Download Computer-integrated finite element modeling of human middle ear

Original paper
Biomechan Model Mechanobiol 1 (2002) 109 – 122 Springer-Verlag 2002
DOI 10.1007/s10237-002-0014-z
Computer-integrated finite element modeling of human
middle ear
Q. Sun, R. Z. Gan, K.-H. Chang, K. J. Dormer
Abstract The objective of this study was to produce an improved finite element (FE) model of the
human middle ear and to compare the model with human data. We began with a systematic and
accurate geometric modeling technique for reconstructing the middle ear from serial sections of a
freshly frozen temporal bone. A geometric model of a human middle ear was constructed in a
computer-aided design (CAD) environment with particular attention to geometry and microanatomy.
Using the geometric model, a working FE model of the human middle ear was created using
previously published material properties of middle ear components. This working FE model was
finalized by a cross-calibration technique, comparing its predicted stapes footplate displacements with
laser Doppler interferometry measurements from fresh temporal bones. The final FE model was shown
to be reasonable in predicting the ossicular mechanics of the human middle ear.
Introduction
Hearing loss is the most common sensory impairment in the United States. More than 28 million
Americans have hearing and/or speech impairments. Conductive hearing loss, involving the conductive pathway, results from external or middle ear disorders and often can be corrected by surgery
(VMBHRC 2000). Advances in otology and bioengineering have led to new surgical reconstruction
techniques and implantable hearing devices that also allow otologist s to restore hearing to sensorineural loss patients. Further advancements of techniques and devices for the middle ear require a
better understanding, however, of middle ear transfer functions and the biomechanics of sound
transmission.
The middle ear consists of the eardrum, three ossicles (malleus, incus and stapes), their suspensions
by ligaments and tendons, and the bony middle ear cavity. Middle ear components have tiny and complex
geometry, and play an essential role in sound transmission. Sounds collected and conveyed in the
external ear canal are first transformed into mechanical vibrations of the eardrum and ossicular chain,
and then transformed into traveling waves in the fluid-filled cochlea (inner ear). Devices for hearing
restoration to middle ear conductive impairments or inner ear (sensorineural loss) are often mechanical
in nature and thus can be studied using mechanical or analog computational models.
Received: 18 February 2002 / Accepted: 6 June 2002
Q. Sun, R. Z. Gan (&), K.-H. Chang
School of Aerospace and Mechanical Engineering,
The University of Oklahoma, 865 Asp Avenue,
Room 200, Norman, OK 73019, USA
e-mail: [email protected]
Tel.: +1-405-3251099
Fax: +1-405-3251088
K. J. Dormer
University of Oklahoma Health Sciences Center,
Oklahoma City, OK 73190, USA
R. Z. Gan, K. J. Dormer
Hough Ear Institute, Oklahoma City,
OK 73112, USA
The preparation of temporal bone histological sections of Robert
K. Dyer, Jr., MD is gratefully recognized. The Whitaker Foundation
supported this work (Research Grant RG-98-0305).
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Several quantitative middle ear models, including an analytical model (Rabbitt and Holmes 1986),
analog circuit model (Hudde and Weistenhöfer 1997) and multibody model (Eiber et al. 2000) sought
to predict normal and pathological mechanics in the middle ear. Yet, the finite element (FE) method, a
general numerical procedure, has distinct advantages in modeling complex biological systems when
compared to other methods. FE modeling is capable of easily modeling the complex geometry, ultrastructural characteristics, and non-homogenous and anisotropic material properties of biological
systems. FE models can also easily determine the detailed vibration shapes, stress distributions, and
dynamic behavior at any location in a system.
Others have performed FE analysis of middle ear mechanics since Funnell and Laszlo (1978) first
published FE model of the cat eardrum. Several FE models have been used to simulate the static or
dynamic behaviors of the middle ear subsets or entire middle ear. Among these models, the following
five three dimensional (3D) FE models for the middle ear represent the state-of-the-art in this area.
Ladak and Funnell (1996)
This model elaborated on a previous model of the cat eardrum (Funnell et al. 1987) by adding explicit
representations of the ossicles and cochlear load. In the 1996 model, the eardrum was modeled as a
curved conical shell. The model was applied to investigate sound transmission in the normal cat
middle ear for frequencies below 1 kHz and modified to simulate the effects of two types of middle ear
surgeries. This model extended Funnell et al.’s early FE work into a new level. However, the geometric
details were not considered in their model. Their cat model also cannot be employed to investigate the
behavior of the human middle ear.
