Middle-ear models

1. Circuit models

1.1 Electrical-mechanical-acoustical analogies

Electrical	Mechanical	Acoustical
voltage	force	pressure
current	velocity	vol. velocity
resistor	dashpot	mesh
inductor	mass	tube
capacitor	spring	volume

The analogies can also be inverted:

voltage ► velocity
current ► force

Each approach has advantages and disadvantages. Equations have same form.

Electrical	Mechanical	Acoustical
$v = i R$	$f = R u$	$p = R U$
$v = L \frac{d i}{d t}$	$f = M \frac{d u}{d t}$	$p = M \frac{d U}{d t}$
$v = \frac{1}{C} \int i d t$	$f = k \int u d t$	$p = \frac{1}{C} \int U d t$

1.2 Middle-ear circuit models

Block diagram of middle ear Block diagram of middle-ear model.

Applies to middle ears of most mammals.
Circuit model

Circuit model.

Note two components for eardrum.
More complex circuit model

More complex circuit model.

1.3 Problems with lumped models

Lack of direct connection between parameter values and anatomical or physiological properties.

For example:

moment of inertia changes if axis of rotation changes
mass parameter for stapes changes if its mode of vibration changes
eardrum parameters change if its vibration pattern changes

After http://xkcd.com/730/

2. Finite-element method

In the finite-element method, a distributed physical system to be analysed is divided into a number (often large) of discrete elements.

The complete system may be complex and irregularly shaped, but the individual elements are easy to analyse.
Eardrum mesh

The division into elements may partly correspond to natural subdivisions of the structure.

For example, the eardrum may be divided into groups of elements corresponding to different material properties.

Most or all of the model parameters have very direct relationships to the structure and material properties of the system.

Relatively few free parameters ...

... if parameters are known a priori.

3. Eardrum model

3.1 Model parameters

Model parameters:

Shape
Thickness
Material properties:
- Young’s modulus
- Poisson’s ratio
- Mass density
Damping
Loads: pressures and/or concentrated loads

In this model, there are different material properties for the triangles in

pars tensa
pars flaccida
manubrium

3.2 Low-frequency simulation results

Qualitatively similar to experimentally observed patterns.

Varying mesh resolution to decide how fine a mesh is required:

Results will converge monotonically if

the element formulation is ‘compatible’, or ‘conforming’
(e.g., Bathe K-J (1982): Finite element procedures in engineering analysis, Prentice-Hall, Englewood Cliffs, xiii+735 pp.; p. 167 etc.)
the geometry doesn’t change as the mesh is refined

Based on maximum-displacement values, there seems to be little advantage here to using a mesh resolution greater than about 15 elements/diameter.

But why not use highest resolution?

Balance between required accuracy and acceptable computation time.

3.3 Higher-frequency simulation results

Undamped natural frequencies and modes of vibration.

Increasing complexity of vibration pattern with frequency.

With damping, different areas of drum have different phases.

Recent gerbil model.

(Maftoon et al., 2015)

Damped frequency response Frequency response of point on manubrium is smoother than those of points on eardrum.

Note very large phase lag at high frequencies.

(Maftoon et al., 2015)

3.4 Insights via numerical experiments

3.4.1 Effects of parameter variations

Varying ossicular parameters Variation of ossicular stiffness and moment of inertia (with fixed axis of rotation)

little effect on lowest natural frequencies

Funnell (1983)

Varying p. tensa parameters Variation of stiffness, density and thickness of pars tensa

large effects

Funnell (1983)

Varying shape parameters Variation of curvature and depth of cone

large effects
suggestion that depth may be ‘optimal’.

Funnell (1983)

Gerbil sensitivity analysis Parameter sensitivity analysis in recent gerbil model

one parameter at a time

(Maftoon et al., 2015)

Sensitivity analysis showing interactions between parameters.

Effects on stapes footplate displacement:

Eardrum and ligaments:
little interaction
Incudomallear and incudostapedial joints:
strong interaction

(Qi et al., 2004)

3.4.2 Effects of asymmetry

Natural frequencies, and increasingly complicated mode shapes.

Spread of natural frequencies:
f₁₀/f₁ = 9.1

Ellipse:
f₁₀/f₁ = 6.0

Ellipse with symmetrical ‘manubrium’:
f₁₀/f₁ = 3.8

Ellipse with asymmetrical ‘manubrium’:
f₁₀/f₁ = 3.4

Cat eardrum:
f₁₀/f₁ = 2.5

Damping smears the closely-space natural modes together.

Increasing the damping (left to right) smooths the curves more and more but leaves the overall levels and slopes unchanged.

Funnell et al. (1987)

3.4.3 Wave propagation

Animation: Wave propagation for point stimulus with strong anisotropy

Spread of response to point stimulus, with anisotropy.
Animation: Wave propagation for point stimulus with isotropic properties

Spread of response to point stimulus, without strong anisotropy.
Animation: Wave propagation for pressure stimulus with strong anisotropy

Actual stimulus is a pressure.

4. Modelling of ossicles and ligaments

Ossicular vibration is not (in general) around a fixed axis.

Depends on

ligaments
eardrum
mass distribution
bending of bone?

4.1 Bending of manubrium

Importance of model predictions
Interaction between modelling and experiment

Funnell WRJ, Khanna SM & Decraemer WF (1992): On the degree of rigidity of the manubrium in a finite-element model of the cat eardrum. J. Acoust. Soc. Am. 91(4): 2082-2090 (doi:10.1121/1.403694)

4.2 Incus/stapes coupling

Very thin bony connection

Shrapnell HJ (1833): On the structure of the os incus. London Medical Gazette 12: 171–173
Lenticular process

Funnell WRJ, Siah TH, McKee MD, Daniel SJ & Decraemer WF (2005): On the coupling between the incus and the stapes in the cat. JARO 6(1): 9-18 (doi:10.1007/s10162-004-5016-3)

5. Non-linear viscoelasticity

Experiments by Cheng et al. (2007) on strips of eardrum:

Uniaxial stress-strain tests
Relaxation tests

Ogden hyperelastic model fitted to loading curves. $W = \sum_{i = 1}^{N} \frac{2 μ_{i}}{α_{i}^{2}} (λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3)$

Motallebzadeh et al. (2013)
Relaxation curves

Prony series used to fit relaxation curves. $G (t) = 1 - \sum_{i = 1}^{N} g_{i} (1 - e^{- t / τ_{i}})$

Motallebzadeh et al. (2013)
Hysteresis loops

iterative process to adjust parameters to fit loading curves
- while taking relaxation into account
model fits both loading and unloading curves with same parameters

Motallebzadeh et al. (2013)

Simulation of tympanometry

Choukir, 2017

6. Simulation-based inference

Make clinical measurements
Adjust model parameters to fit measurements
Use fitted parameter values to make diagnosis

For example: increased cochlear-load parameter suggests otosclerosis (immobile stapes)

The fitting process is computationally expensive.

Run a lot of simulations
Use simulation results to train a neural network
Make clinical measurements
Ask neural network what model parameters would match measurements
Use parameter values to make diagnosis

Running the simulations is computationally expensive.

Motallebzadeh et al., 2024

BMDE-501 Modelling middle-ear mechanics

R. Funnell

Last modified: 2024-10-21 12:43:04