Shape and sound analyses of the human ear-canal geometry

Abstract

The curved centerline of the ear canal, and its spatial variations of area, transverse to this centerline influence sound transmission in the human ear canal at higher frequencies. The area function was directly assessed from images obtained using a hand-held device inserted into the canal and compared to sound area functions indirectly calculated from acoustical measurements. For shape data, areas were calculated by modeling each canal cross section as an ellipse. In a discrete Frenet frame procedure, the shape outputs were the spatial variations of ellipse area and eccentricity, with the curvature and torsion parameters representing the centerline. The resulting shape area functions were compared with sound area functions by finding the best alignment of the functions in each ear. The median shape and sound areas of the ear canal agreed within 0.3 mm² at the probe tip and approximately 6 mm² at 3.6 and 7.2 mm lateral to the probe tip. This supports the potential use of acoustical assessment of ear-canal area in future research.

I. INTRODUCTION

Access to high-frequency sounds is important for (1) speech and language development in children, particularly those with reduced audibility above 5 kHz due to hearing loss (Stelmachowicz et al., 2004); (2) perception of singing and speech in adults above 5.7 kHz (Vitela et al., 2015); and (3) speech localization above 8 kHz (Best et al., 2005). Clinical audiograms are typically limited to frequencies up to 8 kHz, although extended high-frequency audiograms are measured up to 20 kHz. However, calibration in the ear canal is not well specified at frequencies above 8 kHz because sound pressure level varies between the headphones or inserted probe and the tympanic membrane (TM). While the audiological goal is to specify sound input at the TM that drives sound transmission into the cochlea, the ear-canal sound field becomes increasingly more complex at higher frequencies.

Standing wave effects for the sound field in the human ear canal at various locations become more prominent above 2 kHz. Measuring the sound field in the human ear canal near the TM may help interpret audiometric and other behavioral hearing tests, but testing is routinely performed at more lateral locations to simplify test procedures. Knowledge of ear-canal shape is needed to transform a measured sound pressure level from a more lateral location within the canal to near the TM¹ (Keefe, 2020).

The area function, i.e., the area versus distance along the canal centerline, may be measured directly via three-dimensional (3D) imaging or indirectly via acoustical testing. Area functions have been directly measured from earmolds or computed tomography (CT) scans in cadaver ears (Johansen, 1975; Stinson and Lawton, 1989; Egolf et al., 1993) and in live subjects (Voss et al., 2020; Balouch et al., 2023). Area functions have been indirectly estimated from sound data by various methods and laboratories (Hudde, 1983; Okabe et al., 1988; Joswig, 1993; Rasetshwane and Neely, 2011; Keefe, 2020; Keefe et al., 2024).

Voss et al. (2008) compared area functions in nine human-cadaver ears (from subjects ages 41–84 years), which were measured both directly from data obtained from silicone-based earmolds and indirectly by an acoustical method. The areas averaged across all ear-canal locations, which ranged from 3–28 mm from the TM, were reported for each earmold. At an individual canal location, the indirect acoustical measurement of the area (over a bandwidth of 0.2–6 kHz) was typically larger, and often much larger, than the directly measured area.

This study reports direct measurements of canal geometry (shape data) in a group of adults using fast “digital” scanning with a hand-held device inserted into the ear canal (i.e., real-ear scans), and also using scans of earmolds from a subset of ears (i.e., earmold scans). Area functions from these shape data were compared to those measured indirectly using acoustical responses (sound data).

The central axis of the ear canal is curved and the canal area to be measured at a given point on this axis is a flat surface that lies in the plane perpendicular to that axis. Sound propagation in a curved duct along its centerline is similar to that in a straight tube if duct diameter is small compared to the acoustic wavelength and its curvature is not too large (Rayleigh, 1896). Because the tangent to the centerline varies according to its curvature, the orthogonal plane containing this area also varies in orientation in the underlying 3D space.

Stinson and Lawton (1989) described the centerline curve of an ear canal using the Serret–Frenet equations, or Frenet frame, which embeds a non-intersecting, continuously curved, one-dimensional (1D) line in 3D space. A Frenet frame attaches three orthogonal unit vectors to each point, s, at a particular distance along the curve (of the centerline). The tangent vector of unit magnitude is defined at each s to point along the curve, and a normal vector is defined as the unit vector that points in the direction of the rate of change of the tangent vector with distance. The magnitude of the rate of change of the tangent vector is the curvature per unit length along the curve. Curvature is always positive, except that it is zero for a straight centerline. The inverse of curvature is the radius of curvature of a local bend in the curved centerline. The unit vector orthogonal to the tangent and normal vectors is called the binormal vector. The rate of change of the binormal vector with distance is a scalar multiple of the normal vector in the direction of the normal vector. This scalar multiplied by minus one defines the torsion per unit length, or “twisting,” of the curve in 3D space out of the plane formed by the tangent and normal planes. Torsion may be positive, negative, or zero. Thus, the intrinsic geometry of the curvilinear curved centerline is described by curvature and torsion parameters at each location along the centerline. Using this approach, Stinson and Lawton (1989) calculated, in an iterative manner, the shape of the ear-canal centerline and its associated area at each location using geometrical data from silicone rubber molds of cadaver ear canals.

Using these same earmold data from Stinson and Lawton (1989), Stinson and Daigle (2005) calculated the sound field using a 1D plane wave transmission-line approach (with curved centerline) and a 3D boundary element method. The authors reported good agreement for the pressure level along the centerline, although they did point out level differences of about 1.5 dB at 8 kHz and 4.5 dB at 15 kHz. This suggests the potential utility of describing the curvature and torsion parameter values in a 1D description of the ear-canal centerline.

Close links exist between the spatially varying geometry of the ear canal along its centerline and the interpretation of sound measurements using an 1D inhomogeneous transmission-line model. An important advantage of 1D acoustical waveguide models is their ability to estimate the transformation at higher frequencies of sound pressure from a probe inserted into a lateral canal location relative to a location that is near the TM. Such a transformation is calculated more efficiently in a 1D than a 3D model. “Near the TM” ideally refers to a location within about a radius away from the lateral edge of the TM. This radius is on order of 4 mm for a typical adult ear canal, which is 7.5% less than a quarter-wavelength of sound at 20 kHz. While wave effects of sound are important above 8 kHz, an acoustical estimation of sound pressure level at a location termed “near-TM” would better represent the acoustic input acting on the middle ear than does the sound field at the probe tip.

From the models of ear-canal shape of Stinson and Lawton (1989), Xia et al. (2024) introduced a curve twist ratio to quantify the shift in the predicted resonance frequencies of the ear-canal sound field calculated by 3D finite-element models for ear canals having a circular cross section with either a straight or curved centerline. The curve twist ratio was “defined as the total length of the curve divided by its projected length on the z axis of the Cartesian system…,” such that a curvilinear canal with non-zero curvature and torsion values along its length would have a curve twist ratio that exceeds one. The present study reports measurements of the torsion of the centerline of the ear canal.

One complication in constructing models of ear-canal shape is that the spatial data representing the wall of the ear canal are discrete, so that the resulting centerline to be estimated is a set of connected, discrete-line segments. This spatial discretization of the curvilinear centerline is more accurately described using a discrete Frenet frame (Hu et al., 2011; Lu, 2013) rather than the Serret–Frenet equations. The Frenet frame includes curvature and torsion parameters calculated for each segment of the canal.

Using CT scans in ears of adult human subjects, Yu et al. (2015) reported the mean height and width of the canal opening and the depth of the first (or most lateral) bend, with larger dimensions for ears in males than females. They reported that the ear cross section was elliptical rather than circular near its opening, and this shape should be taken into account in fitting earplugs of hearing-protection devices. Databases of 3D models of the ear canal were constructed by magnetic resonance imaging techniques in human subjects (Darkner et al., 2018). These results are relevant to better fit hearing aids and develop 3D finite-element models that couple ear-canal sound to middle ear and cochlear function.

