305

Hraring Research, 54 (1991) 305-318 0 1991 Elsevier Science Publishers B.V. 037%5955/91/$03,50 HEARES 01593

Malleus vibration mode changes with frequency Willem F. Decraemer

‘, Shyam M. Khanna 2 and W. Robert J. Funnel1 3

’ Laboratory qf Biomedical Physics, lJnic,ersity of Antwerp, RijksuniL,ersitair Centrum Antwerpen, Antwerp, Belgium, ’ Department of Otolatyngology, Columbia Unicersity, New York, New York, U.S.A. and .’ Departments of Biomedical Engineering and Otolaiyngology, McGill lJnit,ersily, Montreal, Quebec, Canada (Received 12 June 1990; accepted 13 February 1991)

The mode of vibraton of the cat manubrium is investigated by measuring its vibration in response to sound stimulus at four locations between the umbo and the processus lateralis with a heterodyne interferometer. The determination of mode requires high precision in measurement because amplitude differences between the points are small (about 20% at low audio-frequencies). Changes in the frequency response with time have been reported in an earlier paper. The nature and magnitude of this time change is analysed in detail: over a period of 1 h the average change in amplitude is about 5% and in phase 5’. The malleus vibration at some frequencies is purely translational, it is rotational at others and mixed at most frequencies. When the motion is rotational the position of the axis of rotation shifts with frequency, the shifts are so large that the axis can lie near the umbo so that amplitudes at the processus lateralis are larger than at the umbo. The classical concept of the malleus rotating around a fixed axis running from the anterior mallar to the posterior incudal ligament fits our measurements only at low frequencies Manubrium; Vibration mode: Rotation axis; Interferometer

Introduction

The high sensitivity of hearing in the mammalian ear is achieved by efficient transmission of the acoustical signal received by the external ear to the sensory receptors in the inner ear. According to the classical theories, the pressure acting on the footplate is increased by the lever action of the middle ear ossicles and by the area ratio of the tympanic membrane to the footplate (van BCkCsy, 1960; Wever and Lawrence, 1954). The concept of lever action is based upon the observation that the ossicles rotate around an axis formed by a line between the anterior maliar process and the posterior incudal ligament (Barany, 1938; Wever and Lawrence, 1954;

Correspondence

fo: Willem F. Decraemer, Laboratory of Biomedical Physics, University of Antwerp. Rijksuniversitair Centrum Antwerpen, 171 Groenenborgerlaan, Antwerpen B2020, Belgium

Guinan and Peake, 1967; Khanna, 1970). A force applied to the malfeus will therefore be increased at the footplate by the lever formed by the malleus and the incus (Wever and Lawrence, 1954, also includes a historical review of this concept). The rotation of the malleus around an axis means that the amplitude of malleus vibration will increase linearly with the distance from the axis. At low frequencies (below a few kHz) this proportionality was seen in measurements of malleus vibration using time-average holography (Kharma, 1970). The present experiments are intended to investigate the mode of the malleus vibration and the changes in this mode with frequency. To determine the mode of malleus vibration, vibration amplitude and phase was determined at several points distributed along the manubrium. High accuracy of measurement is necessary because the difference in amplitude between these points is quite small (dividing the length of the manubrium in four equal parts the amplitudes at

low frequencies are approximately in the ratio 6 : 5 : 4 : 3, Khanna, 1970). Our vibration measurements are made with a constant voltage applied to the acoustic transducer. It was shown earlier that the sound pressure produced by the transducer varies with time (Decraemer et al., 1990). In order to take this variation into account, the pressure must be measured repeatedly throughout the experiment. In our experiments vibration measurements were interfeaved with sound pressure measurements at intervals of less than 30 min and the sound pressure nearest to a vibration measurement was used as the reference. Thus all vibration measurements were made within I5 min of a reference sound pressure measurements A complete set of vibration measurements on four points on the malleus can be made in roughly an hour. It was shown previously that the middle ear response changes with time (Decraemer et al., 1990). It is therefore necessary to make sure that the differences in the vibration amplitude measured at the four positions on the malleus are not due to the time changes. If the changes with time were common to all the points measured then the effect of change with time could be compensated by choosing one point as the reference point and by remeasuring its vibration after each of the other points. We have investigated wether such a compensation scheme is feasable. Method

Three cats (left ears> were used in the present series of experiments, which consists of interferometric measurement of malleus vibrations in response to free field sound pressure applied to the tympanic membrane. The sound driver and the microphone were kept at a constant position relative to the cat eardrum. The sound pressure levels used, ranged typically between 70 and 9.5 dB SPL. The frequency was varied between 100 Hz and 25 kHz (sometimes 30 kHz) in steps of 250 Hz. The experiments were carried out with the bulla closed but vented with a capillary tube. The surgical techniques used in exposure of the bulla and for obtaining wide access to the malleus have been described earlier (Decraemer et al., 1989). The techniques of measuring the sound

