Analyses of the Tympanic Membrane Impulse Response Measured with High-Speed Holography

Abstract

The Tympanic Membrane (TM) transforms acoustic energy to ossicular vibration. The shape and the displacement of the TM play an important role in this process. We developed a High-speed Digital Holography (HDH) system to measure the shape and transient displacements of the TM induced by acoustic clicks. The displacements were further normalized by the measured shape to derive surface normal displacements at over 100,000 points on the TM surface. Frequency and impulse response analyses were performed at each TM point, which enable us to describe 2D surface maps of four new TM mechanical parameters. From frequency domain analyses, we describe the (i) dominant frequencies of the displacement per sound pressure based on Frequency Response Function (FRF) at each surface point. From time domain analyses, we describe the (ii) rising time, (iii) exponential decay time, and the (iv) root-mean-square (rms) displacement of the TM based on Impulse Response Function (IRF) at each surface point. The resultant 2D maps show that a majority of the TM surface has a dominant frequency of around 1.5 kHz. The rising times suggest that much of the TM surface is set into motion within 50 μs of an impulsive stimulus. The maps of the exponential decay time of the IRF illustrate spatial variations in damping, the least known TM mechanical property. The damping ratios at locations with varied dominant frequencies are quantified and compared.

1. Introduction

The eardrum or tympanic membrane (TM) is a thin multi-layer cone-shaped structure that separates the outer ear from the middle ear, where TM motion initiates sound transmission to the inner ear. The sound-induced motions of the TM surface are affected by its mechanical properties, shape, thickness, microstructure, and the load from the ossicular chain and inner ear [1–15]. It has been observed that the sound-induced vibration of the TM surface is frequency dependent and the spatial patterns of the vibrations tend to change from simple to complex patterns as frequency increases [16–19]. To quantitatively investigate these motion patterns and better understand the function of the TM for hearing, various none-contact optical methodologies have been developed[20–26].

Our group has been studying the sound-induced motion of the TM using holography-based methodologies: we measured TM motions on cadaveric human temporal bones (TBs) at discrete frequencies using a stroboscopic holographic method. We further implemented a high-speed full field of view holographic shape and displacement measurement system that measured the TM 3D shape and responses under transient acoustic excitations [27, 28] that enabled the estimation of the TM surface normal displacements regardless of the orientation of the TM within the measurement system [29, 30]. The use of transients, which include many frequencies, allowed more broad-band analyses, including frequency and impulse response computations. Recently we introduced a higher power laser and a new holographic shape measurement methodology [29] into the system that reduced the need for sample surface preparation of the naturally non-transparent human TM (e.g., the application of paint to the imaged surface was no longer required). The system allows measurements (at each of more than 100,000 points on the TM surface) of the displacement produced by acoustic transients with 10 nanometer scale resolution and a temporal resolution of less than 20 μsec [30]. At those same points we can measure the TM shape (specifically the depth of the TM cone) with better than 100 μm resolution [29].

In this paper, we present measurements of the TM shape and transient responses induced by acoustic clicks from 8 normal TM samples using our high-speed holographic system in two shape-measurement configurations. The shapes of the first 5 TM samples were obtained through a multi-wavelength method using a tunable 10 mW laser that produced a sweep of wavelengths: a combination of measurements with different wavelengths allowed the generation of synthetic optical wavelengths similar to the depth of the TM cone [29–31]. The lower power of this tunable laser necessitated the use of reflective paint in order to provide enough reflected light from the TM surface for accurate holographic shape and displacement measurements [29–31]. After measurements on the fifth TM sample, we introduced a 200 mW fixed-wavelength laser that did not require paint for accurate holographic measurements on the human TM samples. At the same time we introduced a new multi-angle shape measurement method: controlled changes in the angle of illumination produced changes in the interference between the reference and reflected light beams in sationary TMs that were solved to estimate the depth of the TM at each surface point [29, 30, 32] with a resolution better than 100 μm. The displacement results are measured with a previously described high-speed holographic method [30]. The shape information is used to compensate the variations of the sensitivity vectors of the displacement results due to near-field effects as well as compute the displacement component normal to the TM surface [30]. The displacement results are further analyzed in both frequency and time domains to (a) derive the acoustic normalized frequency response function (FRF), and (b) reconstruct the impulse response (IRF) of the TM. Two-dimensional maps of the dominant frequency of the FRF function were obtained, and maps of the root mean square displacement, rising time and Exponential decay time were obtained from the IRF. The response maps are used to describe features of the mechanics of the TMs relevant to sound transduction.