Thus, knowledge of the shape of an individual ear canal allows prediction of the transformation of the sound field from the probe tip to any canal location up to the near-TM region. The directly determined area function of the entire canal from shape measurements would suffice to calculate this transformation in a 1D acoustical waveguide model, aside from ear-canal wall loss effects on the transformation that can be estimated (Keefe, 2020). Additional measurements of curvature and torsion in an individual ear canal would provide anatomical information more detailed than that of the curve twist ratio, which would be relevant to 1D acoustical models modified by the properties of the non-straight centerline of real-ear canals, as might conceivably be extracted from comparing 1D and 3D model predictions for the same ear. The directly determined shape of an individual ear canal, as encoded through detailed 3D shape measurements of the canal, would suffice to calculate this transformation in a 3D finite-element model, although that model would need to represent middle- and inner-ear function acting over the TM.

Gaps in previous research addressed by this study include the following: (1) the extent to which the cross sectional area of the ear canal departs from a circular shape has not been measured at locations more medial than the lateral entrance; (2) the intrinsic geometry of the curvilinear centerline has not been reported; and (3) the area functions estimated using direct and indirect assessment techniques have not been compared in the same ears of human subjects. The present study addressed those gaps.

Preliminary review of real-ear scan data (described below) showed that the two-dimensional (2D) canal cross section at each location along the centerline had a pattern that was more elliptical than circular (bear in mind that a circle is a special case of an ellipse). The lack of axial symmetry of the canal wall cross section around a centerline might be modeled by a number of shapes, e.g., an ellipse, polygon, or some spline approximation to the shape. This study modeled the 2D cross section by an ellipse parameterized by its area and eccentricity. The eccentricity describes that extent to which an ellipse is non-circular. Its value ranges from zero to one, with zero for the case of a circle and approaching one for the case of a highly elongated ellipse.

The curved centerline of the canal in 3D space was modeled in terms of a discrete Frenet frame having curvature and torsion parameters. Finally, this study compared directly measured area functions with those measured indirectly in the same adult ears based on time-domain analyses of acoustical responses (Keefe et al., 2024). The goal was to examine inter-relationships between the two very different methods of obtaining the geometry of typical ear canals.

II. METHODS

A. Human subjects and clinical measurements

Adults were consented prior to participation as human subjects in this study with the approval of the Internal Review Board at Boys Town National Research Hospital. Any individual with a history of external-ear malformation or ear surgery was excluded from the study. Clinical tests including 226 Hz tympanometry and air-conduction audiometry were performed just before acquiring digital ear scan data. Inclusion criteria were 226 Hz tympanometry peak-compensated admittance magnitude between 0.3 and 1.4 mmhos and a peak pressure between −100 and 20 daPa. The mmho unit is commonly used in clinical tympanometry with the definition that 1 mmho is 10⁻⁸ m⁴ s/kg. The test group for digital ear scans included 58 ears from 30 adult subjects (22 female) of mean age 42 years and standard deviation (SD) of 14 years. All subjects self-identified as of White race, with two as Hispanic or Latino ethnicity and the rest as Not Hispanic or Latino. The sample included 28 left ears and 30 right ears.

B. Digital ear scans and earmolds

A digital scan of the 3D surface geometry of the ear canal (i.e., real-ear scan) was measured using a portable handheld medical device (Lantos Ear Scanning System, Lantos Technologies, Derry, NH) approved for use in adult patients. Fligor and Grenier (2018) have measured scans with this device to design custom ear devices. The device included a movable camera within the canal covered by a membrane that progressively inflated with a fluid conforming to the concha and canal walls. The fluid was delivered by a tank within the device and contained an approved coloring selected for its optical properties. The second author (H.L.P.) was the operator who measured all the real-ear scans. The operator inserted the membrane into the ear through the use of a video otoscope. The maximum insertion depth was identified by the operator's examination of the otoscope image at a location lateral to any point on the TM.

The operator started the 3D scan mode of the device after determining this maximum insertion depth. The device used a laser-induced fluorescence technique based on small light scatterers embedded in the membrane and the optical properties of the fluid to measure the geometry of the canal wall from locations on the concha to the maximum insertion depth within the canal. The operator manually adjusted the hand-held device during the 3D scan mode to new orientations to progressively image the more medial (or deeper) locations in the canal. The membrane was then deflated by the device and removed from the ear. The image of the wall boundaries of concha and ear-canal structures was encoded as a surface triangle language (STL) file that was transferred to a lab computer.

Silicone earmold impressions were made in a subset of ears following clinical audiologic procedures appropriate to custom fitting of some hearing aids and hearing-protection devices. This procedure included placement of a cotton block with a wick into the ear canal past the second bend as deeply as possible followed by filling the ear canal and concha bowl with a catalyzed silicone mixture. After hardening for approximately 3 min, the earmold impression was removed from the ear. Each silicone mold was digitally scanned at Sonova USA, Inc. (Aurora, IL) to produce a STL file imaging the ear geometry (i.e., earmold scan).

C. Image analysis

The entire image analysis of each STL file was processed almost entirely automatically in less than 5 min to output all the results presented in this report. The only manual step in this analysis was a volume segmentation step that is next described. Custom laboratory software was written in matlab (The MathWorks Inc., Natick, MA) to analyze all data.

1. Voxelization and volume segmentation

The STL file represented the surface of the external ear structures comprising part of the concha near the lateral canal opening and ear canal as shown in Fig. 1 for a test ear A (left ear, female, age 33 years).

FIG. 1.

Image of scanned external ear from STL file of test ear A (left ear, female, age 33 years). The color map is relative to the z coordinate of the center of each triangle.

This particular image was a closed surface composed of 15 061 triangles connected at 7647 vertices. The information in the STL file labeled each triangle with an integer and associated with it the 3D Cartesian coordinates of the three vertices of the triangle. Each vertex in the interior of a surface was a vertex of two or more other triangles. These 3D coordinates were defined in terms of the position of the hand-held scanner in the operator's hand when the automated ear scan was begun. As such, the origin of the 3D coordinate system was arbitrary and not referenced to the locations of the head or opposite ear. Distinctive anatomical features are labeled in Fig. 1 for the pinna and ear canal according to earmold terminology (Alvord et al., 1997). Of particular note is the cymba stalk at a maximum height on the z axis relative to the most medial point of the canal.

To enable numerical analyses, the surface triangle image was converted to a voxelized image in terms of a Cartesian 3D coordinate system.² Each voxel was a cube of length 0.25 mm on each site, which set the spatial resolution. The image in Fig. 1 after voxelization was contained in an array with 150 voxels in the x and y directions and 47 voxels in the z direction. Each voxel was coded with value zero if it did not contain any part of any triangle in the STL image, and one otherwise. This operation converted the image into a binary file.

Two values along the z axis of each voxelized scan were defined: z1 at the most medial point of the canal stalk and z2 at the most medial point on the cymba stalk. These values correspond to relative maxima in Fig. 1. A set of 2D slices was extracted from the binary image. Adjacent slices were one voxel apart along the z axis, or 0.25 mm. Slice 1 at z1-1 (in voxel units) extracted a small open region near the most medial end of the canal stalk. Slice 2 at z2 + 1 just above any voxel on the cymba stalk extracted a larger open region through points in the canal and concha cavum (but not the crus helix valley in Fig. 1 that was a ridge in the real ear). The centers of each of the open regions in slice 1 and slice 2 were calculated and joined by a line (not shown) to form an oblique axis terminating near the medial end of the canal.

A new (x′, y′, z′) coordinate system was defined with the z′ axis coincident with the line through the centers of slices 1 and 2. New x′ and y′ coordinates were calculated using the Euler angles between the two coordinate systems (Goldstein, 1950). Dropping the primes on the new coordinates, this centerline was vertically aligned along the (rotated) z axis. This had the effect of approximately aligning the ear-canal stalk along the z axis, but its intrinsic curved shape remained unchanged. This rotation to near vertical of the canal followed previous studies (Khanna and Stinson, 1985; Stinson and Lawton, 1989), and prevented non-monotonicities from occurring in iterative calculations of the curved centerline of the canal.

The next analysis step was to remove those parts of the voxelized image that would lead to multiple regions on the 2D slice at any z value. It was not sufficient to discard parts of the binary image with z less than the new z location of the maximum on the cymba stalk, because that would discard parts of the ear canal closer to its lateral entrance.

