The mechanical behaviour of the eardrum has previously been shown to depend critically on
its shape, but accurate shape measurements have been difficult to make. Phase-shift moiré topography
provides a valuable technique for measuring such shapes, and measurement in the presence of large
static pressures facilitates the determination of the boundaries of the pars tensa, pars flaccida and
manubrium. New measurements of the shape of the cat eardrum are presented. The presence of
hysteresis in the pressure-displacement response is demonstrated. The shapes are incorporated in
individualized finite-element models for four different ears, and the variability between and within
animals is examined. Fixed-manubrium low-frequency displacements are simulated and compared for
the different models.
PACS numbers: 43.64.Bt, 43.64.Ha
1. INTRODUCTION
The mechanical behaviour of the eardrum critically depends on its shape, but until recently
the available shape data (Helmholtz, 1869; Kojo, 1954; Kirikae, 1960) were neither precise nor
detailed. As a consequence, the shape was approximated in our finite-element models by only two
parameters: the depth of the cone, and an average or typical radius of curvature for the sides of the
cone (Funnell & Laszlo, 1978).
Moiré topography is an optical technique involving the projection of a grating of parallel lines
onto the surface being measured, and offers a convenient non-contacting method for measuring the
shapes of small objects. In the first attempts to measure eardrum shape using moiré topography
(Khanna & Tonndorf, 1975a,b) the measurements were made on Silastic castings in order to obtain
adequate optical contrast. This caused the boundary definition to be poor because the castings could
not accurately replicate the very narrow space between the ear-canal wall and parts of the eardrum.
Furthermore, the moiré technique that was used resulted in fringes which had to be counted to
determine z coördinates, which made the analysis very difficult and provided only limited depth
resolution (Funnell, 1981). The analysis is greatly simplified by the use of the phase-shift moiré
technique, which involves the combination of four moiré fringe images, resulting in images in which
the value of each pixel is directly related to the z coördinate (Dirckx et al., 1988; Dirckx &
Decraemer, 1989). The technique also provides greatly improved depth resolution, with an accuracy
of 20 m (Dirckx & Decraemer, 1990).
It also became possible to measure the eardrum directly rather than with castings, and to
measure the shape under the influence of sequences of static pressures applied in the middle-ear air
cavities (Decraemer & Dirckx, 1991). Elimination of the castings removed one source of error in
defining the boundary of the eardrum, but in some regions there is no distinct landmark to demarcate
the boundary. Application of static pressures facilitates identification of the boundary in such regions
because the eardrum moves in response to the pressures while the ear-canal tissue does not. In other
regions, however, the boundary is hidden under overhanging tissue which cannot be removed without
damaging the eardrum. In these cases a different approach is required, as described below.
Unpressurized eardrum shapes measured with the phase-shift moiré technique have previously
been presented for human (Decraemer et al., 1991) and cat (Decraemer & Dirckx, 1991). Shape data
with static pressures applied have previously been presented for the human eardrum (Dirckx &
Decraemer, 1991) and for a single cat (Decraemer & Dirckx, 1991; Funnell et al., 1993).
In this paper the moiré shape-measurement technique as applied to the cat eardrum is briefly
summarized in Section 2, and the procedures used to analyse the shapes and to incorporate them in
finite-element models is described in Section 3. New shape data for the cat eardrum are then
presented in Section 4 and simulation results for finite-element models incorporating the shape data
are presented in Section 5. The simulations include some that use a parametric representation of
eardrum shape similar to that used in previous models.
2. EXPERIMENTAL SHAPE MEASUREMENT
The experimental technique has been described previously (Decraemer et al., 1991; Dirckx
& Decraemer, 1991). The fresh temporal bones to be measured are obtained from cats that have been
used for purposes unrelated to the auditory system. The cats are sacrificed with an intracardiac
injection of pentobarbital Na solution (60 mg/kg bodyweight). The dissection of the temporal bone
starts about 15 minutes post mortem. In order to provide a good view of the eardrum, the ear canal
is resected to within about 0.5 mm of the tympanic ring, as close as possible without damaging the
eardrum. Static pressures of as much as 220 mm H2O are applied to the middle ear through a small
hole drilled in the bulla wall. So that the eardrum does not dry out, it is regularly moistened during
the preparation, but not during the recording time itself which starts about 6 hours post mortem.