Wada et al. (1992)
This human middle ear FE model included eardrum, ossicles, and cochlear impedance, and assumed a
fixed rotational axis from the anterior process of the malleus to the short process of the incus. Middle
ear geometry was based on published geometric data of middle ear components. The model was later
modified (Wada et al. 1996) to include anterior mallear and posterior incudal ligaments, tensor
tympani and posterior stapedial tendons, middle ear cavity (as a rectangular solid), and ear canal (as a
rigid tube). The frequency response to an acoustic excitation was derived for frequencies below 3 kHz.
The model was further modified (Koike et al. 2002) by adding some new features such as the ligaments, tendons, incudostapedial joint, external ear canal, and middle ear cavities. Wada et al.’s work
includes detailed modeling discussion and application analysis, but they did not provide a method to
reconstruct accurate geometry of the entire middle ear.
Williams et al. (1996)
This FE model included the eardrum, ossicles, and outer ear canal, and was a sequel to their preliminary models of the human middle ear (Williams and Lesser 1990). The geometry of the eardrum
and ossicles was constructed using methods similar to those of Funnell and Wada, but the geometry of
the ear canal was determined using nuclear magnetic resonance imaging (NMRI) on a healthy human
subject. The anisotropic property of the eardrum was included in this model by introducing pseudofibers. The most important contribution of Williams et al. falls in the fact that diseases of the middle
ear were first related to model parameters in their FE model. However, the geometric details were not
considered in their model and no suspensory ligaments and tendons were added.
Beer et al. (1996, 1999)
The geometric models of their middle ear components were first constructed based on measurements
of eardrum surfaces and ossicles of human temporal bones using laser-scanning microscopy. The FE
models of middle ear components were generated separately on the basis of the corresponding
geometric models and then assembled into different subsets of the middle ear components. The Beer
FE models also were applied to investigate the dynamic characteristics of the middle ear system. The
geometric modeling method employed by Beer et al. is accurate for each component. However,
although the scaling procedure was performed for assembly, the inconsistency between different
components was unavoidable because each component came from different temporal bones. It is also
hard to keep these components in proper spatial positions and orientations while assembling.
Prendergast et al. (1999); Ferris and Prendergast (2000)
This model of the human middle ear included eardrum, ossicles, anterior mallear ligament, posterior
incudal ligament, posterior stapedial tendon, tensor tympani tendon, incudomalleolar joint, incudostapedial joint, and ear canal. The geometry of the model was same as Williams et al. (1996) and the
material properties were similar to Wada et al. (1996). The difference from other models is that more
ligaments or tendons and joints were included in this model and the frequency response analysis was
performed on a broad frequency range. The model was used to examine the frequency response
characteristics of the normal and reconstructed middle ear.
Development of these FE models was concentrated on improving FE analysis on middle ear mechanical functions and necessarily was preceded by a search for more realistic geometrical reconstructions and material properties of the middle ear. All evaluated the models by comparing
experimental observations on animals or temporal bones. All of these FE models are useful for a better
understanding of human middle ear biomechanics although none is a perfect model.
We now have constructed an FE model of the human middle ear based on serial histological sections of
a human temporal bone. This improved modeling method using computer-aided geometric serial reconstruction (histological sections of a normal human temporal bone) was used to build a geometric
model of the middle ear with ligaments and muscle-tendons. The geometric model and published
material properties of the middle ear components were integrated to create an FE mechanical model.
Finally, the FE model was evaluated by comparing the frequency responses of the model with five sets of
laser Doppler interferometry (LDI) experimental measurements on human subjects or human temporal
bones. Compared to the published FE models of others, the novel attributes of this modeling process
include: (i) developing a practical procedure to build a FE model of the middle ear based on computeraided 3D geometric reconstruction from human temporal bone; (ii) providing a useful 3D human middle
ear FE model with all attached ligaments and muscle tendons; (iii) evaluating the FE model using five sets
of independent laser Doppler interferometry frequency response measurements.
Building the working FE model
The FE modeling approach was divided into three parts: computer-aided geometric modeling, constructing the working FE model, and cross-calibration of the working model to obtain the final FE
model.
Geometric modeling
To determine the morphology of the middle ear, a normal fresh human temporal bone (female, age 52,
right ear) was fixed, decalcified, and embedded in celloidin. The resulting block was removed from the
chloroform, trimmed, mounted on a microtome plate and stored in 70% alcohol. Before the cellodin
block was sectioned, four parallel fiducial holes (perpendicular to the cutting plane) were made in the
celloidin block using a drill press. Permanent ink was injected into the fiducial holes to stain the
marks. Sections (n = 600) were then cut on a sliding microtome (AO 860, American Optical Corporation) at a thickness of 20 lm. Next, in sequence, every tenth section was stained and mounted on
glass slides. Finally, each histology slide was scanned into a computer using a flatbed scanner and
saved as an image file. The images were aligned with a template constructed by a typical section image
using the fiducial marks (Photoshop, Adobe Systems, Inc.). Aligned images were then trimmed as
standard-sized images, brought onto a sketch plane in CAD software (SolidWorks, SolidWorks, Inc.),
and fitted in a reference rectangle with the same size as the image so that the actual size could be
represented accurately in SolidWorks. The images were then digitized by marking points along the
outlines of the middle ear structures as identified by an otologic surgeon. These structures included
eardrum, ossicles, attached ligaments, and muscle tendons. The B-spline curves were constructed to
best fit these points using the curve fitting technique and were quilted to form the closed boundary
surfaces for each middle ear structure using the surface skinning technique. More detailed procedures
have been reported previously (Sun 2001; Sun et al. 2002).