A volume segmentation step was manually performed after data were acquired (using the Volume Segmenter app in matlab). The operator viewed each 2D slice of the voxelized image along the z axis in the rotated coordinates from top to bottom and manually selected the slice with the top-most open region (i.e., with maximum z) corresponding to the most medial end of the canal stalk. The operator used mouse clicks in the software to draw a first polygon on the slice that contained this top-most open region in its interior. The 2D slice above this one sometimes contained a cluster of points in a closed region denoting the extension of the membrane tip into the canal without touching the canal wall. Even an open region might correspond to a point of the membrane not yet conforming to a canal wall, as is further considered below.

Some slices at z values at and below the maximum z of the cymba stalk contained multiple distinct regions, one of which was a further extension of the canal toward its as yet undefined exit into the concha bowl. The operator visually tracked that canal region through lower slices until the bottom-most slice was reached (i.e., with minimum z) that contained an open canal region. This bottom-most slice might contain other regions such as that related to the cymba stalk. The operator again used mouse clicks to draw a second polygon that contained this bottom-most open canal region in its interior, but that did not contain any points associated with other regions of the surface of the pinna. In the volume segmentation step, the two polygons manually drawn in the top-most and bottom-most slices were interpolated to generate polygons in all intervening 2D slices of the canal. The operator confirmed that the polygon in each intermediate slice only contained the open region of the canal, or canal plus concha bowl. These regions were then labeled and stored as an intermediate file. The resulting spatially filtered image is shown in Fig. 2 for test ear A. The voxelized image includes the ear canal and nearby locations in the concha cavum (near $z=0$).

FIG. 2.

Voxelized image of test ear A after volume segmentation.

The above processing is illustrated in Fig. 3 in test ear A by a montage of all 2D slices from the surface in Fig. 2.

FIG. 3.

Montage of 2D ear slices (for test ear A) with the most medial slice in the lower right corner and most lateral slice in the top left corner.

Pronounced changes are evident in the size and shape of each 2D slice, although each slice is perpendicular to the z axis rather than to the real curved centerline of the canal. The location of the canal entrance is not evident in this montage. The area reduction and irregular boundary in the lower right ear slice is likely a result of the most medial region of the membrane not conforming to the canal wall (see below).

2. Iterative procedure to calculate canal geometry along its curved axis

A total of ten iterations were performed to estimate the curved centerline of the ear canal and the canal geometry represented by the model ellipse. Each iteration involved calculations from top to bottom along the canal stalk in increments of 1 mm steps along the z axis. This differed from the voxel dimension of 0.25 mm shown in Figs. 2 and 3 so as to obtain smoother estimates of the curved shape of the canal centerline in 3D space. Nevertheless, the underlying spatial resolution of 0.25 mm was retained in intermediate calculations on each 2D slice.

a. Initial iteration.

The initial iteration began with the most medial 2D slice that is shown for test ear A as the right, bottom closed curve in Fig. 3. Counting from this first slice to the left and upwards in Fig. 3, the initial set of 2D slices for this ear were the first, fifth, ninth, etc., until the upper-left closed curve was reached. There were $M=13$ of these 2D slices for ear test A, corresponding to M values of the z coordinate of the initial model centers. These M values of the canal centerline initialized the iteration procedure.

The points in each 2D slice on the initial and subsequent iterations formed a curve on the plane, from which the “data center” was calculated by averaging across all x and y locations in the slice. Next, a model ellipse was calculated that best fitted the data (see Appendix A). Relevant outputs from the ellipse model were its area, eccentricity, major axis, minor axis, and model center. One measure of the goodness of fit of the ellipse to the data were the error distance between the model and data centers. Another measure described in Appendix A was the magnitude of the residual error of the constrained least squares solution of the model fit. An error-free fit of the data would result in zero values for the error distance and residual error.

A histogram for each 2D slice was constructed of the number of data points in each of 12 azimuthal sectors around the model center. Each sector had a sector angle of 30° (starting counterclockwise from the positive x axis relative to the model center of each 2D slice shown below in Fig. 4). A count of the number of sectors without any data were maintained for each slice in each iteration. The use of this histogram is further discussed below but the gist of it is that any empty sectors (away from the near-TM region) would indicate that the slice lay partially outside the canal entrance.

FIG. 4.

(A) Final iteration detail for test ear A at the most lateral location within the canal (slice $\#10$). (B) Analogous plot for next location (slice $\#11,$ or 1 mm more lateral) that lies partially within the canal and partially outside it. Model ellipse line fitted to data (X markers) shown in voxels. Model ellipse outputs shown in physical units. Also, shown are the mean center of data (open square) and the model ellipse center (closed circle) connected by dotted line.

The model center of the nth slice expressed in the 2D coordinates of the slice was transformed to a point in the original 3D coordinate system of the voxelized image as the center vector ${\overrightarrow{s}}_{n,1}=\{x_{n,1},\hspace{0.17em}{y}_{n,1},\hspace{0.17em}{z}_{n,1}\},$ in which the left subscript denoted slice n and the right subscript denoted the iteration number, in this case iteration 1. Each vector extended from the origin of the 3D coordinate system to a model center. The canal centerline passed through the center vector location of each slice. A line segment vector, $\Delta {\overrightarrow{s}}_{n,1}$, was defined for slice n in iteration 1 as a difference in the position vectors of one slice relative to one or both adjacent slices as follows:

$$\begin{array}{c}\Delta {\overrightarrow{s}}_{1,1}={\overrightarrow{s}}_{2,1}-{\overrightarrow{s}}_{1,1},\hfill \\ \Delta {\overrightarrow{s}}_{n,1}=\left({\overrightarrow{s}}_{n+1,1}-{\overrightarrow{s}}_{n-1,1}\right)/2\hspace{1em}\text{for}\hspace{0.17em}1<n<N-1,\hfill \\ \Delta {\overrightarrow{s}}_{N,1}={\overrightarrow{s}}_{N,1}-{\overrightarrow{s}}_{N-1,1}.\hfill \end{array}$$ (1)

The line segment vector was defined for slice 1 by an equation of forward difference, for slice n with $1<n<N$ by a midpoint difference, and for slice N by a backwards difference. The scalar length of the line-segment vector, $\Delta {\overrightarrow{s}}_{n,1}$ was $\left|\Delta {\overrightarrow{s}}_{n,1}\right|.$ The tangent vector, ${\overrightarrow{T}}_{n,1}$, of unit length along this nth section of the canal was defined for iteration 1 by

$${\overrightarrow{T}}_{n,1}=\frac{\Delta {\overrightarrow{s}}_{n,1}}{\left|\Delta {\overrightarrow{s}}_{n,1}\right|}.$$ (2)

The center vector, ${\overrightarrow{s}}_{n,1}=\{x_{n,1},\hspace{0.17em}{y}_{n,1},\hspace{0.17em}{z}_{n,1}\}$ (but not ${\overrightarrow{T}}_{n,1}$), was then rounded after these calculations to the nearest 3D location on the grid with spatial resolution of 0.25 mm.

b. All other iterations.

The current iteration, k, ranged over $1<k\le 10.$ What differed in all iterations after the initial one was that the 2D slices were not assumed to lie along the z axis, so they did not lie in the x-y plane of the 3D coordinate system. An oblique 2D slice was taken for slice n at its center vector location on the previous iteration that was perpendicular to the tangent vector calculated for that slice on the previous iteration.

The closed curve forming the canal wall was extracted from the slice and an ellipse was modeled to best fit the data. The new model center of the ellipse in general differed from that on the previous iteration because of any changes in the new tangent vector or new model outputs for the ellipse. The model center on iteration k of the nth slice expressed in the 2D slice coordinates was transformed to the original 3D coordinates as the center vector, ${\overrightarrow{s}}_{n,k}=\{x_{n,k},y_{n,k},z_{n,k}\}$. This differed from that on the previous iteration $k-1.$ The histogram of data in each of the 12 sectors was calculated on each iteration across all 2D slices.

