THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA
VOLUME 34, NUMBER 8, PART 2
SEPTEMBER 1962
Analysis of the Middle-Ear Function. Part I: Input Impedance
j. ZWISLOCKI
Bioacoustlcs
Laboratory,SyracuseUnioersity,Syracttse,.¾e•oYork
(Received March 12, 1962)
A quantitativetheoryof themiddle-ear
acoustics
is developed
andexpressed
in termsof an electricanalog.
The analognetworkis basedon the functionalanatomyof the middleear. The numericalvaluesof its elementsarederivedfromimpedance
measurements
on normalandpathological
earsandfromanatomicaldata.
It is shownthat the input impedance
of the analogagreeswithin the experimental
errorwith the acoustic
impedanceat the eardrum,and that changesin analogparameterscorresponding
to knownanatomical
changes
producethe sameeffecton its impedance
characteristics
asmeasured
at the eardrum.
INTRODUCTION
They should vanish gradually, as the experimental
redundancyaccumulates.
FTERfourcenturies
of empirical
research
onthe The knowledgeof the acousticfunctionof the middle
middleear,• our knowledgeof its acousticpropertiesseemsto be approaching
an asymptote.We went ear may be synthesizedin various ways. The most
throughperiodsof pure speculation,pseudotheories,primitive way wouldbe to build an artificialmiddleear
workinghypotheses,
ruggedempiricism,
and it is possi- that would replicate the natural middle ear in every
ble that we reacheda stagewherean integratiretheory detail. This is impractical, however, and the analytic
is within our reach.By theoryis meanthere a formal
systembasedon empiricalevidence
andconcerned
with
interpolation
amongexperimental
dataratherthanwith
extrapolation--inotherwords,a systemthat showshow
variousexperimentalresultsfit together.
This paperdescribes
an attempt to producesucha
information
that could be derived from such an under-
taking would be low. Any more sophisticatedmodel
must be basedon mathematicaltheory, explicitly or
implicitly. As a consequence,
the theory developedin
further portionsof this paper is essentiallymathematical, although it is presentedin the disguiseof an
electric analog. Equations describingthe action of
theory.Theprocedure
usedissimilarto thatof a jigsaw
variousparts of the middleear, as well as of the whole
puzzlewhere,at the end of many trials, a coherent
middleear, are complexand their solutionis tedious.It
pictureemerges.
The piecesof the puzzleare bits of is more economicalto expressthem by meansof elecinformationproducedby variouskinds of research
trical networksand to study the input-output relation4
rangingfrom anatomyto physiology
and acoustics.
ships. In this form, the theory appears reasonably
Experimentson living subjectsas well as on postsimple and straightforward.
mortempreparations
are included.
There is one importantdifferencebetweena jigsaw
ANATOMICAL
puzzleandpiecingtogethera scientific
theory.Whereas,
in the first instance,all piecesare madeto fit exactlyfor
EVIDENCE
The acousticallyimportant parts of the middle ear
all practicalpurposes,
in the second,the fit is usually are shownschematicallyin Fig. 1. Proceedingfrom the
muchlessperfectand somepiecesdo not seemto fit outer-ear canal, the first part is the eardrum. To it is
anywhere.
The effectof a certainnoisein thesystem
is connected the chain of ossicles that consists of the
compensated
to someextentby redundancy.
Neverthe- malleus,incus,and stapes(hammer,anvil, and stirless,someblurredspotsremainin the final picture.
rup). The foot plate of the stapesis imbeddedin the
oval window of the inner ear and its motions are trans-
• G. vonBik6sy and W. A. Rosenblith,"The Early HistoW of
HearingsObservations
and Theories,"J. Acoust.Soc.Am. 20,
727 (1948).
mittedto the inner-earfluids.The volumedisplacements
producedby the stapesare compensated
by nearly
1514
ded 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/
MIDDLE-EAR
PNEUMATIC
CELLS
FUNCTION:
,MALLEUS
•0 d'•'•INCUS
EAR
• • .•.•p•INC
UO0STAPE
DIAL
INPUT
IMPEDANCE
1515
tractedit pullsthe eardrumtowardthe middleear. The
stapediusis attachedto the stapesand pullsthis ossicle
sideways.While the acousticeffect of the tympanic
muscleis uncertain, that of the stapedialmuscleis
noticeable.It producesa change in the impedance
measured at the eardrum
characteristic of the ear. s
and
in
the
transmission
The eardrum,the ligamentsand the musclesof the
middle ear, the ligamentsholdingthe stapedialfoot
plate in the oval window,and the elasticpropertiesof
equally large displacements
of the membraneof the the round-windowmembraneprovide the middle-ear
systemwith stiffnessand introducesome frictional
round window.
resistance.
The main resistivecomponent
is contributed
B•kdsy2 hasshownthat at low and mediumfrequenby the input impedanceof the cochlea.
•
cies the major central portion of the eardrum moves
Motions of the eardrum are transmitted to the ossicunearly as a rigid body and with the sameamplitudeas lar chain as well as to the air-filled cavities of the middle
the part of the malleus attached to it. The eardrum
ear. As a consequence,
thesecavitiesaffectthe acoustic
appears to rotate around its upper edge--a modeof characteristics of the middle ear and should not be
motionfacilitatedby a flexiblefold near its loweredge.
ignored.From the acoustic
pointof view, they may be
Nevertheless,not the entire membranemoveswith the
dividedinto threeparts: the tympaniccavity immedisame amplitude.Toward the edges,the motion deately behind the eardrum, the epitympanumwhich
creasesto zero.Also,due to their flexibility,parts of the
extendsupwardfromthe tympaniccavity,andthemore
eardrumcan moveevenafter a completefixationof the
or lessextensivesystemof pneumaticcellsof the temossicles.At high frequencies,the mode of motion
poral bonewhichcommunicate
with the othercavities
changesand different sectionsvibrate in different
throughtheantrum.Mostof theepitympanum
isfilled
phases.