Finite element modeling
Finite element mesh
All surfaces of the geometric model were translated into HyperMesh (a commercial FE pre- and postprocessing software developed by Altair Computing, Inc.) for finite element modeling of the middle ear.
Based on these surfaces, the FE mesh of the middle ear components and attached ligaments/tendons was
created by the meshing capabilities of HyperMesh, as shown in Fig. 1. Figure 1a shows the meshed three
ossicles and eardrum and Fig. 1b displays the anterior-medial view of the middle ear FE model with
attached ligaments/tendons and cochlear fluid constraints. A total of 1,746 three-noded triangular and
four-noded quadrilateral shell elements were created to mesh the eardrum. Surrounding the eardrum
periphery, the tympanic annulus was modeled using 113 three-noded triangular and four-noded
quadrilateral shell elements. A total of 812 eight-noded hexahedral, six-noded pentahedral and fournoded tetrahedral solid elements were created to mesh three ossicles, two joints (incudomalleolar and
incudostapedial joints) and six ligaments/tendons (superior mallear ligament C1, lateral mallear
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Fig. 1. a Finite element models of human right middle ear components. b Finite element model of human
right middle ear in isometric view. C1 represents superior mallear and incudal ligament, C2 lateral mallear
ligament, C3 posterior incudal ligament, C4 anterior mallear ligament, C5 posterior stapedial muscle tendon, C6
cochlear fluid (cochlear impedance) and C7 tensor tympani tendon
ligament C2, posterior incudal ligament C3, anterior mallear ligament C4, posterior stapedial tendon C5
and tensor tympani tendon C7). Lying in the stapes footplate plane, there were 25 spring elements used to
model the stapedius annular ligament. A total of 49 spring–dashpot elements perpendicular to the
footplate plane were incorporated to model the cochlear fluid (cochlear impedance). The total number of
nodes was 1,497, and the number of degrees of freedom was 4,491 in the model shown in Fig. 1b. The size
of the model was adequate for an FE analysis with a reasonable computation time.
Material properties
The human middle ear functions as a linear system with small vibrations for acoustic-mechanical
transmission from the eardrum to the cochlea. For the middle ear system, the maximum displacement
occurs at the eardrum when a sound pressure is applied on the eardrum. Tonndorf and Khanna (1972)
showed that the maximum displacement of the eardrum is less than 1 lm when a sound pressure of
121 dB (maximum sound level for hearing tolerance) is applied on the eardrum. Based on a simplified
conical eardrum, we can roughly estimate the strains in the eardrum to be in the order of 10)4
corresponding to the maximum displacement. Thus, it is reasonable to assume linear material
properties for the middle ear system.
Mechanical characteristics and functions of the middle ear were incorporated into the FE modeling
by assigning different material properties to different parts of the middle ear system. The Poisson’s
ratio was assumed 0.3 for all materials of the middle ear system based on the fact that all published
Poisson’s ratios of middle ear components are close to this value. In addition, there was no previous
evidence for significant effects of Poisson’s ratio on dynamic behavior of the middle ear system
(Funnell and Laszlo 1982). The element-damping matrix for the solid and shell elements was expressed by
½C ¼ a½M þ b½K
where [M] and [K] are element mass and stiffness matrices of the solid and shell elements, respectively,
and a and b are the damping parameters (SAS IP Inc. 1998). The damping coefficient was employed to
calculate the element-damping matrix of the spring–dashpot element. Modeling considerations for
middle ear components, joints, and connections are described in the following subsections.
Eardrum
The eardrum consists of three layers: non-glandular skin of external meatus covering its outer surface,
intermediate fibrous layer, and mucous membrane continuous with that of the middle ear. An
ultrastructural study of the eardrum shows the regularity of the radial and circumferential fibers
appearing in the pars tensa (Lim 1970). The eardrum in this FE model was modeled as a linear elastic
shell structure with homogeneous and orthotropic material properties. Materials of the eardrum were
assumed to have a circumferential Young’s modulus of 20 MPa in the pars tensa and 10 MPa in the
parsa flaccida, and radial Young’s modulus of 32 MPa in the pars tensa and 10 MPa in the pars
flaccida, respectively (Beer et al. 1996; Prendergast et al. 1999). The annular ligament, which divides
the eardrum into pars tensa and pars flaccida, was assumed to have the same Young’s modulus as the
pars tensa, as shown in Fig. 2. The thickness varied in accordance with the eardrum geometry of the
CAD model from histological sections of the temporal bone.