The model centers of the first slices ( $n=1$ and 2) on iteration k were assigned to be invariant, so they were set equal to the corresponding model centers on iteration 1, i.e., ${\overrightarrow{s}}_{1,k}={\overrightarrow{s}}_{1,1}$ and ${\overrightarrow{s}}_{2,k}={\overrightarrow{s}}_{2,1}.$ This was to prevent any undue oscillations at the end points. Similarly, the model centers of the last two slices ( $n=N$ and $N-1$) on iteration k were assigned to be invariant and thus set equal to the corresponding model centers on iteration 1, i.e., ${\overrightarrow{s}}_{N,k}={\overrightarrow{s}}_{N,1}$ and ${\overrightarrow{s}}_{N-1,k}={\overrightarrow{s}}_{N-1,1}.$ These slices $N-1$ and N were typically outside of the interior of the lateral canal (i.e., they were partly or entirely within the concha bowl) so this latter assignment had little or no effect.

The original idea was that the locations of the curved centerline calculated in each iteration by the rounded values of ${\overrightarrow{s}}_{n,k}$ would converge over the ten iterations. Because of the spatial resolution of 0.25 mm, each location tended to oscillate back and forth over adjacent iterations with nearby grid locations at many of the points along the curved centerline. That is, the canal centerline did not converge.

A relaxation technique (simple exponential smoothing) was used to ensure convergence, in which the center vector on subsequent iterations was calculated as a geometrically smoothed version of center vectors on up to the previous three iterations. The center vectors were smoothed for various k values as

$$\begin{array}{c}{\overrightarrow{s}}_{n,2}=\frac{2}{3}\left({\overrightarrow{s}}_{n,2}+\frac{1}{2}{\overrightarrow{s}}_{n,1}\right),\\ {\overrightarrow{s}}_{n,3}=\frac{4}{7}\left({\overrightarrow{s}}_{n,3}+\frac{1}{2}{\overrightarrow{s}}_{n,2}+\frac{1}{4}{\overrightarrow{s}}_{n,1}\right),\\ {\overrightarrow{s}}_{n,k}=\frac{8}{15}\left({\overrightarrow{s}}_{n,k}+\frac{1}{2}{\overrightarrow{s}}_{n,k-1}+\frac{1}{4}{\overrightarrow{s}}_{n,k-2}+\frac{1}{8}{\overrightarrow{s}}_{n,k-3}\right).\end{array}$$ (3)

Because these center points were fixed, this smoothing would have no effect and was not performed at the end values of n of $1,2,N-1,N$.

The smoothed center vectors were linearly interpolated so that the z coordinate of each was the same as on the initial iteration. This constrained the algorithm to advance in 1 mm steps from the medial to the lateral end of the canal. After interpolation, the x-y coordinates of the center vectors were smoothed along this z coordinate using a cubic spline interpolation.

The line segment vector, $\Delta {\overrightarrow{s}}_{n,k}$, was calculated on each iteration using Eq. (1) with iteration k replacing each occurrence of iteration 1 in the equation. Its scalar length, $\left|\Delta {\overrightarrow{s}}_{n,k}\right|$, and tangent vector, ${\overrightarrow{T}}_{n,k}$, were calculated using Eqs. (1) and (2), respectively, in which iteration k replaced iteration 1. The center vector, ${\overrightarrow{s}}_{n,k}$, was then rounded to the nearest grid location (0.25 mm resolution) to serve as the center point in the oblique slice on the next iteration (with the tangent vector as the other input). The curved canal length was the sum of the lengths of the line-segment vectors, $\left|\Delta {\overrightarrow{s}}_{n,k}\right|$, across all segments until that at the lateral canal entrance (as described below). The square root of the sum of the squared differences of the vector ${\overrightarrow{s}}_{n,k}-{\overrightarrow{s}}_{n,k-1}$ across all n was calculated as a root mean-squared (RMS) measure of the overall change in the curved centerline between the current and previous iterations. Finally, k was incremented to $k+1$ for the next iteration or until all iterations were completed.

c. Example of model fitting.

An intermediate result for ear test A is shown for the final iteration and a pair of 2D slices at values of $n=10$ [Fig. 4(A)] and $n=11$ [Fig. 4(B)].

The methods for calculating the model ellipse, and its center and major and minor axes, are described in Appendix A. Figure 4(A) for slice 10 shows the individual canal-wall data with x markers as well as the ellipse that was fitted to the data. Text in the panel displays calculated values of the major and minor axes, and the area and eccentricity. This was the most lateral slice wholly contained in the interior of the canal. Its eccentricity of 0.72 showed that the canal wall was not circular at this location. The model and mean data centers were close together.

The data in the 2D slice in Fig. 4(B) for slice 11 show an open curve. Despite that property, the model ellipse closely passed through the available data but continued to make a closed curve that was distant from any data. The corresponding model axes and area were much larger in Fig. 4(B) than in Fig. 4(A) and the eccentricity much smaller, i.e., more nearly circular. The histogram for slice $n=10$ would show that all 12 sectors around the model center contained data, whereas that for slice $n=11$ would show that several sectors were empty. The condition that at least one sector was empty was the basis for determining the entrance to the ear canal, for which the location $n=10$ was the most lateral within the canal. The objective definition of canal entrance as the most lateral centerline location of a planar slice of data entirely contained within the canal was analogous to the criterion in Balouch et al. (2023), in which the analysis methods otherwise differed. The larger distance between the mean data and the center of the fitted ellipse in the bottom panel is additional evidence that slice 11 lies partially outside the canal.

d. Processing after iteration.

Upon completing all ten iterations, those iterations were selected for which the largest number of 2D slices was within the canal. If those included more than one iteration, the iteration with the smallest RMS change in the curved centerline from the previous iteration was selected as the best iteration.

As an example, the iterative estimates of the curved centerline of the ear canal of test ear A are shown in Fig. 5.

FIG. 5.

The curved canal centerline of test ear A is shown in 3D on each of ten iterations given in the legend. The top is the near-TM location and the bottom is just within the canal entrance. The best iteration 4 is identified by filled symbols whereas all other symbols are open.

Each iteration began at the most medial slice at the top of each curve and proceeded through all 2D slices to the bottom. Iteration 4 was the best iteration. Convergence to the best iteration was rapid due to the smoothing in Eq. (3).

The curved path was quantitatively described in terms of a discrete curvature and discrete torsion per unit length using the discrete Frenet frame procedure, which is described in Appendix B. The plot range for curvature and torsion was for slice data from 4 to $N-3.$ The plot range for area function and eccentricity of each 2D slice orthogonal to the curved centerline of the canal extended from 2 to $N-1.$ The data in the initial 2D slice 1 were omitted due to the fixation of its center point while the final slice N was always more lateral than the canal entrance in any case.³

Because the line segments of the curved path of each ear were calculated at increments of 1 mm along the z axis, the resulting length, $\left|\Delta {\overrightarrow{s}}_{n,k}\right|$ (of line segment n for best iteration k), was usually larger than 1 mm by an amount unique to each ear. It was convenient for group averaging across ears to adopt a uniform grid spacing along the centerline of l mm. This was accomplished using a spline interpolation of the results for each ear of area, eccentricity, curvature, and torsion.

III. RESULTS

A. Individual shape results

Individual results are shown for test ear A. Figure 6 shows the model error functions for this ear on the best iteration of data in Fig. 5.

FIG. 6.

Model error functions for ear test A for each 2D slice on its best iteration 4. (A) Center distance between model and data centers, (B) residual error of fitting ellipse to data, (C) # of empty sectors around the ellipse center. Four of the sectors were empty.

Appendix A describes the methods to calculate the model error functions, which included the center distance error and residual error, and the number of empty sectors around the model center as functions of the data in each 2D slice from 1 to 13. Figure 6(A) shows the model distance error, which is the distance between the model center and the data center in each slice. This distance was about 0.2 mm or less for slices 2–10 and increased for slices 11–13. The residual error of the model fit in Fig. 6(B) was approximately constant near 0.2 mm for slices 1–9, slightly larger for slice 10, and much larger for slice 11.