= All this meansthat, althoughthe coupling by the bodyof theossicles,
sothat onlya narrowpassage
between the eardrum and the malleusis close,it is not
remainsbetweenthe tympaniccavity and the antrum
perfect,and someof the acousticenergyactivatingthe
with the adjoiningpneumaticcells.Seenfrom the eareardrum is not transmitted to the ossicles.
Fro. 1. Schematic of the middle-ear
mechanism.
Furtherenergylosses
maybeexpected
in theossicular
jointswhichare not completelyrigid.This is particularly true for the frail incudo-stapedial
joint.a Its
pronouncedflexibility becomesapparent from the
followingobservations.
After a completefixationof the
stapesthroughan otosclerotic
ossification,
it is possible
to move the incus without
i
3
5
much effort. In normal
conditions,the incusand the stapesrotate around two
different axes that are almost perpendicularto each
Fro. 2. Blockdiagramof the middle-earmecha•dsm.
other.A contractionof the stapedius
musclemaychange
the plane of stapedialrotation by ninety degrees.
= drum, sucha systemmay be expectedto producea
Finally, the extremelysmallcross-sectional
areaof the resonanceand an antiresonanceeffect, as has been
joint must lead to high stresses
duringsoundtransmission.Such stresseswould producenoticeabledeformationsevenin a materialwith a highconstantof
elasticity.
By contrastto theincudo-stapedial
joint, themalleoincudaljoint appearsrobust,and it may be assumed
recentlydemonstrated
by Onchi.7
On the basisof the functionalanatomy,it is possible
to draw a block diagramof the acousticunits of the
middleear. This hasbeendonein Fig. 2, whichcontains
five functionalunits. Somejustificationfor theseunits
and their placement
appearsnecessary.
First of all, it
that the two ossiclesmove as one unit? They appear to may not be obviouswhy the blockrepresenting
the
rotatearoundtheirpointof gravitysothat the effective middle-earcavitiesprecedes
the blocksassociated
with
massof the systemis reducedto a minimum?.4
the eardrum.This orderis dictatedby the fact that a
The ossicles
are heldin placeby ligamentsand two displacement
ofanypartoftheeardrum
isreflected
in a
muscles,the tensor tympani and the stapedius.The compression
of air enclosed
in the middle-earcavities
tensortympaniis attachedto the hammer;whencon- while it is not necessarily
transmittedto the ossicular
=G. yon B&k(•sy,Experiments
in Hearing (McGraw-Hill Book
Company,New York, 1960).
aj. Zwislocki,
"SomeImpedance
Measurements
onNormaland
Pathological
Ears,"J. Acoust.Soc.Am. 29, 1312(1957).
• E. B,Srftny,"A Contributionto the Physiologyof BoneConduction," Acta Oto-Laryngol.Suppl.26 (1938).
• O. Metz, "The AcousticImpedanceMeasuredon Normal and
Pathological
Ears,"Acta Oto-Laryngol.
Suppl.63 (1946).
• J. Zwislocki,
"Theorieder Schneckenmechanik"
(Theoryof
CochlearMechanics),Acta Oto-Laryngol.Suppl. 72 (1948).
?Y. Onchi, "Mechanismof the Middle Ear," J. Acoust.Soc.
Am. 33, 794 (1961).
d 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/in
1516
J.
ZWISLOCKI
600
Consequently,further experimentswere undertaken.
They included acoustic measurementson the four
temporal
bonesandon patientswith a reflected
'eardrum. Most of thesemeasurements
wereperformedby
meansof an acousticbridgeof the Schustertype) On
one temporalbone, the soundpressureproducedin the
ear canal by means of a high-impedancesourcewas
measureddirectly and comparedto that producedin a
standard cavity.ø In this method, the volume to be
determined is equal to the product of the standard
volume and of the sound-pressure
ratio. From bridge
measurementson the temporal bones, the following
FREQUENCY
IN CYCLES
PER
SECOND
volumesof the middle-earcavitieswere obtained:2.1,
3.4, 11.7, and 17.44 cc. With the exceptionof the
Fro. 3. Acousticinput impedanceof the middle-earcavitiesand
their electricanalog.Heavy linesshowthe impedancecomponents 17.44-ccfigure, thesevalues are reasonablycloseto
obtained by Onchi on one temporal bone; thin lines indicate thoseobtainedby meansof waterfilling.The methodof
analog resultsbasedon average data.
directsound-pressure
measurement
confirmedthe value
of the largest volume obtained with the help of the
chain. The secondblock representsthe parts of the acousticbridge.It is possible,therefore,that in the first
eardrum whose motion differs from that of the ossicles.
method the water did not fill all the cavities. The mean
Sincea portion of the eardrum, the malleusand the
volumeresultingfrom bridgemeasurements
on the four
incus,may be considered
asoneunit vibratingwith the temporal bonesamounts to 8.6 cc.
sameamplitudeand phase,they may be includedin one
Sinceit couldbe suspected
that the largevolumes
block. The next block, number four, accountsfor the obtainedon the temporalbonesweredue to the absence
fact that not all acousticenergyis transmittedacross of soft tissue,bridgemeasurements
wererepeatedon
the incudo-stapedialjoint. Finally, block numberfive otosclerotic
patientsduringthe mobilizationoperation.
standsfor the stapeswith its attachmentto the oval During one phaseof the operation,a portionof the
window, the input impedanceto the cochlea,and the eardrumis reflectedin orderto makethe stapesvisible.
round-window membrane. One block is sufficient for
It then becomespossibleto measureacousticallythe
this complexstructure,sincethe stapesand the round- effective volume of the middle-ear cavities in the lowwindow membrane vibrate with approximately the frequencyrange.Frequencies
of 150, 200, and 300 cps
sameamplitudeand phase.
The next step is an analysisof the network blocksof
Fig. 2 and a determinationof the numericalvaluesof
their elements.
ACTION
OF THE
MIDDLE-EAR
CAVITIES
The acoustic action of the middle-ear cavities has
been ascertained on the basis of static volume deter-
minationsand impedancemeasurements.