Ossicles
Three ossicles (malleus, incus and stapes) were modeled as linear elastic, isotropic, and homogenous
with a Young’s modulus 14.1 GPa (Speirs et al. 1999). The mass densities vary in different portions of
the ossicles (Kirikae 1960) and are listed in Table 1. Although there may exist a relationship between
the Young’s modulus of a material and its density, this relationship was not considered in our work
due to lack of experimental data.
Connection between malleus and eardrum
The primary connection between the malleus and eardrum was modeled as linear elastic, homogeneous and isotropic material (Fig. 2). The connection is softer than the ossicular bones because the
manubrium of the malleus is not tightly attached to the eardrum in human temporal bone (Graham
et al. 1978). Therefore, the Young’s modulus of the connection was assumed to be one third of the
ossicles (4.7 GPa) to simulate the ‘‘softer’’ effect.
Incudomalleolar joint
Data from sound transmission studies on the human ear indicate the incudomalleolar joint, as shown
in Fig. 2, is relatively rigid at physiological sound pressure levels for frequencies below 3 kHz
(Vlaming 1987). This means that there is no relative motion between malleus and incus in low
frequency range even though this may not be the case at high frequencies or high sound pressure
levels. Therefore, the incudomalleolar joint was modeled as a homogeneous and isotropic hard tissue
with a Young’s modulus identical to the ossicles (14.1 GPa).
Incudostapedial joint
The incudostapedial joint, as shown in Fig. 2, does not appear to be rigid (Hüttenbrink 1988)
and there is some relative motion between incus and stapes to protect the inner ear from loud
noise and sudden pressure change. The incudostapedial joint was modeled as homogeneous and
isotropic ligamentous tissue (Ferris and Prendergast 2000) with Young’s modulus of 0.6 MPa
(Wada et al. 1996).
Fig. 2. Illustration for the connection
between malleus and eardrum, incudomalleolar joint, incudostapedial joint,
and the eardrum structure
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Table 1. Material properties of middle ear components used for the final FE model
Data used for FE model
Source
1.2 · 103
Williams et al. (1990)
2.0 · 107 (circumferential), 3.2 · 107 (radial)
1.0 · 107 (circumferential), 1.0 · 107 (radial)
a = 0 s)1, b = 0.0001 s
Beer et al. (1996)
Prendergast et al. (1999)
Sun (2001)
2.55 · 103 (for head), 4.53 · 103 (for neck),
3.70 · 103 (for handle)
1.41 · 1010
Kirikae et al. (1960)
2.36 · 103 (for body),
2.26 · 103 (for short process),
5.08 · 103 (for long process)
1.41 · 1010
a = 0 s)1, b = 0.0001 s
Kirikae et al. (1960)
2.20 · 103
1.41 · 1010
a = 0 s)1, b = 0.0001 s
Kirikae (1960)
Herrmann et al. (1972)
Sun (2001)
3.2 · 103
1.41 · 1010
a = 0 s)1, b = 0.0001 s
Assumed
Assumed
Sun (2001)
Density (kg/m3)
Young’s modulus (N/m2)
1.2 · 103
6.0 · 105 N/m2
Damping
a = 0 s)1, b = 0.0001 s
Assumed
Wada et al. (1996),
Prendergast et al. (1999)
Sun (2001)
1.0 · 103
4.7 · 109
a = 0 s)1, b = 0.0001 s
Assumed
Assumed
Sun (2001)
Eardrum
Density (kg/m3)
Young’s modulus (N/m2)
pars tensa
pars flaccida
Damping
Malleus
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Density (kg/m3)
Young’s modulus (N/m2)
Incus
Density (kg/m3)
Young’s modulus (N/m2)
Damping
Stapes
Density (kg/m3)
Young’s modulus (N/m2)
Damping
Incudomalleolar joint
Density (kg/m3)
Young’s modulus (N/m2)
Damping
Incudostapedial joint
Malleus attachment
on the eardrum
Density (kg/m3)
Young’s modulus (N/m2)
Damping
Speirs et al. (1999)
Speirs et al. (1999)
Sun (2001)
The damping parameters for the eardrum, ossicles, two joints, and connection between the
malleus and eardrum were assumed to be a = 0 s)1, b = 0.0001 s. The connection between the
malleus and eardrum, and the incudomalleolar and incudostapedial joints connect the middle ear
components by coupling the corresponding FE nodes of them. All the material properties used
for the middle ear FE model are summarized in Table 1 with the corresponding sources.