2. Methods

2.1. Specimen Preparation:

Eight fresh human cadaveric temporal bones were used. The cartilaginous and much of the boney external ear canal were removed to better view the surface of the TM. The middle-ear cavity was opened through the facial recess to check the normality of the middle-ear ossicles, and then it was re-sealed using silastic ear-canal impression material (Westone Silicone Singles ®). The inner ears remained intact and fluid filled. A 1 mm diameter, 10 cm length nylon tube sealed in the eustachian tube channel equalized static middle ear pressure with ambient pressure[26]. The lateral surface of the first 5 TMs (TB1 to 5) were sprayed with a thin layer of water-soluble white paint using a modified commercially available airbrush. To ensure the paint layer was both thin and uniform, the airbrush was set to generate the finest particles while the sample was placed >30 cm away from the airbrush. The surfaces of the last 3 TMs (TB 6 to 8) were not painted because of the increase in reflected light produced by the 200 mW laser source.

2.2. Experimental Equipment and Measurements

A detailed description of the high-speed displacement and multi-wavelength shape measurement equipment has already been published [29, 30]. A schematic of the digital holographic system used in the multi-angle shape method is shown in Figure 1. The beam of a single wavelength (532nm) 200 mW laser source is divided into reference beam (blue) and object beam (green) by a tunable beam splitter. A mirror mounted on a piezoelectric actuator (PZT mounted mirror) modulates the length of the reference optical path by multiple wavelengths of the light. A motorized rotating mirror in the object path changes the illumination angle for shape measurements, and an expansion lens spreads the object beam to illuminate much, if not all, of the TM surface. An imaging lens collects the scattered wavefront from the object. The reflected object wavefront and the expanded reference wavefront are combined by the wedged beam splitter in front of the High-speed camera (PhotronSAZ). A speaker (SB Acoustics SB29RDNC-C000–4) provides the sound stimulus to the sample, and a piezo-electric microphone (Knowles FG-23329-P07) monitors the acoustic stimulus within 4 mm of the center of the TM. The single frequency laser described here produced significantly more incident power than the tunable laser (10 mW) used in previous shape and motion measurements [29, 30]. The increased optical power increased the power reflected from the human TM and permitted shape and vibration displacement measurement without applying paint to the TM surface in the 3 TBs whose shape was measured in this manner.

Figure 1.

Schematic of the high-speed digital holographic multi-angle shape and vibration measurement system. During the shape measurement, the illumination angle θ is altered by the motorized rotation mirror. The PZT (piezoelectric ceramic material) positioned mirror is used to shift the phase of the reference beam for shape and displacement measurement [33]. The middle ear sample is excited by transient click-like sounds produced by the speaker, and the resultant sound pressure near the surface of the TM is recorded by a calibrated microphone sensitive to a broad range of frequencies.

2.2.1. Holographic shape measurements: Theory and implementation

We applied the previously described multi-wavelength method [29, 30]to measure the shape of painted TMs in TB1–5. Using this method, the shape of the TM, described by its relative depth at varied (x,y), z(x,y), is related to that of a flat surface with a measured optical phase difference ∆γ_c introduced by changing the wavelength of the illumination in the following equation:

$$z\left(x,y\right)=\frac{\Lambda }{4\pi }\Delta {\gamma }_c\left(x,y\right)$$ (1)

Where Λ is the synthetic wavelength produced by the interference of the multiple wavelengths, and x,y describe the two-dimensional surface of the camera backplane.

In TB6, 7 and 8, we applied a fixed-wavelength multi-angle method [32] to measure the relative depth of the surface of the TM, using the following equation:

$$z\left(x,y\right)=\frac{d{P}_m\left(x,y,d\theta \right)}{dk\left(x,y,d\theta \right)}$$ (2)

where dP_m(x, y, dθ) is the optical phase changes due to the change of the illumination angle dθ. And dk(x, y, dθ) is the change of the sensitivity vector resulting from the dθ obtained from a flat surface sample calibration. In both depth determinationtechniques, a normalization technique defined the plane of the tympanic ring where z=0, and all estimates of z(x,y) were converted to distances from that plane [33].

The multi-angle measurement method was implemented by rapid, repeated measurements of the interference between the reference and reflected object beam while the angle of illumination was varied. The total measurement sequence took 500 ms; the timing of the sequence is shown in Figure 2. The sequence was built around a 500 ms duration 1.5° rotation (the sloped pink line in Figure 2) of the motorized rotation mirror. During the rotation, at ten evenly-spaced 3 ms long periods, the position of the PZT mounted mirror was ramped from 0 to 5 μm (the small blue slopes) while the camera collected 200 frames at 67.2 kHz (the regions of pink background). Each of the ten sets of 200 ramp-associated frames defined an optical interference pattern at a different angle of rotation [27]. These interference patterns defined the change in the optical phase at each point in the collected interferograms dP_m(x, y, dθ) and were used with Eqn 2 to define the relative depth z(x,y) at each of the surface points [32]. The shape measurement was performed 500 ms after the displacement measurements.

Figure 2.

The timing of a multi-angle shape measurement. Ten interference patterns are defined during a 500 ms duration 1.5° mirror rotation. Each pattern is the result of 200 camera frames captured within 3 ms when the PZT mounted mirror in the reference beam is moved from 0 to 5 μm. The motion of the PZT supplies the controlled shift in the optical phase necessary to define the interference pattern at each of the 10 instances [27].