Fig. 1. Photograph positioned

along

of thr eardrum the

manubrium

with the four glass beads of the

malleus

CM). The

beads are seen as four shiny spots. The bead at the umho i? marked

fU)

and the approximate

cesf is marked (Ptf. the

text

its bead

position of the lateral

The bead nesr the umbo is referred 1 and

the other

beads

proto in

in succession

as

numbers 2. 3 and 4. respectively.

pressure and the vibration have been described elsewhere (Decraemer et al., 1989, 1990; Willemin et al., 1988). The malieus, when viewed through the ear canal, is inclined with respect to the axis of the ear canal. The angle is approximately 60”. In order to measure malleus vibrations it was necessary to increase the optical reflectivity. To achieve this. four glass microbeads were placed on the manubrium of the malleus between the umbo and the lateral process with a fine moist paintbrush. A photograph taken to record the position of the beads in one experiment is shown in Fig. 1. The beads appear as four bright spots. Observations Accurncy of sound pressure meusurement

Two successive measurements of sound pressure amplitude taken 4 min apart at the tympanic membrane for constant voltage (0.1 V amplitude) applied to the acoustic transducer are shown in Fig. 2A. The repeatability is good and the open

307

is within ? l”, showing that the phase accuracy over short measuring periods is quite good.

A

At: 4 min

Repeatability of sound pressure longer measuring intervals

c

0.14

0.7 I 0

5

10 15 FREOUENCY

20 kHz

25

I 30

Fig. 2. (A). Sound pressure at the tympanic membrane in Pascal per unit driving voltage applied to the acoustic transducer. Two successive measurements taken roughly 4 min apart are shown. The open field response of the acoustic transducer is excellent and the pressure amplitude remains within 0.4 and 3.6 Pa/V between 3 and 30 kHz. (B)Ratio of two successive sound pressure measurements shown in Fig. 2A. Repeatability is excellent. The ratio varies between 0.99 and 1.028. Variation is less than + 1.2.5%, - 1% at all but two frequencies.

amplitude

over

The ratio of the sound pressure amplitudes measured at an interval of 27 min is shown in Fig. 4A. The variation is within -3% to +5.6% at all but one frequency over the entire frequency range. The ratio of the sound pressure amplitudes measured at an interval of 70 min is shown in Fig. 4B. The variation is less than - 12% to + 15% at all but one frequency. The variability in measurement as a function of measuring time was investigated further. The mean and standard deviation of the ratio between two sets of measurements was calculated for the entire frequency range. These means and standard deviations are shown in Fig. 5A as a function of time interval between the measurements. It is clear that the mean value of the ratio does not significantly deviate from 1 with time. The standard deviation clearly increases with time.

0

field frequency response between 3 and 30 kHz does not contain large amplitude maxima and minima. The ratio of the two sets of measurements is shown in Fig. 2B; it remains within + 1.25% and - 1% at all but two frequencies (on a total of about 100 frequencies). The repeatability of the sound measuring system is therefore adequate for the intended measurements over short periods of time.

-400 -600 -1200

-1600 20,

P

0

I

t At: 4 min

Phase

Two successive measurements of sound pressure phase, corresponding to the amplitude measurements described above, are shown in Fig. 3A. The phase response shown on a linear frequency scale has two distinct slopes. The average slope between 100 Hz and 3 kHz is -0.26”/Hz, and the average slope between 3 and 30 kHz is - O.O29”/Hz. The phase difference between the two successive measurements is shown in Fig. 3B. At all but the first frequency the repeatability of the phase

[ 0

5

10

15

FREWENdY

20 kHz

25

30

Fig. 3. (A) Phase of sound pressure at the tympanic membrane. Two successive measurements corresponding to the amplitude measurements in Fig. 2 are shown. The phase curves have an initial steep slope (average slope -0.26”/Hz) and a final shallow slope (average slope -O.O29”/Hz). (B) Phase difference between two successive measurements are less than + l”, showing excellent accuracy and repeatability of measurement.

1.3'

I'*'. 1.1.'

At: 27 min

This indicates that the fluctuations of the sound pressure response increase with time.

A

P

1.31

t 30

0.7 0

Fig. 4. (A)

Ratio

of the amplit~ldes

responses measured

27 min apart.

0.97 and 1.056. Variation but one frequency. pressure

20 kHZ

10 15 FREQUENCY

5

of two sound pressure

The

is less than

(B) Ratio

responses measured

25

ratio varies between

- 30: to +!i.Wi

of the amplitude 70 min apart.

between 0.98 and 1.015. Variation

at all

of two sound

The

is less than

ratio varies

- 12%.

i 15’;

at all but one frequency.

Phase differences between two measurements made 27 min apart are shown in Fig. 6A. These correspond to the amplitude measurements shown in Fig. 4A. The phase differences arc less than + 4”, -3” over the entire frequency range. Phase differences between two sound pressure measurements made 70 min apart are shown in Fig. 6B. The differences are less than +6.3”, -5”. We also calculated the mean and the standard deviation for the difference in phase response as a function of time interval between the measurements. The results are plotted in Fig. 5B. The variability increases with time while the mean shift remains smaller that 5” over a total time lapse of 325 min. Acc~lrucy of’ 1‘ibrution measurement

. P iit u

3 :

Amplitude responsr Two successive measurements of vibration amplitude are shown in Fig. 7 as a function of frequency at a point midway along the manubri~lm ibead 3). The two curves do not superimpose exactly, showing a small difference.