Boundary conditions
Boundaries of the FE model include suspensory ligaments, intra-aural muscle tendons, tympanic
annulus, stapedius annular ligament, and cochlear fluid. As shown in Fig. 1b, the malleus, incus, and
stapes are attached to the eardrum by the malleus and at the oval window by the stapes footplate.
Suspensory ligaments and intra-aural muscle tendons also supported the ossicles. Four major suspensory ligaments (superior malleus C1, lateral malleus C2, posterior incus C3, and anterior malleus
C4) and two intra-aural muscle tendons (posterior stapedial C5 and tensor tympani C7) were regarded
as elastic constraints. The tympanic annulus was modeled as an elastic ring that connects the periphery of the eardrum and the bony wall of the ear canal. The cochlear fluid was assumed as a
viscoelastic constraint, C6. The detailed modeling considerations for the boundary conditions are
described as follows.
Suspensory ligaments and intra-aural muscle tendons
Suspensory ligaments and intra-aural muscles were modeled as homogeneous, ligamentous tissue. The
published Young’s moduli were chosen for all ligaments and muscles except the anterior mallear
ligament (C4). The published Young’s modulus for ligament C4, 21 MPa (Wada et al. 1996), was not
used in our model because it differs significantly from rest of the ligaments, 0.049–2.6 MPa (Beer et al.
1996; Wada et al. 1996). This consideration was based on the assumption that the materials having
similar histological structures should have similar mechanical properties. The Young’s modulus of
ligament C4 (2.1 MPa) was finalized through the FE model cross-calibration process, a method for
selecting model parameters by comparing the stapes footplate displacements between the FE model
prediction and experimental measurement. Damping parameters for all ligaments and muscle
tendons were assumed to be a = 0 s)1, b = 0.0001 s. Each suspensory ligament or intra-aural
muscle tendon was attached to the ossicles by coupling corresponding FE nodes among them at
one end, and fixed at the other.
Tympanic annulus
To simulate the flexible support around the periphery of the eardrum, the tympanic annulus was
modeled as an elastic ring of 0.2 mm width and 0.2 mm thickness. The annulus was connected to the
periphery of the eardrum at the inner edge by coupling corresponding nodes and was constrained
against translational movements at the outer edge based on the model of Williams et al. (1996). The
Young’s modulus was initially assumed identical to the radial value for the eardrum, and then adjusted
as 0.6 MPa through the FE model cross-calibration process. The Young’s modulus of the part corresponding to the notch of Rivinus (Fig. 2) was assumed to be one third of that of the annulus.
Damping parameters for the tympanic annulus were assumed to be identical to the eardrum
(a = 0 s)1, b = 0.0001 s).
Stapedius annular ligament
The in-plane stiffness action on the stapes footplate due to stapedius annular ligament was represented by 25 linear spring elements distributed evenly around the periphery of the footplate (Fig. 3).
These springs are perpendicular to the side periphery of the footplate. Each spring was attached to the
corresponding node of the side periphery at one end, and fixed at the other, simulating the attachment
to the margin of vestibular fenestra. This arrangement replicates the fact that the stapedius annular
ligament restrains mainly against in-plane motion of the footplate. The out-of-plane stiffness was
combined into the consideration for the stiffness induced by the cochlear impedance. Assuming that
the material of the stapedius annular ligament is uniformly distributed surrounding the side periphery, Lynch et al. (1982) estimated the Young’s modulus of the stapedius annular ligament to be
10 kPa. Based on Lynch’s assumption, the stapedius annular ligament can be considered as a slab
surrounding the side periphery of the footplate. The thickness of the slab was estimated as 0.057 mm
by measuring the average space between the side periphery of the footplate and the wall of the
oval window on histological sections of the human temporal bone. The area of the footplate periphery
was 1.275 mm2 in our middle ear FE model. If the slab is divided uniformly into 25 segments along the
side periphery of the footplate, each segment can be treated as a bar element perpendicular to the
footplate periphery. Thus, each bar has a cross-sectional area of 0.051 mm2 and a length of 0.057 mm.
Based on this calculation, the spring constant for each spring was estimated as 9.0 N/m.