Drying out of the eardrum is fairly slow because the epithelial layer on the lateral side of the eardrum
is relatively thick and is covered by a layer of Chinese ink (which is water-based); and because the
middle-ear air cavity is kept closed, protecting the very thin mucosal layer on the medial side of the
drum. No shape changes were observed when the air in the ear canal was humidified by breathing into
it.
Moiré interferograms of the eardrum are obtained by casting the shadow of a grating of
parallel lines onto a temporal bone placed close behind the grating. The shadow is a set of deformed
lines and forms a moiré interference pattern when observed through the line grating. The moiré
images are recorded using a CCD camera and a frame store. For the optical setup used for the present
work, the scaling of the moiré images was 45.8 m/pixel horizontally and 31.7 m/pixel vertically.
During the recording of each image the grating is moved in its plane in order to average out the
grating lines. Four phase-shifted images are obtained by moving the object slightly along the axis
perpendicular to the grating between images. The four images are then combined pixel by pixel to
form a single image in which the value of each pixel specifies the z coördinate of a point (Dirckx et
al., 1988). The computation of z at each pixel involves an arctangent and the result is therefore
'wrapped' into the range 0 to 2k, where k is a calibration constant (m/rad). The z coördinates must
be 'unwrapped' by adding multiples of 2k where required. For the measurements reported here, the
grating had 4 lines/mm, resulting in a z calibration of 118 m/rad. This corresponds to 0.74 mm for
2 radians, which is the size of the ambiguities that must be resolved by unwrapping as discussed
below.
The moiré technique requires a diffusely reflecting surface in order to obtain good optical
contrast, but the eardrum reflects only poorly, so something must be done to make it more reflective.
Application of white Chinese ink (Pelican Drawing Ink A, 18 White) has been shown to result in a
layer which is only about 10 m thick and quite uniform (Decraemer & Dirckx, 1991; Stoffels, 1993).
The measured eardrum surfaces were generally quite smooth, except in one animal in which there
were artefacts in two locations near the manubrium. The weight of the ink is equivalent (assuming
a density of twice that of water) to a pressure on the order of only 20 m H2O, so its effect on the
shape of the eardrum is negligible.
In the present paper results will be presented for all four cats for which measurements were
made with static pressures applied. Some data from one animal (CAT8) have been presented
previously (Dirckx & Decraemer, 1991; Funnell et al., 1993). The moiré data for the other three
animals (CTM2, CTM3 and CTM4) have not previously been published (Stoffels, 1993). For animals
CAT8 and CTM2, moiré images were obtained for 9 and 11 different pressures, respectively,
including one each at zero pressure and several for negative and positive pressures. For each of
animals CTM3 and CTM4, two sets of pressure measurements were obtained, with each set starting
at zero, increasing in several steps to a maximum, jumping back to zero, then decreasing in several
steps to a minimum. For animal CTM3 there is then a third measurement at zero pressure at the end
of each set. There was a time interval of about two hours from the beginning of one such set to the
beginning of the next. Several minutes are required to acquire the data at each pressure step.
3. DATA-ANALYSIS METHODS
3.1 Phase unwrapping
The phase-shift moiré technique results in an image matrix in which the column and row
numbers of each pixel are proportional to the x and y coördinates, respectively, of a point on the
structure being observed, and the value of the pixel is proportional to the z coördinate of the point.
The pixel values are actually calculated as phase angles; they are therefore subject to a 2 ambiguity
and must be 'unwrapped'. Because there are true abrupt jumps in the z coördinates, due mostly to the
overhang at the edge of the eardrum but sometimes to large displacements, conventional phase-unwrapping algorithms cannot be relied upon, and it is often difficult to unambiguously determine
how to unwrap the data even visually. Figure 1 shows a profile through one particular set of data. A
limited number of point-by-point mechanical position measurements were made in this particular
animal. In spite of their relative inaccuracy these measurements do help to resolve the 2 ambiguities,
but they are extremely time-consuming to make and are not usually performed.