1.i

I.0 0.9 0.B

20 6 z

0.7:

cx

s

I

10

d

At: 27 min

10

At: 70 min

w L

-201 0

100

200 TIME minutes

300

400

Fig. 5. iA) Mean and standard deviation of pressure ratio as a function

of measurement

interval.

In general,

deviation

of time. The

increase,

however,

systematic. (B) Mean

and standard

deviation

of difference

pressure deviation

phase as a function

of measurement

increases as a function

interval.

of time. The increase,

ever, is not systematic.

FREQUENCY

iit-

creases as a function

is not in The how-

Fig, h. (AI

Phase differences

between

measurements

made 77 min apart.

less than

~ 3”. (B) Phase difference

+3”,

pressure measurements

kH2 two sound

pressure

The phase variations between

are

two sound

made 70 min apart. The phase varia-

tions are less than

+ h.3”. - 5.0”.

309

the amplitude responses shown in Fig. 7. The agreement between the two runs is very good. Phase dependence on time

L

\+

t 1.0 FREOUENCY kHZ

10

Fig. 7. Vibration amplitude of a point located midway the length of the malleus, as a function of frequency. successive measurements taken 4 min apart are shown. response amplitude rises slowly with frequency between Hz and 1720 Hz. It displays a minimum due to bulla nance at 3.75 kHz. The amplitude decreases gradually increasing frequency between 5060 Hz and 20 kHz, and steeply above 20 kHz.

30

along Two The 125 resowith then

Amplitude dependence on time

The ratio of two sets of vibration amplitudes (shown in Fig. 7) that were measured 4 min apart is shown in Fig. 8A. The ratio varies over the frequency range between 0.964 and 1.18. The ratio is slightly greater than one over most of the frequency range showing an increase in vibration amplitude. The ratio of another set of amplitudes measured 31 min apart is shown in Fig. 8B. The ratio varies over the frequency range between 0.845 and 1.29. Between 19 and 25 kHz the ratio increases from 1 to 1.2, clearly showing a change in response. The ratio of two sets of measurements taken 69 min apart is shown in Fig. 8C. The ratio varies between 0.85 and 1.29. Mean and variance of amplitude ratio as a function of time

The mean and standard deviation of the amplitude ratio of sets of malleus vibrations measured at different intervals of time is shown as a function of the time in Fig. 9A. In the present experiment, the mean first decreases and then increases with increasing time. The standard deviation increases with time. The mean amplitude response changes about 7% within 60 min. Phase response

Two malleus phase responses measured in direct succession are shown as a function of frequency in Fig. 10. These responses correspond to

Differences between the two sets of phase responses (Fig. 10) 4 min apart are shown in Fig. 11A. The phase differences vary between - 7 o and + 9 o over the frequency range. Phase differences between two sets of measurements made 31 min apart are shown in Fig. 11B. The differences are between - 8” and +7.5” over the frequency range of 25 kHz. Phase differences between two sets measured 69 min apart are shown in Fig. 11C. Differences vary between - 16” and + 7 over the frequency range. These phase differences correspond to the amplitude differences shown in Fig. 8C. Mean and standard deviation of phase difference as a function of time

The mean and standard deviation of the differences between phase responses as a function of 1.31

:

1.2

:

:

:

:

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:

:

:

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:

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At: 4 min

f

0.7

:

:

:

:

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:

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1.31

:

:

:

:

:

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::::I

:

:

:

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_: r

: : 12 FREQUENCY

:

: 16 kHz

:

: 20

:

:T 24

0.7t::::::::::::t

1.31

:

:

:

:

0.77

:

: 4

:

: 8

0

Fig. 8. Variation in malleus vibration amplitude with time. (A) Amplitude ratio for two successive measurements taken 4 min apart. The ratio varies between 0.964 and 1.18. (B) Amplitude ratio for two measurements taken 31 min apart. The ratio varies between 0.8 and 1.2. (0 Amplitude ratio for two measurements taken 69 min apart. The ratio varies between 0.845 and 1.29.

20

:

,o..

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At: 4 min

:

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A

0.

-2ot

0.77

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lo__

At: 31 min

:

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t At:

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Fig. 9. (A) malleus

60 40 TIME minutes

20 Mean

and standard

vibrations

measurements.

as a function

and standard deviation the malleus vibrations measurements.

deviation

100

of time

interval

between

and then in-

increases with time. (B) Mean

of the difference as a function

in phase response of

of time interval

between

The mean value stays at zero. The standard

is quite

small within

-204 0

of the ratio of the

The mean value first decreases

creases. The standard deviation

deviation

00

69 mln

40 min but is considerably

Fig.

i IA.

min

apart.