Cochlear fluid impedance
Cochlear fluid impedance was considered as viscous damping in several FE models (Wada et al. 1996;
Ferris et al. 2000). The damping coefficients were estimated using the acoustic input impedance of the
cochlea. Ladak et al. (1996) modeled the cochlear impedance as a set of springs perpendicular to
Fig. 3. Stapedius annular ligament respresented by 25 spring elements. The spring elements
were oriented perpendicular to the side periphery of the stapes footplate
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Fig. 4. Cochlear fluid impedance represented
by 49 spring–dashpot elements. The spring–
dashpot elements were oriented perpendicular to the footplate plane
the plane of the stapes footplate. The stiffness coefficient was estimated at a specific frequency using
the acoustic impedance or acoustic compliance of the cochlea. In our FE model, the action of the
cochlear fluid on the stapes footplate was modeled as a set of 49 spring–dashpot elements distributed
on the footplate shown in Fig. 4. Note that these spring–dashpot elements are oriented in the
normal direction of the footplate plane. The total stiffness and damping coefficient were determined
as 60 N/m and 0.054 N m/s by the cross-calibration process because the published data were estimated at specific frequencies and not suitable to the FE model in general. Table 2 lists the boundary
conditions of the FE model.
FE model cross-calibration process
Cross-calibration is a process for determining additional model parameters by comparing the stapes
footplate displacements between our working FE model and the experimental data obtained from
human temporal bones. The objective of the process was to minimize the difference of our working
FE model and the experimental data, by selecting appropriate, as yet, undetermined parameters of
the working FE model. Experimental data were the standard of comparison, with which the model
should agree. Mean peak-to-peak stapes footplate displacement data obtained at the input sound
pressure of 90 dB SPL at the eardrum by Gan et al. (1999) were used to derive undetermined
parameters of the final FE model using this cross-calibration process. The experimental measurements were made using laser Doppler interferometry on six normal human temporal bones. To be
consistent with the experimental setup, a uniform pressure of 90 dB SPL (0.632 Pa) was applied to
the lateral side of the eardrum in the FE model. The harmonic analysis was conducted on the FE
model across the frequency range of 250–8,000 Hz using commercial, finite element analysis software (ANSYS, ANSYS Inc.).
The mechanical properties of middle ear components, joints, and connections were assigned based
on data published by others and our estimations from published experimental observations. Most of
Table 2. Boundary conditions used for the final FE model
Young’s modulus or spring constant
Damping
Superior mallear ligament (C1)
4.9 · 104 N/m2 (Beer et al. 1996)
Lateral mallear ligament (C2)
Posterior incudal ligament (C3)
6.7 · 104 N/m2 (Beer et al. 1996)
6.5 · 105 N/m2 (Wada et al. 1996;
Prendergast et al. 1999)
2.1 · 106 N/m2 (determined by FE model
cross-calibration)
5.2 · 105 N/m2 (Wada et al. 1996;
Prendergast et al. 1999)
2.6 · 106 N/m2 (Wada et al. 1996;
Prendergast et al. 1999)
60 N/m (determined by FE model
cross-calibration process)
9 N/m (converted from Lynch et al. 1982)
6.0 · 105 N/m2 (determined by FE model
cross-calibration)
a = 0 s)1,
b = 0.0001 s
Anterior mallear ligament (C4)
Posterior stapedial muscle (C5)
Tensor tympani muscle (C7)
Cochlear fluid (C6)
Stapedius annular ligament
Tympanic annulus
0.054 N s/m
0 N s/m
a = 0 s–1,
b = 0.0001 s
the boundary conditions in our model are based on the published data of others. Four parameters
(Young’s moduli of the anterior mallear ligament and tympanic annulus, stiffness and damping
coefficients of the cochlear fluid) were determined using the FE model cross-calibration process. Thus,
the final FE model (Fig. 1b) with all parameters (Tables 1 and 2) was completed for subsequent model
evaluations or mechanical analyses.
Evaluation of the final FE model
Finally, five independent experimental results on human subjects or human temporal bones were used
to comparatively evaluate the final FE model.
Comparison of frequency response curves of stapes footplate displacements
Experimental data of stapes footplate displacements obtained from 17 human temporal bones by Gan
et al. (2001) were initially selected for the comparison. The measurements were conducted using a
single point laser Doppler interferometer. When pure tone narrow band, filtered sound of 90 dB SPL
was delivered near the eardrum in the ear canal, the displacement of the stapes footplate was
measured across the frequency range of 250–8,000 Hz. Under the same sound pressure applied on the
lateral side of the eardrum in the final FE model, the magnitude of the displacement at the stapes
footplate was calculated using ANSYS. The model-predicted results were plotted with the seventeen
temporal bone experimental curves in Fig. 5. This figure shows that the model-predicted stapes
displacement curve falls into the range of the 17 temporal bone experimental curves. Our FE model
curve is lower than the mean experimental curve, especially in the frequencies of 700–2,000 Hz.
However, the trend is similar to the mean experimental curve. The difference between the FE model
and experimental data may come from the significant variations of individual temporal bones.
Comparison of frequency response curves of umbo displacements
The experimental data from 64 normal human subjects published by Nishihara and Goode (1996) were
also selected for model evaluation. When each of the 34 pure tone sounds of 80 dB SPL were delivered
to the eardrum, the displacement of the umbo (the medial apex of the eardrum) was measured using
a laser Doppler interferometer. Figure 6 shows the upper and lower bounds of 64 experimental curves.