2.2.2. Displacement measurements: Theory and implementation

The generation of displacement-related- interference patterns from high-speed images collected during sound stimulation has been described previously [27, 29, 30]. Figure 3 illustrates the timing of the optical reference and motion measurements with a transient sound stimulus. Each measurement consisted of 1600 frames of continuous acquisition by the high-speed (67.2 kHz) camera – a duration of 23.8 ms. During the initial 3 ms, just prior to stimulus initiation, the PZT mounted mirror moved from 0 to 2.5 μm to provide optical phase shifted images for use as references in the computation of the optical phase of the deformed TM images [29]. The displacement resolution was better than 20 nm.

Figure 3.

The timing of the transient displacement measurement. At time 0, the PZT mirror starts to move, reaching 2.5 μm at 3 ms. The high-speed camera records the alterations in the interference between the changing reference beam and reflected light from the stationary TM to define a 90° shift in the reference phase. On completion of the reference sweep, a brief acoustic click is generated by the speaker which reaches the microphone near the TM and induces vibration of the TM after 0.3 ms, while the camera continues to gather images for another 20 ms.

After the PZT motion was completed, a 50 μs square electrical pulse delivered to the speaker generated an acoustic click. The level of the pulse was chosen to result in motions that were large enough to produce a significant number of interference fringes, but small enough that the fringes were easily resolvable. About 0.3 ms after the pulse, the acoustic click propagated to the TM surface and induced a transient vibration of the TM. The TM displacement from one pixel near the center of the image is plotted in black in Figure 3. The sound-pressure stimulus measured by the microphone is plotted in blue.

2.2.3. Computation of surface normal displacements from shape and motion measurements

Previous measurements [33] have demonstrated that much of the sound-induced motion of the TM can be explained by motion vectors that are normal to the TM surface. We combined our nearly coincident measurements of TM shape and displacement to derive the surface normal displacement at each measurement point. The deformation of a point S(x, y, t) on the surface along the surface normal direction of the sample is described by:

$$S\left(x,y,t\right)=\frac{\Delta \varphi \left(x,y,t\right)}{\mathit{K}\left(x,y\right)\cdot \mathit{n}\left(x,y\right),}$$ (3)

Δϕ(x, y, t) is the optical phase change due to the deformation of the TM. Vector K is the sensitivity vector of the holographic system, which is calculated based on the geometry of the optical setup to compensate for the near field effect due to the close placement of the the sample and camera. The vector n is the unit surface normal vector of the sample which is obtained from the shape measurement [33].

2.3. Description of the motion at each measurement point

2.3.1. Frequency response function

The frequency response function (FRF) is a quantitative representation of the output spectrum of the TM’s transient motion to a broad-band acoustic stimulus like a click that can be used to characterize the broad-band dynamics of the TM. In our measurement, we measured the full-field transient displacement of the entire surface of the TM S(x, y, t), which allows us to compute the FRF at all measured (x,y) points. At each point, the FRF is calculated by the ratio of Fourier transforms of the surface-normalized displacements S(x, y, t) and the stimulus sound pressure p(t) :

$$FRF\left(x,y,f\right)=\frac{FFT\left(S\left(x,y,t\right)\right)}{FFT\left(p\left(t\right)\right)}$$ (4)

where FFT presents the Fast Fourier Transform, t is the time and f is the stimulus frequency. Figure 4 shows an example of the frequency-dependent magnitude of an FRF measured at a single point on the TM surface in TB7. The magnitude is scaled in μm/Pa. The red dot marks the Dominant Frequency, 3.4 kH, where the magnitude of the FRF is maximum.

Figure 4.

The FRF magnitude of a single pixel of the TM (TB7)

2.3.2. Impulse response function

The impulse response function (IRF) of the TM is obtained by taking the Inverse Fast Fourier Transformation (IFFT) of the FRF measured at each point on the TM as shown in equation (5)

$$IRF\left(x,y,t\right)=IFFT\left[FRF\left(x,y,f\right)\right]$$ (5)

An example of an IRF is illustrated in Figure 5. Based on the IRFs of the TM, we developed the following quantitative analyses to gain insight into TM mechanics.

Figure 5.

Reconstructed IRF(x,y,t) (in blue) of a single point at the center of the umbo in TB1. The yellow arrows at the top left of the plot describe the rising time t^R, the time to the first peak of the IRF. The orange line is the absolute value of the IRF. The peaks of the orange line (marked in orange diamonds) are fit by an exponential decay function. The green triangle and line mark the time constant of the fitted exponential decay (the exponential decay time t^D of the IRF)

2.3.2.1. Root mean square of the impulse response

The Root-mean-square of the IRF(x,y,t) at each TM location, IRMS(x,y), is derived by calculating the RMS of the first 3 ms of the IRF at each location. The 3 ms threshold is based on the observation that in most cases the TM impulse response converges to zero before 3 ms.