The
volume
of the middle-ear
cavities
was deter-
minedon four preparationsof temporalbonesafter the
soft tissue had been removed. The bones were sealed
were used. They produced comparable results. The
volume
of the middle-ear
cavities measured on six
patients ranged from 5.5 to 14.0cc with a median of
9.3 ccanda meanof 8.7 cc.Thesevaluesagreewellwith
thoseobtainedon the bonepreparations.
The acoustic action of the middle-ear cavities at
mediumand high frequencies
wasrecentlydetermined
by Onchi7 whoperformedimpedancemeasurements
on
variouspartsof freshear preparations.
Unfortunately,
the impedance
of themiddle-earcavitiesisgivenfor one
specimen only. Onchi estimated its total middle-ear
volume to be 3.45cc, which is below our average.
hermetically
withwaxandconnected
to a vacuumpump Nevertheless,
the generalshapeof the impedance
curves
by means of a perforated earplug securedin the ear of the specimenshouldnot depart in principlefrom the
canal. A valve arrangementmade it possibleto switch average.For this reason,Onchi'simpedancecurvesare
from the vacuumpumpto a containerfilledwith colored reproducedin Fig. 3 by meansof heavy lines.The thin
water. The amount of water that flowed into the bone
could be determined from the level difference in the
linesshowthe realand the imaginarycomponents
of the
impedanceof the electricalanalogshownin the inset.
container. It was observedthat, unless the air was No effort was made to match Onchi's individual curves
evacuated from the bone, the cavities were not filled
completely. The volumes of the middle-ear cavities
exactly.
The electricalanaloghasbeendesignedmainly on the
obtainedon the four bonesamountedto 2.43, 4.1, 8.6, basis of the anatomical evidence. The electrical elements
and 10.5 cc, respectively.Thesevaluesare larger than shownin Fig. 3 have the followingsignificance.
The
the averagevaluesgenerallyacceptedin the literature.s inductanceL• is an analogof the acousticinertancedue
sG. yonB•.k•syand W. A. Rosenblith,"The MechanicalPropertiesof the Ear," in Handbook
of Experimental
Psychology,
edited
by S.S. Stevens(JohnWiley & Sons,Inc., New York, 1951).
• J. Zwislocki,"SomeMeasurements
of the Impedanceat the
Eardrum," J. Acoust.Soc.Am. 29, 349 (1957).
ed 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info
MIDDLE-EAR
FUNCTION:
INPUT
IMPEDANCE
1517
to the narrowpassage
betweenthe tympaniccavityand
the remainder of the cavities. The resistanceRa stands
CO
L0
.1•
0
for the acousticresistance
arisingfrom the samenarrow
passage.
The capacitance
Cv represents
the compliance
due to the air volumeof the antrum and of the pneumaticcells.The pneumaticcellshavea complexstructure and the capacitance
C• can be considered
an adeCd2 Ld
quate analogonly at low frequencies.
Somesort of a
transmission
line, as suggested
by Onchi, would have
been a more likely structure. However, there are no
sufficientdata for the designof sucha line,and a study
of Onchi'simpedancecurveshas shownthat a simple Fro. 4. Electric analog of the middle ear without incus. Elementstienotedby subscriptsa, p, m, and t belongto the middlecapacitancecan satisfyapproximatelythe input condi- ear cavities,thosewith the subscriptd to a portion of the earo to themallealcomplex.
tions. Beyond the resonancepoint, which is situated drum,andthosewith thesubscript
around600 cps,the importanceof the pneumaticcells
decreases.The resistanceR• representsthe sound the analog of Fig. 3 may be regardedas a sufficient
absorptionin the wallsof the tympaniccavity and the approximation.
Eustachiantube. Its valueis alsoaffectedby the sound
EAR WITHOUT
INCUS
absorptionin the pneumaticceils.Finally, the capacitance Ct stands for the acoustic complianceof the
!
<•Cd•
In certain surgical procedures,the incus is removed
middle-earcavities.The sumof the capacitancesCp and
without
otherwise affecting the middle-ear structures.
Ct correspondsto the total volume of the middle-ear
Ears without the incus are acousticallysimpler than
cavities.If we acceptthe acousticcompliance
asdefined
normalears?The effectsof the incudo•tapedialjoint,
by the equation
of the stapes,of the cochlea,and of the round-window
C= V/pd,
membraneare eliminated.The analogcircuit contains
only threeunits of Fig. 2, i.e., middle-earcavities,earwith V meaningthevolumeof air, a the averagedensity drum, and malleus. The acoustic properties of the
of air, and c the speedof sound,then
middle-earcavitiesare knownalready,sothat only the
C•,+C•=
Acceptingfor the total volumeVman approximatevalue
of 8 cc, we obtain C,+C•=5.45X10 -6 cgs units or, in
electrical terms, 5.45 uF. The average volume of the
tympaniccavity may be estimatedto be of the orderof
0.5 cc. This leadsto Ct=0.35uF and Cv=5.1•F. The
value of L, was so chosenthat the zerosof the analog
reactanceagreeapproximately
with the zerosof Onchi's
reactance curve. This value amounts to 14 mH. Other
characteristics of the malleus with its attachments and
of the eardrum remain to be considered.
The malleal structuresconsistof the malleus, the
ligamentsholdingit in place,the tensortympani, and
the portionof the eardrumthat vibrateswith the same
amplitudeand phaseas the malleus.The whole structure can be approximatedin the electricanalogby a
seriesconnectionof a capacitance,a resistance,and an
inductance.
At low frequencies,
the portionof the eardrumthat is
valuesweretried and rejectedon the basisof subsequent not coupleddirectly to the malleuscan be represented
in serieswith a resistance?
At higher
experiments.