Figure 6(C) shows the number of empty sectors of the 12 sectors around the azimuth of the model ellipse, which was the measure used to locate the lateral entrance of the canal. The number was larger for the first slice near the TM, and then zero for slices 2–10. Slice 1 was the most medial slice and sensitive to the onset of contact of the fluid-filled membrane of the device with the canal wall. Thus, a non-empty slice 1 was not relevant to finding the lateral opening the canal wall. The number of empty sectors was three at slice 11 and five at slices 12 and 13. The detailed modeling for slices 10 and 11 of this ear in Fig. 4 paralleled the behavior in all three panels of Fig. 6. The model error functions showed clear evidence of the canal entrance at $n=10,$ which was defined in terms of the most medial slice in the lower panel that had no empty sectors. This example illustrates how the lateral opening of the ear canal was determined. Additional evidence for the location of the canal entrance is indicated by the large increase in the residual error between slices 10 and 11 in Fig. 6(B). This method was insensitive in preliminary analyses to varying the number of sectors (i.e., no change in the canal entrance location using eight or 16 sectors).

The canal shape outputs for test ear A are shown as a function of distance along the curved centerline of the canal in Fig. 7.

FIG. 7.

Ear-canal shape outputs for test ear A versus curved centerline of canal from its medial end. (A) Area, (B) eccentricity, (C) curvature, (D) torsion. The near-TM and entrance locations are marked in each panel.

All shape outputs smoothly varied with centerline location. The area function [Fig. 7(A)] for this ear was among the smallest in the group results presented below. The eccentricity [Fig. 7(B)] varied between 0.8 near the entrance to 0.5 near the TM, so that the cross sectional shape of the ear canal was not circular and especially so near the entrance. The canal length between the near-TM and entrance locations was 7.5 mm.

The spatial extents of the plots of curvature [Fig. 7(C)] and torsion [Fig. 7(D)] were restricted by two fewer positions at each end compared to the area and eccentricity plots, as explained in Appendix B. The first bend of the ear canal occurred at the location (6 mm) of maximum curvature [Fig. 7(C)], with maximum value of 0.045. The torsion [Fig. 7(D)] remained low at most interior points. This digital scan did not extend to regions close to the TM. The near-TM region labeled in Fig. 7 is consequently more aptly called the most medial location of the real-ear scan. Although there was one slice more medial than the one termed near-TM, it was omitted in the Results plots because it involved a 2D slice of data whose model center was constrained as fixed (see Appendix B).

A second set of results are shown for test ear B (right ear, male, 56 years) that had the longest distance in the ear canal scanned by the device. Figure 8 shows the curved centerline on each iteration.⁴ The longer length with two distinct bends is evident.

FIG. 8.

For test ear B (right ear, male, age 56 years), the curved canal is shown in 3D on each of ten iterations given in the legend. The top is the near-TM location and the bottom is just within the canal entrance. The best iteration 9 is identified by filled symbols whereas all other symbols are open.

The area function for test ear B in Fig. 9(A) was largest near the entrance and tapered near the TM while the eccentricity in Fig. 9(B) remained close to 0.8.

FIG. 9.

Ear-canal shape outputs for test ear B versus curved centerline of canal from its medial end. (A) Area, (B) eccentricity, (C) curvature, (D), torsion. The near-TM and entrance locations are marked in each panel.

The first bend of the canal was at the more lateral location of maximum curvature (of 0.026 mm⁻¹) at 8.3 mm in Fig. 9(C), which was about 10 mm from the entrance. The second bend was at the more medial location of maximum curvature (0.035 mm⁻¹) at 3 mm, or about 15.7 mm from the entrance. Thus, this digital ear scan showed both canal bends but did not extend medially to the ideal near-TM region as described in the Introduction. The torsion in Fig. 9(D) had two peaks near locations of minimum curvature, indicating out-of-plane shifts in the canal centerline.

B. Compare shape data of real-ear and earmold scans

This section presents an example, or case study, which compares the ear-canal shape output calculated from the real-ear and earmold scans. The same analysis software analyzed the image file (STL) data whether from a real-ear or earmold scan.

Adult shape data for test ear C (left ear, male, 64 years) are shown in Fig. 10 for the real-ear ear scan (lines with x symbols) and earmold scan (lines with circle symbols).

FIG. 10.

Comparison of shape outputs between a real-ear scan (x symbols) and earmold scan (circle symbols) for test ear C (left ear, male, age 64 years) versus curved centerline of canal from the medial end of the device scan. (A) Area, (B) eccentricity, (C), curvature, (D) torsion. Title shows canal length for the earmold scan versus 13.2 mm for the real-ear scan.

A key property was that the earmold scan data extended only a shorter distance from the entrance (and also more laterally to concha locations). Area data in Fig. 10(A) were similar from the entrance (13.5 mm) to about 8 mm, which was a location 5.5 mm from the entrance. At more medial locations, the earmold areas were much smaller due to the effects of the otoblock placement in the canal. The shape differences between 8 and 13.5 mm between the two sets of scan data may be due to different measurement conditions (fluid-filled membrane at excess pressure versus silicone injection), the fact that datasets were obtained several months apart, or any measurement errors. The eccentricity plots in Fig. 10(B) showed that the ear-canal wall shape was highly non-circular in both scans.

In Fig. 10(C), the first bend in terms of the maximum curvature was observed at the same location (6  or 7.5 mm from the entrance) for real-ear and earmold scan data. This slightly exceeded the mean distance of 6.0 between canal entrance and first bend reported by Balouch et al. (2023). The canal length inferred from the real-ear scan was relatively short (8.3 mm) compared to other ears, and the second bend was not fully imaged although a local maximum in curvature was present at its medial end point (2 mm). In Fig. 10(D), a maximum in torsion was observed at the same location (9 mm) in both scans, which was lateral to the location of maximum curvature. A more medial second maximum in torsion was evident in the real-ear scan at 4 mm.

The earmold scans in other ears are not further described as their calculated area functions had much shorter lengths than the real-ear scans in the same ears. Thus, this study was unable to assess whether there were systematic differences between real-ear and earmold scans.

C. Group analyses of shape data from real-ear scans

A real-ear scan was accepted as valid for the group analyses if its curved centerline length was at least 8 mm. As a result, 50 of 58 test ears were accepted as valid and analyzed as a group. The resulting histogram of lengths is shown in Fig. 11.

FIG. 11.

Histogram of curved centerline lengths for 50 digital scans of adult ears.

The median length was 11.9 mm and the largest length was 17.7 mm (for the test ear B in Fig. 9).

The boxplot of ear-canal areas versus curved centerline distance from the entrance is shown in Fig. 12(A). The median area was largest at the entrance (72 mm²) and decreased at more medial locations, e.g., to 22 mm² at 16 mm.

FIG. 12.

Box plots versus distance from the entrance of (A) cross sectional area, (B) eccentricity.

The boxplot of ear-canal eccentricity versus curved centerline distance from the entrance is shown in Fig. 12(B). The median eccentricity varied from 0.64 at the entrance to a maximum of 0.74 at 4 mm, which thereafter decreased to 0.50 at 15 mm. That is, the ear-canal cross section was non-circular.

The boxplot of ear-canal curvature is shown in Fig. 13(A). The maximum of the median curvature of 0.030 mm⁻¹ occurred at 8 mm. Its inverse of 34 mm was the minimum of the median radius of curvature (see Appendix B). This maximum curvature was approximately equal to the median location of the first canal bend. As explained in Appendix B, the axial range of length values was restricted by 2 mm in both the lateral and medial ends compared to those for ear-canal area.

FIG. 13.

Boxplots versus curved centerline distance from the entrance for (A) ear-canal curvature, (b) torsion.

The boxplot of ear-canal torsion is shown in Fig. 13(B). The maximum of the median torsion of 0.138 mm⁻¹ occurred at 5 mm, or 3 mm more lateral than the maximum of median curvature. Averaging across ears tended to obscure the variations of median curvature and torsion compared to values in individual ears (e.g., in Figs. 7, 9, and 10).