2.3.2.2. Rising time of the IRF

The rising time at each TM location t^R(x,y) is the time required for the IRF(x,y,t) to reach the first peak (marked by the yellow arrows in Figure 5).

2.3.2.3. Exponential decay time

The exponential decay time at each TM location t^D(x,y) was based on a fit of the exponential function [a e^bt] to a series of points in time described by the successive peaks (the yellow diamonds in Figure 5). of the first 2 ms of the absolute value of the IRF(x,y,t). The first peak in the series was the maximum in the IRF(x,y,t). The decay time is the time (t^D =−1/b) when the fitted exponential reaches 1/e (~36.8%) of the maximum displacement.

2.4. Two-dimensional maps of the motion parameters

Our basic analysis results were two-dimensional (2D) maps of the measured and computed motion parameters. Maps of the RMS value of the raw displacements at each x and y position are illustrated on the left in Figure 8. As our camera records a square of 512×512 pixels, which included the elliptical TM area, we needed to define the edges of the TM and also identify the location of the umbo and manubrium within the square of pixels. This was done with the aid of the raw RMS displacement maps. Since the bony ear canal outside the TM surface has zero motion, the edges of the TM can be defined by the sudden drop of the RMS displacement value. Also, the umbo and manubrium are known to have smaller motions compared to most of the TM surface [11,17,28,33], and the outline of the umbo and manubrium can be identified by lower RMS values. Once the edges were determined and the umbo and manubrium were identified, the various measured parameters described above were then quantified on each TM surface map with a color-coded magnitude scale.

Figure 8.

Measured displacements of TB1 and TB7. The 2D maps on the left show the RMS value of the raw displacements during the first 3 ms. Panels i.b and ii.b illustrate the microphone measurement of the stimulus, and panels i.c and ii.c illustrate the measured surface normal displacement at 5 points on the TM during the 3 ms after stimulus initiation.

2.4.1. Spatially Averaged TM response functions

To help us perform quantitative comparisons of the complex and varied 2D patterns of TM motion parameters, we also computed spatial averages of the measured parameters. To compare the total stimulus-induced motion of different TM samples, we use Eqn 6 to define the spatially averaged magnitude of the FRF at each of the spectral frequencies:

$$Spatially\hspace{0.33em}average\hspace{0.33em}magnitude\left(f\right)=mean\left[abs\left(FRF\left(x,y,f\right)\right)\right]$$ (6)

where the abs function describes the magnitude of a complex argument and mean is the 2D averaging operator. Only points determined to be on the TM surface were included in the average. We also applied 2D-averaging to different fractions of the TM area. For example, we have defined the average of parameters in different quadrants of the TM, and over the umbo and manubrium (More details are included below).

3. Results

3.1. Shape results

The 3D shape measurement results from eight fresh cadaveric human TMs are shown in Figure 6, with a depth resolution of less than 0.15 mm. The shape results are spatially filtered via 5 by 5 pixel moving average filter to remove any discontinuities. All the results are oriented such that the manubrial axis of each TM is at 12 o’clock, and the results from right ears are flipped about the vertical axis so that the anterior TM is always on the left. The TM annulus is set at z=0 mm, and the medial direction is directed towards the negative z-direction. Within each plot, the spoon-shaped umbo and the more columnar manubrium are outlined in black, the red line runs along the manubrial axis, and the blue line is positioned at the center of the umbo perpendicular to the manubrial axis. All eight TMs show similar conical shapes with a depth of the umbo of about −1.5 mm, consistent with published human TM depths [34].

Figure 6.

Shape measurement from 8 TMs. Manubrium and umbo are outlined by the solid black line. Within each TM, a red line is plotted along the manubrial axis and a blue line perpendicular to the manubrial axis at the center of the umbo. The TM shape profiles along these two lines are compared in Figure 7.

In Figure 7, the depth of the TM cone along and perpendicular to the manubrial axis are compared in the 8 TBs. (The lengths of the individual lines are limited by variations in the exposed surface of the TM.) In all 8 specimens the depth of the TM, from the highest point (often above the TM ring) to the center of the umbo, is between 1.25 and 1.75 mm. The depth profiles are generally similar in the different bones, though there are differences. For example, in the bottom plot, the depth (z) of the umbo along the anterior-posterior direction is similar in all TBs, but the distance between the umbo and the anterior rim is much smaller (2 vs 4 mm) in TB4. (This variation in TB4 is also visible in the 3D images of Figure 6). In most specimens, there is a superior-inferior asymmetry where the fraction of the superior-inferior distance (top plot) that is superior to the umbo is about 0.60. There is also anterior-posterior asymmetry, where the fraction of the anterior-posterior distance that is posterior to the manubrium umbo is about 0.55. Associated with this asymmetry is a steeper slope on the anterior side.

Figure 7.