The resistance
valuesof R• and R,• were by a capacitance
the massof the eardrumbecomes
effective,
determinedfrom the phase relationshipsthat result frequencies,
from Onchi'simpedancemeasurements
on the middle- and an inductance must be added to the circuit. At still
the eardrumvibratesin sections,
ear cavities and on the basis of impedancemeasure- higherfrequencies,
mentsat the eardrumof normal and pathologicalears. and a complexelectricalnetworkis required.A transIt mustbe emphasized
that Onchi'simpedance
curves missionline wouldprobablyconstitutethe bestanalog.
couldhavebeenduplicatedconsiderably
betterthan in However, it could be demonstratedthat the simplified
Fig. 3. A smallercapacitance
Cv, a largerinductance circuit of Fig. 4 is sufficientfor low and medium
La, and resistance
R, wouldbring the analogand real frequencies.
Figure4 showsthe completeanalognetworkof the
ear-impedance
curves to closecoincidence.Such an
to the
undertakingwouldbe in conflictwith otherempirical ear without incus.The elementscorresponding
evidence,however,particularly with the determination middle-ear cavities have been describedalready. The
of the averagevolumeand with the impedancemeasure~ elementswith the subscripto refer to the malleal commentsat the eardrumof normal and pathologicalears. plex, those with the subscriptd to the remaining
Sincethe impedanceof the middle-earcavitiesis low by portionsof the eardrum.
The numerical values of the elements in the two
comparisonto other partsof the middleear, its effecton
the impedanceat the eardrumand on the soundtrans- latter groups,with the exceptionof La, Ca:, and Ra•,
missionto the inner ear is not critical. As a consequence, were determinedfrom impedancemeasurementsat t_h•
d 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/in
1518
J.
ZWISLOCK[
R0= 70 •,
La= 15 mH,
Ral= 40 •,
Rd== 220 fl.
The agreementbetweenthe experimentaldata and
the analogresultsmaybe considered
satisfactory.
This
is particularlytrue for the reactance.In the resistance
data, a discrepancy
appearsin the vicinityof 1000cps.
_z
While the resistance measured in the ear seems to deI00
FREQUENCY
FIG. 5. Reactance
IN
I000
CYCLES
at the eardrum
I0000
PER
SECOND
of an ear without
incus.
Pointsshowdata obtainedon two patients;curvesindicateanalog
results.
eardrum. The experimentalmethod was describedin
two previouspapers? and only small technicalim-
creasewith frequency,the resistance
of the analog
increases.
The latter effectis in agreement
with anatomical considerations,
andit is possible
that the decrement
in the experimentalresistancedata is due to an artifact.
Onchi7pointsout that soundabsorptionin the ear canal
may be the cause.In his experiments,the effectof the
ear canal couldbe eliminated,and the resistanceshows
provementswere introducedfor the purposesof the an upwardtrendsimilarto that of the analog.
presentinvestigation2
ø These improvementsconcern A methodof determiningthe networkelementswas
in a previous
paperin whicha highlysimthe couplingbetweenthe probetubesand the ear canal, described
and the measurement of the volume of the ear canal.
plifiedelectricanalogof the middleear wasdiscussed.
amountof informationit
First, an otologicalspeculumwas securedin the ear Although,with increasing
canalby meansof a plasticearplugand soorientedthat becamepossibleto devisemore refinedmodels,the
the same.As a consequence,
the eardrumcouldbe seen.Second,the probetubeswere basicmethodremained
discussion
at thisplace.
connected.tothe speculumthrougha tightly fitting thereis noneedfor its renewed
adapter.In thisway, an obstructionof the tubesby the It may be of interest,however,to comparethe magniwalls of the ear canal could be avoided. After the termitude of the analogelementswith the availableanatomiis possiblefor the inducnation of the acousticmeasurements,
the tubeswere cal data. Sucha comparison
to the effectivemass
withdrawn from the speculumand the ear canal was tancesL0 andLa whichcorrespond
one-halfof the earfilledwithalcoholup to thetip of thespeculum,
usinga of the malleusandof approximately
The elements
of the analogare
calibratedsyringeand a blunt hypodermicneedle.The drum,respectively.
elements.In
amount of injected alcoholservedas a measureof the numericallyequalto acousticimpedance
values,theyhave
volumeof air betweenthe tips of the probetubesand orderto obtaintheactualmechanical
the eardrum. Since the volume of the ear canal is con- to bemultipliedby the squareof the effectiveareaof the
2 this area amountsto
tainedin thecomputational
formulasfor theimpedance eardrum.Accordingto B•kdsy,
the inductance
at the eardrum, its determinationwas undertaken with 0.55cm= on the average.Consequently,
L0 is equivalentto a massof 12mg andthe inductance
great care.
Typicalexperimental
resultsobtainedontwopatients La to oneof 4.5 mg. Accordingto B•k•sy and Rosenwith removedincusare shownin Figs.5 and 6. The blith,a the total massof the malleusis of the order of
approximately
twiceasgreatas
curvesindicatethe input impedanceof the electrical 23mgandis,therefore,
theeffective
mass
determined
by means
of theanalog.
analogwith the followingnumericalvalues:
This agreeswith the observationthat the ossicles
tend
Co= 1.4•F, Ca•=0.23/•F,
to rotate aroundtheir centerof gravity, so that the
effectivemassis reduced.Althoughextractionof the
incusmay be expectedto throw the systemout of
L0= 40 mH, Ca==0.40 3*F,
balance,a the effectivemassshouldremain smaller than
the total mass of the malleus. The total mass of the
eardrummaybecalculated
fromthegeometrical
dimensionsand the specificdensity.B•k•sy= estimates
the
600
total area of the eardrum to be 0.85 cm= and the thick-
200
Ioo
I000
FREQUENCY
IN
CYCLES
I0000
PER
SECOND
Fro. 6. Resistanceat the eardrum of an ear without incus.