D. Area function comparisons from shape and sound data

Results from 24 adult ears in the present shape study were also tested acoustically in a companion sound study (Keefe et al., 2024). The companion study indirectly assessed the area function of the ear canal through measurements of the acoustical reflection function. It was of interest to compare the area function results obtained in the same ears in the shape and sound studies.

As described above, the shape area functions of the ear canal were measured at increments of 1 mm from the concha bowl to the canal entrance and medially at locations into the ear canal. The sound area functions were measured longitudinally at increments of $\Delta s=cT/2\approx 3.6$ mm along the curvilinear central axis of the canal from the tip of a probe inserted into the canal in a leak-free manner to a location near the TM. Here, c was the free-space phase velocity of sound and T was the sample period, i.e., the inverse of the sample rate of 48 kHz used in the companion study. The probe contained a sound source and microphone coupled to the ear canal at its tip.

Based on visual inspection of the inserted probe during subject testing, the average insertion distance of the probe tip was initially estimated to be $8\pm 3$ mm, because the probe insertion was unique for each test ear with slight variations between ears. While the inclusion criterion for group analyses above of shape data were a minimum canal length of 8 mm, shorter canal lengths were accepted from digital scans analyzed in this group comparison to better assess possible variation of probe-insertion depths in the sound recordings.

To match the area functions calculated from sound and shape measurements in the same ear, the shape data had to extend a medial distance into the canal that was at least equal to the insertion depth of the probe tip in the sound data. The sound area function data were spline interpolated to provide values at the 1 mm increment used in the shape data. The probe tip was constrained to reside at the canal location in the range from 5 to 11 mm that minimized the RMS difference per canal location between the sound and shape area functions (best match). The number of locations in the RMS difference calculation varied across ears, because of the varying length of the ear canals across subjects and the more limited depth of the digital scans of ear-canal shape relative to the sound measurements.

Of the 24 ears with shape and sound data, six ear tests had shape data extending less than 5 mm medially past the canal entrance so that these ear tests were discarded from the group comparisons. Statistics were calculated for the remaining 18 ears at the canal locations common to both the shape and sound data. Because the shape data were at increments of 1 mm, group statistics were calculated for each sound location, i.e., at multiples of $\Delta s=3.6$ mm, at which the shape data were present within 0.5 mm of the sound location. These common locations extended medially into the canal from the probe tip at distances of 0, 3.6, 7.2, and 10.8 mm (no digital ear scan extended to 14.6 mm).

An example of best-matched shape and sound area functions is shown in Fig. 14 for test ear D (right ear, female, 38 years). The sound area function is plotted over a length of 28.8 mm from near-TM to probe-tip locations. The shape area function is plotted at 1 mm increments from a mid-canal location to the canal entrance (farthest right open circle at 34 mm). The two most medial locations in the shape area function (19 and 20 mm) are biased and omitted from the possible matches because these are where the fluid-filled membrane of the device may not have adhered to the canal wall. Shape data were matched within 0.5 mm of corresponding sound data at the three most lateral points of the sound area function (28.8, 25.2, and 21.6 mm). The total canal length was calculated as the sum of the probe-tip insertion depth and the distance between probe-tip and near-TM locations, or 34 mm for test ear D.

FIG. 14.

Area functions are plotted for best-matched sound data (open circles) and shape data (filled circles) for test ear D (right ear, female, age 38 years). Distance measured from medial end of sound area function with vertical dotted line at the probe tip distance.

For the group of 18 test ears, the median distance from the lateral entrance with interquartile-range (IQR) in parentheses were 9 mm (1 mm) for the probe-tip insertion depth, 21.5 mm (14.4 mm) for curvilinear distance between probe-tip and near-TM locations (from the sound data), and 30.5 mm (13.8 mm) for total canal length.

Median and IQR values of area are listed in Table I at each location relative to the probe tip for the shape area function, the sound area function, and the difference area function $\Delta A$ defined by subtracting the shape area function from the sound area function.

TABLE I.

Comparison of median (and IQR) values of the sound and shape area functions, their area difference $\Delta A$ (sound area minus shape area) at canal locations medial to the probe tip (as listed in the Location column with 0 mm at the probe tip), and the percentage of $\Delta A$ relative to the median shape area. The number N of ear tests is shown for each location.

Location		Sound area	Shape area	$\Delta A$	Relative
(mm)	N	(mm2)	(mm2)	(mm2)	(%)
0	18	38.3 (31.3)	43.0 (18.3)	−0.3 (8.1)	−1
3.6	5	54.9 (11.9)	48.8 (9.4)	6.1 (8.4)	12
7.2	2	49.7 (4.9)	44.2 (3.9)	5.5 (2.0)	12
10.8	1	93.1	32.2	60.9	189

The median areas were similar in the shape and sound measurements at the probe tip (0 mm) and at locations 3.6 and 7.2 mm from the probe tip. However, the one test ear with shape data present at 10.8 mm medial to the probe tip had a much larger sound area than its shape area. As an alterative example of dispersion around the centroid at the probe tip, the mean (SD) of the areas were 41.4 (16.3) mm² for sound area and 42.1 (14.2) mm² for shape area, and the mean (SD) of $\Delta A$ was −0.7 (14.0) mm².

IV. DISCUSSION

A. Measurement duration and limitations

The time to calculate shape responses from image-file data was a few minutes, having only one manual step to classify image locations within the ear canal from more peripheral locations external to the ear canal (i.e., the concha, cymba, etc.). In contrast, analyses of direct area function data in past studies have been more time-consuming, e.g., from 3 to 6 h in Balouch et al. (2023). The centerline of the ear canal was iteratively calculated in the present study using a procedure that converged in several iterations. The canal entrance was calculated to within 1 mm along this centerline, and it served as the spatial origin for interpreting the shape functions on the centerline.

The main limitation of this study was that the real-ear scans of the ear canal rarely extended past, or even as deep as, the second bend of the ear canal. This would limit the clinical potential of using this imaging device, for which a scan of the complete ear canal up to the TM region would be desirable, as in Balouch et al. (2023). Despite this limitation, useful ear-canal shape data were obtained at more lateral locations, and, in particular, at locations near the tip of a probe that might be inserted into the canal for acoustical reflectance measurements.

1. Individual ears

The location of the most lateral maxima in curvature in individual real-ear scans [e.g., Figs. 7(C) and 9(C)] may not coincide with the first bend identified by a particular 2D view from a high-resolution CT imaging of the skull of a human subject. For example, Balouch et al. (2023) identified the first bend as a sharp change in canal location relative to transverse and sagittal views, and noted from Voss et al. (2020) that the first bend was likely close to the canal depth to which a soft foam tip of a probe extended. The depth of the soft foam tip was reported to be about 12 mm from the lateral canal entrance (Voss et al., 2008). This insertion depth was larger than the median depth of 9 mm in the present study for a rigid probe with an attached plastic eartip, which was much less compressible than a foam tip. The measured mean distance of 6.0 mm of the first bend from the entrance (Balouch et al., 2023) was less than these probe-insertion depths, so the areas at the first bend would be expected to exceed those at the probe-insertion location in the canal. In common practice, the first bend of the canal extends over a region rather than a single location.

General features of ear-canal shape inferred from individual device scans are: (1) scans did not extend medially to locations near the actual TM; (2) the cross sectional shape of the canal was more accurately approximated as an ellipse rather than a circle, with area functions and non-zero eccentricities that were relatively constant over the scan locations; and (3) properties of alternating maxima and minima occurred for curvature that showed the major canal bends and for torsion whose locations did not coincide with the locations of the extrema of curvature.

The custom software used in the present study to analyze ear-canal shape in individual ears is available for download (https://github.com/BoysTownOrg/phal-shape). This software has the potential use to analyze STL files obtained in non-human mammalian ear canals. That possibility has not been explored in this study.

2. Group analyses

The present area function results in ears of adults were compared with those of previous studies measuring the area function along a curvilinear centerline of the ear canal. The discrete Frenet frame used to calculate the curved canal centerline in the present study was an outgrowth of the Serret–Frenet equations used in Stinson and Lawton (1989) for cadaver ears. Balouch et al. (2023) study measured area functions in ears of human subjects from CT scan data in which the location of the entrance into the curved ear canal from the concha was objectively determined. The present study calculated the canal entrance using an alternative procedure.