The relative depth z(x,y) along and perpendicular to the manubrial axis of 8 measured TBs (a) The manubrial axis from superior to inferior. (b) The anterior to the posterior axis. The center of the umbo is at position 0 on both axes.

3.2. Displacement results

Examples of measured surface-normal displacement data S(x,y,t) from TB1 and TB7 are presented in Figure 8. The left panels (i.a and ii.a) show maps of the RMS of the initial 3 ms of the raw displacement measured at all points on the surface of the two TMs. Panels i.b and ii.b show the measured microphone signals over 3 ms, and panels i.c and ii.c show the displacement waveforms at 5 different points that are marked on the RMS surface maps. The maps and the point data illustrate that the amplitude of motion varies across the TM surface in a complex manner. The point data illustrate variations in the timing of the click-induced displacements and complicated variations in the waveforms of the induced displacements at different positions on the TM surface, including differences in phase, frequency content and the time needed for the displacement to return to zero. The maps and waveforms illustrate that the umbo and manubrium motions are of lower amplitude than much of the rest of the membrane. Comparison of the top (i.a and i.c) and bottom (ii.a and ii.c) panels demonstrate significant differences in the response of the TMs of TB1 and TB7. The displacement waveforms of TB7 (ii.c) show many more cycles of vibration within the first 3 ms compared to TB1 (i.c). While the measured displacements (maps and waveforms) are of similar amplitude, the microphone measurements display a significant difference in maximum stimulus amplitude (10 Pa for TB1 and 3 Pa for TB7).

3.3. Frequency and impulse response analysis

3.3.1. Spatially averaged frequency response functions

Equation 6 was used to define the spatially averaged FRF of the eight TMs, and the results are presented in Figure 9, with the median spatially averaged FRF plotted in solid black. Figure 9 suggests the FRF varies considerably (by more than an order of magnitude below 2 kHz) across the different bones. Such variation is consistent with large variations reported in LDV measured umbo displacement across subjects [35].

Figure 9.

Magnitude of the surface averaged FRF of 8 TMs. The median of the 8 measurements is plotted in black.

3.3.2. Representative results from one temporal bone

One set of complete results (the impulse responses, dominant frequencies, IRMS, t^R, t^D and the correlation coefficient of the curve fitting for the decay time calculation) of TB 1 is shown in Figure 10. The dominant frequency map (10a) suggests the umbo has a dominant frequency of 1.7 kHz. There are four areas on the surface of the TM that have dominant frequencies of less than 2 kHz. For the rest of the TM, the dominant frequencies are similar. Panel 10b maps the RMS of the IRF (relative to the RMS amplitude at the umbo P1); the umbo has the lowest amplitude (a relative amplitude of 1), and portions of the TM have RMS amplitudes that are 4 to 5 times larger. Panel 10 c is the impluse response waveform at the umbo and at the point of maximum RMS within each quadrant. Panel 10d illustrates that the rising time ( t^R ) varies between 0.1 to 0.2 ms across the TM surface with a value of 0.15 at the umbo, consistent with the rising time shown in black markers in the impulse responses plot (Panel 10 c). Panel 10e shows large variations in decay time ( t^D ) across the TM’s surface, ranging from 0.2 to 1.5 ms. The map also shows spatial clusters in the t^D value, as t^D is a parameter closely related to the mechanical properties of the membrane and the significance of these spatial variations will be discussed later. As the estimation of t^D depends on curve fitting, we also include a map of the correlation coefficient (Panel 10f) between the fit and the data demonstrating high correlations over much of the TM surface. It is worthwhile noting that some of the regions of poorer correlation (R ~ 0.6) correspond to regions where the RMS value of the IRF is low.

Figure 10.

Frequency and impulse analysis results of TB1:(a) dominat frequency map, (b) impluse RMS response map, the RMS values of the impulse response are normalized by their umbo value, and the colorbars show the ratio to the umbo value, (c) individual impluse responses of selected six points on the TM surface (marked by points in each of the maps). P1 is the umbo, and the other points are placed in each of the four quadrants of the TM. The dots on the IRFs mark the time to the first response maximum t^R. (d) rising time map, (e) decay time map, and (f) correlation coffiecient of the decay time.

3.3.3. Grouped maps and spatially averaged responses

We now focus on the individual descriptors of our displacement data in all of the 8 bones. The initial evaluation takes the form of comparisons of the 2D maps. Such comparisons are primarily qualitative. To enable more quantitative comparisons, we used spatial averaging of the data sets to reduce the spatial patterns to a series of numbers. To maintain some of the spatial information in this averaging, we averaged over five separate regions of the TM, as described in the insets of the next four figures. These regions include (R1) the region posterior to the manubrium and superior to the umbo, (R2) the region posterior to the manubrium and inferior to the umbo, (R3) inferior to the umbo and anterior to the manubrium and (R4) superior to the umbo and anterior to the manubrium. A fifth region is the area defined by the spoon shape terminal region of the manubrium that includes the umbo.