With a uniform harmonic pressure stimulus of 80 dB SPL (0.2 Pa) applied to the lateral side of the
eardrum in the final FE model, the harmonic analysis was conducted across the frequency range of
250–8,000 Hz using ANSYS. The predicted umbo displacements were converted to frequency response
curve of peak-to-peak displacement and shown in Fig. 6 with Nishihara’s data. As can be seen, the
model-predicted umbo displacements are within the bounds of the 64 experimental curves across
the frequency range of 250–8,000 Hz. Likewise, the FE model-predicted umbo displacement is close to
the lower bound of the experimental results, consistent to the case of the stapes footplate displacement
in comparison with Gan et al.’s results.
Fig. 5. Comparison of stapes footplate displacements between FE model prediction and Gan et al.’s (2001) 17
experimental measurements under sound pressure of 90 dB SPL applied at lateral side of the eardrum. The thick
solid line represents the FE model prediction. Each thin solid line represents one temporal bone experimental data.
The dashed line represents the mean of 17 experimental results
117
118
Fig. 6. Comparison of umbo displacements between FE model prediction and Nishihara and Goode’s (1996) 64
human subject experimental measurements under sound pressure of 80 dB SPL applied at lateral side of the
eardrum. The thick solid line represents FE model prediction. The other lines represent the mean, upper and lower
bounds of the 64 experimental results
Comparison of middle ear sound transfer function
The ratio of stapes footplate velocity to the sound pressure in the external ear canal is termed the ear
canal sound pressure to the stapes footplate velocity transfer function (STF). The STF curves reported
by Aibara et al. (2001) were also used to evaluate our final FE model. When a pure tone sound of
90–120 dB SPL was delivered to the eardrum, the stapes footplate velocities were measured using LDI
on 11 fresh temporal bones. The sound pressure in the ear canal was monitored near the eardrum
using a probe-tube microphone. The STFs were calculated based on the measurements.
Because of linear characteristics of the FE model, the ratio of stapes footplate velocity to the pressure in
the ear canal does not depend on the magnitude of the sound pressure applied to the eardrum. Therefore,
a uniform pressure of 90 dB SPL on the lateral side of the eardrum was applied to the final model. A
harmonic analysis was conducted on the model across the frequency range of 250–8,000 Hz using
ANSYS. The calculated STF was plotted with the mean and the upper and lower bounds of the eleven
experimental curves in Fig. 7. This figure shows that our model-predicted STF curve lies close to the
lower limits of the experimental curves. Again, it is consistent with previous evaluations.
Fig. 7. Comparison of the stapes footplate velocity transfer function (STF) between the FE model-predicted result
and 11 experimental measurements by Aibara et al. (2000). The thick solid line represents FE model prediction.
The other lines represent the mean, upper and lower bounds of the 11 experimental results
119
Fig. 8. Comparison of ratio of umbo to stapes footplate displacements between FE model prediction and
Nishihara et al.’s (1993) measurements under sound pressure of 80 dB SPL applied at lateral side of the eardrum.
The solid thick line represents FE model prediction and the thin line represents the experimental result
Comparison of the ratio of umbo to stapes footplate displacements
The ratio of umbo to stapes footplate displacements reflects the effectiveness of the middle ear system
in coupling sound pressure into the inner ear. This ratio for our FE model may not fall within a
reasonable range even though the displacements of both the stapes footplate and the umbo are within
the reasonable ranges. Therefore, it is meaningful to test the final model by comparing the ratio
between experimental measurements and FE model prediction. Experimental data for the evaluation
were calculated from measurements of Nishihara et al. (1993). When a pure tone sound of 80 dB SPL
was delivered to the eardrum, displacements of the stapes footplate and the umbo caused by the sound
pressure on the eardrum were measured using laser Doppler interferometer. Our FE model prediction
was calculated from the harmonic analysis under the same sound pressure level of 80 dB SPL using
ANSYS. Both experimental and FE model results are plotted in Fig. 8. As can be seen, the FE prediction is, in general, consistent with the experimental results of Nishihara.
Comparison of the ratio of umbo to malleus (short-process) displacements
The ratio of umbo to malleus short-process displacements reflects the vibration pattern of the malleus
that affects the sound transmission in the ossicular chain. The umbo and malleus short process are
indicated in Fig. 2. Experimental data for the comparative evaluation were calculated from measurements by Goode et al. (1994) and shown in Fig. 9. When a pure tone sound of 94 dB SPL was
delivered to the eardrum, displacements of the short process and umbo were measured using laser
Doppler interferometry. The FE model prediction was calculated from the harmonic analysis under
uniform sound pressure stimulus of 94 dB SPL and plotted in Fig. 9. Results show that the FE
prediction is close to the experimental result at frequencies greater than 1,000 Hz. However, the model
prediction is lower than the experimental data at low frequencies. The difference at lower frequencies
is probably because we used the uniform Young’s modulus but non-uniform density in malleus in the
final FE model. The inconsistency of Young’s modulus and the density may affect the transmission
characteristics in low frequencies.