:

: 4

:: e

Phase difference The

phase

between

data

and +(I”. (B) Phase differences

made 31 min apart.

to the

between

Phase data correspond

ratios shown in Fig. 813. The

are between measurements are between

between measurements

taken 69 mm apart. The phase data correspond

to amplitude

ratios shown in Fig. XC. Phase variations are within

the time interval between the measurements is shown in Fig. 9B The results correspond to the amplitude data shown in Fig. 9A. The mean vaIue does not deviate significantly. Up to intervals of 40 min the standard deviation remains smaller than 5”. At larger intervals it increases considerably and reaches a value of 40” between 70 and 90 min.

0’

%

li

E

-500

..

::

I 1 a

-lOOO-

i

; -1300 . 0.1 Fig.

10. Phase

1.0 FREGUENCY ktir of the

malleus

frequency. Two measurements

vibration

10 as a function

t 30 of

taken 4 mitt apart are shown.

-I

amplitude

to the amplitude

phase variations

- 8” and + 7.5”. (C) Phase differences

larger after 60 min.

nle~~surements made

correspond

response shown in Fig. 7. The phase differences -7”

:L

24

- 16” and

i7”.

In order to determine the mode with which the malleus vibrates we measured the amplitude and phase at 4 points along the length of the manubrium: the points (or ‘beads’) were numbered 1, 2, 3, 4 in the direction from the umbo to the lateral process. The total range of the points spanned about 3/4th of the manubrium. As described in the introduction, accurate measurement of frequency response at each point takes time. During this time the vibratory response at each point continues to change and therefore spatial differences and temporal differences are mixed together. lf the two types of differences are comparable in magnitude, it will be necessary to take the temporal changes into account or else to minimize them.

311

One possible way to correct for the time change was tried out for 1 animal. Bead 3 was used as a reference point and the following measuring sequence was used: bead3,, bead2 1, bead3,, bead4,, bead3,, beadl,, bead2,, bead3,, bead 4,, bead3,, beadl, (i.e. one set over the 4 beads and a repeat set). The total time for one complete set (set 1: measurements 2, to 1,; set 2: 2, to 1,) was approximately 50 min. This method of compensation using a reference bead would work only if we may assume that the temporal changes affect the vibration of each of the points equally at all frequencies. Mathematically this may be expressed as:

Ai(w> t, A;(w,

t=O)

=f(o,

t)

(1)

and

Y,(W, t> - Y,(W, t = 0) =g(w, t)

(2)

where AJo, t) and y&w, t) stand for the amplitude and phase of the frequency response of bead number i at a given frequency w measured at a given instant of time t and where the functions f(w, t) and g(w, t) are independent of the bead index i. From the section on repeatability of the measurements we know that these f and g functions, if they exist, must be slowly varying with time. Therefore the nearest measurement on bead 3 could be used to reduce the amplitude Ai and phase yi of the response measured at t to Ai and -y: at t = 0 using A:(w, t = 0) =AJ

w, t>

.A3(0,

t =

O)/A,( w, t)

0.77

:

: 4

:

I. :

0

: e

: : 12 FREQUENCY

:

: 16 kHz

:

: 20

:

:’ 24

Fig. 12. Ratio of the second measurement and the first measurement on bead 2 (solid line); the same ratio after using the time change compensation on the second measurement (dotted line). The time change compensation on the second measurement did not make the ratio come closer to 1.

the amplitudes measured at time t are reduced to t = 0. The dotted line shows the A’Jw, t = O)/A,(w, t = 0). If the assumption behind eq. (1) and (2) holds, this second ratio has to be ‘exactly’ equal to 1. The results shown in Fig. 12 show clearly however that the time compensation scheme used did not make the ratio come closer to 1, on the contrary! The corresponding phase differences, y&w, t> - y2(w, t = 0) and y;(w, t = 0) - y2(w, t = 01, are shown in Fig. 13. The phase differences also did not get any closer to 0 by the proposed compensation scheme. It is clear from this example that the frequency response measured at each point changes with time and that the changes differ at different points. Therefore the compensation using a reference point will not work. For comparison of responses at different observations points the best we can do is to keep the time between the observations small. Comparison four points

of frequency

responses measured

at

Vibration amplitudes measured as functions of frequency on beads 1 (umbo), 2, 3 and 4 are

(3) 201

:

:

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: 4

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:

and

-2ot 0

Compensation for change in response with time

In Fig. 12 the ratio of amplitudes of the measurements bead2, and bead2, is shown with a solid line, i.e. A,(w, t)/A,(w, t = 0). Using eq.(2)

: : 12 FREQUENCY

:

: 16 kHZ

:. 24

Fig. 13. Phase differences between the second and first measurements on bead 2 (solid line); the same phase difference after time change compensation on the second measurement (dotted line). The compensation did not bring the phase difference closer to 0.

