The Influence of Fiber Orientation of the Conus Elasticus in Vocal Fold Modeling

Abstract

While the conus elasticus is generally considered a part of continuation of the vocal ligament, histological studies have revealed different fiber orientations that fibers are primarily aligned in the superior-inferior direction in the conus elasticus and in the anterior-posterior direction in the vocal ligament. In this work, two continuum vocal fold models are constructed with two different fiber orientations in the conus elasticus: the superior-inferior direction and the anterior-posterior direction. Flow-structure interaction simulations are conducted at different subglottal pressures to investigate the effects of fiber orientation in the conus elasticus on vocal fold vibrations, aerodynamic and acoustic measures of voice production. The results reveal that including the realistic fiber orientation (superior-inferior) in the conus elasticus yields smaller stiffness and larger deflection in the coronal plane at the junction of the conus elasticus and ligament and subsequently leads to a greater vibration amplitude and larger mucosal wave amplitude of the vocal fold. The smaller coronal-plane stiffness also causes a larger peak flow rate and higher skewing quotient. Furthermore, the voice generated by the vocal fold model with a realistic conus elasticus has a lower fundamental frequency, smaller first harmonic amplitude, and smaller spectral slope.

1 Introduction

The conus elasticus, also known as the cricothyroid ligament, cricothyroid or cricovocal membrane, denotes the fibroelastic layer that lies under the mucosal of the subglottic region of the larynx [1]. While the median part of the conus elasticus connects the anterior arch of cricoid cartilage to the inferior rim of thyroid cartilage and does not directly affect vocal fold vibration, the lateral parts of the conus elasticus extend from the upper rim of the cricoid cartilage to the inferior edge of the vocal ligament (Fig. 1), where the structure could have influence on vocal fold vibration. Due to their firm anchoring to the cricoid cartilage, the lateral parts of the conus elasticus are believed to be important for restricting the vertical movements of the vocal folds [2].

Fig. 1.

Coronal section of the human larynx

The tissue biomechanical anisotropy is known to be closely related to the orientation of tissue fibers [3,4]. The conus elasticus is generally considered to be composed of densely arranged elastic and collagen fibers [5,6]. As the fiber orientations in the conus elasticus are primarily along the superior-inferior direction [2], the anisotropy of the conus elasticus is believed to be different from that of the vocal ligament, in which the fibers are aligned predominantly in the anterior-posterior direction [7,8]. However, the role of this material difference in vocal fold vibration and voice production has seldom been investigated. While the true material anisotropy of the conus elasticus is naturally included in the in vivo/excised larynx, most of current computer or physical (synthetic) vocal fold models have largely overlooked this difference. The conus elasticus layer was either simply not considered or integrated with the ligament layer with the same material [9–14]. A number of mechanics studies have shown that fiber orientation can play a significant role in determining the mechanical properties of anisotropic materials [3,15,16]. Therefore, it is of interest to study how the fiber-orientation-related material anisotropy of the conus elasticus affects the biomechanical response of vocal folds during the interaction with glottal aerodynamic forces. Furthermore, if the anisotropic direction of the conus elasticus has a significant influence on vocal fold vibration, it is worth investigating how it will affect the aerodynamic and acoustic parameters that reflect voice type and quality.

In this work, we study the influence of fiber orientation of the conus elasticus on the flow-induced vibration of a three-dimensional computational vocal fold model. One focus of this work is to understand the relationship between the mechanical anisotropy and the stiffness of the conus elasticus in the coronal plane. It is expected that such knowledge would help explain the vibratory characteristics of vocal fold observed in the fluid-structure interaction simulation. Output measures including vocal fold motion, flow rate waveform, aerodynamic and acoustic parameters are analyzed to determine the effect of the conus elasticus on vocal fold vibration and voice production. The findings of this study could provide deeper insights into the mechanism of vocal fold vibration and contribute to the development of more accurate voice simulation models.