Pointsshowdata obtainedon two patients;curvesindicate
analog results.
nessto be 0.01cm.Assuming
a specific
densityof 1.1
g/cma,whichis slightly
largerthanthatof water,we
obtain9.35mg for the total mass.If an areaof 0.55cm=
is rigidlycoupled
to themalleus,
theremaining
area,
whichaccountsfor La, amountsto 0.3 cmz and its mass
to 3.3mg.In viewof thedifficulties
in estimating
the
of the eardrum,this value must be con•0j. Zwislocki,
"Electrical
Model
oftheMiddle
Ear,"'J.Acoust. dimensions
Soc.Am. 31, 841(A) (1959),
sidered
inagreement
withthatobtained
fromtheanalog.
ed 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info
MIDDLE-EAR
FUNCTION:
INPUT
IMPEDANCE
1519
u• 15oc
It maybe mentioned
that WeverandLawrence
TM
give
•
for the massof the tympanic membranea value of
•
of 4.6mg.
s 500
14mgwhich
would
leadtoanequivalent
mass
ofLa •o•-•5oc
o
OTOSCLEROTIC
EAR
oo
• m 1500
In the otoscleroticear, the stapesis immobilizedin
2 •o
the oval windowand, as a consequence,
the cochlea
disconnected
from the middle-earsystem?In termsof
•25DO
gß
the analognetwork,thismeansan interrupted
circuit
• 3•oo
betweenthe blocks3 and 5. Block4, whichcorresponds
to the incudo-stapedialjoint, remainsin the network,
-
and its elements have to be determined
obvious
that
the
m
•5oo
from anatomical
considerations
and from impedancemeasurements
at
the eardrum.
It is rather
z
elastic
and
fricative
[000
FREQUENCY
IN
CYCLES
I0000
PER
SECOND
l:m. 8. Average reactance and resistanceat the eardrum of
otoscleroticears. Closedcirclesindicate experimentalresultsof an
earlier series; cressesthose of a more recent serieswhere an improvedtechniquewas used;curvesshowanalogresults.
forces in a joint between two bones should play a
preponderantrole.The simplestelectricalanalogof such
joint at all frequencies,
and becauseof the low coupling
a systemis a capacitancein serieswith a resistance.
a It impedanceof the eardrumat high frequencies.
It was
could be demonstrated
that these two elements suffice
alsopossibleto duplicatethe otosclerotic
impedanceat
to accountfor the effectof the incudo-stapedial
joint on the eardrumwithout any changesin the analogcircuit
the impedanceat the eardrum.Their numericalvalues of the eardrum. For thesereasons,the analognetwork
of the middleear without the incuswasacceptedas the
La
Ra Cp
analogof the otosclerotic
ear after the analogcircuitfor
Co
Lo
Re
the incudo-stapedial
joint had beenadded.The analog
is shownin Fig. 7. Its numericalvaluesare C•=0.25 uF
and R•=3000 •, in addition to thosedeterminedpreCd•
----C s
viously.The valueof L0 canbe doubledor eventripled
withoutproducingany appreciable
effect.
The experimental
and the analogimpedance
data are
Rd2
shownin Figs.8 and9. Figure8 containsmediansof two
seriesof experiments
performed
on 14ears(14 patients)a
and 17 ears (9 patients),1ørespectively.Closedcircles
Fro. 7. Electric analogof the otoscleroticear. Elementsdenoted
by subscrlpt s belong to the incudo-stapedial joint. The other
denote the results of the former, crossesof the latter
seriesin which the morerecent,improvedtechniquewas
elements are the same as in l:ig. 4.
used. The curvescorrespondto the analog data. The
agreementbetweenthe experimentaland the analog
canbe easilydeterminedfrom impedancemeasurements resultsshouldbe consideredsatisfactory,althoughsome
at the eardrum,providedit is possibleto assumethat discrepancies
are apparent.The absolutevalue of the
the parts of the network determinedon ears without analogreactanceis somewhattoo small at low and
incusdo not requireany modifications.
Suchan assump- somewhattoo largeat mediumfrequencies.
The analog
tion is opento criticism,as has beenpointedout in a
preceding
papersandelaboratedon by M•ller.12First of
all, the incushasbeenaddedto the malleus.Second,the
motionof the two ossicles
is probablyaffectedby the
fixation of the stapes,and, accordingto M•ller, their
effectivemassshouldbe increased.The changein the
ossicularmodeof motionmay, in turn, affect the acoustic propertiesof the eardrum.
Studieson the electric analog have demonstrated
that, although the effective mass of the ossiclesmay
well be increasedby a fixationof the stapes,the effecton
the impedanceat the eardrumis negligible.This is so
becauseof the high impedanceof the incudo-stapedial
500
o
•2S00
450C
FREQUENCY
hE.
G. Wever and M. Lawrence, Physiological Acoustics
(PrincetonUniversityPress,Princeton,New Jersey,1954).
n A. R. M•ller, "Network Model of the Middle Ear," J. Acoust.
Soc.Am. 33, 168 (1961).
IN
CYCLES
PER
SECOND
Fro. 9. Reactance and resistance at the eardrum of otosclerotic
ears. Circles and crossesindicate data obtained on 3 patients;
curvesshowanalogresults(sameas in Fig. 8).
d 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/in
1520
J. ZWISLOCKI
resistance
is somewhat
toosmalloverthe entirerange.
In general,the agreementis better with the second
experimental
series.The discrepancies
couldbe easily
correctedby a slight modificationof the eardrum
circuit.Althoughsucha modification
couldbe justified
in view of probablechangesin the modeof ossicular
o
motion,its physicalmeaningcouldnot be ascertained
at
present. Consequently,it was consideredsuperfluous
_z 1500
0
500
ß METZ
for•thepurposes
of thisarticle.The effectof various
• ASPeNALL, MORTON, AND JONES
parameterson the impedanceat the eardrum will be
o ZWlSLOCKI
(SECOND
ß ZWISLOCKI
(THIRD
SEBIES)
* M©LLER
(FIRGT
SERIES)
SERIES)
x HELLER
(SECOND
discussed
in a subsequent
paper.
