Technical noteMagnitude and phase of three-dimensional (3D) velocity vector: Application to measurement of cochlear promontory motion during bone conduction sound transmission
Introduction
Vibrational motions have been measured to assess sound transmission through the ear structures and skull contents in normal and reconstructed middle ears, for both air conduction (AC) and bone conduction (BC) stimulation. For example, motions of the stapes are frequently measured together with the ear-canal pressure to obtain the middle-ear transfer function (METF) for transmission of AC sound via the middle ear; vibrations of the cochlear promontory are measured assuming that it reflects transmission of BC sound to the cochlea and can thus predict the BC stimulation required to induce hearing sensations (Stenfelt and Goode, 2005; Eeg-Olofsson et al., 2008). Measurements of the vibrational motions of the ear structures were made in a one-dimensional (1D) manner in the early stages and were later extended to include three-dimensional (3D) behavior. Laser Doppler vibrometry (LDV) systems are one of the standard tools used to measure small vibrations of the ear structures, and 3D measurements are typically based on a method introduced by Decraemer et al. (1994). They mounted temporal bones on two stacked goniometers and measured velocities at a point on the middle-ear ossicles using an LDV system with several different laser beam angles that were obtained by rotating the two goniometers. Recently, measurement of the three translational components at a specified measurement point has become easier with the development of 3D LDV systems, which have three built-in laser beams with three different measurement angles.
The 3D behavior of a point has generally been described with the three orthogonal components of the motion at the specified point. However, when the 3D components have different phases, they do not show the magnitude and phase of the resultant directly. The magnitude and phase of the resultant may have important meanings in some cases, for example, indication of BC sound transmission by measuring vibration of the cochlear promontory. In this case, we hypothesize that the resultant of the promontory motion rather than any orthogonal component of the promontory motion is a better descriptor of BC hearing. This article describes a method for calculating the magnitude and phase of the resultant from the three orthogonal motion components. The method was applied to measurements of cochlear promontory motion on the ipsilateral and contralateral sides with excitation by a bone-anchored hearing aid (BAHA) in cadaver heads. The transcranial attenuation and phase delay of the contralateral side relative to the ipsilateral side were obtained based on the attenuation of the maximum magnitude and phase delay of the resultant. The results were compared with interaural threshold difference between the ipsilateral and contralateral sides and expected transcranial phase delay data reported in the literature.
This article provides a way to represent 3D velocity with the magnitude and phase of the resultant, which have been previously described with the three orthogonal components in the Cartesian coordinate system. Such an approach could be useful in cases for which the resultants are presumed to provide a more direct indication of sound transmission, for example measuring vibration of the cochlear promontory to assess transcranial attenuation and phase delay in BC sound transmission.
Let's consider a velocity vector V of a harmonic motion in the 3D space, which has Vx, Vy and Vz as the xyz components. The vector V can be written as,where the i, j, and k are the unit vectors in the x, y, and z direction, and n is the unit vector in the direction of the vector V. Note that the direction of the unit vector n changes during a cycle whereas the directions of the i, j, and k are fixed in the 3D space. If the xyz components Vx, Vy and Vz have phases of θx, θy, and θz, respectively, at an angular frequency ω, the Vx, Vy and Vz can be represented as a function of time t as,Then,
By applying a formula ,
By applying another formula ,with
The last two terms in Eq. (5) can be combined as follows:with
Since has the magnitude of , the maximum magnitude of the spatial velocity vector V can be obtained as,
Consequently, the minimum magnitude of the spatial velocity vector V becomes,
The phase of the spatial velocity vector V can be calculated from Eq. (7). Note that two values for are obtained from Eq. (7). That is, once is calculated in the range of using Eq. (7), then, the two solutions for are:
Since the phase is calculated from the square of the velocity in Eq. (6), two phases with a difference of are obtained for two opposite directions. That is, the unit vector , which represents the direction of the maximum magnitude of the vector V, can be defined in either of two opposite directions, and the phase is determined by choice of the direction of the maximum magnitude. Considering the fact that V has its positive maximum at , the two unit vectors and for and can be obtained by,
In Eq. (11), scales the magnitude of the directional vectors and to a unit. Since the phase of the minimum magnitude is obtained by adding π/2 to the phase of the maximum magnitude, the two unit vectors and for the minimum magnitude can be obtained by,
Fig. 1 illustrates the trajectory of a 3D velocity vector V with the two points of the maximum magnitude (P1 and P2) and the two points of the minimum magnitude (Q1 and Q2) during a cycle. Note that the two unit vectors and in the directions of the maximum magnitude are in opposite directions from each other (i.e., ), and so are the two unit vectors and indicating the directions of the minimum magnitude.
While the trajectory generally has a shape of an ellipse, the trajectory has a shape of a straight line (i.e., the 3D velocity vector V becomes one-dimensional) or s circle (i.e., the magnitude of the 3D velocity vector V is constant during a cycle) for some cases.