2 Methods

2.1 Vocal Fold and Conus Elasticus Model.

In this study, the geometric shape of the vocal fold is based on the mathematical description by Titze and Talkin [17]. The dimensions and layered structure of the vocal fold model are shown in Fig. 2(a). In the model, the cover layer is the outermost layer covering the entire vocal fold. The ligament and conus elasticus are located between the cover and body layer. The ligament layer starts at the superior margin of the medial surface and connects to the conus elasticus at the inferior margin of the medial surface. The conus elasticus goes through the subglottic region and ends at the inferior aspect of the lateral surface of vocal fold. The thickness of the cover and ligament layers are determined from the averaged value of the histological measurements [18], which are 0.33 mm and 1.11 mm, respectively. The thickness of the conus elasticus in the coronal plane is assumed to be the same as that of the ligament. Along the anterior-posterior direction, the thickness of the cover, ligament, and conus elasticus are assumed to be invariant.

Fig. 2.

(a) The dimensions and layered structure of the vocal fold in the simulation. (b) Loading and boundary conditions in the finite element model of conus elasticus. (c) superior-inferior (left) and anterior-posterior (right) fiber orientation in the conus elasticus.

All the layers in the model are modeled as nearly incompressible and transversely isotropic linear elastic materials. For the cover, ligament, and body layer, the transverse isotropy is in the coronal plane, and the material properties of the three layers are taken from previous studies [9,19]. To investigate the effect of anisotropic direction of the conus elasticus, two vocal fold models with different anisotropic directions in the conus elasticus are generated: In model 1, the fiber orientation in the conus elasticus is consistent with the histologic observation which is aligned with the inferior angle of the vocal fold (40 degrees with respect to the horizontal plane in current model); in model 2, it is assumed that the fiber orientation in the conus elasticus is the same as that in the ligament layer, which is along the anterior-posterior direction. Due to the lack of measurements of the material properties of the conus elasticus and the fact that the conus elasticus has similar composition of elastic and collagen fibers as the ligament [5,7], the material parameters of the conus elasticus are assumed to be the same as those in the ligament layer. The adopted material parameters of each vocal fold layer are listed in Table 1.

Table 1.

Material properties of each vocal fold layer

	ρ (g/cm3)	E (kPa)	ν	E′ (kPa)	G′ (kPa)	v′	η (poise)
Cover	1.043	2.014	0.9	40	10	0.0	5.0
Ligament	1.043	3.306	0.9	66	40	0.0	7.5
Conus elasticus	1.043	3.306	0.9	66	40	0.0	7.5
Body	1.043	3.990	0.9	80	20	0.0	12.5

Notes: ρ–tissue density, E–transverse Young's modulus, ν–transverse Poisson ratio, E′, longitudinal Young's modulus, G′, longitudinal shear modulus, v′, longitudinal Poisson ratio, η, damping coefficient.

The dynamics of the vocal fold are characterized by the Navier equation

$${\nabla }_j{\sigma }_{ij}+f_i=\rho \frac{{\partial }^2{d}_i}{\partial t^2}$$

where the indices $i$ and $j$ range from 1 to 3, ${\nabla }_j=\partial /\partial x_j$ is the partial derivative with respect to the $j_{th}$ coordinate, $\sigma$ is the stress, $f$ is the body force, $\rho$ is the tissue density, and $d$ is the displacement. The Navier equation is solved using an in-house finite element code.

2.2 Bernoulli Flow Model.

In this study, the glottal airflow is modeled by a 1-D quasi-steady Bernoulli flow until it separates from the glottal wall at the minimum glottal area. Downstream of the flow separation point, the flow pressure is assumed to be equal to the atmospheric pressure. At a cross section $i$ upstream of the flow separation point, the intraglottal pressure ( $p_i$) is given by

$$p_i=\hspace{0.17em}{p}_{\mathrm{sub}}+\frac{1}{2}{\rho }_{\mathrm{air}}{Q}^2\left(\frac{1}{{A_{\mathrm{sub}}}^2}-\frac{1}{{A_i}^2}\right)$$

where $p_{\mathrm{sub}}$ is the subglottal pressure, ${\rho }_{\mathrm{air}}$ is the air density, $Q$ is the flow rate, $A_{\mathrm{sub}}$ and $A_i$ are the cross-sectional area at the subglottal region and $i_{\mathrm{th}}$ cross section, respectively. The flow rate $Q$ is solved by applying the Bernoulli equation between the subglottal area and the minimum glottal area

$$Q=\sqrt{\frac{2p_{\mathrm{sub}}}{\rho }}{A}_{\min }$$

where $A_{\min }$ is the minimum area of the glottis. The viscous loss in the glottal flow is neglected in this study.