Figure 9 comparesempiricaldata obtainedon three
Iooo
IOOOO
FREQUENCY
IN CYCLES
PER
SECOND
typicalpatientsto the analogcharacteristics.
It can be
seenthat the agreement
is at leastas goodas with the
Fro. 11. Average reactance at the eardrum of normal ears.
mediandata, and that little informationwaslost in the Symbolsindicatedata obtainedby severalinvestigators;curve
showsanalogresults.
preceding
figureasa resultof the averaging
procedure.
2500
NORMAL
SERIES)
EAR
dicatethe presenceof a masscomponentin the input
of the innerear.This masscomponent
is not
Functionally, the otoscleroticcar contains all the impedance
dueto the entirecolumnof perilymphin the cochlea,as
elements
of thenormalear,excepttheinputimpedance
by severalauthors,but
of the cochlea
and the impedances
of the stapesandof has beenassumederroneously
is producedby a shortcolumnof fluid in the vestibule,
the round-window
membrane.
A simplecircuitfor the
betweenthe cochleaproperand the oval window.The
input impedance
of the cochleaitselfis resistive,due to
the interactionbetweenthe massof the perilymphand
the compliance
of the basilarmembrane.
This is clearly
evident from the generalhydrodynamicaltheory of
wave propagation,
6 and has been confirmedexperimentallyby Mundiet• on the guinea-pigear.
The mass component of the inner ear cannot be
determineddirectly from BCk6sy'sexperiments,since
the conditionsprevailingat the oval windowmay have
beenalteredduring the experimentalprocedure.HowFro. 10. Electric analog of the normal ear. Elements with subscriptc are addedto thoseof Figs.4 and 7; they belongto the ever,it may be estimatedfrom anatomicalsectionsthat
cochlearcomplex.
the columnof perilymphbetweenthe oval windowand
the cochleaproper does not exceed2 mm. Since the
average
areaof the stapesfoot plate hasbeenestimated
missingcomplexmay be derivedfrom the theoryof
electroacoustic
analogies.
a It consists
of a capacitance to be 3.2 mm2, the massof the effectiveperilymph
6.4 mg. This
and a resistance
for eachof the two windows,and of a columnshouldamountto approximately
resistance and an inductance for the cochlea. All these valueis increased
by themassof thestapes(2.5mg)• to
elementsare connectedin series,so that the capaci- 8.9 mg. In order to obtain the acousticmass,the latter
tances and the resistances that stand for the two
figuremustbe dividedby the areaof the stapessquared.
windowsmay be lumped together.Furthermore,the The calculationyields8.7 g/cm4.The equivalentacouspracticallynegligible
massof the stapesmay be added tic massreferredto the impedanceat the eardrumis
to the massresultingfrom the inner-earimpedance
to obtainedby divisionby the squareof the acoustic
constituteone inductance.All the resistances
may be
lumpedtogetheralso.Asa result,thenetworkof Fig. 10
is obtained.It is an analogof the normalmiddleear,in
•o
which all the elementsexcept C,, Lc, and R, have
½F-SOD
already beendetermined.
The elementsof the cochlearcomplexhave been so
chosen
that theinputimpedance
of the analogmatches
F•EOUENCY
iN CYCLES
F'E•
SECOND
the averageimpedanceat the eardrum.
Fro. 12. Average resistance at the eardrum of normal ears.
During the matchingexperiments,it wasfound that
Symbols indicate data obtained by several investigators; curve
the input impedanceis so insensitiveto the inductance showsanalogresults.
L• that this elementmay be completelyeliminated.
Nevertheless,L, has been left in becauseof anatomical
considerations,
and becauseB8k&y's experiments
in-
•aj. R. Mundie and D. F. Henges,"SomeFactorsInfluencing
the AcousticalImpedanceof Guinea Pig Ear," J. Acoust.Sec.
Am. 32, 1495(A) (1960).
ed 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info
MIDDLE-EAR
FUNCTION:
INPUT
IMPEDANCE
1521
of the ear canal. Morton and Jonesmeasuredthe distancefrom the probeto the eardrumand multipliedit.
by the averagecross-sectional
area of the ear canal.In
this way, they obtained a volume of 0.48 cc which is
smallerby 0.07cc than the averagevolume usually
obtained under similar conditions.Since, on the one
hand,theirimpedance
valuesat the entranceto the ear
canalagreedrather well with data discussed
in a previous paper,'ø and, on the other, the calculatedimpedanceat the eardrumappearedsomewhattoo small,
it was decided to adjust their data to a volume of
o
,• iooo
z_ 15oo
• 2ooo
0.55 co.
•: 2500
I00
I000
FREQUENCY
IN
CYCLES
PER
SECOND
Figuresi1 and 12 showthat the experimental
results
are in a fairly goodagreementwith eachother. The
largest discrepancies
appear above 1000 cps for the
reactanceand below 300 cps for the resistance.The
probablereasonfor this effect can be found in the
magnituderatio of the two components.At low frequencies,
the reactanceis largeand the resistance
small;
transformer ratio between the oval window and the
above 1000 cps, the reverseis true. The smallercomeardrum. B•k&y arrives at a theoreticaltransformer ponent is very sensitiveto the phase measurement.
ratio of 22. Consequently,the equivalentacousticmass
Consequently,
an errorin phasedeterminationproduces
at the eardrumamountsto approximately20 mg. This
a magnifiederrorin the smallerimpedancecomponent.
corresponds
to an inductanceof 20 mH.
The curvesof Figs. 11 and 12 indicate the results
The othervaluesof the cochlearcomplexthat fit the
obtained on the electrical analog. In the frequency
measuredimpedanceat the eardrumare
Fzc. 13. Reactance
at the eardrum
of normal
ears. Circles and
crossesindicate data obtained on three subjects; curve shows
analogresults(sameas in Fig. 11).
