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Basic Multifrequency Tympanometry:

The Physical Background

by

Joachim Gruber

(a pdf version of this document can be found at this address)

In tympanometry the mobility of the tympanic membrane is measured while the membrane is exposed to a (sinusoidal) tone of frequency f.

In the ear, the tympanic membrane is mechanically coupled with the middle ear ossicles to the oval window -the interface between middle and inner ear. It is this entire system (membrane, middle ear, oval window) that is forced into oscillation. The oscillation is detected by a microphone. (A more detailed description is given in "Tympanometry in just seconds".)

A linear theory used to evaluate the signal from the microphone is presented here. The response of a linear system when driven by a periodic oscillation can be expressed in terms of the resistance with which the system responds to the excitation (called "impedance") or in terms of the ease with which it is set into motion (called "admittance"). Both expressions of the response are presented here next to each other in a table.

An excellent summary of the linear theory, and a review of its practical applications and the reliability of multifrequency tympanometry in diagnosing middle ear diseases is:
Robert H. Margolis, Lisa L. Hunter, Acoustic Immittance Measurements, Chapter 17 of Audiology: Diagnosis, by Ross J. Roeser, Michael Valente, Holly Hosford-Dunn (eds), Thieme 2000.


I. Forced Linear Oscillations

Mechanical Impedance

Assumption 1 (Hooke's law)
Let m be a mass on a spring and F the force resulting from an elongation z of the spring. Then Hooke's law approximates the force F as being proportional to z 

(1) 

D is the Hook spring constant (compliance).
 

Mechanical Admittance

Let t be the time variable. .
Assumption 2 (velocity proportional friction)
When the mass moves, it is slowed down by friction R. Let the friction force be proportional to the speed v of the movement 

(2) 

where R is the friction coefficient.
 

.
Definition 1 (periodic force)
Let 

(3) 

be a force wiggling at mass m with 

is the imaginary unit. F0 is the amplitude of the force keeping the mass m in oscillatory motion.
 

.
To wiggle at mass m, force F has to be the sum of the following forces
  • (force to overcome the inertia of the system),
  •   (force to counteract the friction),
  •   (force necessary to overcome the tension of the spring).
.
Theorem 1 (equation of motion)
The resulting movement of the mass can be calculated from the force balance 
F = Fm + FR + FH,  i.e.

(4) 
 

.
Assumption 3 (periodic movement).
Let mass m oscillate with frequency  and let this oscillation be out of phase (in comparison with the oscillation of force F) by 

(5) 
 

.
The velocity v of mass m is then

(6) 

Abbreviate 

(7) 
 

.
Theorem 2 (system response)
The response of system (4) can be characterized by the ratio between force F and velocity vf

(8) 



Proof: 
Plugging (5) into (4), and then dividing both sides of (4) with 

yields

(9) 

Substituting (3) and (7) in (9) gives us (8).
 

Theorem 2' (alternative system response)
Alternatively, the system response can be characterized by the ratio between  velocity vf and force F

(8') 



Proof:
The reciprocal of (9) is

Substituting (3) and (7), this can be written as

Definition 2 (impedance)

(10) 

This ratio (10) will be called mechanical impedance Zm. The real and imaginary parts of the sum have the following names: 
 
 
 
 
 

resistance R
 

reactance 
 
 

mass reactance 
 
 

compliant reactance 

Definition 2' (admittance)

(10') 

In (10') the following abbreviations are used:

admittance 

conductance 
 

susceptance 

mass susceptance 

compliant susceptance 

II. Acoustics

Acoustic impedance 

Acoustic admittance

Let V be a fast (adiabatic, i.e. heat non-dissipating) change of a volume V of air and P the corresponding pressure change.  .
Definition 3 (compressibility )
The adiabatic compressibility of air is defined as

(11) 
 

.
Theorem 3
The compressibility can be expressed in terms of the density  of the air and the speed of sound c in air:

(12) 

Proof can be found in textbooks of physics.
 

.
Let volume V be approximated by a cylinder with base A (and a height h).  .
Definition 4 (cross section A of air volume)
Then volume change V can be expressed as change z of the cylinder height

(13) 
 

.
The corresponding pressure change P can be written in terms of the force F on A

(14) 
 

.
Definition 5 (Hooke's constant D for air, acoustic stiffness Ka)
Combining (11) - (14) the force F resulting from the volume change V can be written similarly as Hooke's law

(15) 

with the abbreviation

(16) 

where (see (11), (12)) 
 

.
Assumption 4 (friction R)
Let the volume V of air dissipate energy similarly as the mass m on a spring in (2):


 

.
Assumption (rigid body of oscillating masses)
The periodic oscillation of the air in the ear canal wiggles at the tympanic membrane, the middle ear ossicles etc. This has been ignored in the system dealt with until now.