2.3 Fluid-Structure Interaction Model and Simulation Setup.

The solid solver and the Bernoulli flow solver are explicitly coupled to perform the fluid-structure interaction (FSI) simulations. The vocal fold and the flow dynamics are alternately solved with the latest flow pressure as the load being applied on the vocal fold surfaces and the updated vocal fold position providing the boundary for the flow. More details about this simulation method can be found in Geng et al. [20]. The vocal fold vibrations are assumed to be symmetric about the glottal midplane so that only the left vocal fold is simulated in the current study. The glottal midplane is set at x = 0.0 mm, which creates a nearly zero initial glottal gap. The anterior, posterior, and lateral surfaces of the vocal folds are fixed to mimic the attachment to the thyroid and arytenoid cartilage, while the remaining surfaces are free to move. A penalty coefficient contact model is employed to model the vocal fold collision, in which contact force is proportional to the distance of surface nodes across the midplane and is calculated using the same equation in Geng et al. [20]. The vocal fold is discretized into 9767 ten-node quadratic tetrahedral elements, and grid independence is achieved with this mesh. Along the flow direction, the glottis is discretized into 100 equidistant sections where the Bernoulli's equation is solved and glottal pressures are evaluated. Air density of 1.225 kg/m³ is used. For each vocal fold model, five FSI simulations with five different subglottal pressures (0.2, 0.4, 0.6, 0.8, and 1.0 kPa) are conducted. A small time-step of 2.27 × 10⁻⁵ s is used for all the simulations. Each simulation runs for a period of 0.5 s, which is sufficient to pass the transient state and reach the limit-cycle steady-state vibration.

3 Results and Discussion

3.1 Effect of Material Anisotropy on Coronal-Plane Stiffness of Conus Elasticus.

The conus elasticus is superiorly connected to the bottom of the vocal ligament. Its coronal-plane stiffness will affect the pliability of the ligament by providing constraints at the junction of the two tissues, which can further change the overall coronal-plane stiffness of the vocal fold and influence its vibrations. As tissue is much stiffer in the fiber direction than in the transverse plane, different arrangements of the fiber orientation in the conus elasticus will alter its stiffness in the coronal plane which is also the vibration plane. Therefore, it is of interest to first evaluate the effect of material anisotropy on the coronal-plane stiffness of the conus elasticus itself.

For the evaluation, the cover and body layers are removed from the vocal fold model. Static finite element analysis is performed with a uniform pressure load applied on the medial surface of the conus elasticus (Fig. 2(b)) to evaluate its coronal-plane stiffness through the force–displacement response. Such pressure loading would deform the conus elasticus in a similar way that the subglottal pressure deforms the conus elasticus during phonation. The ligament layer is kept in the model in order to provide a more realistic boundary condition for the superior surface of the conus elasticus. The anterior and posterior surface of the conus elasticus and ligament are fixed, as well as the bottom surface of the conus elasticus. These boundary conditions are consistent with those in the full vocal fold model. The fiber orientations in the two conus elasticus models are illustrated in Fig. 2(c), where the superior-inferior and anterior-posterior orientations correspond to model 1 and model 2, respectively. Figure 3(a) shows the deformation of the two models under a pressure load of 0.4 kPa. The maximum deflection of both models occurs below the superior surface of the conus elasticus, where the conus elasticus bulges laterally into a cone shape. The superior edge of the conus elasticus in model 1 is deflected more in the sideways direction than that in model 2. Force-displacement relationships are obtained for both models by varying the pressure load from 0.2 to 1.0 kPa with an increment of 0.2 kPa, in which the force is the total resultant force of the applied pressure load, and the displacement is the average displacement of the entire conus elasticus. The force–displacement relationships for the two models is shown in Fig. 3(b). Both models exhibit linear relations between force and displacement due to the linear elastic materials used in this study. The slopes of the force–displacement curves indicate the coronal-plane stiffness of the conus elasticus under this uniform pressure load. The slopes corresponding to model 1 and model 2 are 0.504 N/cm and 0.591 N/cm, respectively, suggesting the anterior-posterior fiber orientation makes the coronal-plane stiffness of the conus elasticus 17.3% larger than that of the superior-inferior fiber orientation. To compare the deflection at the junction of the conus elasticus and ligament between the two models under the applied pressure load, force versus the average displacement of the superior surface of the conus elasticus are shown in Fig. 3(c). The relations between the force and displacement are also linear. The average displacement of the superior surface of the conus elasticus in model 1 is 58.1% larger than that in model 2 throughout all the loads applied.

