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Errors in measurement of three-dimensional motions of the stapes using a Laser Doppler Vibrometer system
Hearing Research 270 (2010) 4e14
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Methodological paper
Errors in measurement of three-dimensional motions of the stapes using a Laser Doppler Vibrometer system Jae Hoon Sim a, *, Michael Lauxmann b,1, Michail Chatzimichalis a, 2, Christof Röösli a, 3, Albrecht Eiber b, 4, Alexander M. Huber a, 5 a b
University Hospital of Zurich, Department of Otorhinolaryngology, Head and Neck Surgery Frauenklinikstrasse 24, 8091 Zurich, Switzerland University of Stuttgart, Institute of Engineering and Computational Mechanics (ITM) Pfaffenwaldring 9, 70569 Stuttgart, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 May 2010 Received in revised form 9 August 2010 Accepted 18 August 2010 Available online 27 August 2010
Previous studies have suggested complex modes of physiological stapes motions based upon various measurements. The goal of this study was to analyze the detailed errors in measurement of the complex stapes motions using Laser Doppler Vibrometer (LDV) systems, which are highly sensitive to the stimulation intensity and the exact angulations of the stapes. Stapes motions were measured with acoustic stimuli as well as mechanical stimuli using a custom-made three-axis piezoelectric actuator, and errors in the motion components were analyzed. The ratio of error in each motion component was reduced by increasing the magnitude of the stimuli, but the improvement was limited when the motion component was small relative to other components. This problem was solved with an improved reflectivity on the measurement surface. Errors in estimating the position of the stapes also caused errors on the coordinates of the measurement points and the laser beam direction relative to the stapes footplate, thus producing errors in the 3-D motion components. This effect was small when the position error of the stapes footplate did not exceed 5 degrees. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction It has been revealed by previous works that motions of the stapes involve piston-like motions as well as other components such as rotational motions of the stapes footplate in human and mammals. However, the precise components of stapes motion patterns and the relative sizes and the significance of the non piston-like motions are still unclear. Vlaming and Feenstra (1986) measured motions of four points on the stapes footplate in human temporal bones and observed no significant differences between motions at the four points. They concluded that the motion of the stapes is piston-like. Similar observations and conclusions were also made in animal measurements by Guinan
Abbreviations: LDV, Laser Doppler Vibrometer; SLDV, Scanning Laser Doppler Vibrometer; MPE, maximum possible error; ER, error ratio; EC, ear canal; GP, guinea pig. * Corresponding author. Tel.: þ41 44 255 36 80; fax: þ41 44 255 41 64. E-mail addresses: [email protected] (J.H. Sim), [email protected]. de (M. Lauxmann). 1 Tel.: þ49 711 685 66821; fax: þ49 711 685 66400. 2 Tel.: þ41 44 255 58 05; fax: þ41 44 255 41 64. 3 Tel.: þ41 44 255 58 62; fax: þ41 44 255 41 64. 4 Tel.: þ49 711 685 66393; fax: þ49 711 685 66400. 5 Tel.: þ41 44 255 58 63; fax: þ41 44 255 41 64. 0378-5955/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.heares.2010.08.009
and Peake (1967) and Dankbaar (1970). By contrast, Békésy (1960), based on his anatomical observation on the annular ligament of the stapes, noted that the anterior portion of the footplate had larger displacement than did the posterior part in human. He also predicted that the modes of stapes motion would differ with sound pressure level in the ear canal. Kirikae (1960) measured stapes motion with a drained cochlea and observed rotational motions of the footplate as well as the piston-like motion. It is now generally accepted that the stapes motions are piston-like at low frequencies and include rotational motions at high frequencies in human (Heiland et al., 1999; Voss et al., 2000; Huber et al., 2001; Hato et al., 2003) as well as experimental animals, such as cat (Decraemer et al., 2000) and gerbil (Decraemer et al., 2007; Ravicz et al., 2008). Various techniques have been developed to measure the dynamic motions of the stapes and other middle-ear structures. Békésy (1960) used capacitive probes, and Gilad et al. (1967) applied the Mossbauer method to measure motions of the ear structures. Optical methods using imaging such as holography (Tonndorf and Khanna, 1968; Gundersen and Hogmoen, 1976; Bally, 1978), video stroboscopy (Helms, 1974; Gyo et al., 1987), and electronic speckle pattern interferometry (Lokberg et al., 1980) have also been used to measure dynamic motions as well as static motions of the middle-ear structures. Merchant et al. (1996) used
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an optical motion sensor to measure stapes motions. Recently, Laser Doppler Vibrometer (LDV) systems have been widely used due to their reliability and accuracy in measurements of middle-ear vibrations. The measurements using the LDV system have been applied to temporal bones from human (Gan et al., 2004; Stenfelt and Goode, 2005) and animal cadavers (Sueur et al., 2006; Akache et al., 2007), and intra-operative measurements of live human (Goode et al., 1993; Huber et al., 2001; Whittemore et al., 2004; Chien et al., 2009). The measurements using the LDV system have been in one dimension, and several methods of obtaining 3-D components of the stapes motions have been developed recently. They have also been applied to measurements in human (Heiland et al., 1999; Hato et al., 2003; Sim et al., 2010) and experimental animals such as cat (Decraemer et al., 2000), gerbil (Decraemer et al., 2007; Ravicz et al., 2008), and guinea pig (Huber et al., 2008). In measurements of the 3-D motion components using LDV systems, artifact in measured velocities and error in estimating the position of the stapes are present, and they are reflected in the 3-D components obtained. Signal-to-noise level in the measured velocities is presumed to be increased by increasing stimulation intensity and magnitude of the stapes motions, resulting in a reduced ratio of the error of the 3-D motion component to the corresponding 3-D motion component. However, the relation between the stimulation intensity and the error in measured velocities has not been quantitatively analyzed, and improvement of the measurement by increasing the stimulation intensity may have a limitation. In most previous measurements and resulting calculations of the 3-D motion components of the stapes (Hato et al., 2003; Eiber et al., 2007; Huber et al., 2008), the position and direction of the stapes in the measurement frame were estimated, and location of the measurement points on the stapes and the relative angular position of the laser beam with respect to the stapes were obtained based on the estimation. Efforts to obtain a more precise position of the stapes with respect to the LDV measurement frame have been made (Decraemer et al., 2007; Sim et al., 2010), but the methods require complicated and delicate setups and procedures. Furthermore, these procedures are limited under specific measurement conditions such as intra-operative and live-animal measurements. In this study, the motions of the stapes of guinea pig ears were measured at multiple points on the footplate using a scanning Laser Doppler Vibrometer (SLDV) system in response to acoustic stimuli in the ear canal as well as mechanical stimuli using a custom-made three-axis piezoelectric actuator. Effects of the stimulation intensity on errors of the resulting 3-D motion components were examined using the maximum possible error (MPE) and error ratio (ER) introduced in our previous work (Sim et al., 2010). Effects of the stapes position error on the 3-D motion components were mathematically formulated and were simulated for several different magnitudes of angular position error of the stapes footplate. 2. Material and methods 2.1. Specimen preparation and mounting Right ears from four guinea pigs were used: two for measurements with mechanical stimuli by our custom-made three-axis piezoelectric actuator; two for measurements of physiological motions with acoustical stimuli. Guinea pigs were sacrificed by the provider, and were delivered and immediately refrigerated (4 C). All measurements were performed within 2 days after the sacrifice. Surgical access to the footplate of the stapes was undertaken within 4 h and was accessed from the posterior part of the bulla without
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damaging the middle ear and inner ear structures. The facial nerve and surrounding bone were partially removed. During the surgical access, the ear was moistened periodically with a physiological saline solution. In order to place a three-axis piezoelectric stimulator on the stapes head, parts of the middle-ear structures, including the tympanic membrane and the malleus-incus complex, were removed without damaging the stapes and its annular ligament at the oval window. After surgical opening, the guinea pig head was mounted in a custom-made head-holder, alignment of the stapes footplate was estimated with a surgical microscope and was indicated by “direction-indicator” in the head-holder, and the head-holder was placed in the test rig such that the long and short axes of the footplate (x and y axes) were parallel and perpendicular to the three axes of the test rig frame. The relative position of the scanning Laser Doppler Vibrometer (SLDV) system with respect to the test rig frame was measured to get the relation between the SLDV measurement frame (XYZ coordinate system) and the footplatefixed frame (xyz coordinate system). The XYZ coordinate system of the SLDV measurement frame was set such that the laser beam was along the Z direction, and the XY plane was perpendicular to the laser beam. In the footplate-fixed frame, the long axis of the footplate was set as the x axis, the short axis of the footplate as the y axis, and the direction normal to the xy plane as the z axis footplatefixed frame. The posterior, inferior, and lateral directions were set as positive directions for the x, y, and z axes for the right ear. 2.2. Measurement with mechanical stimuli With acoustical stimuli applied to the ear canal, relative magnitudes of the 3-D components in motions of the stapes cannot be controlled. To enhance the desired 3-D motion components and observe effects of measurement errors on each of the decoupled 3-D motion components, the preparation was stimulated by our custom-made three-axis piezoelectric actuator (Fig. 1). The actuator, mounted on stacks of micromanipulators, was advanced to the stapes head, and firm contact between the needle tip of the actuator and the stapes head was obtained through microscopic control (Eiber et al., 2007; Huber et al., 2008). Next, the contact was fastened with a monofilament nylon thread of 0.03 mm diameter (9-0 Nylon, S&T AG, Switzerland), which was enclosed in the needle, looped around the stapes head, and pulled by a 0.5 N of a force meter. With the three-axis piezoelectric actuator, the excitation in the zh direction mainly was expected to generate a piston-like motion (translation in z direction at the footplate center), while the excitations in the xh and yh directions were expected to mainly generate rocking motions about the long and short axes of the footplate (rotations about x and y axes). Though small amounts of undesired motion components were always contained in the induced vibrations, the measured displacements showed a predominant motion in the desired direction of excitations (Eiber et al., 2007). Specimens GP1 and GP2 were excited by the three-axis piezoelectric actuator with harmonic stimuli at frequencies of 0.5, 1, and 2 kHz. The stapes of GP1 was excited by three different modes of stimuli, on which one motion component (either the piston-like or one of the two rocking motions) was dominant compared to the other two components. Magnitudes of the non-dominant motions were less than a third of magnitude of the dominant motion, as magnitudes measured on the stapes head (measured velocity on the stapes head in xh, yh, and zh directions). Several different magnitudes of the stimuli were applied at each mode to see the relation between the stimulation magnitude and errors on the decoupled 3-D components. GP 2 was stimulated by the same amount of excitation for the xh, yh, and zh directions, and this
