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 = i R f = R u p = R U
v = L d i d t f = M d u d t p = M d U d t
v = 1 C i d t f = k u d t p = 1 C 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:

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:

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

3.2 Low-frequency simulation results

Qualitatively similar to experimentally observed patterns.

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

Convergence and times

Results will converge monotonically if

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 reasonable computation time.

3.3 Higher-frequency simulation results

Natural frequencies

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 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: suggestion that shape is ‘optimal’.

Funnell (1983)

Gerbil sensitivity analysis Parameter sensitivity analysis in recent gerbil model.

(Maftoon et al., 2015)

Sensitivity analysis showing interactions between parameters.
Sensitivity analysis Sensitivity analysis

(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

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

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

4.1 Bending of manubrium

Manubrial bending

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

Shrapnell (1833)

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

Experimental measurement on TM strip

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

Stress-strain curves

Ogden hyperelastic model fitted to loading curves. W = i = 1 N 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 i = 1 N g i ( 1 e t / τ i )

Motallebzadeh et al. (2013)
Hysteresis loops

After 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. More realistic models

Finite-element model of gerbil middle ear

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
Last modified: 2018-11-27 18:49:16