Summary
A working FE model, constructed using a realistic geometric model from computer-aided reconstruction, was cross-calibrated with middle ear displacement data to obtain a final FE model of middle
ear mechanics. This final FE model was evaluated against five independent experimental measurements: the stapes footplate displacement, the umbo displacement, the stapes footplate velocity transfer
function, the ratio of umbo to stapes footplate displacements, and the ratio of umbo to malleus shortprocess displacements. The results were compared for three different locations, umbo, stapes footplate, and short process of the malleus. These comparative evaluations showed that the FE model
predictions, in general, matched empirical results obtained from human subjects or human temporal
bones using laser Doppler interferometry. The FE model was able to predict dynamic behaviors of the
human middle ear, the main concern of the middle ear modeling.
120
Fig. 9. Comparison of ratio of umbo to malleus short-process displacements between FE model prediction and
Goode et al.’s (1994) measurements under sound pressure of 94 dB SPL applied at lateral side of the eardrum. The
solid thick line represents FE model prediction and the thin line represents experimental results
It is also noticed that the FE model predictions for the stapes footplate displacement, the umbo
displacement, and the stapes velocity transfer function are consistently lower than the corresponding
mean experimental measurements, especially in the frequency range of 700–2,000 Hz. The difference
may be explained as follows:
1.
2.
3.
4.
5.
6.
The individual temporal bone difference in geometry and material properties resulted in the
difference between the FE model predictions and experimental results. Our FE model is based on
only one temporal bone and one set of mechanical properties of middle ear components and
boundary conditions. It has been stated that anatomical and physiologic individual differences of
the external ear and middle ear can induce up to 25 dB individual variation in hearing thresholds,
particularly at frequencies of 1–4 kHz (Goode et al. 1994).
Some of the material properties adopted in the final FE model were assumed or determined by a
cross-calibration process on the working FE model, which may not be the actual values.
The ear canal was not included in the FE model. It has been reported that the ear canal acts as a
pipe resonator that boosts hearing sensitivity in the range of 2–5 kHz (Rossing 1990).
The action of cochlear fluid was simplified as linear springs and dashpots with constant stiffness
and damping coefficients in the FE model. In reality, the stiffness of springs and the damping
coefficient of dashpots are dependent on frequency if they were calculated directly from the
experimentally measured cochlear input impedance.
The uniform Young’s modulus and non-uniform density were employed for ossicles in the model.
In reality, the inconsistency of Young’s modulus and the mass density may affect the characteristics of middle ear transmission at low frequencies.
Sound pressure at the eardrum was simplified as uniform pressure load. In reality, the load is
non-uniform due to the particular shapes of the eardrum and the ear canal (Stinson 1985).
Although there are some differences between the FE model predictions and experimental data, we
can still conclude that the final FE model is useful for predicting the dynamic behaviors of the middle
ear, the major concern in middle ear sound transmission. We have suggested several clinical applications of the FE model elsewhere (Gan et al. 2002). The applications include prediction of the effect of
mass loading in ossicular chain on sound transmission and the effect of the eardrum stiffness on
stapes movement.
Two different masses were placed on the incudostapedial joint to simulate two models of the
implant for a passive middle ear device, such as the Soundtec Direct (Soundtec, Inc., Oklahoma City,
Okla., USA). A stable enhancement, approximately 3.5 dB caused by 15 mg reduced mass loading was
predicted by this FE model (Fig. 10, Gan et al. 2002). The result was compared with published
temporal bone experimental data. The effect of eardrum stiffness change on stapes footplate movement was examined using the FE model. An increased stiffness (Young’s modulus above normal) of
the eardrum resulted in reduced stapes displacements at low frequencies (f<1,000 Hz) and increased
displacements at high frequencies (f = 1,000 Hz). A decreased stiffness of the eardrum resulted in
reduced stapes displacements at both low and high frequencies (Fig. 11, Gan et al. 2002).
We have constructed an FE model of the human middle ear based on histological section images of
a human temporal bone. The model was evaluated with five independent experimental results and
some preliminary clinical applications of the model were also conducted. However, improvement of
the FE model is still necessary in future studies, such as including the external ear canal and middle
ear cavity to the model. This will enhance the potential for physiological and pathological applications
of the model, for instance, simulating acoustic effects of eardrum perforations on mechanical transmission of the sound from the external ear canal to the cochlea.
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