Let us assume that all those masses comprise a rigid entity meff that oscillates as a whole and in phase with the air in the ear canal. In other words, the masses of which meff is composed do not oscillate separately and out of phase with the air.

Definition 7 (oscillating mass m)
The total oscillating mass m is therefore the mass V of the air plus the effective mass:

m = V + meff

Thus, the force to overcome the inertia of m is 

(17) 
 

.
Theorem 4 (equation of motion)
As in the case of the mechanical oscillator, the resulting movement of the air particles in volume V can be calculated from the force balance F = Fm + FR + FH, where

(18) 

(18a) 
 

.
For a periodic pressure being applied by a loudspeaker to the ear canal air and mass meff (assumption 3), the system's response is analogous to (8) (note that again F = A p): 

(19) 
 

.
Definition 6 (volume velocity U)
Volume velocity U is defined as the volume that flows through the air canal cross section per unit time:


 

.
It is customary to replace vf in (19) with iU/A. Deviding both sides of (19) by A2 we get the following expression chaaracterizing the system response

(20) 



Definitions 7 (acoustic resistance Ra, acoustic inertance M)

(1) To simplify the form of the equations, we will  introduce the acoustic resistance 

(2) Likewise, Kinsler and Frey (1962,  p. 190 (Eq. 8.14)) introduced the definition of acoustic inertance 

.



Using (16), the last term on the right hand side can be simplified:

(21) 
 

.
Theorem 5 (system response)
The final expression for the system response is (20). In analogy with (10) the ratio (22) is called acoustic impedance Za

(22) 
 

Theorem 5a (alternative system response)
Alternatively, the system response can be characterized by the inverse of ratio (22)

(22') 

Definition 8 (acoustic impedance Za)
The impedance Za given in (22) has a real and an imaginary part (see (20)). 

With definitions 7.1 (acoustic resistance Ra) and 7.2 (acoustic inertance M) and definition 5 (acoustic stiffness Ka) the acoustic impedance can be written in analogy with definition 2, and the following names are given:

(23)

resistance Ra

reactance 

mass reactance 

compliant reactance 

 

Definition 8' (acoustic admittance Ya, eqs. (23'))
 


 
 
 

(23')

conductance 

susceptance 

mass susceptance 

compliant or stiffness susceptance 

 

.

Fig. 1: GaRa and BaRa as a funtion of Xa/Ra. At |Xa|/Ra = 1 Ga Ra and Ba Ra have the same size. At resonance GaRa = 1 and BaRa = 0.


Fig. 2: Oscillation plotted in {GaRa, BaRa} plane lies on a circle with radius 1/2, because GaRa2+ BaRa2 = (1/2)2 for all Xa/Ra. The angle  will be used to calculate Ra from tympanographically meaasured Ga and Ba.
 

. Fit of Ra to multifrequency tympanogram Ga(f) and Ba(f)

Definition of  (see Fig. 2)

(24)

(25) 

From (23), (24) follows

(26) 



Proof:



Multifrequency tympanogramm gives Ga(f) and Ba(f) . Thus (26) is a function of the immission frequency f. (26) can be solved for  as a function of f.

With (25) Ra can be fitted to the tympanogram

(27) 
 

Definition 9 (resonance frequency r)
Let the frequency at which the reactance Xa and susceptance Ba vanish be called resonance frequency r of the system
.

Solving for r

(28) 

At resonance r conductance and resistance are simple reciprocals of each other:


 
 
Data:

(d1) 

At f = 226 Hz

(d2) 

Plugging (d1) and (d2) into the definition of Xca above 

(d3) 

A volume V = 1 cm3 of air has a compliant reactance 

At high positive or negative ear canal pressures the tympanic membrane is almost fixed and the middle ear is nearly motionless (meff approx. 0, Ra approx. 0) the admittance Ya = Ba = Bca (the latter because Xma << Xca) with


Since Bca can be determined experimentally, the ear canal volume V can be calculated from this equation. At f = 226 Hz

A volume V = 1 cm3 of air has a compliant susceptance

III. Parameter determination from multifrequency tympanometry

III.1 Single resonance frequency system

Fit of V and m/A2 to Xa/Ra

Use definitions 8 (23) of Xma and Xca:

(23.1) 

(23.2) 

In detail (see Fig. 3):

  1. Plot log|Xa| as a funtion of logf as shown below.
  2. Intersections of asymptotic lines with y-axis at logf = 0  give

Fig. 3: Extrapolation of Xca(f) and Xm(f) yields V and m/A2.
 