In Fig. 12, the median area function in a group of ears of adults varied from 72 mm² at the entrance to values of 22–36 mm² at medial distances of 15–17 mm from the canal entrance. To compare with previous studies, the median (IQR) values of area were 47.0 (25.6) mm², 40.1 (18.9) mm², and 35.3 (20.3) mm² at distances from the canal entrance of 6, 9, and 12 mm, respectively.

The mean area at the canal entrance in adult ears was 72 mm² in Stinson and Lawton (1989) and 97 mm² (with SD of 24 mm²) in Balouch et al. (2023). Stinson and Lawton (1989) measured the average areas 6, 9, and 12 mm at distances from the canal entrance to be 53, 45, and 45 mm², respectively [average areas were estimated in Stinson and Lawton (1989) from their Fig. 12]. These average areas were within 4–5 mm² of the median areas in the present study and were within the IQRs in Fig. 12 of the present study. There was a larger difference at the 12 mm distance, which is difficult to interpret because their plotted data showed only the maximum and minimum areas at each distance.

The area at the canal entrance was larger in Balouch et al. (2023) than in the present study. This difference may be due to methodological differences in how the entrance location and its area were measured. Because ear-canal area monotonically decreased from the concha bowl into the lateral parts of the canal up to the first bend, the area data suggest that the location of the entrance may have been more lateral in Balouch et al. (2023) (or, equivalently, more medial in the present study). Alternatively, a more oblique orientation of the cross section would increase its area based on a difference in the path of the centerline in the two studies. Such differences are important to resolve in future research as these are the only two studies that used an objective method to measure the location of the canal entrance and its area.

Balouch et al. (2023) measured the location of the first bend of the canal at an average distance of 6 mm from the canal entrance, where the area was 53 mm² with a 95% confidence interval (CI) of 48–58 mm². The median area in the present study of 49 mm² was slightly smaller than this mean, but these values do not appear to differ with respect to this CI or the IQR in the present study.

Any differences in the area function data found in this study relative to past studies may be due, in part, to differences in measurement techniques, sample populations, or how anatomical landmarks were defined in each real-ear scan for the canal entrance and the location of the probe tip.

Group results on eccentricity showed that the cross sectional area of the human ear canal was better described as elliptical than circular, with a median eccentricity of 0.64 at the entrance. Yu et al. (2015) reported maximum and minimum diameter values of the ear canal at its entrance of 9.6 and 6.75 mm, respectively, for males and 9.15 and 6.3 mm, respectively, for females (these diameters are averages of their left- and right-ear data). Interpreting the maximum diameter as the diameter of the major axis of a fitting ellipse and the minimum diameter as the diameter of the minor axis of that ellipse, the corresponding areas would be 50.9 mm² in males and 45.3 mm² in females. This compares to 72 mm² in the present study. The corresponding eccentricities calculated from data in Yu et al. (2015) would be 0.71 for males and 0.73 for females, contrasted with the median of 0.64 in the present study. The IQR range in the present study included the values inferred from Yu et al. (2015). This is satisfactory agreement given the procedural differences and possible subject differences in the Taiwanese subject population examined by Yu et al. (2015) relative to subjects in the present study.

No attempt was made to interpret the group analyses of curvature and torsion in Fig. 13 because of the relative constancy of the median responses with distance. The first bend location was predicted to align with the location of the relative maximum in curvature that was most lateral in the ear canal. Rather than seeking a local maximum in the median curvature (as shown in Fig. 13), an alternative approach would be to determine the local maximum curvature, and hence, the estimated first-bend location, in each individual ear. This was not done in the present study because so many ear scans were too short to fully resolve the location of the most lateral maximum (see distribution of canal lengths shown in Fig. 11). Scan data in future studies are needed at more medial canal locations.

B. Shape and sound comparisons

The total length of the ear canal estimated above by combining data from 18 ears with shape and sound data were 30.5 mm. This slightly differs from the total length of 31.6 mm in 40 adult ears Keefe et al. (2024) from acoustical data after adding an average probe-tip distance of 9 mm from the entrance. This length difference is attributed to random error. These values are similar to the estimated total length of 30 mm in Stinson and Lawton (1989) and 31.4 in Balouch et al. (2023).

The area data at the probe-tip location and at more medial distances of 3.6 and 7.2 mm from the probe tip agreed within 12% between sound and shape measurements. Moreover, the area data in this present imaging study were similar to data in previous imaging studies at similar mid-canal locations. These two findings lend confidence that acoustical measurements of the area function may be useful in providing information on the shape of the ear canal that can be used to estimate the TM at locations closer to the TM than the probe-lip location. Unexamined in this study is a comparison of shape and sound areas at closer distances from the TM.

The finding that the ear canal geometry was more accurately represented as having an elliptical rather than a circular cross section limits the potential accuracy of 1D waveguide models of ear-canal acoustics, especially at high frequencies for which the effective canal diameter is not small compared to the quarter-wavelength of sound.⁵ In addition to the elliptical cross section, the 1D centerlines of ear canals were also curved in 3D space with specific patterns of curvature and torsion, for which no exact 1D waveguide theory can be constructed to predict sound along the centerline. A 3D finite-element model of the ear-canal sound field can be used to model acoustical effects in ear canals having elliptical (or other) cross sections and non-zero curvature and torsion of the centerline. Such validated 3D theories would be useful in understanding, and possibly improving, the relative accuracy of 1D waveguide theories of ear-canal acoustics.

V. CONCLUSIONS

A clinically approved, hand-held device was used to rapidly acquire and analyze 3D images of the geometry of the ear canal in human subjects. A novel algorithm was used to calculate the ear-canal area using an elliptical model of the cross sectional area of the canal at each location, and the curvature and torsion of the curved centerline of the canal in 3D space. The 2D cross section of the ear canal was non-circular along its central axis, which was quantified in terms of the eccentricity of the ellipse that best fitted the shape of the canal wall. The main limitation of this shape study was that it was only possible to scan the most medial 8–18 mm into the canal from the canal entrance. Hence, the shape of the ear canal closer to the TM was not assessed. The estimated canal areas were generally similar to those previously reported in the literature for distances over which scan data were acquired in the present study. Area measurements in this imaging study were combined with area measurements in the same ears acquired by an indirect acoustical inference method. The estimated areas agreed within 12% in the shape and sound experiments for canal distances close to the insertion distance of the probe used in the acoustical experiments, and for more medial locations, 3.6 and 7.2 mm from the probe tip. This provides evidence that the information derived from imaging studies of ear-canal geometry may help interpret results from acoustical measurements using an ear-canal probe.

SUPPLEMENTARY MATERIAL

See the supplementary material for data supporting the findings of this study.

APPENDIX A: MODELING A 2D SLICE AS AN ELLIPSE

The scan data on each 2D slice were fitted to an ellipse. Each 2D slice was a set of points $(x_n,\hspace{0.17em}{y}_n)$ in a local x-y coordinate system, in which index n varied over all N points on the slice. These coordinates differed from the x-y coordinates in the 3D system. The following equation embodies the condition that all points lie on a conic section in a plane

$$A\hspace{0.17em}{x}_n^2+B\hspace{0.17em}{x}_n\hspace{0.17em}{y}_n+C\hspace{0.17em}{y}_n^2+D\hspace{0.17em}{x}_n+E\hspace{0.17em}{y}_n+F=0.$$ (A1)

The real data are such that all points do not generally lie on the conic section. This is accommodated by introducing a non-zero error term, ${\mathcal{E}}_n$, on the right-side of the above equation as follows:

$$A\hspace{0.17em}{x}_n^2+B\hspace{0.17em}{x}_n\hspace{0.17em}{y}_n+C\hspace{0.17em}{y}_n^2+D\hspace{0.17em}{x}_n+E\hspace{0.17em}{y}_n+F={\mathcal{E}}_n.$$ (A2)

The ordinary least squares approximation to solve Eq. (A2) to calculate the model parameter values $(A,B,C,D,E,$ and F) that minimize the total mean-squared error, $\mathcal{E}$, over all N points defined by

$$\mathcal{E}=\frac{1}{N}\sqrt{\sum _{n=1}^N{\left|{\mathcal{E}}_n\right|}^2}.$$ (A3)

The limitation of this ordinary least squares procedure is that while it fits a conic section, it does not guarantee that the modeled conic section is an ellipse.

