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

Equations have same form.

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.

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

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:

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

But why not use highest resolution?
Convergence and times

Balance between required accuracy and reasonable 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.

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.

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.