Another possibility, using (28), see Fig. 4:
Ear canal cross section A together with oscillating mass m can be fitted to the resonance frequency fr.


Fig. 4: Plot of contours of constant resonance frequency fr as a funtion of the ear canal radius r and the oscillating effective mass meff.
Example marked by arrows: for r = 0.37 cm and meff = 0.002 g the resonance frequency is fr = 1140 Hz.

As the contour plot Fig. 4 shows, a possible choice for fr = 1140 Hz is:


= 0.00129 g/cm3
V = 1.36 cm3
m = V + meff = (0.0018 + 0.002) g = 0.0038 g.
 

III.1.1 Example


Definitions and data

Definitions of frequencies characteristic of system Choice of dependence of V and meff on air pressure p established by tympanometer in ear canal:
(29) 
(30) 

Data used in Example:

(31)    V0 = 1.36 cm3, TW = 40 daPa = 400 Pa  (with 1daPa = 10 Pa)

(31a)    meff = 0.002 g, Ra = 1000 ohm, r = 0.37 cm,  = 0.00129 g/cm3.



 

Fig. 5: Plot of the two components of the reactance as functions of immission frequency f. Heavy curves represent mass reactances, light curves compliant reactances. Curve parameter is the ear canal pressure p. Curves are plotted for p = 0 and p = 400 daPa. (for implementation of p see (29) and (30)).
 

 .

Fig. 6: Plot of total reactance as a function of immission frequency f for fixed ear canal pressures p = 0 and p = 400 daPa. At resonance Xa = 0.

.

Fig. 7: Plot of total reactance as a function of ear canal pressure p. Curve parameter is the immission frequency f. Curves are plotted for f =113 Hz and the following 6 octaves above 113 Hz.

.
Conductance GaRa

Fig. 8: GaRa as a function of both ear canal pressure p (daPa) and immission frequency f (log f is used, with f in Hz).
3D-plot: p is plotted along the x-axis (range: -400 daPa <p < 400 daPa), log f is plotted along the y-axis (range: 2 < log f < 3.7).
2D-plot: GaRa(p) is plotted with f as parameter, i.e for f fixed at 226 Hz and the 5 following octaves above 226 Hz.

Fig. 9: Detail of Fig. 8 near resonance at zero ear canal pressure, p = 0.

Susceptance BaRa

Fig. 10:  BaRa as a function of both ear canal pressure p (daPa) and immission frequency f (log f is used, with f in Hz).
Left: 3D-plot, p is plotted along the x-axis (range: -400 daPa < p < 400 daPa), log f is plotted along the y-axis (range: 2 < log f < 3.7).
Right: 2D-plot BaRa(p) with f as parameter.

Fig. 11:  Detail of Fig. 10 near equivalence frequency at zero ear canal pressure p = 0.

Admitttance YaRa

Fig. 12:  YaRa as a function of both ear canal pressure p (daPa) and immission frequency f (log f is used, with f in Hz).
Left: 3D-plot, p is plotted along the x-axis (range: -400 daPa < p < 400 daPa), log f is plotted along the y-axis (range: 2 < log f < 3.7).
Right: 2D-plot YaRa(p) with f as parameter.

Fig. 13:  Detail of Fig. 12 near resonance at zero ear canal pressure p = 0.
 

Graphical Construction of GaRa(log f), BaRa(logf)

Fig. 14: Graphical explanation of shapes of curves in Figs. 8 - 11:
Lower plots: Xa/Ra as functions of f for fixed p = 0 and p = 400 daPa (see Fig. 6).
Upper plots: GaRa and BaRa as functions of Xa/Ra (see Fig. 1).
To obtain a value GaRa for a given immission frequency f
(1) choose f and read Xa/Ra from lower plot (follow line 1 in direction of arrow),
(2) then read GaRa for Xa/Ra (follow line 2 in direction of arrow).
 

III.2 Coupled Systems

(fit of acoustical behavior with an electrical network model after Zwislocki)

Fig. 15 Electrical system composed of 2 subsystems (1) and (2) arranged in series.
 


Fig. 15': Electrical system composed of 2 subsystems (1) and (2) arranged in parallel.

Definition 10 (complex electrical resistance -"impedance", Z) 

(32) 
 

.
Observation (Ohm's Law for complex resistance)

Let U be the voltage between entrance and exit terminals of a system i and qi the electric charge in system i. Then the charge qi is proportional to the applied voltage U:

(33) 

The same is true for a composite circuit:

(34) 
 

.
Observation
The flow of charges through subsystems arranged in series is the same in each subsystem:

(35) 
 

Observation

Charges in parallel subsystems add up in composite circuit:

(35') 

Theorem 6 (Composite resistances)

The composite resistance Z of a system composed of subsystems arranged in series is:

(36) 
 



Proof:
Definition of Z:

(37) 

From (35) follows: . Comparison with (37), the definition of Z ( i.e. with U = q Z), follows Z = Z1 + Z2.