The modeled curve is an ellipse only if the discriminant below is negative:

$$B^2-4A\hspace{0.17em}C<0.$$ (A4)

An approximate solution to fit a model ellipse was obtained by noticing that Eq. (A1) is satisfied if each model coefficient is multiplied by the same (non-zero) constant, which is chosen to transform Eq. (A4) into the equality, $4A\hspace{0.17em}C-B^2=1.$ The model coefficients of the ellipse were calculated using a constrained least squares fit of the data in which this discriminant equality was added with a Lagrange multiplier to the error function (Fitzgibbon et al., 1996, 1999). The present study used the procedure as simplified by Hal´ıř and Flusser (1998). Once the parameters values were determined, the error terms, ${\mathcal{E}}_n$, were calculated using Eq. (A2) for all n, and the total residual error was calculated from Eq. (A3). While this residual error was not the error minimized in the constrained least squares approximation, it nonetheless served to quantify the approximation error between the fitted ellipse and the data.

In terms of this approximate solution, the model center $\{X_0,Y_0\}$ of the ellipse was

$$\begin{array}{c}{X}_0=-2C\hspace{0.17em}D+B\hspace{0.17em}E,\hfill \\ Y_0=-2A\hspace{0.17em}E+B\hspace{0.17em}D.\hfill \end{array}$$ (A5)

The major axis a and minor axis $b\le a$ of the ellipse were

$$\{a,b\}=\sqrt{2(A\hspace{0.17em}{E}^2+C\hspace{0.17em}{D}^2-\mathit{BDE}-F)\hspace{0.17em}(A+C\pm \sqrt{{(A-C)}^2+B^2})},$$ (A6)

in which a was the larger of the two expressions on the right side and b the smaller. The area S and eccentricity $ϵ$ of the ellipse were

$$\begin{array}{c}S=\pi ab,\hfill \\ ϵ=\sqrt{1-{(b/a)}^2}.\hfill \end{array}$$ (A7)

The eccentricity measured the extent to which the model ellipse was non-circular.

1. Correspondence with examples in the main text

For test ear A considered in the main text, the model ellipse center given by Eq. (A5) is shown with the data center in Fig. 4 for each of a pair of adjacent 2D slices. The center distance error plotted for test ear A in Fig. 6(A) was calculated as the distance between the model and data centers. The residual error plotted in Fig. 6(B) was calculated using Eq. (A3). For scans such as Fig. 4(B), three empty sectors were found and plotted as such in Fig. 6(C) for slice 11. For such slices with one or more empty sectors, the conic section fitted by ordinary least squares was often a parabola while the constrained least squares approximation always calculated the best-fitting ellipse.

APPENDIX B: DISCRETE FRENET FRAME OF CANAL CENTERLINE

A discrete Frenet frame was used to calculate the intrinsic curved path of a connected set of discrete-line segments in a 3D space. It did so by calculating two new variables, curvature and torsion, which were allowed to differ in each segment. For the best iteration of the canal centerline, the unit tangent vector, ${\overrightarrow{T}}_n$, of each slice was defined in Eq. (2) in terms of the discrete distance vector between the points (in which the best iteration number is suppressed). All variables in the discrete Frenet frame were calculated in terms of these tangent vectors for all points on the discrete-line segments for n between 1 to N.

The unit binormal vector, ${\overrightarrow{B}}_n$, and normal vector, ${\overrightarrow{N}}_n$, were defined by cross-products to form an orthonormal set of vectors with the tangent vector

$${\overrightarrow{B}}_n=\frac{{\overrightarrow{T}}_{n-1}\times {\overrightarrow{T}}_n}{\left|{\overrightarrow{T}}_{n-1}\times {\overrightarrow{T}}_n)\right|},$$ (B1)

$${\overrightarrow{N}}_n={\overrightarrow{B}}_n\times {\overrightarrow{T}}_n.$$ (B2)

Because ${\overrightarrow{B}}_n$ depended on both ${\overrightarrow{T}}_n$ and ${\overrightarrow{T}}_{n-1}$ and because each of these was defined in Eq. (2) in terms of a pair of points, then ${\overrightarrow{B}}_n$ was only defined for n from 3 to N (for a tangent vector defined by a forward difference). The matrix equation connecting the frames between $n-1$ and n is (Hu et al., 2011; Lu, 2013)

$$\left(\begin{array}{c}{\overrightarrow{N}}_n\\ {\overrightarrow{B}}_n\\ {\overrightarrow{T}}_n\end{array}\right)=\left(\begin{array}{ccc}\hspace{0.17em}\text{cos}\hspace{0.17em}{\psi }_{n,n-1}\hspace{0.17em}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta }_{n,n-1}& \hspace{0.17em}\text{cos}\hspace{0.17em}{\psi }_{n,n-1}\hspace{0.17em}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta }_{n,n-1}& -\text{sin}\hspace{0.17em}{\psi }_{n,n-1}\\ -\text{sin}\hspace{0.17em}{\theta }_{n,n-1}& \hspace{0.17em}\text{cos}\hspace{0.17em}{\theta }_{n,n-1}& 0\\ \hspace{0.17em}\text{sin}\hspace{0.17em}{\psi }_{n,n-1}\hspace{0.17em}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta }_{n,n-1}& \hspace{0.17em}\text{sin}\hspace{0.17em}{\psi }_{n,n-1}\hspace{0.17em}\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta }_{n,n-1}& \hspace{0.17em}\text{cos}\hspace{0.17em}{\psi }_{n,n-1}\end{array}\right)\left(\begin{array}{c}{\overrightarrow{N}}_{n-1}\\ {\overrightarrow{B}}_{n-1}\\ {\overrightarrow{T}}_{n-1}\end{array}\right),$$ (B3)

with angles ${\psi }_{n,n-1}$ and ${\theta }_{n,n-1}$ implicitly defined by

$$\text{cos}\hspace{0.17em}{\psi }_{n,n-1}={\overrightarrow{T}}_n·{\overrightarrow{T}}_{n-1},\hspace{1em}\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta }_{n,n-1}={\overrightarrow{B}}_n·{\overrightarrow{B}}_{n-1}.$$ (B4)

The angle ${\psi }_{n,n-1}$ measures the change in direction of the tangent vector between segments $n-1$ and n, and the angle ${\theta }_{n,n-1}$ measures the change in direction of the binormal vector. The curvature ${\kappa }_n$ and torsion ${\tau }_n$ (per unit length) are defined in terms of these angles by

$${\kappa }_n=\frac{{\psi }_{n,n-1}}{\left|\Delta {\overrightarrow{s}}_n\right|},\hspace{1em}{\tau }_n=\frac{{\theta }_{n,n-1}}{\left|\Delta {\overrightarrow{s}}_n\right|}.$$ (B5)

Because ${\theta }_{n,n-1}$ in Eq. (B4), and thus ${\tau }_n$ in Eq. (B5), depend on both ${\overrightarrow{B}}_n$ and ${\overrightarrow{B}}_{n-1},$ then the torsion was only calculated for n from 4 to $N-3.$

Because Eq. (1) was defined using both forward and backwards differences, the torsion was also omitted for the final three points. Thus, the plot range for torsion is from 4 to $N-3$ and a similar plot range was used for curvature.

A simple example of a discrete Frenet frame is for a discretized helix in 3D, for which its calculated curvature and torsion are constant. Its curvature describes the inverse of its local radius of curvature while its torsion describes a uniform translation along the curved path in the direction perpendicular to the plane in which the projected curve is circular.⁶ Generalizing from this helix example, a useful definition of the radius of curvature of a curve in 3D space is the inverse of the curvature ${\kappa }_n$ at a particular location, as long as the torsion is not too large.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material. In addition, the raw STL files for each of the 58 ears included in group analyses or real-ear scans are available to download with the shared custom software.