# 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.
Electrical Mechanical Acoustical
$v=iR$ $f=Ru$ $p=RU$
$v=L\frac{di}{dt}$ $f=M\frac{du}{dt}$ $p=M\frac{dU}{dt}$
$v=\frac{1}{C}\int idt$ $f=k\int udt$ $p=\frac{1}{C}\int Udt$

### 1.2 Middle-ear circuit models

Block diagram of middle-ear model.

Applies to middle ears of most mammals.

Circuit model.

Note two components for eardrum.

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.

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

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 (or 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.

Recent gerbil model.

(Maftoon et al., 2015)

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

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

• little effect on lowest natural frequencies

Funnell (1983)

Variation of stiffness, density and thickness of pars tensa

• large effects

Funnell (1983)

Variation of curvature and depth of cone

• large effects
• suggestion that shape is ‘optimal’.

Funnell (1983)

Parameter sensitivity analysis in recent gerbil model

• one parameter at a time

(Maftoon et al., 2015)

Sensitivity analysis showing interactions between parameters.

• 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.

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

Spread of response to point stimulus, with anisotropy.

Spread of response to point stimulus, without 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

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. $W=\sum _{i=1}^{N}\frac{2{\mu }_{i}}{{\alpha }_{i}^{2}}\left({\lambda }_{1}^{{\alpha }_{i}}+{\lambda }_{2}^{{\alpha }_{i}}+{\lambda }_{3}^{{\alpha }_{i}}-3\right)$

Prony series used to fit relaxation curves. $G\left(t\right)=1-\sum _{i=1}^{N}{g}_{i}\left(1-{e}^{-t/{\tau }_{i}}\right)$

Simulation of tympanometry

Choukir, 2017

## 6. More realistic models

Maftoon et al. (2015): Finite-element modelling of the response of the gerbil middle ear to sound. JARO 16(5): 547–567

BMDE-501 Modelling middle-ear mechanics

R. Funnell