The phase unwrapping has been carried out with a combination of manual and semi-automatic
methods. A special interactive programme, MUM (Modelling Using Moiré), has been written for this
purpose. The methods used include (1) automatic unwrapping of the column in the centre of the
image, working upward and downward from its centre, followed by unwrapping of each row working
leftward and rightward from the centre; (2) addition or subtraction of 2 to a single manually
specified pixel, or to all pixels above, below, to the left of or to the right of such a pixel; (3) automatic
addition of multiples of 2 to all pixels in a row to minimize the differences between it and some
other, previously unwrapped row; and (4) automatic image unwrapping by pairwise matching, by
addition of multiples of 2 to all pixels in an image to minimize the differences between it and some
other, previously unwrapped image.
When the last method is applied to a set of images corresponding to a range of static
pressures, rather than matching all of the images to be unwrapped to a single previously unwrapped
image, the images are first sorted in order of their pressures and then automatically unwrapped by
comparison with one another in consecutive pairs, starting with a previously unwrapped image and
working upward and downward through the range of pressures. For example, if images had been
measured at pressures of -20, 0, 20, -40 and 40 mm H2O, and the image for p=0 had been unwrapped
first, then the other images would be unwrapped by pairwise matching in the order (0,-20), (-20,40),
(0,20), (20,40). This is done because the z-coördinate differences betweeen two images with different
pressures may well be greater than 2 unless the pressures are close together.
3.2 Determination of boundaries
Once a set of images has been unwrapped, it is necessary to determine the boundaries of the
pars tensa, pars flaccida and manubrium. In some regions this is quite straightforward, but in others
it is not. Figure 2 shows a set of profiles, for pressures of 220, 0 and -220 mm H2O, along a line
roughly perpendicular to the manubrium and running through a point approximately halfway between
the short process and the umbo. Anteriorly all of the profiles converge to a point (indicated by an
open circle) clearly corresponding to the edge of the eardrum. Posteriorly, however, a large jump
occurs before the different profiles converge to a single z value. This is because the eardrum passes
underneath the overhanging ear-canal tissue, so the actual edge cannot be seen. In such cases the
position of the edge was determined by visually extrapolating the profiles. (Many different pressurized
profiles are used, but only two are shown in the Figure for clarity.) An open circle is shown at the
boundary position so estimated. The radius of the circle is 0.1 mm, an overestimate of the possible
error in the boundary position.
A determination of the boundary of the manubrium can generally be made based on the angle
with which the eardrum meets it, and on the invariance of the cross-sectional shape of the manubrium
as different static pressures are applied. In Figure 2 the posterior edge of the manubrium is quite clear
and is indicated by a small open circle. The anterior edge is somewhat less clear, and its estimated
position is indicated by an open circle of radius 0.1 mm.
In almost all cases the x, y and z coördinates of the edges of the pars tensa and manubrium
can be estimated with confidence to within 0.1 mm, and usually to within 0.05 mm or less. In many
cases the superior edge of the pars flaccida is more difficult to identify, but this is assumed to be
relatively unimportant for the purposes of modelling the mechanical behaviour of the eardrum.
It is not possible to determine the exact boundary between the pars flaccida and the pars tensa
based on the moiré data. There is often a displacement minimum in the profiles in that region, and in
the present work the position of the minimum has been taken to indicate the position of the boundary.
As will be seen below, however, the position in which decreased displacements are predicted by the
finite-element model does not generally correspond to the position of the boundary between the pars
flaccida and pars tensa, but rather is determined by the local geometry of the model.