3.3.3.1. Dominant frequencies and the RMS of the impulse response

Figure 11 (a) shows the maps of dominant frequency from all eight TBs. As mentioned previously, all the right ears are flipped horizontally and rotated, so the manubrium is at 12 o’clock and anterior is to the left. While TB7 shows relatively higher dominant frequencies (~3.6 kHz on a majority of the TM surface), the rest of the TBs have large sections where the dominant frequencies range from about 500Hz to 2 kHz. The pattern of the dominant frequency distribution across the TM surface varies significantly across the bones. Figure 11 (b) shows the spatially averaged value of the 4 quadrants (area R1 to R4) and the manubrium and umbo of each bone. Many of the quadrant averaged dominant frequency values vary between 0.5 and 2 kHz and are reasonably approximated by the median value from the 8 TBs (the blue line in the plot). However, TB7 shows dominant frequencies that are significantly higher (between 3 and 3. 5 kHz). Excluding TB7, our analysis suggests the tuning of the TM is similar across the surface of individuals and between individual TMs.

Figure 11.

(a) Dominant frequency maps of TB1 to TB8. (b) the four quadrants and umbo averaged value and their median value.

Figure 12(a) shows maps of the ratio of IRMS(x,y) normalized to the umbo. In many of the specimens, we see that much of the area of the TM moves more than the umbo (relative magnitudes >1), though there are regions where the umbo normalized IRMS is less than 1 (this is particularly true in TB5 and TB8). Figure 12(b) shows the quadrant and umbo averaged values for 8 bones and their median value. These averages vary between 0.7 and 2.8. As noted above TB5 and TB8 show average IRMS values that are smaller than the umbo (except for R3 in TB5), while the average quadrant averaged IRMS in TB1 and TB6 are some of the largest with values of 2 or larger. While the medians of the quadrant averages are relatively stable (between 1.1 and 1.6), the individual quadrant averages do vary about the median (total range of 0.7 to 2.7).

Figure 12.

(a) Amplitude ratio of the RMS of first 3 ms of the calculated impulse response. The IRMS maps are normalized by the value at the umbo, and the colorbar shows the ratio to the umbo value. (b) The four quadrant averaged values and their median.

3.3.4. Rising time and decay time

Figure 13(a) shows maps of t^R, the rising time of the impulse response, and Figure 13(b) shows the quadrant and umbo spatial average values and their medians. The t^Rs in TB7 are significantly shorter than those in the other TBs, and by excluding TB7 the t^R values range between 0.12 and 0.19 ms. The median value of the rising time is between 0.14 to 0.16 ms, and the umbo has a median t^R that is 0.02 to 0.03 ms longer than the 2 anterior quadrants (R3&4). Region R2 (the posterior-inferior quadrant) in TB4, 6 and 8 have t^R values larger than that at the umbo.

Figure 13.

(a) Maps of the rising time t^R of the reconstructed impulse response of TB1 to TB8, (b) the averaged quadrants and umbo value and their median value.

Figure 14 displays maps of the decay time t^D computed from the exponential curve fitting to the peaks in the IRFs along with the quadrant and umbo averaged values. The spatially averaged quadrant and umbo values from all the results range from 0.2 ms to 1.5 ms, while the median values of each area are between 0.46 to 0.6 ms. The decay times in TB4 are significantly longer than in the other ears. In quadrant R2 TB4 and TB6 have an overlapped decay time of 0.9 ms. The median t^D at the umbo is shorter than that of the quadrants, and in general, the t^D at the umbo is shorter than that observed in at least 3 of the four quadrants. Since decay time is inversely related to the damping in a system, the shorter t^D at the umbo may be related to the direct load from the cochlea and ossicles at the umbo, and any variation in the t^D across the four quadrants may be related to spatial differences in the damping within the mechanical properties of the TM itself. Maps of the correlation between the fitted exponential and the fitted data are included in Appendix A. These maps illustrate that the exponential fits were highly significant R> 0.85 over most of the TM surface. However, there are isolated regions of the TM where the correlation value is lower. Whether the decrease in correlation is due to small motion amplitudes in those locations (as suggested from Figure 10), a need for a more complicated description (e.g. multiple exponentials) of decay time, or something else is not clear.

Figure 14.

(a) Maps of the decay time t^D of the impulse response, (b) the quadrant and umbo spatial averages and their median values.

4. Discussion

We used a newly developed high speed holographic system to quantify both the shape and acoustic click induced full-field transient responses of the human TM within a few seconds of time. We performed frequency and impulse analysis on the measurement results to describe the function of the TM for sound transmission, with several parameters that have not been described beforec, including dominant frequency, rising time, and decay time.