Fig. 3.

(a) Deformation of the two conus elasticus models under the pressure load of 0.4 kPa. Dashed lines indicate the location of the superior edges that are not visible in the figure. The gray, transparent shape indicates the original vocal fold model from which the conus elasticus is separated. (b) Force versus average displacement of the entire conus elasticus under uniform pressure loads. The average displacement is calculated by taking the arithmetic mean of the nodal displacements normal to the medial surface of the conus elasticus (y' direction shown in Figure (a)). (c) Force versus average displacement (in y' direction) of the superior surface of the conus elasticus under uniform pressure loads.

As the conus elasticus is integrated into the vocal fold model, its coronal-plane stiffness will affect the pliability of the tissues connected to it and the overall vocal fold coronal-plane stiffness. With the superior-inferior fiber orientation, the superior edge of the conus elasticus is more flexible in the lateral direction, which might provide less resistance at the junction of the conus elasticus and ligament and allow larger vibrations of vocal fold in the lateral direction. It suggests that the superior-inferior arrangement of the fibers in the conus elasticus may be important not only for preventing the excessive vertical motion [2] but also for promoting the lateral vibration of vocal folds.

3.2 Vocal Fold Vibration.

To further evaluate the effects of anisotropic direction of the conus elasticus on vocal fold vibration, FSI simulations were conducted on the two vocal fold models with five different subglottal pressures (0.2, 0.4, 0.6, 0.8, and 1.0 kPa). Figure 4(a) shows the midcoronal profiles of the two models under 1.0 kPa subglottal pressure at five different phases within one oscillatory cycle. The medial surface of both vocal fold models exhibits a convergent shape during glottal opening (t/T = 0.25) and a divergent shape during glottal closing (t/T = 0.68), which are the typical vibratory characteristics of vocal fold during normal phonation. The third phase (t/T = 0.46) is around the time at which the maximum flow rate of model 1 is reached. To compare the kinematics between the two models, the medial-lateral displacements are extracted at three points (denoted by the markers in Fig. 4(a)) located at the superior and inferior aspect of the medial surface and the junction of the conus elasticus and ligament, respectively. The corresponding phase-averaged displacement waveforms of the two models at 1.0 kPa subglottal pressure are plotted in Fig. 4(b). The superior, inferior, and junction displacement amplitudes of model 1 are larger than those of model 2, suggesting a larger overall vibration amplitude with model 1. At the junction of the conus elasticus and ligament, the displacement amplitude of model 1 (0.363 mm) is 24.3% larger than that of model 2 (0.292 mm). The inferior displacement amplitude of model 1 is 0.503 mm, while the corresponding value of model 2 is 0.429 mm, decreasing by 14.7%. Compared to the junction and inferior displacement amplitude, the difference in superior displacement amplitude between the two models is relatively small. The superior displacement amplitude of model 2 is 6.6% smaller than that of model 1. The displacement amplitude versus subglottal pressure for the two models is plotted in Fig. 5 for the inferior and superior point, separately. Overall, model 1 has a larger vibration amplitude than model 2 at both inferior and superior aspects with a larger difference occurring at the inferior aspect, suggesting that the overall vibration amplitude of the vocal fold increases with the decrease of the coronal-plane stiffness of the conus elasticus and the effect is more prominent at the inferior aspect where the conus elasticus is close. The difference in displacement amplitude between model 1 and model 2 increases with the increasing of subglottal pressure.

Fig. 4.