shown in Fig. 14A. The amplitude at bead 1 is highest only in a frequency region below 8 kHz. At frequencies above 14 kHz the amplitude at the umbo is the lowest of the four points measured. The response curves measured at the four positions interweave over the frequency range. Cum~ffris~n of phase responses rne~~~red at j&r paints The phase responses measured at the four points are shown in Fig. 14B. These correspond to the ampiitude responses shown in Fig. 14A. The four points vibrate nearly in phase below 8 kHz. Above this frequency the phase differences between the four points start to increase. Above

10-7

0

:

:

:

4

:

:

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8

:

12 FREQUENCY

-500J 0

a

4

12 FREQUENCY

Fig. 14A. Vibration

amplitude

corresponding

:

20

:

n

:

24

kHz

16

20

24

kHZ

bead

1 (long

dash dot line).

dot line), bead 3 (dotted

bead 4 (solid line). (BI Vibration the mallets

is

:

in meters per Pascal measured

at four points on the malleus: head 2 (long dish-d~~uble

!

line) and

phase of the four points on

to the amplitude

responses shown

above. The same line symbols arc used.

I3 kHz the phases at points 2, 3 and 4 Lag behind the phase at point I. Phase differences are as large as 72” at 22 kHz and increase at higher frequencies.

Amplitude w tios Vibration amplitudes measured at points 4, 3 and 2 are compared to that measured at point 1. Point 1 is used here as a reference. The ratios A ,/A, are shown as functions of frequency in Fig. lSA, A,/A, in Fig. 15B and A,/A, in Fig. 1SC. The ratio was calculated twice for measurements in two separate series of measurements on beads I,23 and 4 roughly I h apart (first ratio solid line, second dotted line). The ratio of vibration amplitudes at positions 2 and 1 is greater than 1 between 12 and 23.5 kHz (Fig. 15A). The amplitude of point 2 is about 20% higher than that of the point at the umbo in this frequency region. The results are roughly the same after a 1 h period. The ratio of vibration amplitudes at point 3 and point I (Fig. 133) is generally smaller than 1 below 8 kHz. It is greater than I between 12 and 25 kHz. From 18 to 33 kHz the ratio is grcatcr than 1.4. The shape of the curve is basically repeatable over a I h interval. The ratio of amplitudes at points 4 and I is shown in Fig. 1SC. The ratio is generally smaller than 1 below I2 kHz and greater than 1 above 1.3 kHz.

Phase differences Phase angles of points 4. 3, and 2 with respect to point I arc shown in Figs. IfiA, B. and C. These curves have been calculated from data shown in Fig. 14B. Two curves measured an hour apart are shown (dotted and solid lines). The phase difference y? - yr between positions 2 and I (Fig. 16A) increases with frequency above 3 kHz. At 20 kHz the differcnccs are about 40”. The phase difference y3 -- y, (Fig. 16B) increases with frequency above 12 kHz. The differences reach 100” at 25 kHz. Differences measured an hour apart arc repeatable, showing that the differences are not due to time dependent changes. The phase difference between positions 4 and 1. yJ - y, (Fig. 16C). increases with frequency above about 12 kHz. Differences at 25 kHz reach 110”.

OJ 2.01

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6

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d k

-40

;I I

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16 kHz

20

-00

z -120

a -120

Fig. 15. Ratio of the amplitude of vibration of four points on the malleus. (A) Points 2 and 1; (B) Points 3 and 1; (C) Points 4 and 1. The first set of data is shown with a solid line, a second set obtained an hour later is shown with a dotted line. Point 1 is on the umbo; therefore its vibration should be highest and all ratios should be less than 1. The ratios are less than 1 only at frequencies below about 12 kHz. They become greater than 1 between 12 and 25 kHz.

: 4

:

: e

:

: 12 FREQUENCY

:

: 16 kHz

:

: 20

:

... ..._ Y

24

Fig. 16. Phase differences between vibrations of four points on the malleus. (A) Points 2 and 1; (B) Points 3 and 1; (C) Points 4 and 1. The first set of data is shown with a solid line, a second set obtained an hour later is shown with a dotted line. If all points on the malleus vibrated in phase, the phase difference should be zero. Below 6 kHz, the phase differences are small. Above 13 kHz the phase lag increases with frequency. The lag is progressively higher for bead 2, bead 3 and bead 4, respectively.

l.EI

Comparison of amplitude ratios The ratio between vibration amplitude measured at points 4, 3, 2 and that measured at point 1 (umbo) is shown in Fig. 17. This ratio is calculated from data shown in Fig. 14A. A ratio smaller than 1 indicates that the vibration amplitude of the measured point is less than that at the umbo. This is the case for each of the points 2, 3 and 4 at frequencies below 8 kHz. Above about 12.5 kHz the ratios for points 2, 3 and 4 are greater than 1. Between 14 and 22 kHz these ratios are approximately 1.3 and they reach a peak for each point in a different frequency region. The peak value for point 3 is about 1.8 and occurs in the frequency region around 19 kHz. The peak value for point 4 is about 1.6 and occurs in the frequency region around 23 kHz. These ratios are well above the level of fluctuations caused by temporal changes in response.