C• = 0.6 •F, R• = 600 •.
rangebetween100 and 1500cpsfor the reactanceand
between200and 1500cpsfor the resistance,
theyagree
with the experimental
data withinthe errorof measure-
The resistance
agreesverywellwith thevaluepredicted ment.
in earlierpapers.
a.6
Figures 13 and 14 comparethe analogimpedance
The experimentaldata obtainedby severalinvesticharacteristicsto typical data obtained on three subgatorsas well as the analogresultsare shownin Figs.
jectswith normalhearing.The agreementis as goodas
11 and 12. The data of M•ller •.'• and Zwislockia.•øwere
with the averagedata, which leads to the conclusion
takendirectlyfromoriginalpapers.The valuesgivenby
that the analogis a valid representationof a typical
Metz s had to be convertedfrom specificimpedance
middle
ear.
expressed
in pcunitsto acousticimpedance
expressed
in
cgsunits.The conversion
wasbasedon a characteristic
COMPARISON
WITH
PREVIOUS
MODELS
acousticimpedance
of air of 130 cgsunitswhichcould
be derivedfrom the description
of Metz'sexperimental The electricalanalogdiscussed
in this paper evolved
setup.The resultsof Morton and Jones
]• wereadjusted as a resultof successive
approximations.
The first crude
because of their somewhat crude estimates of the volume
modelwaspublished
in 1957,a somewhat
iraprovedone
was discussed
beforethe AcousticalSocietyof America
in 1959. Severalother modelsremainedunpublished.It
_m
• I000
may be of interestto comparethe numericalvaluesof
the principalelementsin the threepublishedanalogsin
order to seewhetherthere is someconsistency.
Using
• 5oo,, o
the nomenclatureof this paper,we can make the following comparisons.
The capacitance
Corepresenting
the
complianceof the malleo-incudalsystemhad in the
0100
I000
first model a value somewhathigher than 0.56 uF, in
FREQUENCY
IN
CYCLES
PER
SECONO
the secondit grewto 1.64uF, and in the third it deFro. 14. Resistance at the eardrum of normal cars. Circles and
crossesindicate data obtained on three subjects; curve shows creaseda little to 1.4uF. The inductancestandingfor
analogresults(sameas in t'ig. 12).
the acousticmassof the ossicularchain changedfrom
60 to 50 to 40 mH. The capacitancein the complexof
• A. R. MMller, "Improved Techniquefor Detailed Measurementsof the Middle Ear Impedance,"J. Acoust.Soc. Am. 32, the incudo-stapedialjoint varied irregularly from 0.35
2S0(A) (1960).
a•J. ¾. Morton and R. A. Jones,"The AcousticalImpedance
Presentedby SomeHuman Ears to HearingAid Earphonesof the
Insert Type," Acustica6, 339 (1956).
and 0.52 to 0.25 uF. The analog capacitance of the
complianceof the cochlearwindowshad an unreasonably high value of 2.6 #F in the first model, but de-
d 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/in
1522
J.
ZWISLOCKI
creasedto 0.7 #F in the second,and further to 0.6 #F in
the third. The input resistance
of the cochlearemained
reasonablystable changingonly from 540 to 400 to
600•2. It is evident that, despite the growing complexity of the analog,the principal elementskept their
order of magnitude.It is probable,therefore,that they
are representative
of the actualconditionsin the middle
in this paper. At tirst glance, M½ller'sassumption
appearsquite attractive. Closerscrutinyleadsto some
sclerotic fixation. While, in the first situation, the
agreement would result. These data, obtained on ear
doubts. Rotation
of the two ossicles around the incudo-
stapedialjoint wouldnecessitate
a motionof the portion
of the malleus imbedded in the eardrum in a plane
tangent to the surfaceof the eardrum. Such a motion
would be opposedby an extremelyhigh mechanical
ear.
impedancewhich is not borne out by experiments.
In this context,it may not be superfluousto mention Rotationof the twoossicles
aroundthe incudo-stapedial
that the inductance L0 obtained for the malleo-incudal
joint wouldalsoproducea high masscomponentin the
complexin the latestanalogis exactlyequivalentto the impedance
at the eardrum.This masswouldbe greater
effective mass of these ossiclesas determined by than the total massof the malleo-incudalcomplexand,
Frank.•6 By letting the ossiclesoscillatearound their as a consequence,
wouldexceed50 mg. In the electrical
axis of gravity, Frank obtained an effectivemass of analog,this masswould be reflectedby an inductance
12 mg. When the valueof 40 mg/cm4, equivalentto the of 167mH insteadof about 50 mH, as calculatedby
inductanceL0=40 mH, is multipliedby the squareof M½ller for his analog.
the effective area of the eardrum, an effective mass of
From the analysisof M½ller's work, there emerges
12 mg results.
an interestingconclusion
that the middle-earstructure
A considerable
numberof electricalanalogshavebeen effectivelypreventsany motion of the ossiclesother
suggestedby various investigators.The older ones than rotationaroundan axis passingthrough,or at
becameobsoletein the light of accumulatingexperi- leastnear, the centerof gravity of the malleo-incudal
mentalevidence.The morerecentones,designed
on the complex?
basisof quantitativeinvestigations,
needto bediscussed. Anothermodelof the middleear has beensuggested
M½ller• hassuggested
an analogbasedon impedance by Onchi.7 It was presentedfor the first time in 1949
measurements
at the eardrumof clinicallynormal ears. asa mechanicalanalog? In a recentpaper,Onchi shows
In order to determinethe circuit elements,he varied the a conversion
of the mechanicalanaloginto an electrical
responseof the ear by coating the eardrum with col- one. The electricalversion is based on essentiallythe
lodionand othersubstances
and by lettingthe stapedius sameassumptions
that underlythe analogof thispaper,
musclecontract.This methodis somewhatanalogous
to and, as a consequence,
the generalstructuresof the two
the onediscussed
in this and a preceeding
paper.How- analogsare similar. Nevertheless,there are somedifferever, changesin the middle-ear characteristicswere encesthat shouldbe mentioned.They concernmainly
producedin a differentway.
the portionof the eardrumthat is not rigidlycoupledto
There are three aspectsof M•ller's approachthat the malleus, and the incudo-stapedialjoint. Both
appearopento criticism.First of all, the complexeffect systemsare representedby a capacitancein Onchi's
of the middle-ear cavities is ignored; second,it is paper. Sucha simplificationshouldbe permissible
for
assumedthat the stapediusmuscleeffectivelyimmobi- the eardrumat lowfrequencies.