4.1. Effect of painting the TM on the shape and displacement measurements

Our measurements were done on 5 painted and 3 unpainted TMs. To evaluate the effect of painting the TM on its response and shape, we compare the shape measurement results and the spatially averaged FRF in the painted and unpainted samples. Figure 15(a) and (b) compare the mean and standard deviations of the depth of the TM along our two orthogonal axes. The shapes of the painted TMs are illustrated by the solid black line, and shapes of the unpainted TMs are illustrated by the dashed blue lines. The center of the umbo is set at zero along the x-axis, and the annulus of the TM is set at zero in the Y-axis. The overall shapes of the painted and unpainted TMs are similar, with a similar level of standard deviation (less than ±0.25 mm). Please note that in Figure 15(b), the mean shape of the unpainted TMs shows a sharp increase at the position of −2 mm, because of the decreased length and much different depth of TB4 along the anterior-posterior axis (Figure 9b). The comparison suggests that the thin layer of paint applied to increase the TM’s reflectivity has little effect on the measured shape.

Figure 15.

Mean (±STD) position along and perpendicular to the manubrial axis of painted specimens (TB1 to TB5) and unpainted specimens (TB6 to TB8).

Figure 16 compares the median and range of surface averaged magnitudes of FRF of painted and unpainted specimens. At frequencies less than 2 kHz, the unpainted median FRF is lower in magnitude. At frequencies between 3 and 7 kHz, the median spatially averaged FRF of the unpainted TMs is of higher magnitude. The small number of individuals in each group complicates statistical analyses. However, the less than a factor of 3 (10dB) difference of the medians in the two groups is well within the ±10 dB range of individual differences observed by [35] from LDV measurements.

Figure 16.

Median and range of the surface averaged magnitude of FRF of painted specimens (TB1 to TB5) and unpainted specimens (TB6 to TB8).

4.2. Comparison of our shape measurements with other published results

Our shape measurements have features in common with other human TM shape measurements made in small numbers of specimens using holographic techniques, including phase-shifted holography[33] and moire interferometry[34, 36]. All show depths of the TM cones that vary between 1.25 and 2.25 mm, with similar 8–9 mm diameters. All also show asymmetries in the cone shape, including larger superior fractions of the anterior-superior distance, as well as larger posterior fractions of the anterior-posterior distance. The steeper anterior slope is also in common. Similar features are observed in the shape of the TM of chinchillas [37] and gerbils [38].

4.3. Comparison of our displacement results with published umbo displacements

Figure 17 compares the mean value of 8 averaged surface FRFs and the mean value of 8 averaged umbo FRFs from our results with the mean umbo response published by Gan et al. 2004. In Gan’s results, the umbo response peaks at around 1 kHz and decreases with a slope of near −40dB / decade as frequency increases. Our results show the averaged surface and averaged umbo have similar decreasing trends, though the peak in our umbo and spatially averaged FRFs occur at about 1.5 kHz. Consistent with previous observations that significant sections of the TM move more than the umbo [39], the mean of our averaged umbo FRF is generally ~20% lower in magnitude than the mean averaged surface FRF.

Figure 17.

Comparison of averaged umbo and surface FRFs (normalized by sound pressure) with the pressure normalized umbo motion from Gan et al. 2004 [22]. Mean(±SE) magnitudes are plotted and compared.

4.4. New insights into the role of the TM in sound transmission

The maps of the dominant frequency and the quadrant analysis of Figure 11 illustrate spatial variations and inter-specimen differences in dominant frequency where the median value is about 1.25 kHz (with TB7 as an outlier). The dominant frequency is a measure of the tuning of the TM to a broad-band stimulus, and it is a complex function of the structural features that define the localized modulus and mass of the TM as well as more generalized stiffness and mass due to the material properties of the fibers within the membrane and any tension within those fibers. The spatially varying rising and decay time of the IRF for the different TBs also address TM mechanics, where t^R describes relative delay in the growth of the response along the surface, t^D is inversely related to the magnitude of damping within the TM and the coupled ossicles and cochlea.

Maps of the rising time showed distributions with somewhat regular gradients in t^R (Figure 13a) with a small number of regions of different values. Excluding TB7, the distribution of averaged t^R on the TM surface varied from 0.12 ms to 0.20 ms (Figure 13b), where the rising time at the umbo was slightly longer than at adjacent areas with a typical time difference of 15 to 25 μs. This difference is significantly smaller than the 83 μs middle-ear delays described for cochlear sound pressure measurements in human temporal bone [40]. It is, however, similar to the delay between TM and umbo motion in the much smaller gerbil TM [15]. [15] suggested that the longer rising time at the umbo can be explained by the larger mass of the umbo and manubrium compared to the localized membrane by itself, where the larger mass slows the time to maximum displacement. An alternative hypothesis is that the rising time is the result of a combination of different modal and traveling wave patterns and that the observed umbo lag is due to the superposition of multiple waves acting on the surface of the TM, some of which cancel. In such a model, the cancelation of early waves will occur less often in parts of the TM that are not centrally located.