(a) Midcoronal profiles of model 1 (solid line) and model 2 (dashed line) at five phases within one oscillatory cycle (Due to the much smaller displacement difference between the two models as compared to the vertical dimension of vocal fold, the x-axis scale is enlarged to better show the difference). From top to bottom, the dot and cross markers indicate the superior and inferior aspect of the vocal fold and the junction of the conus elasticus and ligament. (b) Superior (dashed–dotted line), inferior (solid line), and junction (dash line) medial-lateral (x-direction) displacement waveforms over steady-state cycles of the two models. Left, model 1; right, model 2. Data correspond to the cases of 1.0 kPa subglottal pressure.

Fig. 5.

Medial-lateral displacement amplitude at inferior and superior vocal fold versus subglottal pressure

Proper orthogonal decomposition (POD) method is exploited to further disclose the vocal fold dynamics of the two models [21,22]. In all current cases, the first two POD modes account for more than 99% of the total kinetic energy, therefore the dominant coherent vibration patterns can be sufficiently identified by just using these two modes. The midcoronal profile of the first two modes of model 1 and model 2 at 1.0 kPa subglottal pressure is shown in Fig. 6(a). The first mode mostly captures the medial-lateral motion while the second mode captures the propagation of mucosal wave along the medial surface. To quantify the mode similarity between model 1 and 2, dot-products are conducted between the corresponding normalized POD modes from the two models. A value of 1 represents two identical modes while a value of 0 represents two orthogonal modes. Figure 6(b) shows the mode dot-products at different subglottal pressures for the first and second modes, respectively. The overall mode similarities for all the cases are above 0.86, indicating that POD modes between the two models are highly similar and the material anisotropy of the conus elasticus has little effect on the POD modes. The energy distribution between the two modes at different subglottal pressures is further plotted in Fig. 7(a). In general, with the increase of subglottal pressure, the energy percentage of the first mode increases while the energy percentage of the second mode decreases. This observation indicates that for both models, the lateral motion becomes relatively stronger with the increase of subglottal pressure. It is also found that compared with model 2, the energy percentage of the second mode is about 14% higher in model 1, suggesting that model 1 has a relatively stronger mucosal type of motion. Figure 7(b) shows the maximum divergent glottal angle of the two models as a function of subglottal pressure. The maximum divergent glottal angle is observed to be larger in model 1 than model 2, and the difference increases from 0.18 degrees at 0.2 kPa subglottal pressure to 1.42 degrees at 1.0 kPa subglottal pressure, confirming that a stronger mucosal type of motion occurs with model 1. These results suggest that a decrease in the coronal-plane stiffness of the conus elasticus would promote a stronger mucosal wave motion of the vocal fold. It needs to be pointed out that the increase of the maximum divergent angle with subglottal pressure in Fig. 7(b) does not contradict the decrease of the energy percentage of the second mode observed in Fig. 7(a), as the actual value of the mode energy can still increase with subglottal pressure, resulting in the increase of the maximum divergent angle.

Fig. 6.

(a) The midcoronal profile of the first two POD modes at two-extreme phases for model 1 (solid line) and model 2 (dashed line) at 1.0 kPa subglottal pressure. The dot and cross marker denote the junction of the conus elasticus and ligament in model 1 and model 2, respectively. The vertical dotted line (x = 0.0 mm) is the reference line for comparing mode shapes at different phases. (b) Dot-products of the first and second POD modes between model 1 and 2 at different subglottal pressures.

Fig. 7.

(a) Mode energy of the first two POD modes versus subglottal pressure. (b) Maximum divergent glottal angle versus subglottal pressure.

3.3 Flow Rate Waveform and Aerodynamic Measures.

The phase-averaged flow rate waveforms of the two models at 1.0 kPa subglottal pressure are shown in Fig. 8(a). The flow rate of the two models is almost identical during the opening stage. While model 2 reaches its maximum flow rate of 268.1 ml/s at t/T = 0.402, the flow rate in model 1 continues increasing until t/T = 0.444, leading to a higher peak flow rate of 298.2 ml/s, which is 11.2% higher than model 2. For all other different subglottal pressures, the flow rate is consistently higher in model 1 with increasing difference with the increase of subglottal pressure. The variation of peak flow rate with subglottal pressure for the two models is shown in Fig. 8(b). Figure 8(c) shows the variation of average flow rate with subglottal pressure for the two models. Compared with the peak flow rate, the difference in average flow rate between the two models is small. Similar to the peak flow rate, the average flow rate difference increases with the increase of subglottal pressure.