:

0

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: 0

: 4

:

: E

: : 12 FREOUENCY

A4

/%

:

: 16 kHz

:_:

:

:I

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: 24

-.-.-.:

: 20

Fig. 17. Ratios of vibration amplitude of four points on the malleus. (A) Points 2 and 1 (solid line); (B) Points 3 and 1 (dotted line); (C) Points 4 and 1 (dashed dotted line). Data are replotted from Fig. 15. Each of the three ratios vary with frequency. The ratios are generally smaller than 1 below 12.5 kHz and greater than 1 above this frequency. Ratios smaller than 1 indicate that the vibrations at the umbo are larger than at points higher up on the malleus. Ratios larger than 1 indicate that umbo vibrations are smaller than those of the points higher up on the malleus. The changes in the ratios with frequency indicate a change in the mode of the malleus vibration.

314

-40

‘.

-80

..

-1201

: 0

v3 -

J‘l

r4-

I’ i

--.--

: 4

:

: B

Fig. IX. Phase differences

ii:

: 12 FREWENCY

:

: 16

:

: 20

:

:’ 24

kHZ

between vibrations

at four points on

the malleus. (A) Points 2 and 1 (solid line); (B) Points 3 and I (dotted

line); (0

are replotted four

points

Points 4 and I (dashed

dotted

from Fig. Ih. The phase differences of measurement

increase

changes in the phase differences

with

line).

frequency.

with frequency

Data

between the between

The the

four points on the malleus are indicative of the changes in the mode of malleus vibration.

Comparison

of phase differences

The phase differences between the vibrations of points 4, 3, 2 and the vibration of point 1 arc shown in Fig. 18. These differences are calculated from data shown in Fig. 14B. A zero phase difference indicates that all points on the malleus vibrate in phase. This condition is approximated at frequencies below 8.0 kHz. Above 8.0 kHz the phase differences increase with frequency. The phase differences between points 1 and 2 reach a maximum (55”) at 20 kHz. The differences between point 1 and points 3 and 4 continue to increase with frequency reaching a magnitude of about - 110” at 25 kHz. The four points on the malleus do not vibrate in phase at high frequencies. Mode of malleus

vibration

The mode of vibration of the malleus can be visualized more clearly if the position of the four points on the malleus can be seen at selected instants during the vibration cycle. The displacement dJt,w) of a point i at an angular frequency of w and time t is given by: d,(t,

OJ) =A,(o).sin(wt+y,(w))

i=

1, 2, 3, 4

(5) where Ai

is the peak amplitude

at point i and

frequency w, and y,(w) is the phase angle at the same point and frequency. The amplitude and phase, Ai( yi(w>, and the coordinates of the four points have been measured, therefore the displacement of the malleus can be calculated for any instant t. This displacement is shown for seven instants of time 30” apart during half a cycle ( - 90 to +90”) in Fig. 19. The center line in each set represents the zero position of the first point (umbo). the top line its extreme up position and the bottom line its extreme down position. Four additional lines show the intermediate positions. The vibration patterns for the second half of the cycle are mirror images of the first half. They are not included in Fig. 19. The mode at the lowest frequency is shown at the top of the column and that at the highest frequency at the bottom of the column. Each column shows data from one animal at seven selected frequencies. Data are shown for three animals. For ear 1 the total measuring time (240 min) was not minimized because we were following a different experimental protocol. The changes with time in frequency response make the data somewhat more noisy. but still good enough to illustrate the changes in malleus vibration mode. Measurements for ear 2 were completed within 1 h. The data for ear 3 were recorded within 40 min. In this animal bead 3 was used as a reference to keep a track of time changes. Here we also recorded a second set of measurements directly after the first set. Figures 15 and 16 show that repeatability of the two sets of data is good. Therefore the corresponding results shown in Fig. 19 are reproducible. The bottom of each column shows the positions of the four points on the manubrium projected in a plane perpendicular to the observation direction. Position 1 is on the umbo while position 4 is nearest to the lateral process. The angle between the observation direction and the upper straight part of the manubrium (beyond the umbo) for the 3 animals is listed under each column in Fig. 19. As shown earlier (Fig. 14B) the phase differences between the vibration at four points are small at low frequencies. Therefore, the outermost lines of a set represent the peak excursion of each of the four points. This. however. is not the cast at higher frequencies where phase differ-

315

-

4

3

2

1

I

I i mm

ms.