In order for the trajectory to become a straight line, the minimum magnitude of the 3D velocity vector V should have zero value () and from Eq. (9), it results in
By squaring both sides of Eq. (13), we obtain,
With formulas and , Eq. (14) is simplified as,
From Eq. (15), if the phases θx, θy, and θz of the xyz components are the same or have difference each other, and thus the trajectory becomes a straight line, independently of the magnitude of the xyz components.
In order for the trajectory to have a circular shape, the magnitude of the 3D velocity vector V should be constant during a cycle, and from Eq. (6), it results in . That is,
Eq. (16) can be simplified as,
Eq. (17) is satisfied, for example, in the case that , , and . However, a condition of the phases of the xyz components that satisfies Eq. (17) independently of the magnitudes of the xyz components does not exist.
Now, choice of the phase of the velocity vector V among two possible phases of and is of concern. Two suggestions can be provided for choice of the phase of the velocity vector V:
- 1)
Choose either of two possible phases arbitrarily at a specific frequency. Then, at the adjacent frequency, choose the phase such that the direction of the corresponding positive maximum changes with a smaller angle from the direction of the positive maximum at the previous frequency. In this case, a sufficiently small frequency step is necessary.
- 2)
The side of the positive maximum can be defined anatomically. For example, when sound wave propagation on the skull surface with bone conduction stimulation is measured (Dobrev et al., 2017), either of outward side or inward side of the skull surface can be chosen as the side of the positive maximum consistently through the considered frequency range.
The cochlear promontory motions of the ipsilateral and contralateral sides were measured in four cadaver heads with stimulation using a BAHA transducer. A BAHA Cordelle II (Cochlear AG, Australia) transducer was placed on the mastoid (5 cm behind the external auditory canal opening) using a 5-N steel headband (BAHA headband 90138, Cochlear AG, Australia), and the coupling force was maintained at approximately 5 N (Dobrev et al., 2016).
The cadaver heads were positioned in a natural upright position. A metal rod of 12-mm diameter was inserted in the remaining spinal column (∼5 cm) of the cadaver head, and the metal rod was mounted on the vibration-isolation table (M-INT1-36-6-A, Newport Corp., CA, USA) with vibration-isolation legs (sorbothane legs, AV3, Thorlabs Inc., NJ, USA). Additional support with rubber bands was made for horizontal stability of the cadaver heads. Four tensional forces spaced by 90° were made horizontally by four rubber bands, which are supported by four metal rods on the vibration-isolation table and are connected to four points of the cadaver head via a BAHA Softband (Cochlear AG, Australia) on the cadaver head, at approximately 2-cm above the pinna.
The BAHA transducer was driven sequentially by 81 stepped sine signals, distributed logarithmically in the frequency range of 0.1–10 kHz, resulting in approximately 40 frequency points per decade. An endaural incision between the helix and tragus was performed, and tympanomeatal flap was elevated, to expose the cochlear promontory (Fisch et al., 2008; Huber et al., 2013). The cochlear promontory motions near the round window (2–4 mm from the boundary of the round window) were measured using a 3D laser Doppler vibrometry (3D LDV) system (CLV-3D, Polytec GmbH, Germany). The 3D LDV system was positioned such that the laser beam was oriented along the lateral-medial direction. The signal generation and velocity measurement were handled via a data acquisition system (APx585, Audio Precision Inc., USA), with a sampling frequency of 96 kHz and sampling time of 200 ms. The measurement at each frequency was repeated five times, and the five measurements were recorded as complex numbers to represent the magnitude and phase. Then, the median value across the five measurements was taken for each of the real and imaginary parts of the data for further analysis.
Section snippets
Results and discussion
The vector with the maximum magnitude and the corresponding direction is defined as the maximum velocity vector in the remaining parts of this article, for convenience.
Fig. 2 displays trajectories of velocities of the cochlear promontory on the ipsilateral (red) and contralateral (blue) sides during one cycle, at 0.25, 0.5, 1, 2, 4, and 8 kHz in one cadaver head (CH1). The positive x-, y- and z-directions indicate the anterior, superior, and lateral directions, respectively (A right-handed
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2022, Hearing ResearchCitation Excerpt :The VSTAP_DIFF_NORM metric is aimed at scaling the differential stapes motion, relative to the surrounding promontory motion, in order to reduce the effects of the actuator's performance and highlight the physiologically governed behavior of the heads (Stenfelt et al., 2002; Dobrev et al., 2020a). While both individual responses and calculated metrics were available for all 3 orthogonal components and the combined motion, all velocity data (unless otherwise specified) is presented only for the combined motion, as it is assumed to be more representative of the total kinematics of the stapes and promontory (Sim et al., 2010; Dobrev et al., al.,2016a; Dobrev and Sim, 2018). It has also been shown to be a better descriptor of hearing sensation than any individual motion component under BC (Dobrev and Sim, 2018).