Fig. 8.

(a) Flow rate waveforms over steady-state cycles at 1.0 kPa subglottal pressure. (b) Variation of peak flow rate with subglottal pressure. (c) Variation of average flow rate with subglottal pressure.

In the flow rate waveforms, the rising phase corresponds to a convergent glottal shape, under which the superior aspect of the glottis dictates the minimum glottal area and therefore glottal flow rate, while the falling phase is accompanied by a divergent glottal shape, under which the inferior aspect of the glottis dictates the minimum glottal area and flow rate. The peak flow rate occurs at the moment when the superior glottis has the same opening area as the inferior, and this moment is usually during the closing phase of inferior aspect of vocal fold and the opening phase of superior aspect of vocal fold. A detailed analysis of vocal fold vibration reveals that the maximum medial-lateral displacement of inferior aspect of vocal fold happens around 0.396 T for both of the two models at 1.0 kPa subglottal pressure. However, the peak flow rate occurs at different phases: 0.444 T for model 1 and 0.402 T for model 2, suggesting it takes a longer time for the inferior aspect to recoil back to the same glottal area of the superior aspect in model 1. This is likely due to the softer inferior aspect (smaller coronal-plane stiffness) and larger medial-lateral displacement at the inferior aspect in model 1, which prolongs the opening phase and leads to a higher peak glottal flow rate.

Figures 9(a) and 9(b) show the open quotient (the ratio of glottal open time to the period T) and skewing quotient (the ratio of glottal opening time to closing time) of each model versus the subglottal pressure, respectively. The relationships of open quotient and skewing quotient with subglottal pressure are nonmonotonic. Overall, the open quotient is nearly constant with a small fluctuation for each model. The open quotient of model 2 is consistently larger than that of model 1, with the largest difference of 3.7% observed at subglottal pressures of 0.8 and 1.0 kPa. In contrast to the small difference in open quotient between the two models, the skewing quotient of model 1 is much higher (e.g., 24.4% higher at 0.6 kPa subglottal pressure) than that of model 2 for all the simulated subglottal pressures, resulting from the relatively longer opening phase and shorter closing phase of the glottis in model 1. Thus, it suggests that the superior-inferior fiber orientation in the conus elasticus can slightly decrease the open quotient and significantly increase the skewing quotient.

Fig. 9.

(a) Open quotient versus subglottal pressure. (b) Skewing quotient versus subglottal pressure.

3.4 Spectrum Analysis and Acoustic Measures.

Figure 10 presents the plot of fundamental frequency (f_o) versus the subglottal pressure of the two models. f_o is determined from the largest harmonic (also the first harmonic in this study) in the spectrum of flow rate waveform (Fig. 11(a)). As the linear elastic material is adopted, the frequencies almost do not change with the subglottal pressure. f_o of model 2 is higher than that of model 1, with a minimum difference of 3.7% at 0.2 kPa and a maximum difference of 6.6% at 1.0 kPa. This finding is not surprising considering the conus elasticus in model 2 is stiffer in the coronal plane than that in model 1.

Fig. 10.

f_o versus subglottal pressure

Fig. 11.

(a) Spectrum of flow rate waveform of model 1 at 1.0 kPa subglottal pressure. f_o, fundamental frequency; H1, the first harmonic; H2, the second harmonic; H4, the fourth harmonic. (b) Spectral slope is calculated from the line fitting the spectral peaks. Spectral slope in dB/Oct is obtained by multiplying the slope of fitted line by log10 of 2.