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obtained by the relative positions of the stapes and the SLDV system with respect to the frame of the test rig. 2.3. Measurement with acoustical stimuli To see the effects of stimulation magnitude on errors in the 3-D components of physiological stapes motions, two guinea pig ears were acoustically stimulated with several different magnitudes of stimuli (GP3 and GP4). The stapes motions of GP3 were measured on the footplate of natural surface condition, while retro-reflective glass beads of 50-micron diameter were coated on the lateral side (side to the stapes head) of the footplate of GP4 to improve the reflectivity of the laser beam on the footplate’s surface. The loudspeaker (ER-2, Etymotic Research, USA) and the microphone probe (ER-7C, Etymotic Research, USA) were placed in an ear canal near the tympanic membrane (TM). Harmonic sound waves of 0.5, 1, 2, and 4 kHz at different levels of magnitude were generated by a function generator (HP 33120A, Hewlett Packard, USA) and an amplifier (RMS 850, QSC Audio Products, USA), and were delivered via the loudspeaker. The motions of multiple points on the lateral side of the footplate were also measured using the SLDV system. With our surgical access, the measurement points were located on the posterior part of the footplate surface. The sampling frequency and frequency resolution were set the same as those for the measurements with mechanical stimuli. The phase of the motions at the points on the footplate was obtained relative to the sound pressure in the ear canal measured by the probe microphone. 2.4. Three-dimensional components of stapes motion Assuming that a translational motion in the longitudinal direction of the stapes at the footplate center (piston-like motion, voz) and two rotational motions about the long and short axes of the footplate (rocking motions, ux and uy) are dominant components of the 3-D stapes motion (Hato et al., 2003; Huber et al., 2008; Sim et al., 2010), these three components were calculated from measured motions at multiple points on the footplate and coordinates of the measurement points.
vr ¼
Fig. 1. Stimulation of the stapes by the three-axis piezoelectric actuator. A The actuator needle attached to the stapes head from the top and B excitation of the 3-D motion components on the stapes head. For proper coupling, a fine thread was looped between the crura, and a pretension of about 0.5 N was given by a force meter.
measurement of the combined motion components was used as a reference for simulating effects of the stapes position error on the 3-D motion components. The magnitude and phase of the velocities at multiple points (about 150 points) on the posterior part of the footplate were measured using an OFV-3001 Scanning Laser Doppler Vibrometer system (Polytec GmbH, Germany). The phase was measured relative to the excitation signal to the actuator. The sampling frequency and frequency resolution for fast Fourier transform (FFT) analysis were set to 25.6 kHz and 12.5 Hz, respectively. The X and Y coordinates of the measurement points in the SLDV measurement frame were recorded by PSV V8.5 software (Polytec GmbH, Germany). These were converted into x and y coordinates in the footplate-fixed frame using the relation between the two frames, which was
1 T 1 T A A A vm ; cos qL
(1)
where vm is a vector with measured velocities, A is the matrix determined from x and y coordinates of the measurement points on the footplate, qL is an angle between the laser beam direction and the z axis, and vr is a vector with the 3-D motion components voz, ux, and uy. Details about the calculation of these three 3-D motion components from the measurements at multiple points on the footplate were described in our previous article (Sim et al., 2010). 2.5. Maximum possible error (MPE) and error ratio (ER) In our previous work (Sim et al., 2010), the maximum possible error (MPE, Eqs. (2)e(4)) and the error ratio (ER, Eqs. (5)e(7)) were introduced as an estimation of the maximum possible error of a motion component and a ratio of the MPE to the magnitude of the corresponding rigid body motion component.
jux jMPE ¼
n 1 X jðlxÞm jjem j; cos qL m ¼ 1
(2)
juy jMPE ¼
n 1 X jðlyÞm jjem j; cos qL m ¼ 1
(3)
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0
jvoz jMPE
n P jem j n B X 1 Bm ¼ 1 ¼ jðlxÞm jjem j þ jyc j B n cos qL @ m¼1
1 þ jxc j
n X m¼1
C C jðlyÞm jjek jC; A
n P
jux jMPE ¼ ðux ÞER ¼ j ux j
uy
ER
¼
ðvoz ÞER ¼
¼
juEy jMPE j uy j
m¼1
¼
jðlxÞm jjem j
jux jcos qL
n P k¼1
(4)
;
(5)
;
(6)
jðlyÞm jjem j
juy jcos qL
jvEoz jMPE jvoz j n n n P P P jem j þ njyc j jðlxÞm jjem j þ njxc j jðlyÞm jjem j m¼1
k¼1
jvoz jncos qL
m¼1
;
(7)
where em indicates measurement error measure at a measurement point, which is calculated from the difference between the original measured velocity and the reversely calculated velocity with the obtained 3-D motion components. Other details of these equations are described in our previous work (Sim et al., 2010). The MPE and ER were used in this study to see the effects of stimulation magnitude on errors of the 3-D motion components. 2.6. Errors on the 3-D motion components caused by the footplate position error The effects of angular position errors of the stapes footplate on calculation of the 3-D motion components of the stapes motion were mathematically formulated and appended. The footplate position errors generate errors on the coordinates of the measurement point (Eq. (A.7)), the angle between the laser beam and the axis normal to the footplate plane (Eq. (A.8)), and thus 3-D motion components of the stapes motions. Angular position errors of 5, 10, and 20 degrees were given in each of the x, y, and z axes, respectively, and the effects of the angular position errors were simulated for the measurements of GP2, where the same amount of excitation for the xh, yh, and zh directions was applied by the threeaxis piezoelectric actuator. 3. Results 3.1. Stimulation magnitude vs. error on the motion components Fig. 2 illustrates the magnitudes of the 3-D motion components (jvozj, juxj, and juyj) and their MPE (jvozjMPE, juxjMPE, and juyjMPE) with the mean magnitude of the measured velocities (jvmjmean) (left), and the error ratios ((voz)ER, (ux)ER, and (uy)ER) with the magnitudes of the corresponding elementary motion components (right), in the measurement with the stapes mechanically driven by the threeaxis actuator. In Fig. 2, the magnitudes of the 3-D motion components were proportional to the mean magnitude of the measured velocities at
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multiple points, and therefore, the mean magnitude of the measured velocity can be considered as a measure of stimulation intensity. The MPE in each elementary motion component was also proportional to the mean magnitude of the measured velocities, but it had an additional level-independent component that dominated the MPE at low stimulus and response levels (approximately 0.005e0.02 mm/s for translational component and 0.01e0.03 rad/s for rotational components). The ER was reduced by increasing the magnitude of the corresponding elementary motion component when the mean magnitude of the measured velocities was small. However, the ER was not reduced by increasing the magnitude of the corresponding elementary motion component when the mean magnitude of the measured velocities was large enough. Fig. 3 illustrates the effects of stimulation magnitude on errors of the 3-D motion component in the physiological stapes motions (GP3) with acoustical stimuli. While the magnitudes of the 3-D motion components and the corresponding MPEs were proportional to pressure in the ear canal, the ER showed different pattern depending on the 3-D motion components. In Fig. 3A, the ER of the translational velocity ((voz)ER) at 4 kHz was reduced by increasing sound pressure level, while the ERs at other frequencies were not reduced. These trends were also shown in the rotational motion component about the long axis of the footplate (Fig. 3B). The magnitude of the rotational component about the short axis of the footplate was small relative to the magnitudes of the other components, and the large error on the rotational motion component about the short axis of the footplate was not reduced by increasing sound pressure level (Fig. 3C). Fig. 4 also illustrates the effects of stimulation magnitude on physiological stapes motions in GP4, in which retro-reflective beads were coated on the measurement area to improve reflectivity of the laser beam. Considering the weight of the glass bead (about 0.001 mg/bead), effects of the glass beads on inertia of the stapes are presumed to be negligible. Compared to the results in GP3, the ER was reduced, especially on the rotational components. The ER on the rotational motion component about the short axis of the footplate, which was not reduced by increasing sound pressure level in GP3, was also reduced by increasing sound pressure level (Fig. 4C). 3.2. Footplate position error vs. error of the 3-D motion components Fig. 5 illustrates how the magnitude of the elementary motion components changed with the angular position errors of the footplate in the measurement with an excitation mode at 0.5, 1, and 2 kHz. With an angular position error of 5 degrees, the elementary motion components of the stapes had errors of less than 10% for all three motion components. The errors in the elementary motion components increased nonlinearly with the angular position error of the footplate. The effect of the angular position errors on the rotational motion component about the short axis of the footplate was relatively small, while other components were sensitive to the angular position errors. These results were caused by the fact that the amount of error induced in the motion components was also related to the position of the SLDV system relative to the footplate because the matrix T in Eq. (A.2) was determined by the relative position of the SLDV system with respect to the footplate. The relative position of the SLDV system in our measurement setup caused significant errors in the translational component and rotational component about the long axis of the footplate when the angular position error was present. 4. Discussion Errors in results of measurements are a priori not known, and an estimation of errors is necessary for the interpretation of
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Fig. 2. Magnitudes of the 3-D motion components and their maximum possible error (MPE) with the mean magnitude of the measured velocities (jvmjmean) (left), and the error ratios (ER) with the magnitudes of the corresponding elementary motion components (right), in the measurement with the stapes mechanically driven by the three-axis actuator (GP1). A Translational velocities, B rotational velocities about the long axis of the footplate, and C rotational velocities about the short axis of the footplate.