3.3 Finite-element model generation
3.3.1 Definition of generic model
The generation of finite-element models for one set of moiré data is based on a hierarchical
specification of a generic model, based on points, lines and regions. Names and interrelationships are
specified for boundary lines, including the outer boundary of the pars tensa; that of the pars flaccida;
the anterior and posterior boundaries between the pars tensa and pars flaccida; the manubrium; and
the short process. Each line definition contains a set of six boundary conditions, corresponding to x,
y and z displacements and to rotations about the x, y and z axes. These boundary conditions are used
for each node to be generated along that line for the finite-element mesh. For the models discussed
here, the six boundary conditions are either all unconstrained or all fully clamped. Each line definition
may include the x, y and z coördinates of points that have been interactively defined for that line.
Different regions of the model are defined by concatenating directed named lines. For
example, the pars tensa is defined as the region delimited by moving forward around the pars tensa
outline, backward along the posterior boundary between the pars tensa and pars flaccida, backward
around the manubrium, and forward along the anterior boundary between the pars tensa and the pars
flaccida. For each such region are also specified (1) a set of six boundary conditions which will apply
to finite-element nodes that will be generated for the interior of that region; (2) what material type
the region is made of; (3) the thickness of the constituent triangular elements; and optionally (4) an
element-load multiplier which by default is 1 but which may be set to zero for regions (like the head
of the malleus) behind the eardrum and therefore not exposed to the sound pressure in the ear canal.
A region definition may also contain parameters controlling the generation of a parametrically defined
3-D shape using radii of curvature as described below.
3.3.2 Generation of finite-element mesh
Once the coördinates of the boundary nodes have been interactively determined as discussed
above, the model definition is written by the MUM programme into another file which contains the
same information except that the point coördinates are associated with named points, and the line
definitions are converted to explicit sequences of named points. This file is used by a hierarchical
mesh generator programme (FUD) to create a finite-element model. The resolution of the finite-element mesh is specified as a nominal number of elements/diameter (Funnell, 1983).
The finite-element model is then read back in by MUM, and the new internal nodes generated
by FUD are assigned z coördinates by using the appropriate pixel values from a moiré image
corresponding to a static pressure of zero. Occasionally one of the nodes of the finite-element mesh
will be located in the part of the image corresponding to the overhang of the ear canal. In such cases
the z coördinate must be set manually by visual extrapolation of the profiles.
Once the z coördinates have been extracted from the moiré image, each finite-element model
is rotated in three dimensions so as to make the tympanic ring roughly parallel to the x-y plane, to
facilitate comparisons among different models.
In previous finite-element models of the cat eardrum, the three-dimensional curved conical
shape was generated using a normalized radius-of-curvature parameter (Funnell, 1983). For each
internal node created by the 2-D mesh generator, a z coördinate was computed so that the node
would lie on a circular arc whose plane lay perpendicular to the plane of the tympanic ring and whose
two ends lay on the tympanic ring and on the manubrium, respectively. The radius of curvature of the
arc was taken to be the product of the chord length of the arc and a normalized radius parameter. In
the present work the same approach has been used to produce a parametrically defined shape for
comparison with the moiré-derived shapes. The method has been enhanced so that (1) the centre of
each arc is taken to lie below the tympanic ring if the manubrial end is above it, but above the ring
if the manubrial end is below it, so as to produce more realistic shapes near the mallear lateral
process; and (2) the z coördinate computed from each arc is not permitted to lie on the side of the
plane of the tympanic ring opposite that of the centre of curvature, which is a concern only for quite
small radii of curvature. Furthermore, the normalized radius-of-curvature parameter, rather than
necessarily being a constant, is now permitted to vary either linearly or quadratically as a function of
the distance around the tympanic ring of the outer end of the circular arc. It will be seen below that
such a variation is required to match the behaviour of models using the moiré-derived shapes.
Previous finite-element models for the cat included a fixed axis of ossicular rotation which lay
in the plane of the tympanic ring (Funnell & Laszlo, 1978, etc.). In order to represent the axis more
realistically when using moiré data to define the shapes (Funnell et al., 1993), the position of the axis
was established based on the approximate positions of the posterior incudal ligament and anterior
mallear process in a 3-D reconstruction of the cat middle ear (Funnell & Phelan, 1981; Funnell &
Funnell, 1989). Because of the additional complexity that this introduces to a comparison of different
individual animals, in the present work the issue of the position and nature of the axis of rotation is
sidestepped by simulating the condition of a fixed manubrium. This is directly comparable to a feasible
experimental situation and permits concentration on the characteristics of the eardrum and its shape.
