to develop a model for the cupula. The cupula has often been modeled as an elastic or viscoelastic material impermeable to endolymph.30, 33, 46 These models neglect the intrinsic porosity of the cupula’s mucopolysaccharide matrix, resulting in a 1:1 relationship between endolymph and cupula displacement. In this model, we represent cupula porosity by assuming that the structure is composed of a fluid and a solid phase. The solid portion responds to pressure gradients according to the Equation

$$ {\mathbf M}^c\frac{d^2\vec {Q}^c }{dt^2}+{\mathbf C}^c\frac{d\vec {Q}^c }{dt}+{\mathbf K}^c\vec {Q}^c =\Delta \vec {P}+\vec {F}^c, $$

(14)

in which ${{\mathbf M}^c,}$ ${{\mathbf C}^c,}$ and ${{\mathbf K}^c}$ are diagonal matrices corresponding to the equivalent mass, stiffness, and viscosity of each cupula. For these calculations, the upper limit of integration is cupula thickness, h _c . Inertial forcing, ${\vec {F}^c}$, is calculated from Eq. (5), where ρ refers to the total density of the solid and fluid phase components of the cupula. The pressure gradient, ${\Updelta \vec {P}}$, is the interaction force, where

$$ \Updelta \vec {P}=\Upgamma \left({\frac{d\vec {Q}^e}{dt}-\frac{d\vec {Q}^c}{dt}} \right). $$

(15)

The constant Γ is related to Darcy’s constant (D _a ), a term used to describe fluid flow through a porous medium, the cupula solid phase volume fraction (ψ), and the cross-sectional area of the cupula (A ^c) by

$$ \Upgamma =(D_a \psi A^c)^{-1}. $$

(16)

The equations governing endolymph and cupula responses were combined to produce the following differential equation describing the entire system:

$${\mathbf {M}}\frac{d^2\vec {Q}}{dt^2}+{\mathbf C}\frac{d\vec {Q}}{dt}+{\mathbf K}\vec {Q}=\vec {F}. $$

(17)

Note, the forcing vector ${\vec {F}}$ has units of pressure and the displacement ${\vec {Q}}$ has units of volume. The effective mass, stiffness, and damping matrices are

$$ {\mathbf M}=\left[ {\begin{array}{ll} {\mathbf M}^e & 0 \\ 0 & {\mathbf M}^c \\ \end{array} } \right], $$

(18)

$$ {\mathbf C}=\left[ {{\begin{array}{ll} {\mathbf C}^e+\Upgamma & -\Upgamma \\ -\Upgamma & {\mathbf C}^c+\Upgamma \\ \end{array} }} \right] $$

(19)

and

$$ {\mathbf K}=\left[ {\begin{array}{ll}{\mathbf K}^e & 0 \\ 0 & {\mathbf K}^c \\ \end{array} } \right], $$

(20)

respectively, where the matrix ${\Upgamma=\Upgamma {\mathbf I}}$ .

The inertial forcing vector is condensed to

$$\vec {F}=\left[ {\begin{array}{l} \vec {F}^e \\ \vec {F}^c \\ \end{array} } \right] $$

(21)

and endolymph, and cupula volume displacements are simply

$$ \vec {Q}=\left[ {\begin{array}{l} \vec {Q}^e \\ \vec {Q}^c \\ \end{array} } \right]. $$

(22)

Thus any three-dimensional head movement may be described in terms of cupula volume displacements in the three canals. The physical parameter values used in these equations are listed in Table A.1.

Table A.1. Model parameter values.

The mass, damping, and stiffness matrix elements were computed using the geometry and the physical parameters listed in Table A.1. For labyrinth 1, the effective mass matrix (g/cm⁴) was

$${\mathbf M}=\left[ {\begin{array}{llllll} 1134 & 18.6 & 0 & 0 & 0 & 0 \\ 18.6 & 1165 & 38.2 & 0 & 0 & 0 \\ 0 & 38.2 & 1517 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2.39 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2.37 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2.98 \\ \end{array} } \right], $$

(23)

the damping matrix (dyn s/cm⁵) was

$${\mathbf C}=\left[ {\begin{array}{llllll} 2.19e6 & 287 & 0 & -2e6 & 0 & 0\\ 287 & 2.21e6 & 1105 & 0 & -2e6 &0 \\ 0 & 1105 & 2.26e6 & 0 & 0 & -2e6\\ -2e6 & 0 & 0 & 2e6 & 0 & 0 \\ 0 & -2e6 & 0 & 0 & 2e6 & 0 \\ 0 & 0 & -2e6 & 0 & 0 & 2e6 \\ \end{array} } \right], $$

(24)

and the stiffness matrix (dyn/cm⁵) was

$$ {\mathbf K}=\left[{{\begin{array}{llllll} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 17016 & 0 & 0 \\ 0 & 0 & 0 & 0 & 16715 & 0 \\ 0 & 0 & 0 & 0 & 0 & 26522 \\ \end{array} }} \right]. $$

(25)

The pressure forcing vector (dyn/cm²) was determined from

$$\vec {F}=\left[ {\begin{array}{l} -0.6107 \\ -0.00757 \\ -0.00480 \\ 0.00990 \\ 0.00606 \\ 0.00423 \\ \end{array}} \right] {\ddot {\Upomega}}_{{\rm HC}^{\prime}}+ \left[{\begin{array}{l} -0.000348 \\ 0.6399 \\ 0.00981 \\ 0.00115 \\ 0.00393 \\ -0.00559 \\ \end{array}} \right] {\ddot {\Upomega}}_{{\rm AC}^{\prime}} +\left[{\begin{array}{l} -0.000348 \\ 0.00906 \\ 0.7991 \\ 0.00364 \\ -0.00522 \\ -0.0106 \\ \end{array}} \right] {\ddot {\Upomega}}_{{\rm PC}^{\prime}}, $$

(26)

where ${\ddot {\Upomega}_{\rm HC}^{\prime} }$ , ${\ddot {\Upomega}_{\rm AC}^{\prime}}$ and ${\ddot {\Upomega}_{\rm PC}^{\prime}}$ are the projected components of head angular acceleration (rad/s²) in the prime directions ${{n}_{\rm HC}^{\prime} =[-0.064,0.042,-0.997]}$, ${{n}_{\rm AC}^{\prime} =[0.702,0.699,-0.136]}$ and ${{n}_{\rm PC}^{\prime} =[0.857,-0.557,-0.103]}$, respectively.