measurements. In this study, the maximum possible error and the error ratio were used as an estimation of error boundary in the 3-D motion components of the stapes motion. In obtaining equations for the MPE and ER, a strict condition that measurement errors at all measured points have directions to increase errors in the 3-D components was assumed (Sim et al., 2010), and therefore, it is supposed that actual errors in the 3-D components are smaller than the values from the MPE and ER.
The influence of different types of errors on the motion components depends on how the errors vary with level. Fig. 6A is a diagram of different types of error growth with stimulus level. One type of error (Type I) has a magnitude that does not vary with level. These errors are presumed to be related to performance of the measurement system such as the finite resolution of the LDV system. This type of error can explain a large portion of the measured errors at low-response levels, and the size of these errors
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Fig. 3. Effects of the stimulation magnitude on the motion component error in physiological stapes motions with acoustical stimuli (GP3). A translational velocity at the footplate center, B rotational velocity about the long axis, and C rotational velocity about the short axis of the footplate. Velocity components and the maximum possible error (MPE, left) and the corresponding error ratios (ER, right) were calculated from measurements with the magnitude of the ear-canal pressure varied.
should be reduced by increasing the reflectivity of the laser beam on the measurement targets. Errors growing with response levels also exist. These errors are expected to increase proportionally with input levels when the measured motions occur within the linear range of the ear and measurement system (Type II). Errors in estimating measurement position and angular position of the
measured object can be sources of this type of error. When the measured motions exceed the linear range of the middle ear and/or motions begin to saturate, the errors can grow nonlinearly with stimulus level (Type III). Assuming for simplicity that response level increases proportionally with input level, the ratios of the errors with respect to
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Fig. 4. Effects of the stimulation magnitude on the motion component error in physiological stapes motions with acoustical stimuli with measured surface coated with retroreflective beads of 50-micron diameter (GP4). A translational velocity at the footplate center, B rotational velocity about the long axis, and C rotational velocity about the short axis of the footplate. Velocity components and the maximum possible error (MPE, left) and the corresponding error ratios (ER, right) were calculated from measurements with the magnitude of the ear-canal pressure varied.
the response level are shown schematically in Fig. 6B. Dealing with a Type I error, an excitation with low input levels leads to a high portion of this type of error. Type I resolution errors can be decreased by increasing the reflectivity of the target, thereby
shifting the error ratio (Fig. 6B) down and to the left. In case of a Type II error, the error ratio is constant and independent of the input level. The absolute value of the error ratio is determined by the slope of the corresponding error curve, as illustrated in
Fig. 5. Change (left) and percentage error (right) in magnitude of the 3-D components of the stapes motions due to the footplate position errors (GP2). A Translational velocity at the footplate center, B rotational velocity around the long axis, and C rotational velocity around the short axis of the footplate. The stapes motions were measured with the same magnitudes of excitation in xh, yh, and zh directions, delivered by the three-axis piezoelectric actuator at 0.5, 1, and 2 kHz. Angles of 5, 10, and 20 degrees with respect to each of z, y, and x axes from the original footplate position were given as the footplate position errors.
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the stapes motion at each of measurement frequencies was not changed with the excitation magnitude. The MPE in each 3-D motion component was also proportional to the mean magnitude of the measured velocities, but it had an additional level-independent value (k2 in Eq. (9)), depending on frequency.
jve jMPE zk2 þ k3 jvm jmean ;
(9)
where jve jMPE indicates the MPE of the elementary motion component (Eqs. (2)e(4)), and k2 and k3 , as functions of frequency, are determined by measurement conditions such as the surface condition at the measurement points, laser beam focusing, and performance of the SLDV system. k2 and k3 correspond to Type I and Type II errors in Fig. 6. From Eqs. (8) and (9),
ðve ÞER ¼
jve jMPE k2 þ k3 jvm jmean ; z k1 jvm jmean jve j
(10)
where ðve ÞER indicates the ER of the 3-D motion component. For the small mean magnitude of velocities, Eq. (10) is dominated by Type I errors, which are large at low-response levels.
ðve ÞER z
k2 ; k1 jvm jmean
(11)
while for the large mean magnitude of velocities, Eq. (10) is dominated by Type II errors and the error ratio is independent of response level.
ðve ÞER z
Fig. 6. Classification of the measurement error by their dependence on input level. A Types of errors with input levels and B ratios of the errors to response level.
Fig. 6A. A nonlinear error (Type III) is most likely present with higheintensity excitation, and in combination with Type I and Type II errors, makes a minimum point in the curve of the error ratio. The input level around which this minimum occurs can be considered as an ideal input level. The magnitudes of the 3-D motion components at each measurement frequency were proportional to the mean magnitude of the measured velocities at multiple points in the measurements with both acoustical and mechanical stimulations. From this observation, the following equation can be derived.
jve jzk1 jvm jmean ;
(8)
where jve j and jvm jmean indicate magnitude of a 3-D motion component and the mean magnitude of the measured velocities. The term k1 indicates the ratio of the magnitude of the 3-D motion component to the mean magnitude of the measured velocities. When the stapes moves only with piston-like pattern, jvm jmean are generated only by the translation motion voz, such that k1 for voz is 1, and k1 for the two rotational components ux and uy is zero. In our measurement with acoustical excitation in the considered frequencies, motions at the edge of the stapes footplate by the rotational components were about 5e15% of the translational component. With mechanical excitation by the three-axis piezoelectric actuator, this ratio could be controlled by changing ratios between amplitudes of excitations in the xh, yh, and zh directions. In acoustically driven motions, the ratios between the different response components varied with frequency. That is, the mode of
k3 : k1
(12)
From Eq. (11), the ER is reduced by increasing the mean magnitude of the measured velocities when the mean magnitude of the measured velocities is small. However, when the mean magnitude of the measured velocities is large enough, the improvement is limited by k3 =k1 in Eq. (12). This limit value k3 =k1 depends on the relative size of the magnitude of each 3-D motion component with respect to the mean magnitude of the 3 components. When the relative magnitude of the particular component is small k1 is small, and the ratio k3 =k1 is large. Because the mean magnitude of the measured velocities (jvm jmean ) is proportional to magnitudes of the stimuli, it can be concluded that the ratio of the error on the 3-D component to the magnitudes of the corresponding elementary component can be reduced by increasing the magnitudes of the stimuli, but it is limited by the relative size of the magnitude of the 3-D motion component with respect to the mean magnitude of the measured velocities. If this limitation value is large, then reliable measurements cannot be obtained only by increasing the magnitude of stimulation. The ER of the rotational component about the short axis of the footplate ((uy)ER) in the GP3 measurement was not sufficiently decreased by increasing the magnitude of stimulation. To reduce this limitation value, other measurement conditions such as reflectivity of the measurement surface should be improved. In the GP4 measurement, the reflectivity of the measurement surface was improved by coating retro-reflective beads on the surface, and reliable results for the rotational component about the short axis of the footplate were obtained by increasing the magnitude of stimulation. It was also shown in Section 3.2 that the footplate position error caused errors on the coordinates of the measurement points and the laser beam direction, and thus errors on the 3-D motion components (Fig. 5). The errors on the 3-D components were controlled below 10% of the corresponding motion components when the error in orientation of the footplate was less than 5 degrees, and were increased nonlinearly with the angular position
J.H. Sim et al. / Hearing Research 270 (2010) 4e14
error of the footplate. The accuracy of less than 5 degrees in estimating the angular position of the footplate is difficult to obtain using a rough estimation with a microscopic view, and more accurate ways are needed to get precise values of the 3-D motion components. In this study, the effects of stimulation magnitude and angular position error of the stapes on the resultant 3-D motion components were examined. Considering Eqs. (2)e(7), it is presumed that measurement area and the laser beam direction with respect to the stapes footplate also affect the accuracy of the resultant 3-D motion components. A wider range of measurement area closer to the footplate center and a laser beam direction more perpendicular to the footplate are desirable for increasing accuracy, by making jðlxÞm j and jðlyÞm j small and cos qL approximate 1 (Please refer to Sim et al., 2010 for details of jðlxÞm j, jðlyÞm j, and cos qL). However, with small dimensions of guinea pig ears, such changes could not be made while maintaining the eardrum, the ossicular chain, and the inner ear structure intact. The laser beam of the SLDV system was from the posterior-superior direction with an angle of about 45 degrees between the laser beam and the footplate plane in our measurement setup.