In all of our previous finite-element models of the cat eardrum the boundary between the pars
flaccida and pars tensa was modelled as a thickened 'ligament'. No such thickening was reported by
Lim (1968), however, and examination of serial sections has indicated that none occurs in the cat. In
fact, it seems to be hardly possible to distinguish sharply between the pars flaccida and the pars tensa
with light microscopy in histological sections: the characteristic lack of a lamina propria in the pars
flaccida (Lim, 1968) is not evident at low magnifications, and the change in thickness is gradual. In
the present models the boundary has been taken, for lack of a better choice, to coincide with the
region of smaller static displacements between the larger ones seen in the pars flaccida and pars tensa,
respectively.
In previous models the thickness of the pars flaccida was unimportant because no internal
nodes were generated for it. For the present models we have included internal nodes, and have taken
the thickness of the pars flaccida to be 80 m. For the Young's modulus being used (107 dyn cm-2)
this results in more or less reasonable pars-flaccida displacements. The characteristics of the pars
flaccida have little effect on the mechanical behaviour of the rest of the eardrum.
Except as discussed above, the model parameters used for the calculations presented here are
the same as those used in our previous models. Inertial and damping effects are ignored, so the results
are applicable for low frequencies, up to at least 300 Hz. The thickness, Young's modulus and
Poisson's ratio of the pars tensa are 40 m, 2108 dyn cm-2 and 0.3, respectively. The stimulus is a
uniform pressure of 100 dB SPL.
As always in finite-element modelling, the fineness of the mesh must be chosen as a
compromise between decreased accuracy and increased computational expense. For one of the
eardrums considered here, models were generated with nominal mesh resolutions of 20, 30 and 40
elements/diameter. The maximal eardrum displacements were computed to be 518.6, 526.4 and 526.7
nm, respectively, and the overall displacement patterns were very similar for 30 and 40
elements/diameter. This indicates that 30 is an adequate resolution and that is the value used for the
following.
4. MEASURED SHAPES
Since the focus of this paper is on the resting (p=0) shapes, and since the measurements with
static pressures applied were intended primarily to aid in determining boundaries for the models, no
detailed presentation will be made of the displaced shapes. It is important, however, to take a brief
look at the eardrum displacements as a function of pressure in order to understand the variations in
the resting shapes within individual animals. Figure 3 shows, for the animal (CTM3) in which the
pressure variations were measured most systematically, pressure-displacement curves for a single
point in the posterior region of the pars tensa, roughly midway between the tympanic ring and the
manubrium. Positive displacements are outward into the ear canal, and correspond to a more highly
curved eardrum surface.
Note that the pressure-displacement curves exhibit hysteresis, that is, the eardrum does not
return to the same position when the pressure returns to zero from a positive excursion as when the
pressure returns to zero from a negative excursion. This is a common feature of the behaviour of
biological materials (Fung, 1993). It is reflected in the well-known fact that impedance tympanometry
measurements give different results depending on the order in which the pressure is varied
(Osguthorpe & Lam, 1981), and has also been reported in moiré measurements in the gerbil (von
Unge et al., 1993). The behaviour of such a material depends in general upon its history, and the
shape of the hysteretic force-displacement loop may change from one cycle to the next. If the material
is taken through the same load cycle repeatedly, the force-displacement curve will normally converge
to a stable shape; the material is said to have been 'preconditioned'. The fact that the second loop in
Figure 3 exhibits much smaller displacements than the first loop may indicate that more
preconditioning cycles were required, or it may reflect gradual post-mortem changes. Similar
behaviour was found for the other animal (CTM4) in which two pressure cycles were measured.