Angle:

5o”

Fig. 19. Reconstruction of the vibration pattern of the malleus. Vibration pattern reconstructed from amplitude and phase measurements at four points (1, 2, 3, and 4). The position of the malleus at a given instant is shown by a line connecting the four points. The position is shown for 7 instants during a half cycle (+90”, +60”, +30”, o”, -3o”, -60”. -90” phase angle at point 1). Point 1 is on the umbo, point 4 is closest to the lateral process. Data for three animals are shown. The positions of the points of measurement were different in each of the three experiments, as shown. The mode of malleus vibration is shown for 7 frequencies in each case. The frequency is indicated at the top left of each set and the maximum vibration amplitude of the set is indicated by the number on the top right. To obtain absolute amplitude in meters/Pascal, this number should be multiplied by lo-“. The mode of vibration can be seen to change with frequency and bending seems to be present near the manubrium tip at the highest frequencies.

ences become appreciable. The changing pattern of vibration from low to high frequencies shows that the mode of the malleus vibration is changing. At low frequencies the vibration pattern resembles that of a hinge. The approximate location of the axis of rotation can be found visually by extending the outermost lines to their point of

intersection. The axis of rotation is to the left (in an anatomical superior position with respect to the manubrium) and the amplitude of vibration increases from the axis of rotation progressively toward the umbo (point 1). The position of this axis shifts with frequency (compare f.i. 3255 Hz and 6901 Hz, column 2). At frequencies around 10026 Hz for column 2 and 7682 Hz for column 3, the malleus positions are almost parallel to each other and the amplitudes at points 1 and 4 are approximately equal. The axis of rotation therefore is located at infinity, or one could describe the motion as a pure translation. The mode changes further at higher frequencies where vibration at point 4 becomes greater than that at the umbo (12630 Hz, column 2; 13672 Hz, column 3). In this case the system behaves as if a hinge were present to the right of the malleus, i.e. in a position inferior to the umbo. At still higher frequencies (18880 Hz, column 1; 19922 Hz, 22005 Hz, column 2; 19922 Hz, 21484 Hz, column 3) different parts of the malleus vibrate with different phases and the motion becomes even more complex. The vibration mode seems to indicate a marked bending of the malleus near the umbo. Although results at lower frequencies (f.i. 10807 Hz, column 1, 12630 Hz, column 2) also suggest bending it becomes especially prominent at the highest frequencies. The low frequency vibration pattern at 391 Hz of ear 3 was analysed in detail. The instantaneous position of the rotation axis was determined by fitting a straight line through the displacement of the four points and extending it to the point where it intersects with the zero displacement line. This point, indicated with an open circle on Fig. 20 shifts by a large amount during one cycle. Therefore the rotation axis is not fixed. These results can be explained by adding a translational component to the rotational component. To find the magnitudes and phases of the rotational and translational components we need a kinematic model with translational and rotational degree of freedom and fit the model to the experimental data using a numerical minimization procedure. Development of the model, fitting procedure, model predictions and compari-

DISPLACEMENT

EAR

Fig.

20.

inrtant\ circles

391

Instantaneous of time

Hz

position

during

one

on the abscissa.

Data

3Yl Hz (Fig. than

3

Imm

19. top column showing

that

of

half

the

cycle

rotation is shown

axis with

at

are for ear 3 at a frequency 3). The position

even

during

axih does not remain

son with the experimental in a separate paper.

one

7

open ot

shifts over more cycle

the

rotation

fixed.

data will be described

Discussion Chunges of’ the l2wution with time

und pressure

responses

In order to determine the mode of vibration of the malleus amplitude and phase at four points were measured. The amplitude and phase differences between these points are small therefore high accuracy of measurement is required. The measurement sequence requires about an hour to complete therefore it is important to know if the responses at each point are repeatable during this period. The vibrations are measured in response to acoustic stimuli and the sound pressure produced by the acoustic transducer for the same driving voltage, was found to change with time. The effect of this change was minimized by determining the sound pressure within 15 min of measuring the vibratory response. Time dependent changes in vibratory response were observed after taking care of the sound pressure changes. Attempts were made to record these changes by repeatedly measuring the vibrations of a selected reference point and using this information to correct for changes at all other

points. The time dependent changes were found to be different for each point so that this compensation scheme could not be used. These observations indicate that the mode of vibration was changing with time. The reason for this change is not clear. (iI The middle ear muscles were intact in our experiments. If the tension of these muscles changes (for example, due to change in depth of anesthesia) the middle ear response may be altered. (ii) The ear canal was cut short to widely expose the tympanic membrane and the malleus. The tympanic membrane may be drying out due to wide exposure and thus changing its mechanical characteristics. It should be possible in the future to test this hypothesis by changing the moisture in the tympanic membrane cavity. The average change in frequency response amplitude and phase during the recording time on the 4 points was found to amount to 5% for the amplitude and 5” on the phase. Larger differences (up to 50%’ and 100°) were observed in the responses at the 4 points (Fig. 15 and Fig. 16). The accuracy of measurement was therefore sufficiently high to indicate that the mode of malleus vibration changed with frequency. Mode of‘ malleus Lihrution Our present concepts of the mode of malleus vibration are based upon indirect (Wever and Lawrence, 1954: Barany. 193X) and direct measurements (Khanna, 1970) at low frequencies. In cat the long axis of the malleus is tilted by 30” with respect to the ligament axis formed by a line between the anterior mallar process and the posterior incudal ligament. The malleus is believed to rotate around the ligament axis so that vibration amplitude increases linearly with distance from the rotation axis. In the absence of data at high frequencies it was generally assumed that the mode remains the same. The mode is defined by how different parts of the malleus vibrate with respect to each other. When this relationship changes the mode changes. Among the factors that determine the mode are (i) the mechanical characteristics of the ossicular suspension, (ii) the incudal load acting upon the malleus, and (iii) the