5. Conclusion In this study, the maximum possible error was used as a reference for error boundaries of the elementary motion components of the stapes, and effects of measurement errors on the 3-D motion components of the stapes were analyzed. The rocking motion components contain a large error when their relative amplitudes, with respect to the piston-like motion, are small. Ratios of these errors to the motion components can be decreased by increasing the ear-canal pressure up to a certain level for the physiological motions, but the improvement is limited by their relative ratio to the other stapes motion components. To overcome the small magnitudes of the motion components and get valid results, the measurement conditions, such as reflectivity of the laser beam on the measured surface and performance of the SLDV system should be improved. In measuring and calculating complex stapes motions, errors in estimating the footplate position affect the results. The effect of the position errors on the 3-D motion components is small when the magnitude of the error is small, but it increases nonlinearly with the amount of error.
Appendix. Errors in coordinates of measurement points and angular position of the laser beam caused by the footplate position error If we define the xyz coordinate system as a coordinate system of the true position of the footplate and thex0 y0 z0 coordinate system as a coordinate system of the estimated position of the footplate (Fig. A1), then there exists a rotational matrix that relates the two coordinate systems
13
Fig. A1. Estimated footplate plane position and actual footplate plane position. x, y, and z indicate coordinates of a measurement point in the true footplate-fixed frame, and x0 , y0 , and z0 indicate coordinates of the point in the estimated footplate-fixed frame.
8 09 8 9
(A.1)
The matrix R has three rotation parameters and RL1¼ RT as a property of the rotational matrix. If we let T be a matrix that relates the coordinates XYZ of the SLDV system to the estimated footplate coordinatesx0 y0 z0 , it is
8 09 8 9
(A.2)
Given that a measurement point m is on xy plane of the true footplate-fixed frame (zm ¼ 0), the correct coordinates of this point in the estimated footplate-fixed frame are according to Eq. (A.1),
8 0 9 8 9 < xm = < R11 xm þ R12 ym = y0 R x þ R22 ym ; ¼ : 0m ; : 21 m ; zm R31 xm þ R32 ym
(A.3)
where Rij indicates an element of the matrix R in the ith row and jth column. To present x0 y0 z0 coordinates of the measurement point in relation to XYZ coordinates of the SLDV system, z0m from Eq. (A.3) is substituted into Eq. (A.2) to obtain Zm coordinate value which cannot be measured from the SLDV system. It follows where Tij also indicates an element of the matrix T in the ith row and jth column. From Eqs. (A.3) and (A.4), z0 m in the estimated footplate-fixed frame is not zero due to the difference between the true and estimated positions of the footplate. However, in
9 8 > > T T > > 13 13 > 8 0 9 ðT31 Xm þ T32 Ym Þ þ ðR31 xm þ R32 ym Þ > T11 Xm þ T12 Ym > > > > > > T33 T33 < xm = = < y0m ¼ ; T T : 0 ; > T Xm þ T Ym 23 ðT Xm þ T Ym Þ þ 23 ðR xm þ R ym Þ > > > 22 31 32 31 32 zm > > > > 21 T T > > 33 33 > > ; : R31 xm þ R32 ym
(A.4)
14
J.H. Sim et al. / Hearing Research 270 (2010) 4e14
converting XYZ coordinate values of the measurement point m into the x0 y0 z0 coordinate values, the incorrect assumption of z0m ¼ 0 is induced and then incorrect x0 y0 z0 coordinate values ðx0m ; y0m ; z0m ÞIC of the measurement point m are obtained as follows
8 0 9 < xm = y0 ¼ : 0m ; zm IC
9 8 T > T X þ T12 Ym 13 ðT31 Xm þ T32 Ym Þ > > > > > 11 m T33 = < : T > T X þ T22 Ym 23 ðT31 Xm þ T32 Ym Þ > > > > > ; : 21 m T33 0
(A.5)
By comparison of Eqs. (A.3)e(A.5), the incorrect x0 y0 z0 coordinate values of the measurement point m can be expressed with respect to the actual footplate frame coordinates (xm, ym)
8 0 9 < xm = y0 ¼ : 0m ; zm IC
9 8 T > R11 xm þ R12 ym 13 ðR31 xm þ R32 ym Þ > > > > > T33 = < : T 23 > > > > > R21 xm þ R22 ym T ðR31 xm þ R32 ym Þ > ; : 33 0
(A.6)
Finally, coordinate errors ðxm ; ym ; zm ÞE of measurement point m due to the footplate position error and incorrect assumption of z0m ¼ 0 in the estimated footplate plane can be calculated as follows
8 0 9 8 9 8 9 < xm = < xm = < xm = ym ¼ y0m ym : 0 ; : ; : ; zm E zm IC zm 9 8 T > R11 xm þ R12 ym 13 ðR31 xm þ R32 ym Þ xm > > > > > T33 = < : ¼ T 23 > R21 xm þ R22 ym ðR31 xm þ R32 ym Þ ym > > > > > ; : T33 0
(A.7)
An error is also induced to the calculation of the angle between the laser beam and z-axis as
0 ðcos qL ÞE ¼ cos qL IC cos qL ¼ T33 ðT13 R13 þ T23 R23 þ R33 T33 Þ;
(A.8) 0
where ðcos qL ÞE indicates the error on cos qL and ðcos qL ÞIC indicates the incorrect value of cos qL resulted from the incorrect estimation of the footplate position.
References Akache, F., Funnell, W.R.J., Daniel, S.J., 2007. An experimental study of tympanic membrane and manubrium vibrations in rats. Audiol. Neurootol. 12, 49e58. Bally, G., 1978. Holographic vibration analysis of the tympanic membrane. Laryngol. Rhinol. Otol. 57 (5), 444e451. Békésy, G., 1960. Experiments in Hearing. McGraw-Hill, New York.