One of the resting (p=0) eardrum shapes for animal CTM4 is represented in Figure 4 by means
of (a) the x-y projections of the finite-element model with superimposed constant-z contours; and (b)
the x-z projection of the model. The contour lines appear relatively evenly spaced, except near the
tympanic ring where they are further apart. This indicates a small curvature in the inner part of the
eardrum. The same thing can be seen in the x-z view: the profile between the umbo and the tympanic
ring is almost a straight line except near the ring.
The various resting shapes measured for all four animals are summarized in Figure 5 as
constant-z contours. The different shapes measured for a given animal are visually very similar, but
there are visible differences among animals. In animals CAT8, CTM2 and CTM3, the spacing
between contours tends to increase gradually from the manubrium outward, rather than remaining
more or less constant until near the tympanic ring as it does in animal CTM4. The eardrums of all four
animals are approximately the same diameter and approximately the same depth.
5. SIMULATION RESULTS
Figure 6 shows low-frequency displacement-amplitude contours of the finite-element model
generated for animal CTM4, corresponding to the mesh shown in Figure 4. The maximal
displacement on the eardrum is 526 nm and occurs in the posterior part of the pars tensa. In the
anterior region of the pars tensa there is a secondary maximum approximately one quarter as large.
Figure 7 shows similar displacement contours corresponding to all of the models shown in Figure 5.
Just as for the shapes themselves, the simulated vibration patterns are similar to each other within
animals but are significantly different in different animals. The maximal displacements for animal
CTM4 range from 526 to 620 nm, and the anterior maximum is always 20 to 25% as large as the
posterior maximum. In animal CTM3 the posterior maximum is located more superiorly than in
CTM4, and the anterior displacement maximum is only about 10% as large as the posterior one. The
displacements just posterior to the manubrium are also very small, and are actually slightly negative
in some cases. In animal CAT8 the posterior displacements are small, with a maximum of only
487 nm, but the anterior displacements are relatively large (up to about 30% of the posterior
maximum). In animal CTM2 the posterior displacements are large (maximum = 782 nm) but the
anterior displacements are relatively small.
Among the four resting shapes measured for animal CTM4, the first resulted in the smallest
simulated maximal displacement. For a model based on the first shape, decreasing the Young's
modulus of the pars tensa by about 15% increases the maximal displacement to the largest value
found for that animal. Decreasing the Young's modulus in the same model by about 33% increases
the maximal displacement to the largest value found among all four animals, while increasing it by
about 8% decreases the maximal displacement to the smallest value found among all four animals.
Similarly, decreasing the thickness of the pars tensa by about 22%, or increasing it by about 5%,
changes the maximal displacement so it equals the largest or the smallest values, respectively, among
all four animals. Unlike changes in the eardrum shape, changes in either the Young's modulus or the
thickness have very little effect on the overall shapes of the simulated vibration patterns.
In animal CTM2 there is a displacement maximum near the putative boundary between the
pars flaccida and pars tensa which is almost as large as the displacement maximum of the pars tensa
proper. In the first two models for CTM4 there is also a maximum near that boundary but it is
smaller. In the other models there is no displacement maximum in the neighbourhood of the pars
flaccida. Recall that the boundary between the pars flaccida and pars tensa was assumed to lie at the
location of the reduced displacements that occur between the pars-flaccida maximum and the pars-tensa maximum in the response to large static pressures. The simulation results clearly do not support
such an approach, but there is no other way of determining the boundary based on the moiré data.
There is a noteworthy pattern within the simulation results for the animals (CTM3 and CTM4)
for which more than one resting shape was measured: within each cycle of pressures from zero to
positive to zero to negative, the model corresponding to the shape after the positive excursion
produces a greater maximal eardrum displacement than the model corresponding to the shape before
the positive excursion. The increase was 17% and 8% for the two pressure cycles for animal CTM3,
and 6% and 5% for CTM4. For animal CTM3, for which a third resting shape was measured after
the negative pressure excursion, the simulated displacement for the shape after the negative excursion
was slightly less (2% and 1%) than that for the shape before the excursion. These results are
somewhat surprising since (except for the last-mentioned, 1% change) the increased displacements
correspond to eardrum shapes which are more highly curved. A more highly curved structure is
usually stiffer, and displaces less, than a similar structure which is less curved. A more detailed
examination of the different configurations of the eardrum and manubrium would be required to try
to explain the apparent discrepancies. Experiments with the manubrium fixed would be useful in
removing one confounding factor.