 

Theorem 6' (Composite resistances)

The composite resistance Z of a system composed of subsystems arranged in parallel is calculated as:

(36') 



Proof:
Plugging in observation (35') into Ohm's Law (34) 

. Division of both sides with U yields 

.

III.2.1 Example: 2-Component system with subsystems arranged in parallel

(Fig. 15')



Data used in calculations
(Mathematica MR 1, 2 "2-Component System")

meff1 := 0.1 g;  R1 := 1000 ohm;
meff2 := 0.01 g;  R2 := 300 ohm;

r1 := 0.4 cm;   r2 := 0.37 cm;
V1 := 0.9 cm3; V2 := 0.2 cm3;
1 :=  0.001 g/cm3; 2 := 0.00129 g/cm3;
TW = 40 daPa.




Fig. 16: Conductance Ga and susceptance Ba plotted as functions of immission frequency f. Because of (36') Ga = Ga1+ Ga2, and Ba = Ba1 + Ba2.

Fig. 17: Resistance R of the composite system as a function of immission frequency f. By (36') 1/R = 1/R1 + 1/R2.

Fig. 18: Oscillation of composite system in {Ga, Ba} plane. The point {Ga,(f) Ba(f)} runs on the curve in the direction indicated by the arrows, when f runs from 100 Hz to 4111 Hz. The circle has been drawn to emphasize non-circular form of curve.


Fig. 19: Oscillation of composite system plotted in {GaRa, BaRa} plane lies on circle with radius 1/2. The reason for this is the linearity of the composite system: The oscillation of each subsystem lies on this circle (see  Fig. 2), thus the linear composition of these oscillations lies on that circle, too. The curve drawn by hand indicates how the point {GaRa, BaRa} runs on the circle when f runs from 110 Hz (arrow near {0.6, 0.4}) to 4060 Hz (arrow ending near {0.1, -0.1}).
 

III.3 Fit of measured tympanometric data with linear model

In Fig. 17-17, R.H. Margolis and L.L. Hunter present a multifrequency tympanogram. In that particular caase, the tympanic membrane had the highest mobility at an ear canal pressure p = - 250 daPa. Ga and Ba measured at this ear canal pressure are plotted as functions of the immission frequency f in the following Figs. 20 and 21.

Fig. 22 results when these Ba are plotted vs. Ga.

The data are then analysed with a linear model. This means that the deviation of the curve in Fig. 22 from a circle will be interpreted as resulting from a frequency dependent resistance Ra(f) according to (27). This may or may not be justified. It is simply a method of condensing the measured data into a set of equations (the ones developed in this paper) and corresponding parameters (necessary to evaluate the equations).

After calculating Ra(f) with (27) (Fig. 23), BaRa is plotted vs. GaRa, resulting in the circle presented in Fig. 24. The curve in Fig. 24 drawn by hand indicates how the point {GaRa, BaRa} runs first clockwise and finally counterclockwise on the circle when f runs between 230 Hz (arrow at beginning of clockwise part) and 1930 Hz (arrow at end of counterclockwise part). The circle crosses the abscissa (GaRa-axis) at f = 1350 Hz (by definition the resonance fr of the system).

Fig. 20: Conductance Ga as a function of the immission frequency f. The ear canal pressure is p = - 250 daPa. Data from Margolis and Hunter.

Fig. 21: Susceptance Ba as a function of the immission frequency f. The ear canal pressure is - 250 daPa. Resonance frequency fr is defined here as the frequency at which Ba = 0 (fr = 1350 Hz, dashed line). Data from Margolis and Hunter.

Fig. 22: Ba(f) plotted vs. Ga(f). Ba(f) and Ga(f) as presented in Fig. 20, 21. The curve starts at fi =  226 Hz  and ends at fi = 2000 Hz.

Fig. 23: Resistance extracted from oscillation presented in Fig. 22 with method given by eq. (27). Immission frequencies fi used by multifrequency tympanometer are marked as dots in lower part of graph. They start at fi = 226 Hz and end at fi = 2000 Hz. Dashed line marks resonance frequency fr = 1350 Hz. Sampled frequncies fi miss resonance fr.

Fig. 24: Ba(f)Ra(f) plotted vs. Ga(f)Ra(f). Data from Margolis and Hunter.



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