Chien, W., Rosowski, J.J., Ravicz, M.E., Rauch, S.D., Smullen, J., Merchant, S.N., 2009. Measurements of stapes velocity in live human ears. Hear. Res. 249, 54e61. Dankbaar, W.A., 1970. The pattern of stapedial vibration. J. Acoust. Soc. Am. 48 (4B), 1021e1022. Decraemer, W.F., Khanna, S.M., Funnell, W.R.J., 2000. Measurement and modeling of the three-dimensional vibration of the stapes in cat. In: Wada, H., Takasaka, K., Ikeda, K., Phyama, K., Koike, T. (Eds.), Proceeding of the Symposium on Recent Developments in Auditory Mechanics. World Scientific, pp. 36e43. Decraemer, W.F., de La Rochefoucauld, O., Dong, W., Khanna, S.M., Dirckx, J.J., Olson, E.S., 2007. Scala vestibuli pressure and three-dimensional stapes velocity measured in direct succession in gerbil. J. Acoust. Soc. Am. 121 (5), 2774e2791. Eiber, A., Breuninger, C., Sequeira, D., Huber, A., 2007. Mechanical excitation of complex stapes motion in guinea pigs. Zurich, Switzerland. In: Huber, A., Eiber, A. (Eds.), Middle Ear Mechanics in Research and Otology. World Scientific Press, Singapore, pp. 123e129. Gan, R.Z., Wood, M.W., Dormer, K.J., 2004. Human middle ear transfer function measured by double laser interferometry system. Otol. Neurotol. 25 (4), 423e435. Gilad, P., Shtrikman, S., Hillman, P., Rubinstein, M., Eviatar, A., 1967. Application of the Mossbauer method to ear vibrations. J. Acoust. Soc. Am. 41 (5), 1232e1236. Guinan, J.J., Peake, W.T., 1967. Middle-ear characteristics of anesthetized cats. J. Acoust. Soc. Am. 41 (5), 1237e1261. Goode, R.L., Ball, G., Nishihara, S., 1993. Measurement of umbo vibration in human subjects-method and possible clinical applications. Am. J. Otol. 14 (3), 247e251. Gundersen, T., Hogmoen, K., 1976. Holographic vibration analysis of the ossicular chain. Acta Otolaryngol. 82 (1), 16e25. Gyo, K., Aritomo, H., Goode, R.L., 1987. Measurement of the ossicular vibration ratio in human temporal bones by use of a video measuring system. Acta Otolaryngol. 103, 87e95. Hato, N., Stenfelt, S., Goode, R.L., 2003. Three-dimensional stapes footplate motion in human temporal bones. Audiol. Neurootol. 8, 140e152. Heiland, K.E., Goode, R.L., Asai, M., Huber, A.M., 1999. A human temporal bone study of stapes footplate movement. Am. J. Otol. 20, 81e86. Helms, J., 1974. Mechanics of the middle ear. Acta Otorhinolaryngol. Belg. 28, 581e585. Huber, A.M., Schwab, C., Linder, T., Stoeckli, S.J., Ferrazzini, M., Dillier, N., Fisch, U., 2001. Evaluation of eardrum laser Doppler interferometry as a diagnostic tool. Laryngoscope 111, 501e507. Huber, A.M., Sequeira, D., Breuninger, C., Eiber, A., 2008. The effect of complex stapes motion on response of the cochlea. Otol. Neurotol. 29 (8), 1187e1192. Kirikae, I., 1960. The Structure and Function of the Middle Ear. University of Tokyo Press, Tokyo. Lokberg, O.J., Hogmoen, K., Gundersen, T., 1980. Vibration measurement of the human tympanic membrane-in vivo. Acta Otolaryngol. 89, 37e42. Merchant, S.N., Ravicz, M.E., Rosowski, J.J., 1996. Acoustic input impedance of the stapes and cochlea in human temporal bones. Hear. Res. 97, 30e45. Ravicz, M.E., Cooper, N.P., Rosowski, J.J., July 2008. Gerbil middle-ear sound transmission from 100 Hz to 60 kHz. J. Acoust. Soc. Am. 124, 363e380. Sim, J.H., Chatzimichalis, M., Lauxmann, M., Röösli, C., Eiber, A., Huber, A.M., 2010. Complex stapes motions in human ears. J. Asso. Res. Otolaryngol. 11 (3), 329e341. Stenfelt, S., Goode, R.L., 2005. Transmission properties of bone conducted sound: measurements in cadaver heads. J. Acoust. Soc. Am. 118 (4), 2373e2391. Sueur, J., Windmill, J.F.C., Robert, D., 2006. Tuning the drum: the mechanical basis for frequency discrimination in a Mediterranean cicada. J. Exp. Biol. 209, 4115e4128. Tonndorf, J., Khanna, S.M., 1968. Submicroscopic displacement amplitudes of the tympanic membrane (cat) measured by a laser interferometer. J. Acoust. Soc. Am. 44 (6), 5e29. Vlaming, M.S., Feenstra, L., 1986. Studies of the mechanics of the normal human middle ear. Clin. Otolaryngol. 11, 353e363. Voss, S.E., Rosowski, J.J., Merchant, S.N., Peake, W.T., 2000. Acoustic responses of the human middle ear. Hear. Res. 150, 43e69. Whittemore, K.R., Merchant, S.N., Poon, B.B., Rosowski, J.J., 2004. A normative study of tympanic membrane motion in humans using a Laser Doppler Vibrometer (LDV). Hear. Res. 187, 85e104.
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Methodological paper
Errors in measurement of three-dimensional motions of the stapes using a Laser Doppler Vibrometer system Jae Hoon Sim a, *, Michael Lauxmann b,1, Michail Chatzimichalis a, 2, Christof Röösli a, 3, Albrecht Eiber b, 4, Alexander M. Huber a, 5 a b
University Hospital of Zurich, Department of Otorhinolaryngology, Head and Neck Surgery Frauenklinikstrasse 24, 8091 Zurich, Switzerland University of Stuttgart, Institute of Engineering and Computational Mechanics (ITM) Pfaffenwaldring 9, 70569 Stuttgart, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 May 2010 Received in revised form 9 August 2010 Accepted 18 August 2010 Available online 27 August 2010
Previous studies have suggested complex modes of physiological stapes motions based upon various measurements. The goal of this study was to analyze the detailed errors in measurement of the complex stapes motions using Laser Doppler Vibrometer (LDV) systems, which are highly sensitive to the stimulation intensity and the exact angulations of the stapes. Stapes motions were measured with acoustic stimuli as well as mechanical stimuli using a custom-made three-axis piezoelectric actuator, and errors in the motion components were analyzed. The ratio of error in each motion component was reduced by increasing the magnitude of the stimuli, but the improvement was limited when the motion component was small relative to other components. This problem was solved with an improved reflectivity on the measurement surface. Errors in estimating the position of the stapes also caused errors on the coordinates of the measurement points and the laser beam direction relative to the stapes footplate, thus producing errors in the 3-D motion components. This effect was small when the position error of the stapes footplate did not exceed 5 degrees. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction It has been revealed by previous works that motions of the stapes involve piston-like motions as well as other components such as rotational motions of the stapes footplate in human and mammals. However, the precise components of stapes motion patterns and the relative sizes and the significance of the non piston-like motions are still unclear. Vlaming and Feenstra (1986) measured motions of four points on the stapes footplate in human temporal bones and observed no significant differences between motions at the four points. They concluded that the motion of the stapes is piston-like. Similar observations and conclusions were also made in animal measurements by Guinan
Abbreviations: LDV, Laser Doppler Vibrometer; SLDV, Scanning Laser Doppler Vibrometer; MPE, maximum possible error; ER, error ratio; EC, ear canal; GP, guinea pig. * Corresponding author. Tel.: þ41 44 255 36 80; fax: þ41 44 255 41 64. E-mail addresses: [email protected] (J.H. Sim), [email protected]. de (M. Lauxmann). 1 Tel.: þ49 711 685 66821; fax: þ49 711 685 66400. 2 Tel.: þ41 44 255 58 05; fax: þ41 44 255 41 64. 3 Tel.: þ41 44 255 58 62; fax: þ41 44 255 41 64. 4 Tel.: þ49 711 685 66393; fax: þ49 711 685 66400. 5 Tel.: þ41 44 255 58 63; fax: þ41 44 255 41 64. 0378-5955/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.heares.2010.08.009
and Peake (1967) and Dankbaar (1970). By contrast, Békésy (1960), based on his anatomical observation on the annular ligament of the stapes, noted that the anterior portion of the footplate had larger displacement than did the posterior part in human. He also predicted that the modes of stapes motion would differ with sound pressure level in the ear canal. Kirikae (1960) measured stapes motion with a drained cochlea and observed rotational motions of the footplate as well as the piston-like motion. It is now generally accepted that the stapes motions are piston-like at low frequencies and include rotational motions at high frequencies in human (Heiland et al., 1999; Voss et al., 2000; Huber et al., 2001; Hato et al., 2003) as well as experimental animals, such as cat (Decraemer et al., 2000) and gerbil (Decraemer et al., 2007; Ravicz et al., 2008). Various techniques have been developed to measure the dynamic motions of the stapes and other middle-ear structures. Békésy (1960) used capacitive probes, and Gilad et al. (1967) applied the Mossbauer method to measure motions of the ear structures. Optical methods using imaging such as holography (Tonndorf and Khanna, 1968; Gundersen and Hogmoen, 1976; Bally, 1978), video stroboscopy (Helms, 1974; Gyo et al., 1987), and electronic speckle pattern interferometry (Lokberg et al., 1980) have also been used to measure dynamic motions as well as static motions of the middle-ear structures. Merchant et al. (1996) used
J.H. Sim et al. / Hearing Research 270 (2010) 4e14