An evaluation of the sensitivity of the models to errors in determining the exact coördinates
of the eardrum boundary was performed (Funnell et al., 1993) by generating several models (with
mobile manubrium) for the CAT8 data with pars-tensa and pars-flaccida boundaries that were
expanded or contracted by various amounts. Over a range of expansion/contraction of 5%, which
corresponds to a boundary shift of approximately 0.12 mm (for the shorter diameter of the eardrum)
to 0.2 mm (for the longer diameter) and is more than the likely range of inaccuracy in our boundary
estimates, the maximal eardrum displacement changed by +10/-14%. Several models were also
generated by shifting the pars-tensa and pars-flaccida boundaries in the z direction by varying
amounts. Shifts of 0.1 mm caused the maximal eardrum displacement to change by +4/-15%. These
numbers suggest that 15% is a reasonable upper limit for errors in simulated eardrum displacements
due to errors in boundary determination. The manubrial displacement varied much more (+51/-41%
for boundary expansion and +40/-39% for vertical shifts), probably because of the short distance
between the tip of the manubrium and the tympanic ring, but this is not relevant for the present paper
since a fixed-manubrium condition is being modelled here.
In our previous models, the curvature of the sides of the eardrum was modelled by a single normalized radius-of-curvature parameter, as discussed above. For comparison, several models (with mobile manubrium) were generated that way using the boundaries obtained from the moiré data for animal CAT8. Simulations with a range of radii showed that no such model could reproduce the behaviour of the real shape, in the sense of producing reasonable displacements for both the eardrum and the manubrium in the same model (Funnell et al., 1993).
Similarly, models (with fixed manubrium) have been generated for animal CTM4 using
normalized radii of curvature ranging from 1.0 to 15. A model with a parameter of 15 produced the
correct magnitude of displacements in the posterior region of the pars tensa but the anterior
displacements were about twice as large as with the moiré-derived shape. Conversely, a parameter
of 2.5 produces acceptable anterior displacements but the posterior displacements are about 40% too
small.
With a normalized radius-of-curvature parameter which varies linearly from 15
posterosuperiorly to 1.0 anterosuperiorly, the model produces maximal posterior and anterior
displacements which match those produced by the model with moiré-derived shape. The shapes of
the displacement patterns do not, however, match those for the moiré-derived shape so closely, and
some other method would be required to adequately parameterize the shape of the eardrum.
6. CONCLUSIONS
The phase-shift moiré technique, combined with the analysis methods outlined here, makes
it feasible to obtain detailed shape data from experimental animals and to incorporate the data in
models of individual animals. The behaviour of the model, especially the displacement of the
manubrium, is quite sensitive to errors in the position of the eardrum boundary. The application of
static pressures to the eardrum facilitates the definition of the boundary. The ear canal can, with care,
be resected to within about 0.5 mm of the eardrum without damaging the latter, but even so parts of
the boundary of the eardrum will be hidden behind overhanging ear-canal tissue. A visual
extrapolation of the displaced curves permits estimation of the position of the boundary to within less
than 0.1 mm. The simulation results suggest that this error in boundary position would not cause an
error in maximum predicted eardrum displacement of more than 15%.
A significant amount of effort is required to unwrap the z-coördinate measurements, which
are computed as phases and are subject to 2 ambiguities. Software has been written which
implements a number of automatic, semi-automatic and manual unwrapping techniques. One
technique which is being investigated to facilitate the process of unwrapping is the use of two gratings
with different line spacings. For example, a grating with 3 lines/mm could be used to supplement the
4-line/mm grating currently used. The 2 ambiguities for the two gratings would correspond to
different z values, which would help to resolve ambiguities definitively.