As will be discussed more later, the decay time t^D can be related to spatial variations in the damping of the TM. Figure 14a points out multiple regions of roughly uniform t^D that are surrounded by discontinuities, as well as large variations in the t^Ds observed on the TM surfaces in our sample. Some of the uniform regions coincide to regions of relatively uniform RMS displacement (e.g. TB1, 5, and 6 in Figures 12a and 14a) and the regions of discontinuity are often marked by narrow regions where the correlation coefficient of the exponential fit that describes t^D is small, and may point to narrow regions on the TM surface where the motion is small (Figure 10 and Appendix A). The decay time of the eight specimens ranges from 0.1 ms to 2.5 ms. The longer t^Ds point to regions on the surface of the TMs that oscillate more after impulsive stimulation, while shorter t^Ds describe regions on the TM surface where oscillations damp out faster. We might expect that some of the shortest t^Ds would be associated with the umbo and manubrium where the damping of the ossicles and cochlea should reduce the motion of the TM. However, this is not always the case (e.g. TB 4 and 7 in Figure 14a).

4.5. Estimate of damping of the TM

If we assume a simple second-order mass-spring-damping system, with the measured dominant frequency as the natural frequency of the system, the damping ratio ζ can be estimated from the following equation[41]:

$$\zeta =\frac{\alpha }{{\omega }_n}$$ (7)

The exponential decay rate (α) is the reciprocal of the decay time, and the natural frequency(ω_n) can be approximated by our estimate of dominant frequency. Given those definitions ζ varies across the TM’s surface as an inverse function of both the dominant frequency and decay time. Figure 18 illustrates plots of ζ as a function of the dominant frequency (from 500 to 4000 Hz) averaged over the whole TM surface of TB1–8. The spatial distribution of the dominant frequency in each bone varies, but ζ generally decreases from 0.6~0.8 with dominant frequencies < 700Hz (observed in TB1 & 5) to 0.05~0.3 with dominant frequencies > 2.1 kHz (observed in TB3, 6, 7 &8). Note the damping ratio is derived from the TM’s surface motion, but it shows the overall system damping of the middle ear system, including the damping from the mechanical property of the TM material, the ossicles, the middle ear air space and the cochlea. As the damping ratio is one of the least understood mechanical properties of the middle ear, the data of Figure 18 and its spatial distribution of the damping ratio (results are shown in Appendix B) should be of great use to middle-ear theorists and modelers. For example, while previous publications have suggested a frequency dependence in the damping ratio[42, 43], none have demonstrated how this parameter might vary across the surface of the TM.

Figure 18.

Plots the mean and standard deviation of the frequency dependence of the damping ratio from all locations on the 8 individual specimens. Each plotted point is an average of the damping ratio at all (x,y) locations with identical dominant frequency in individual TBs, where the number of (x,y) locations that contribute to each plotted plot varies from a few thousand to tens of thousands.

5. Conclusions and future works

In this paper, we describe new procedures to perform high-speed holographic shape and transient displacement measurements that are completed in a few seconds. Shape measurements show 7 of the 8 TBs have very similarly shaped TMs (except for TB 4) with less than 0.3mm of variation in TM depth vs. position. Displacement results in the time domain (Figure 8) show we can distinguish spatial differences in the amplitude and frequency of the displacement response as well as variations in the timing of the motion in different areas. We also applied frequency and impulse response analysis to the measured motions normalized by the sound stimulus. These analyses point out significant spatial differences in the motion of different locations on the TM surfaces and illustrate general quantitative similarity across the quadrant averaged values in the different ears. The averaged dominant frequencies and magnitudes of motion are similar to previous reports. The rising and decay time of the impulse response show similarity to other measurements but also result in newly described spatial variations in the damping ratio derived from transient stimulated TM motions, as well as new estimates of the frequency dependence of this important feature of TM mechanics.

In the future, we will apply the same analysis method to TBs in which different pathological conditions are experimentally simulated, including fluid injection to the middle ear cavity, stapes fixation, and IS joint interruption. We hope to identify trends in the data associated with different pathologies that would suggest a clinical utility for our techniques. Artificial Intelligence (AI) and Data Mining can be applied to automate the analysis process. Engineering design to implement a holographic otoscope will be investigated to measure such motions in intact live human ears.

Highlights.

• Highspeed holography system measures the acoustically induced transient displacement and shape of the Tympanic membrane.

• Impulse analysis shows the most dominant frequencies around 1.5 kHz.

• The rising times of the impulse response are spatially varied and often less than 50 μs.

• The exponential decaying and the damping ratio of the TM are spatially varied.

Appendix A: correlation values for exponential fitting

Figure 19.

Correlation coefficients of the exponential fitting for TB1- TB8 colorcoded from 0 to 1.

Appendix B: Damping ratio maps

Figure 20.

Damping ratio for TB1- TB8 colorcoded from 0 to 1. TM area with correlation coefficient of the exponential fitting less than 85% are masked out.