an optical motion sensor to measure stapes motions. Recently, Laser Doppler Vibrometer (LDV) systems have been widely used due to their reliability and accuracy in measurements of middle-ear vibrations. The measurements using the LDV system have been applied to temporal bones from human (Gan et al., 2004; Stenfelt and Goode, 2005) and animal cadavers (Sueur et al., 2006; Akache et al., 2007), and intra-operative measurements of live human (Goode et al., 1993; Huber et al., 2001; Whittemore et al., 2004; Chien et al., 2009). The measurements using the LDV system have been in one dimension, and several methods of obtaining 3-D components of the stapes motions have been developed recently. They have also been applied to measurements in human (Heiland et al., 1999; Hato et al., 2003; Sim et al., 2010) and experimental animals such as cat (Decraemer et al., 2000), gerbil (Decraemer et al., 2007; Ravicz et al., 2008), and guinea pig (Huber et al., 2008). In measurements of the 3-D motion components using LDV systems, artifact in measured velocities and error in estimating the position of the stapes are present, and they are reflected in the 3-D components obtained. Signal-to-noise level in the measured velocities is presumed to be increased by increasing stimulation intensity and magnitude of the stapes motions, resulting in a reduced ratio of the error of the 3-D motion component to the corresponding 3-D motion component. However, the relation between the stimulation intensity and the error in measured velocities has not been quantitatively analyzed, and improvement of the measurement by increasing the stimulation intensity may have a limitation. In most previous measurements and resulting calculations of the 3-D motion components of the stapes (Hato et al., 2003; Eiber et al., 2007; Huber et al., 2008), the position and direction of the stapes in the measurement frame were estimated, and location of the measurement points on the stapes and the relative angular position of the laser beam with respect to the stapes were obtained based on the estimation. Efforts to obtain a more precise position of the stapes with respect to the LDV measurement frame have been made (Decraemer et al., 2007; Sim et al., 2010), but the methods require complicated and delicate setups and procedures. Furthermore, these procedures are limited under specific measurement conditions such as intra-operative and live-animal measurements. In this study, the motions of the stapes of guinea pig ears were measured at multiple points on the footplate using a scanning Laser Doppler Vibrometer (SLDV) system in response to acoustic stimuli in the ear canal as well as mechanical stimuli using a custom-made three-axis piezoelectric actuator. Effects of the stimulation intensity on errors of the resulting 3-D motion components were examined using the maximum possible error (MPE) and error ratio (ER) introduced in our previous work (Sim et al., 2010). Effects of the stapes position error on the 3-D motion components were mathematically formulated and were simulated for several different magnitudes of angular position error of the stapes footplate. 2. Material and methods 2.1. Specimen preparation and mounting Right ears from four guinea pigs were used: two for measurements with mechanical stimuli by our custom-made three-axis piezoelectric actuator; two for measurements of physiological motions with acoustical stimuli. Guinea pigs were sacrificed by the provider, and were delivered and immediately refrigerated (4 C). All measurements were performed within 2 days after the sacrifice. Surgical access to the footplate of the stapes was undertaken within 4 h and was accessed from the posterior part of the bulla without
5
damaging the middle ear and inner ear structures. The facial nerve and surrounding bone were partially removed. During the surgical access, the ear was moistened periodically with a physiological saline solution. In order to place a three-axis piezoelectric stimulator on the stapes head, parts of the middle-ear structures, including the tympanic membrane and the malleus-incus complex, were removed without damaging the stapes and its annular ligament at the oval window. After surgical opening, the guinea pig head was mounted in a custom-made head-holder, alignment of the stapes footplate was estimated with a surgical microscope and was indicated by “direction-indicator” in the head-holder, and the head-holder was placed in the test rig such that the long and short axes of the footplate (x and y axes) were parallel and perpendicular to the three axes of the test rig frame. The relative position of the scanning Laser Doppler Vibrometer (SLDV) system with respect to the test rig frame was measured to get the relation between the SLDV measurement frame (XYZ coordinate system) and the footplatefixed frame (xyz coordinate system). The XYZ coordinate system of the SLDV measurement frame was set such that the laser beam was along the Z direction, and the XY plane was perpendicular to the laser beam. In the footplate-fixed frame, the long axis of the footplate was set as the x axis, the short axis of the footplate as the y axis, and the direction normal to the xy plane as the z axis footplatefixed frame. The posterior, inferior, and lateral directions were set as positive directions for the x, y, and z axes for the right ear. 2.2. Measurement with mechanical stimuli With acoustical stimuli applied to the ear canal, relative magnitudes of the 3-D components in motions of the stapes cannot be controlled. To enhance the desired 3-D motion components and observe effects of measurement errors on each of the decoupled 3-D motion components, the preparation was stimulated by our custom-made three-axis piezoelectric actuator (Fig. 1). The actuator, mounted on stacks of micromanipulators, was advanced to the stapes head, and firm contact between the needle tip of the actuator and the stapes head was obtained through microscopic control (Eiber et al., 2007; Huber et al., 2008). Next, the contact was fastened with a monofilament nylon thread of 0.03 mm diameter (9-0 Nylon, S&T AG, Switzerland), which was enclosed in the needle, looped around the stapes head, and pulled by a 0.5 N of a force meter. With the three-axis piezoelectric actuator, the excitation in the zh direction mainly was expected to generate a piston-like motion (translation in z direction at the footplate center), while the excitations in the xh and yh directions were expected to mainly generate rocking motions about the long and short axes of the footplate (rotations about x and y axes). Though small amounts of undesired motion components were always contained in the induced vibrations, the measured displacements showed a predominant motion in the desired direction of excitations (Eiber et al., 2007). Specimens GP1 and GP2 were excited by the three-axis piezoelectric actuator with harmonic stimuli at frequencies of 0.5, 1, and 2 kHz. The stapes of GP1 was excited by three different modes of stimuli, on which one motion component (either the piston-like or one of the two rocking motions) was dominant compared to the other two components. Magnitudes of the non-dominant motions were less than a third of magnitude of the dominant motion, as magnitudes measured on the stapes head (measured velocity on the stapes head in xh, yh, and zh directions). Several different magnitudes of the stimuli were applied at each mode to see the relation between the stimulation magnitude and errors on the decoupled 3-D components. GP 2 was stimulated by the same amount of excitation for the xh, yh, and zh directions, and this
6
J.H. Sim et al. / Hearing Research 270 (2010) 4e14
obtained by the relative positions of the stapes and the SLDV system with respect to the frame of the test rig. 2.3. Measurement with acoustical stimuli To see the effects of stimulation magnitude on errors in the 3-D components of physiological stapes motions, two guinea pig ears were acoustically stimulated with several different magnitudes of stimuli (GP3 and GP4). The stapes motions of GP3 were measured on the footplate of natural surface condition, while retro-reflective glass beads of 50-micron diameter were coated on the lateral side (side to the stapes head) of the footplate of GP4 to improve the reflectivity of the laser beam on the footplate’s surface. The loudspeaker (ER-2, Etymotic Research, USA) and the microphone probe (ER-7C, Etymotic Research, USA) were placed in an ear canal near the tympanic membrane (TM). Harmonic sound waves of 0.5, 1, 2, and 4 kHz at different levels of magnitude were generated by a function generator (HP 33120A, Hewlett Packard, USA) and an amplifier (RMS 850, QSC Audio Products, USA), and were delivered via the loudspeaker. The motions of multiple points on the lateral side of the footplate were also measured using the SLDV system. With our surgical access, the measurement points were located on the posterior part of the footplate surface. The sampling frequency and frequency resolution were set the same as those for the measurements with mechanical stimuli. The phase of the motions at the points on the footplate was obtained relative to the sound pressure in the ear canal measured by the probe microphone. 2.4. Three-dimensional components of stapes motion Assuming that a translational motion in the longitudinal direction of the stapes at the footplate center (piston-like motion, voz) and two rotational motions about the long and short axes of the footplate (rocking motions, ux and uy) are dominant components of the 3-D stapes motion (Hato et al., 2003; Huber et al., 2008; Sim et al., 2010), these three components were calculated from measured motions at multiple points on the footplate and coordinates of the measurement points.
vr ¼
Fig. 1. Stimulation of the stapes by the three-axis piezoelectric actuator. A The actuator needle attached to the stapes head from the top and B excitation of the 3-D motion components on the stapes head. For proper coupling, a fine thread was looped between the crura, and a pretension of about 0.5 N was given by a force meter.
measurement of the combined motion components was used as a reference for simulating effects of the stapes position error on the 3-D motion components. The magnitude and phase of the velocities at multiple points (about 150 points) on the posterior part of the footplate were measured using an OFV-3001 Scanning Laser Doppler Vibrometer system (Polytec GmbH, Germany). The phase was measured relative to the excitation signal to the actuator. The sampling frequency and frequency resolution for fast Fourier transform (FFT) analysis were set to 25.6 kHz and 12.5 Hz, respectively. The X and Y coordinates of the measurement points in the SLDV measurement frame were recorded by PSV V8.5 software (Polytec GmbH, Germany). These were converted into x and y coordinates in the footplate-fixed frame using the relation between the two frames, which was
1 T 1 T A A A vm ; cos qL
(1)
where vm is a vector with measured velocities, A is the matrix determined from x and y coordinates of the measurement points on the footplate, qL is an angle between the laser beam direction and the z axis, and vr is a vector with the 3-D motion components voz, ux, and uy. Details about the calculation of these three 3-D motion components from the measurements at multiple points on the footplate were described in our previous article (Sim et al., 2010). 2.5. Maximum possible error (MPE) and error ratio (ER) In our previous work (Sim et al., 2010), the maximum possible error (MPE, Eqs. (2)e(4)) and the error ratio (ER, Eqs. (5)e(7)) were introduced as an estimation of the maximum possible error of a motion component and a ratio of the MPE to the magnitude of the corresponding rigid body motion component.