The results presented here quantify the observation that the shape of the eardrum, and hence
its mechanical behaviour, depends upon its displacement history. This fact implies that care must be
taken to properly precondition the eardrum in order to obtain repeatable measurements, whether of
point displacements or of volume displacements. More work will be required to determine what
preconditioning procedures are adequate for different experimental situations.
Although all four eardrums considered here are approximately the same size and depth, there
are significant differences in the detailed shapes and curvatures, and also in the displacements
predicted by the corresponding finite-element models. The differences within animals are smaller in
spite of the effects of the eardrum's stress-strain history on its resting shape.
The simulation results suggest that the experimentally observed decreases in displacement in
the neighbourhood of the boundary between the pars tensa and pars flaccida do not in fact correspond
to the position of that boundary, and are determined by the local geometry and curvatures rather than
by the structural properties of the boundary.
The method previously used to parametrically summarize the shape of the eardrum, involving
a single radius-of-curvature parameter, has been shown to be inadequate to represent the true shape
of the eardrum. Use of a variable radius-of-curvature parameter improves the representation of the
shape but further work will be required to evaluate its accuracy and usefulness.
ACKNOWLEDGMENTS
Work supported by the Medical Research Council of Canada and the University of Antwerp.
We thank J. Lauzière for editing the manuscript.
DEDICATION
This paper is dedicated to H. John Funnell, CD, 1922-1995.
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Fig. 2. Moiré image for animal CAT8, static pressure = 0, with two profiles through the moiré data and superimposed mechanically measured points. The gray levels in the moiré vary between black (for the smallest z values) and white (for the largest z values). The profile at the left is along a line perpendicular to the manubrium, in the region of the umbo (as indicated by the vertical line in the bottom box). The profile on the bottom is along a line posterior to the manubrium (as indicated by the horizontal line in the left-hand box) and roughly parallel to it. The gray line shows the moiré profiles before phase unwrapping. The interval between tick marks is 1 mm.
Fig. 4. Profiles through moiré data for animal CTM4, for pressures -220 (bottom line), 0 (middle) and 220 (top) mm H2O, along a line perpendicular to the manubrium, approximately midway between the lateral process and the umbo. The interval between tick marks is 1 mm. The circles indicate the estimated locations of the boundaries of the pars tensa and manubrium; each circle is also shown magnified, together with the curves in its neighbourhood. The sizes of the circles overestimate the possible errors of the estimates. Note that the posterior boundary of the eardrum (on the right) is beneath the overhanging ear-canal tissue.
Fig. 6. Pressure-displacement curves for a location in the posterior pars tensa of animal CTM3. The zero reference for the vertical scale is arbitrary. The arrows indicate the temporal order of the measurements. The circles () correspond to the first cycle of pressures and the crosses () correspond to the second. The gray lines are intended to suggest the forms of the hysteresis loops and do not correspond to actual measurements.
Fig. 8. Shape of finite-element model for animal CTM4. (a) Contour lines of constant z, with an interval between contours of 0.1 mm. The gray lines show the triangles of the finite-element mesh. (b) Side view of finite-element model.
Fig. 10. Shapes of finite-element models for all animals, as in Fig. 4a. There is one model each for animals CAT8 and CTM2; there are six models for CTM3 and four for CTM4, corresponding to different resting shapes. Since the models differ slightly in size, a separate 1-mm scale bar is shown for each. As in Fig. 4, the z between contour lines is 0.1 mm.
Fig. 12. Simulated displacements for the model shown in Fig. 4. The contours are lines of constant displacement amplitude, evenly spaced between 0 and the maximum displacement.
Fig. 14. Displacements for the models shown in Fig. 5, displayed as in Fig. 6. The maximum displacements are 487 nm for CAT8; 782 nm for CTM2; 528, 617, 603, 574, 619 and 614 nm, respectively, for the six CTM3 cases; and 526, 559, 591 and 620, respectively, for the four CTM4 cases.