317

distribution of the driving forces along the length of the malleus. These in turn are related to the vibration pattern of the tympanic membrane and the coupling between the tympanic membrane and the malleus. The mode changes with frequency because all the three factors above are frequency dependent. In studies of the vibration mode with time-average holography only the differences in the extreme positions of vibration are recorded therefore the effects within the cycle can not be seen. The fringes on the malleus become very fuzzy above a few kHz and it was not possible to determine their orientation Khanna, 1970). We now understand that the effect was caused by changes in the mode of vibration within one cycle. Changes in the location of the rotation axis with frequency were also seen in human temporal bones preparations by Gundersen and Hogmoen (1976) at low frequencies. Mode changes are more pronounced at high frequencies, and at some frequencies the axis of rotation lies near the umbo or even beyond it. Understanding of the mode of malleus vibration tells us how the vibrations are transmitted through the middle ear.

Classical concept of the lerler action of the ossicles In the classical concept the acoustic force acting upon the tympanic membrane is transmitted to the malleus. The middle ear ossicles are believed to rotate around a fixed axis to form a lever system so that the force applied to the foot plate is increased by the lever ratio (Wever and Lawrence, 19.54). The lever ratio is calculated by measuring (i) the distance 1, between the umbo of the malleus and the axis of rotation of the lever system; and (ii) the distance 1, between the axis of rotation and the point of action of the incus. The lever ratio is then given by 1,/l,. For cat the lever ratio has been calculated to be 2.5 (Wever and Lawrence, 1954). Our experiments show that the axis of rotation is not fixed and that its position depends on frequency and phase within the cycle. Therefore the force transformation ratio between the malleus and the incus will also be variable. For

example when the axis is at infinity be unity.

the ratio will

Transfer function of the middle ear In studies on the transfer function of the middle ear (Moller, 1963; Guinan and Peake, 1967; Wilson and Johnstone, 1975; Vlaming, 1987) the input to the middle ear is generally determined by measuring the amplitude of the malleus vibration (preferably near the umbo) for a constant sound pressure applied to the tympanic membrane. Our present measurements indicate that the shape of the middle ear input characteristic varies with position on the malleus. It is not clear which position is most appropriate to describe the input function. In view of the mode changes seen with the single measuring axis on a series of colinear points on the manubrium, it is quite likely that malleus vibrations are even more complicated and measurements from several directions may be necessary to understand its three dimensional vibrations. Such measurements are underway now. Acknowledgement This research was supported by program project grant NS 22334 from NIDCD and by the Emil Capita Foundation. References Barany. E.A. (1938) Contribution to the physiology of bone conduction. Acta Otolaryngol. (Stockh). Suppl. 26. BekCsy. G. von (1960) Experiments in Hearing. McGraw-Hill, New York, p, 745. Decraemer, W.F., Khanna, SM. and Funnell, W.R.J. (1989) Interferometric measurement of the amplitude and phase of tympanic membrane vibrations in cat. Hear. Res. 38, 1-18. Decraemer, W.F., Khanna, SM. and Funnell, W.R.J. (1990) Heterodyne interferometer measurements of the frequency response of the manubrium tip in cat. Hear. Res. 47. 205-218. Khanna, S.M. and Stinson, M.R. (1985) Specification of the acoustical input to the ear at high frequencies. J. Acoust. Sot. Am. 77, 577-589. Khanna, SM. (1970) A holographic study of tympanic membrane vibrations in cats. University Microfilm, Ann Arbor, MI 48104. Guinan, J.J. Jr, and Peake, W.T. (1967) Middle-ear character-

318 istics of anesthetized cats J. Acoust. Sot. Am. 41, 12371261. Gundersen and Hogmoen (1976) Holographic analysis of the ossicular chain. Acta Otolaryngol. 82. 16-25. Moller. A.R. (1963) Transfer function of the middle ear. J. Acoust. Sot. Am. 35, 15261534. Vlaming, M.S.M.G. (1987) Middle ear mechanics studied by laser doppler interferometry. Doctoral Thesis, Technical University Delft. Delft, The Netherlands.

Wever. E.G. and M. Lawrence (1954) Physiological Acoustics. Princeton University Press, Princeton. NJ. Willemin, J.F., Dandhker. R. and Khanna, SM. (1988) Heterodyne interferometer for submicroscopic vibration measurements in the inner ear. J. Acoust Sot. Am. 83.787-795. Wilson, J.P. and Johnstone J.R. ( 1975) Basilar membrane and middle-car vibration in guinea pig measured by capacitive probe. J. Acoust. Sot. Am. 75, 705723.