jux jMPE ¼
n 1 X jðlxÞm jjem j; cos qL m ¼ 1
(2)
juy jMPE ¼
n 1 X jðlyÞm jjem j; cos qL m ¼ 1
(3)
J.H. Sim et al. / Hearing Research 270 (2010) 4e14
0
jvoz jMPE
n P jem j n B X 1 Bm ¼ 1 ¼ jðlxÞm jjem j þ jyc j B n cos qL @ m¼1
1 þ jxc j
n X m¼1
C C jðlyÞm jjek jC; A
n P
jux jMPE ¼ ðux ÞER ¼ j ux j
uy
ER
¼
ðvoz ÞER ¼
¼
juEy jMPE j uy j
m¼1
¼
jðlxÞm jjem j
jux jcos qL
n P k¼1
(4)
;
(5)
;
(6)
jðlyÞm jjem j
juy jcos qL
jvEoz jMPE jvoz j n n n P P P jem j þ njyc j jðlxÞm jjem j þ njxc j jðlyÞm jjem j m¼1
k¼1
jvoz jncos qL
m¼1
;
(7)
where em indicates measurement error measure at a measurement point, which is calculated from the difference between the original measured velocity and the reversely calculated velocity with the obtained 3-D motion components. Other details of these equations are described in our previous work (Sim et al., 2010). The MPE and ER were used in this study to see the effects of stimulation magnitude on errors of the 3-D motion components. 2.6. Errors on the 3-D motion components caused by the footplate position error The effects of angular position errors of the stapes footplate on calculation of the 3-D motion components of the stapes motion were mathematically formulated and appended. The footplate position errors generate errors on the coordinates of the measurement point (Eq. (A.7)), the angle between the laser beam and the axis normal to the footplate plane (Eq. (A.8)), and thus 3-D motion components of the stapes motions. Angular position errors of 5, 10, and 20 degrees were given in each of the x, y, and z axes, respectively, and the effects of the angular position errors were simulated for the measurements of GP2, where the same amount of excitation for the xh, yh, and zh directions was applied by the threeaxis piezoelectric actuator. 3. Results 3.1. Stimulation magnitude vs. error on the motion components Fig. 2 illustrates the magnitudes of the 3-D motion components (jvozj, juxj, and juyj) and their MPE (jvozjMPE, juxjMPE, and juyjMPE) with the mean magnitude of the measured velocities (jvmjmean) (left), and the error ratios ((voz)ER, (ux)ER, and (uy)ER) with the magnitudes of the corresponding elementary motion components (right), in the measurement with the stapes mechanically driven by the threeaxis actuator. In Fig. 2, the magnitudes of the 3-D motion components were proportional to the mean magnitude of the measured velocities at
7
multiple points, and therefore, the mean magnitude of the measured velocity can be considered as a measure of stimulation intensity. The MPE in each elementary motion component was also proportional to the mean magnitude of the measured velocities, but it had an additional level-independent component that dominated the MPE at low stimulus and response levels (approximately 0.005e0.02 mm/s for translational component and 0.01e0.03 rad/s for rotational components). The ER was reduced by increasing the magnitude of the corresponding elementary motion component when the mean magnitude of the measured velocities was small. However, the ER was not reduced by increasing the magnitude of the corresponding elementary motion component when the mean magnitude of the measured velocities was large enough. Fig. 3 illustrates the effects of stimulation magnitude on errors of the 3-D motion component in the physiological stapes motions (GP3) with acoustical stimuli. While the magnitudes of the 3-D motion components and the corresponding MPEs were proportional to pressure in the ear canal, the ER showed different pattern depending on the 3-D motion components. In Fig. 3A, the ER of the translational velocity ((voz)ER) at 4 kHz was reduced by increasing sound pressure level, while the ERs at other frequencies were not reduced. These trends were also shown in the rotational motion component about the long axis of the footplate (Fig. 3B). The magnitude of the rotational component about the short axis of the footplate was small relative to the magnitudes of the other components, and the large error on the rotational motion component about the short axis of the footplate was not reduced by increasing sound pressure level (Fig. 3C). Fig. 4 also illustrates the effects of stimulation magnitude on physiological stapes motions in GP4, in which retro-reflective beads were coated on the measurement area to improve reflectivity of the laser beam. Considering the weight of the glass bead (about 0.001 mg/bead), effects of the glass beads on inertia of the stapes are presumed to be negligible. Compared to the results in GP3, the ER was reduced, especially on the rotational components. The ER on the rotational motion component about the short axis of the footplate, which was not reduced by increasing sound pressure level in GP3, was also reduced by increasing sound pressure level (Fig. 4C). 3.2. Footplate position error vs. error of the 3-D motion components Fig. 5 illustrates how the magnitude of the elementary motion components changed with the angular position errors of the footplate in the measurement with an excitation mode at 0.5, 1, and 2 kHz. With an angular position error of 5 degrees, the elementary motion components of the stapes had errors of less than 10% for all three motion components. The errors in the elementary motion components increased nonlinearly with the angular position error of the footplate. The effect of the angular position errors on the rotational motion component about the short axis of the footplate was relatively small, while other components were sensitive to the angular position errors. These results were caused by the fact that the amount of error induced in the motion components was also related to the position of the SLDV system relative to the footplate because the matrix T in Eq. (A.2) was determined by the relative position of the SLDV system with respect to the footplate. The relative position of the SLDV system in our measurement setup caused significant errors in the translational component and rotational component about the long axis of the footplate when the angular position error was present. 4. Discussion Errors in results of measurements are a priori not known, and an estimation of errors is necessary for the interpretation of
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Fig. 2. Magnitudes of the 3-D motion components and their maximum possible error (MPE) with the mean magnitude of the measured velocities (jvmjmean) (left), and the error ratios (ER) with the magnitudes of the corresponding elementary motion components (right), in the measurement with the stapes mechanically driven by the three-axis actuator (GP1). A Translational velocities, B rotational velocities about the long axis of the footplate, and C rotational velocities about the short axis of the footplate.
measurements. In this study, the maximum possible error and the error ratio were used as an estimation of error boundary in the 3-D motion components of the stapes motion. In obtaining equations for the MPE and ER, a strict condition that measurement errors at all measured points have directions to increase errors in the 3-D components was assumed (Sim et al., 2010), and therefore, it is supposed that actual errors in the 3-D components are smaller than the values from the MPE and ER.
The influence of different types of errors on the motion components depends on how the errors vary with level. Fig. 6A is a diagram of different types of error growth with stimulus level. One type of error (Type I) has a magnitude that does not vary with level. These errors are presumed to be related to performance of the measurement system such as the finite resolution of the LDV system. This type of error can explain a large portion of the measured errors at low-response levels, and the size of these errors
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9
Fig. 3. Effects of the stimulation magnitude on the motion component error in physiological stapes motions with acoustical stimuli (GP3). A translational velocity at the footplate center, B rotational velocity about the long axis, and C rotational velocity about the short axis of the footplate. Velocity components and the maximum possible error (MPE, left) and the corresponding error ratios (ER, right) were calculated from measurements with the magnitude of the ear-canal pressure varied.
should be reduced by increasing the reflectivity of the laser beam on the measurement targets. Errors growing with response levels also exist. These errors are expected to increase proportionally with input levels when the measured motions occur within the linear range of the ear and measurement system (Type II). Errors in estimating measurement position and angular position of the
measured object can be sources of this type of error. When the measured motions exceed the linear range of the middle ear and/or motions begin to saturate, the errors can grow nonlinearly with stimulus level (Type III). Assuming for simplicity that response level increases proportionally with input level, the ratios of the errors with respect to
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Fig. 4. Effects of the stimulation magnitude on the motion component error in physiological stapes motions with acoustical stimuli with measured surface coated with retroreflective beads of 50-micron diameter (GP4). A translational velocity at the footplate center, B rotational velocity about the long axis, and C rotational velocity about the short axis of the footplate. Velocity components and the maximum possible error (MPE, left) and the corresponding error ratios (ER, right) were calculated from measurements with the magnitude of the ear-canal pressure varied.
the response level are shown schematically in Fig. 6B. Dealing with a Type I error, an excitation with low input levels leads to a high portion of this type of error. Type I resolution errors can be decreased by increasing the reflectivity of the target, thereby
shifting the error ratio (Fig. 6B) down and to the left. In case of a Type II error, the error ratio is constant and independent of the input level. The absolute value of the error ratio is determined by the slope of the corresponding error curve, as illustrated in
Fig. 5. Change (left) and percentage error (right) in magnitude of the 3-D components of the stapes motions due to the footplate position errors (GP2). A Translational velocity at the footplate center, B rotational velocity around the long axis, and C rotational velocity around the short axis of the footplate. The stapes motions were measured with the same magnitudes of excitation in xh, yh, and zh directions, delivered by the three-axis piezoelectric actuator at 0.5, 1, and 2 kHz. Angles of 5, 10, and 20 degrees with respect to each of z, y, and x axes from the original footplate position were given as the footplate position errors.
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the stapes motion at each of measurement frequencies was not changed with the excitation magnitude. The MPE in each 3-D motion component was also proportional to the mean magnitude of the measured velocities, but it had an additional level-independent value (k2 in Eq. (9)), depending on frequency.
jve jMPE zk2 þ k3 jvm jmean ;
(9)
where jve jMPE indicates the MPE of the elementary motion component (Eqs. (2)e(4)), and k2 and k3 , as functions of frequency, are determined by measurement conditions such as the surface condition at the measurement points, laser beam focusing, and performance of the SLDV system. k2 and k3 correspond to Type I and Type II errors in Fig. 6. From Eqs. (8) and (9),
ðve ÞER ¼
jve jMPE k2 þ k3 jvm jmean ; z k1 jvm jmean jve j
(10)
where ðve ÞER indicates the ER of the 3-D motion component. For the small mean magnitude of velocities, Eq. (10) is dominated by Type I errors, which are large at low-response levels.
ðve ÞER z
k2 ; k1 jvm jmean
(11)
while for the large mean magnitude of velocities, Eq. (10) is dominated by Type II errors and the error ratio is independent of response level.
ðve ÞER z
Fig. 6. Classification of the measurement error by their dependence on input level. A Types of errors with input levels and B ratios of the errors to response level.