Spectra of the glottal source contain important acoustic cues to the variation of voice quality [23]. Figure 11(a) presents the spectrum of flow rate waveform of model 1 at 1.0 kPa subglottal pressure. This spectrum is obtained by performing fast Fourier transform on the last 0.5 s period of the flow rate waveform. Important acoustic measures including H1–H2, H1–H4 (spectral amplitude difference between the first two harmonics, first and fourth harmonics), and spectral slope are extracted from the spectrum. The spectral slope is obtained by using the linear regression in matlab to fit the harmonics between 0 and 2 kHz. The slope of the fitted line is multiplied by log10 of 2 to get the slope in dB per octave. Figure 11(b) illustrates the spectral peaks and the fitted line of model 1 at 1.0 kPa subglottal pressure. H1–H2, H1–H4 and spectral slope of the two models versus subglottal pressure are shown in Fig. 12. As can be seen, both H1–H2 and H1–H4 of model 2 are apparently higher than model 1. Take the values at 1.0 kPa subglottal pressure, for example, H1–H2 and H1–H4 of model 2 are , respectively, 21.6% and 7.6% greater than that of model 1. Considering model 2 has comparatively smaller flow deceleration rate (Fig. 8(a)), the phenomenon observed here is consistent with the finding of Holmberg et al. [24], where they pointed out that relatively gradual vocal fold closures induced a higher amplitude of the first harmonic. In addition, Holmberg et al. [24] also found from their measurements that a higher H1–H2 is mostly accompanied by a small adduction (closed) quotient. Our results show the same relationship as open quotient of model 2 is larger than that of model 1. In Fig. 12, it is also observed that at most subglottal pressures, model 2 has steeper spectral slopes. A steeper spectral slope indicates a weak excitation of higher-order harmonics, which is compatible with the lower flow deceleration rate observed in model 2.

Fig. 12.

Spectral amplitude difference between the first two harmonics (H1–H2), first and fourth harmonics (H1–H4), and spectral slope versus subglottal pressure

4 Conclusion and Limitation

In this study, two vocal fold models with two different fiber orientations in the conus elasticus are built to examine the influence of the direction of material anisotropy on vocal fold vibrations, as well as aerodynamic and acoustic measures of voice production. It is demonstrated that the separate conus elasticus model with a realistic fiber direction, superior-inferior fiber orientation yields smaller coronal-plane stiffness and larger deflection at the junction of the conus elasticus and ligament than the model with the anterior-posterior fiber orientation, which yields a greater vibration amplitude and stronger mucosal wave of the vocal fold. Different directions of material anisotropy in the conus elasticus between the two models also induce differences in aerodynamic and acoustic measures. The flow rate waveform with a superior-inferior fiber orientation in the conus elasticus is observed to have a larger peak flow rate and a higher skewing quotient, owing to the increased inferior vibration amplitude which prolongs the opening phase. Spectrum analysis of the flow rate waveform has shown a decreased f_o of vocal fold vibration when a realistic conus elasticus is considered. Finally, the vocal fold model with a realistic conus elasticus generates a voice with relatively smaller first harmonic amplitude and spectral slope. It is worth pointing out that compared with the spectral slope variations in human voice, the difference in spectral slope between the two models in current study is small and may not be perceived.

One limitation of this study is that the glottal flow is assumed to be inviscid and quasi-steady. Neglecting the viscous and inertia effects may be acceptable in the middle stage of the phonation cycle, but it can cause non-negligible errors when the glottis is nearly closed. Another limitation of this study is that the material parameters of the conus elasticus are simply assumed to be the same as those of vocal ligament, due to the lack of measurement of the mechanical properties of the conus elasticus. Future studies are needed to evaluate a range of property parameters to improve the generalizability of the conclusions. In addition, the anterior, posterior, and lateral surface of the vocal fold are fixed to mimic the attachment to the cartilages, which can only be thought of as an approximate treatment because, in the real situation, the attachment of vocal fold tissues to the cartilages is complex. For example, the anterior part of the conus elasticus was reported to be only attached to the thyroid cartilage midway between the notch and the inferior border of the thyroid cartilage [1,6]. From the inferior border of the thyroid cartilage down, the conus elasticus extends to the upper rim of the cricoid cartilage and is not attached to any cartilage anteriorly. To simulate the boundary conditions applied on the vocal fold more truthfully, a more realistic, complete larynx model which includes the cartilages is needed. Furthermore, including material and geometric nonlinearities in vocal fold models may predict more realistic vibrations of vocal folds.

Funding Data

• National Institute on Deafness and Other Communication Disorders (NIDCD) (Grant No. 2R01DC009435; Funder ID: 10.13039/100000055).

• National Institute of Health (Grant No. 1R03DC014562; Funder ID: 10.13039/100000002).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.