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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). 109 110 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 111 112 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 113 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 114 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 115 116 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. 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