Fig. 6A. A nonlinear error (Type III) is most likely present with higheintensity excitation, and in combination with Type I and Type II errors, makes a minimum point in the curve of the error ratio. The input level around which this minimum occurs can be considered as an ideal input level. The magnitudes of the 3-D motion components at each measurement frequency were proportional to the mean magnitude of the measured velocities at multiple points in the measurements with both acoustical and mechanical stimulations. From this observation, the following equation can be derived.
jve jzk1 jvm jmean ;
(8)
where jve j and jvm jmean indicate magnitude of a 3-D motion component and the mean magnitude of the measured velocities. The term k1 indicates the ratio of the magnitude of the 3-D motion component to the mean magnitude of the measured velocities. When the stapes moves only with piston-like pattern, jvm jmean are generated only by the translation motion voz, such that k1 for voz is 1, and k1 for the two rotational components ux and uy is zero. In our measurement with acoustical excitation in the considered frequencies, motions at the edge of the stapes footplate by the rotational components were about 5e15% of the translational component. With mechanical excitation by the three-axis piezoelectric actuator, this ratio could be controlled by changing ratios between amplitudes of excitations in the xh, yh, and zh directions. In acoustically driven motions, the ratios between the different response components varied with frequency. That is, the mode of
k3 : k1
(12)
From Eq. (11), the ER is reduced by increasing the mean magnitude of the measured velocities when the mean magnitude of the measured velocities is small. However, when the mean magnitude of the measured velocities is large enough, the improvement is limited by k3 =k1 in Eq. (12). This limit value k3 =k1 depends on the relative size of the magnitude of each 3-D motion component with respect to the mean magnitude of the 3 components. When the relative magnitude of the particular component is small k1 is small, and the ratio k3 =k1 is large. Because the mean magnitude of the measured velocities (jvm jmean ) is proportional to magnitudes of the stimuli, it can be concluded that the ratio of the error on the 3-D component to the magnitudes of the corresponding elementary component can be reduced by increasing the magnitudes of the stimuli, but it is limited by the relative size of the magnitude of the 3-D motion component with respect to the mean magnitude of the measured velocities. If this limitation value is large, then reliable measurements cannot be obtained only by increasing the magnitude of stimulation. The ER of the rotational component about the short axis of the footplate ((uy)ER) in the GP3 measurement was not sufficiently decreased by increasing the magnitude of stimulation. To reduce this limitation value, other measurement conditions such as reflectivity of the measurement surface should be improved. In the GP4 measurement, the reflectivity of the measurement surface was improved by coating retro-reflective beads on the surface, and reliable results for the rotational component about the short axis of the footplate were obtained by increasing the magnitude of stimulation. It was also shown in Section 3.2 that the footplate position error caused errors on the coordinates of the measurement points and the laser beam direction, and thus errors on the 3-D motion components (Fig. 5). The errors on the 3-D components were controlled below 10% of the corresponding motion components when the error in orientation of the footplate was less than 5 degrees, and were increased nonlinearly with the angular position
J.H. Sim et al. / Hearing Research 270 (2010) 4e14
error of the footplate. The accuracy of less than 5 degrees in estimating the angular position of the footplate is difficult to obtain using a rough estimation with a microscopic view, and more accurate ways are needed to get precise values of the 3-D motion components. In this study, the effects of stimulation magnitude and angular position error of the stapes on the resultant 3-D motion components were examined. Considering Eqs. (2)e(7), it is presumed that measurement area and the laser beam direction with respect to the stapes footplate also affect the accuracy of the resultant 3-D motion components. A wider range of measurement area closer to the footplate center and a laser beam direction more perpendicular to the footplate are desirable for increasing accuracy, by making jðlxÞm j and jðlyÞm j small and cos qL approximate 1 (Please refer to Sim et al., 2010 for details of jðlxÞm j, jðlyÞm j, and cos qL). However, with small dimensions of guinea pig ears, such changes could not be made while maintaining the eardrum, the ossicular chain, and the inner ear structure intact. The laser beam of the SLDV system was from the posterior-superior direction with an angle of about 45 degrees between the laser beam and the footplate plane in our measurement setup.
5. Conclusion In this study, the maximum possible error was used as a reference for error boundaries of the elementary motion components of the stapes, and effects of measurement errors on the 3-D motion components of the stapes were analyzed. The rocking motion components contain a large error when their relative amplitudes, with respect to the piston-like motion, are small. Ratios of these errors to the motion components can be decreased by increasing the ear-canal pressure up to a certain level for the physiological motions, but the improvement is limited by their relative ratio to the other stapes motion components. To overcome the small magnitudes of the motion components and get valid results, the measurement conditions, such as reflectivity of the laser beam on the measured surface and performance of the SLDV system should be improved. In measuring and calculating complex stapes motions, errors in estimating the footplate position affect the results. The effect of the position errors on the 3-D motion components is small when the magnitude of the error is small, but it increases nonlinearly with the amount of error.
Appendix. Errors in coordinates of measurement points and angular position of the laser beam caused by the footplate position error If we define the xyz coordinate system as a coordinate system of the true position of the footplate and thex0 y0 z0 coordinate system as a coordinate system of the estimated position of the footplate (Fig. A1), then there exists a rotational matrix that relates the two coordinate systems
13
Fig. A1. Estimated footplate plane position and actual footplate plane position. x, y, and z indicate coordinates of a measurement point in the true footplate-fixed frame, and x0 , y0 , and z0 indicate coordinates of the point in the estimated footplate-fixed frame.
8 09 8 9
(A.1)
The matrix R has three rotation parameters and RL1¼ RT as a property of the rotational matrix. If we let T be a matrix that relates the coordinates XYZ of the SLDV system to the estimated footplate coordinatesx0 y0 z0 , it is
8 09 8 9
(A.2)
Given that a measurement point m is on xy plane of the true footplate-fixed frame (zm ¼ 0), the correct coordinates of this point in the estimated footplate-fixed frame are according to Eq. (A.1),
8 0 9 8 9 < xm = < R11 xm þ R12 ym = y0 R x þ R22 ym ; ¼ : 0m ; : 21 m ; zm R31 xm þ R32 ym
(A.3)
where Rij indicates an element of the matrix R in the ith row and jth column. To present x0 y0 z0 coordinates of the measurement point in relation to XYZ coordinates of the SLDV system, z0m from Eq. (A.3) is substituted into Eq. (A.2) to obtain Zm coordinate value which cannot be measured from the SLDV system. It follows where Tij also indicates an element of the matrix T in the ith row and jth column. From Eqs. (A.3) and (A.4), z0 m in the estimated footplate-fixed frame is not zero due to the difference between the true and estimated positions of the footplate. However, in
9 8 > > T T > > 13 13 > 8 0 9 ðT31 Xm þ T32 Ym Þ þ ðR31 xm þ R32 ym Þ > T11 Xm þ T12 Ym > > > > > > T33 T33 < xm = = < y0m ¼ ; T T : 0 ; > T Xm þ T Ym 23 ðT Xm þ T Ym Þ þ 23 ðR xm þ R ym Þ > > > 22 31 32 31 32 zm > > > > 21 T T > > 33 33 > > ; : R31 xm þ R32 ym
(A.4)
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converting XYZ coordinate values of the measurement point m into the x0 y0 z0 coordinate values, the incorrect assumption of z0m ¼ 0 is induced and then incorrect x0 y0 z0 coordinate values ðx0m ; y0m ; z0m ÞIC of the measurement point m are obtained as follows
8 0 9 < xm = y0 ¼ : 0m ; zm IC
9 8 T > T X þ T12 Ym 13 ðT31 Xm þ T32 Ym Þ > > > > > 11 m T33 = < : T > T X þ T22 Ym 23 ðT31 Xm þ T32 Ym Þ > > > > > ; : 21 m T33 0
(A.5)
By comparison of Eqs. (A.3)e(A.5), the incorrect x0 y0 z0 coordinate values of the measurement point m can be expressed with respect to the actual footplate frame coordinates (xm, ym)
8 0 9 < xm = y0 ¼ : 0m ; zm IC
9 8 T > R11 xm þ R12 ym 13 ðR31 xm þ R32 ym Þ > > > > > T33 = < : T 23 > > > > > R21 xm þ R22 ym T ðR31 xm þ R32 ym Þ > ; : 33 0
(A.6)
Finally, coordinate errors ðxm ; ym ; zm ÞE of measurement point m due to the footplate position error and incorrect assumption of z0m ¼ 0 in the estimated footplate plane can be calculated as follows
8 0 9 8 9 8 9 < xm = < xm = < xm = ym ¼ y0m ym : 0 ; : ; : ; zm E zm IC zm 9 8 T > R11 xm þ R12 ym 13 ðR31 xm þ R32 ym Þ xm > > > > > T33 = < : ¼ T 23 > R21 xm þ R22 ym ðR31 xm þ R32 ym Þ ym > > > > > ; : T33 0
(A.7)
An error is also induced to the calculation of the angle between the laser beam and z-axis as
0 ðcos qL ÞE ¼ cos qL IC cos qL ¼ T33 ðT13 R13 þ T23 R23 þ R33 T33 Þ;
(A.8) 0
where ðcos qL ÞE indicates the error on cos qL and ðcos qL ÞIC indicates the incorrect value of cos qL resulted from the incorrect estimation of the footplate position.
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