In the frequencyrange
lizesthe stapes;and third, it is assumedthat the ossicles above 1000 cps, a more complexnetwork appears
rotate around the incudo-stapedial joint when the necessary.In the incudo-stapedialjoint, where the
stapesbecomesimmobile.The effect of the middle-ear lenticularprocessof the incusmovesrelative to the head
cavities is small up to about 1000 cps, and, if the of the stapes,frictionmust arise.It shouldbe reprevalidity of the modelis restrictedto thelowfrequencies, sentedin the electricalanalogby a resistance.
Impethe omission
of thiseffectis not critical.The assumption dance measurements on otosclerotic ears indicate that
of stapedialimmobility due to stapediuscontraction this resistance is substantial.
may be of greater consequences.
A comparisonof
It is not possibleto makea comparison
with Onchi's
M½ller's impedancecurves with analogouscurves ob- analog on a quantitative basis, since Onchi omitted
tainedon otosclerotic
patientsindicatesthat the stape- giving the numericalvaluesof his elements.However,
dius action is considerablyless effective than oto- judging from his impedancedata, a considerabledisimpedanceat lowfrequencies
increases
by a few percent, preparations,showmuchhigherimpedancevaluesthan
in the second,the changeamountsto 100% and often those encounteredin the ears of living subjectswith
more. The mode of ossicular motion is of primary normal hearing. Onchi's explanation that the low imimportancein the designof an electricalanalog,and pedancemeasuredon living subjectsis due to the acousM•ller's assumptionthat the malleusand the incuscan tic conditions
in the ear canaldoesnot seemacceptable,
rotate around the incudo-stapedialjoint makes his sincea highimpedance
is foundin otosclerotic
patients.
analogsubstantiallydifferentfrom the analogdescribed It appears
far moreprobablethat post-mortem
changes
• O. Frank, "Die Leitung des Schallesim Ohr (Sound Trans-
missionin the Ear)," Sitzber. math-physik. KI. bayer. Akad.
Wiss. Miinchen õ$, 11 (1923).
accountfor the high impedance
of Onchi'sspecimens.
•7¾. Onchi,"A Study of the Mechanismof the Middle Ear,"
J. Acoust.Soc.Am. 21,404 (1949).
ed 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info
MIDDI.
E
EAR
I:UNCTION:
INPUT
IMPEDANCE
1523
Thispointwill befurtherelaborated
on in a subsequentear in a quantitativeway. It correlateschangesin the
acoustic impedanceat the eardrum to anatomical
changes
in the middle-earstructures.
It alsoshowsthat
the numericalvalues of its parameters,which were
In an effort to synthesizethe available empirical derivedfrom averagedata, are a true representation
of
information on the acoustic function of the middle ear
conditionsin typical ears. In anticipation of a subseinto a quantitativetheory,an electricanalogof the quentpaper,it is possibleto statefurther that the anamiddleear has beendevised.The analogmay be con- logprovidesinformationon the transmission
properties
sideredan expression
of the underlyingmathematical of the middle ear, on the correlation of these transtheory.Althoughit wouldhave beenpossibleto obtain missionpropertiesto the impedanceat the eardrumfor
the sameresultby meansof a setof algebraicequations, various anatomicalconditions,and finally, on the
theseequationswouldhavebeensocomplexthat their transientresponses.
meaningwouldnot becomeimmediatelyapparent.An
The frequencyrangeof validity for the analogof the
electricanalogprovidesa simplevisualimageof the
normalear extendsfrom at least 100 to 2000 cps.Its
system,and makesit possibleto recognizefunctional
responsecharacteristics
were recordedup to 8000 cps.
relationships
almoston sight.Anotheradvantageis the
However, the empirical data beyond 2000 cps are
time economyin obtaining data for various kinds of
meagerand in poor agreementwith each other. For
input signalsand for variouschangesin the parameters
pathologicalears,no data seemto be availablebeyond
of the system.
1400cps.It is hopedthat thesegapswill be filledin the
It is true that an electricalanalogis an imperfect
near future, althoughexperimentaldifficultiesincrease
paper.
CONCLUSIONS
model and that its validity is restricted to a limited
rangeof the variablesinvolved.The sameis true for all
possiblemodels or theories, however, and cannot be
considereda specificdisadvantage.All the simplifica-
rapidly at frequencieshigher than 1000 cps.
ACKNOWLEDGMENTS
tionsdeemednecessary
in the electricanalogdescribed I shouldlike to expressmy gratitudeto Dr. G. D.
in this paper have been checkedcarefully against Hoople,to Dr. W. H. Bradley,andto Dr. A. S. Feldman
empiricalevidence,and they were not acceptedunless for their help in obtaining data on patients and on
the responseof the analogmatchedthat of the real ear
within the experimentaluncertainty.It is expectedthat
future
research will
narrow
down
the tolerances
and
make certain modificationsnecessary.This should be
viewedas scientificinevitability, however,rather than
as an inherent weakness of the method.
temporal bones. My thanks also go to Mr. V. H.
Sandoval,to Mr. S. Madonna,and to Mr. R. Gardlnier
for their helpin computationof the impedancedata and
in the constructionof the electricalanalogs.
This work was supportedby a researchgrant from
the National Institute of NeurologicalDiseasesand
Blindness,National Institutes of Health, Public Health
For the present,it is probablysafe to state that the
devisedelectricanaloganswersthe followingquestions. Service,U.S. Departmentof Health, Education,and
It specifies
the functionof variouspartsof the middle Welfare.
d 13 Sep 2011 to 156.145.57.157. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/in