applied sciences

Article Effect of Subglottic Stenosis on Vocal Fold Vibration and Production Using Fluid–Structure–Acoustics Interaction Simulation

Dariush Bodaghi 1, Qian Xue 1 , Xudong Zheng 1,* and Scott Thomson 2

1 Department of Mechanical Engineering, University of Maine, Orono, ME 04473, USA; [email protected] (D.B.); [email protected] (Q.X.) 2 Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-207-581-2180

Abstract: An in-house 3D fluid–structure–acoustic interaction numerical solver was employed to investigate the effect of subglottic stenosis (SGS) on dynamics of glottal flow, vocal fold vibration and acoustics during voice production. The investigation focused on two SGS properties, including sever- ity defined as the percentage of area reduction and location. The results show that SGS affects voice production only when its severity is beyond a threshold, which is at 75% for the glottal flow rate and acoustics, and at 90% for the vocal fold vibrations. Beyond the threshold, the flow rate, vocal fold vibration amplitude and vocal efficiency decrease rapidly with SGS severity, while the skewness quotient, vibration frequency, signal-to-noise ratio and vocal intensity decrease slightly, and the open quotient increases slightly. Changing the location of SGS shows no effect on the dynamics.   Further analysis reveals that the effect of SGS on the dynamics is primarily due to its effect on the flow resistance in the entire airway, which is found to be related to the area ratio of glottis to SGS. Below Citation: Bodaghi, D.; Xue, Q.; the SGS severity of 75%, which corresponds to an area ratio of glottis to SGS of 0.1, changing the SGS Zheng, X.; Thomson, S. Effect of severity only causes very small changes in the area ratio; therefore, its effect on the flow resistance Subglottic Stenosis on Vocal Fold and dynamics is very small. Beyond the SGS severity of 75%, increasing the SGS severity, leads to Vibration and Voice Production Using rapid increases of the area ratio, resulting in rapid changes in the flow resistance and dynamics. Fluid–Structure–Acoustics Interaction Simulation. Appl. Sci. 2021, 11, 1221. Keywords: subglottic stenosis; vocal fold; voice production; fluid–structure–acoustic interaction; hy- https://doi.org/10.3390/ app11031221 drodynamic/acoustics splitting method; linearized perturbed compressible equation; computational fluid dynamics; flow resistance Academic Editor: Michael Döllinger Received: 24 December 2020 Accepted: 26 January 2021 Published: 29 January 2021 1. Introduction Subglottic stenosis (SGS) is a narrowing of the airway between the and . Publisher’s Note: MDPI stays neutral The most frequent cause of SGS is post-intubation injury in which edema and inflammation with regard to jurisdictional claims in cause the formation of ulcer and granulation tissues [1,2]. SGS has also been reported published maps and institutional affil- in patients having congenital diseases or traumas, treated by tracheotomy method, etc. iations. The narrowing due to SGS can affect both and voice with the symptoms, includ- ing noisy breathing, shortness of breath, , and dysphonia [3–5]. The severity of SGS is generally categorized into four grades by the level of obstruction of the airway: Grade I, below 50% obstruction; Grade II, 51–70% obstruction; Grade III, Copyright: © 2021 by the authors. 71–99% obstruction; and Grade IV, 100% obstruction [6]. Treatment is different depending Licensee MDPI, Basel, Switzerland. on the severity. For lower grades (Grades I and II), little to no treatment may be needed. For This article is an open access article higher grades (Grades III and IV), open airway reconstructive procedures may be needed distributed under the terms and to restore airway patency; however, voice changes before and after the procedures were conditions of the Creative Commons reported in clinical studies [7–10]. For example, Ettema et al. [11] did a perceptual study Attribution (CC BY) license (https:// on the voice change due to SGS and found that about 50% of the patients’ voices were creativecommons.org/licenses/by/ moderately to extremely affected. More interestingly, their results revealed that the effect 4.0/).

Appl. Sci. 2021, 11, 1221. https://doi.org/10.3390/app11031221 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 1221 2 of 17

of SGS on voice was not correlated with its severity, because some patients with a higher grade SGS retained normal voice quality. However, no detailed qualitative or quantitative data were available for a systematic understanding of the underlying reason. Voice production is driven by a fluid–structure interaction process in which the air forced from the interacts with the vocal folds, to generate self-sustained vibration of the vocal folds. Vocal fold vibration, in turn, modulates the airflow into a pulsatile flow, to form the source. Voice quality is indirectly related to the dynamics of the glottal flow and vocal fold vibrations. To understand how SGS affects voice production, it is necessary to understand how SGS affects the dynamics of the glottal flow and vocal fold vibration. Several studies used computational fluid dynamics models to evaluate the effect of the severity of SGS on airway pressure drop [12–15] and flow resistance [16,17] (defined as ∆P/Q or ∆P/Q2, in which ∆P is the pressure drop and Q is the mean flow rate) during respiration; however, the effect of SGS on vocal fold vibration was not studied. To the best of the authors’ knowledge, the only work focusing on the effect of SGS on vocal fold vibration was a study by Smith and Thomson [18]. In this work, an idealized SGS was incorporated into a two-dimensional airway and fluid–structure interaction simulations were performed, to examine the effect of SGS on vocal fold vibration. The severity of SGS was measured as the percentage of the width reduction of the airway and varied from 0% to 99%, to cover the possible pathological range. The study found no change in vocal fold vibration and glottal flow rate up to 60% of SGS severity. By increasing the SGS severity to 90%, a decrease in vocal fold vibration was observed. By further increasing the SGS severities, both vocal fold vibration and the flow rate amplitude were decreased. However, the decrease in the maximum flow declination rate (MFDR, defined as the maximum negative slope of the glottal flow rate waveform [19]) and fundamental frequency (f 0) was observed only when the SGS severity was above 95%. The study also suggested that these effects were primarily related to the effect of SGS on flow resistance. While the study provides quantitative measurements of the effect of SGS on vocal fold vibration, it is limited by the underlying two-dimensional (2D) assumption. It is unclear if the effects remain the same in three-dimensional (3D) geometries, and more importantly, how the effects are related to the 3D properties of SGS. In the current study, we used an in-house 3D fluid–structure–acoustic interaction numerical solver to quantify the effect of SGS on the dynamics of glottal flow and vocal fold vibration in an idealized 3D model. Two SGS properties, the percentage of area occlusion and the location of the SGS, were varied to investigate the effect of these two properties on the glottal flow rate and its waveform; vocal fold vibration amplitude, primary vibratory modes, and voice acoustics were quantified. In the context below, the modeling method, including the numerical algorithms of the computational solver, geometric models, and simulation setup, is presented in Section2; the results and discussions are presented in Section3, and the conclusion is presented in Section4.

2. Models 2.1. Numerical Algorithms The glottal flow dynamics was governed by the three-dimensional, unsteady, incom- pressible Navier–Stokes equations, as shown below:

∂Vi = 0 ∂xi ∂V V 2 (1) ∂Vi + i j = − 1 ∂P + υ ∂ Vi ∂t ∂xj ρ0 ∂xi 0 ∂xj∂xj

where ρ0, V, P, and υ0 are incompressible flow density, velocity, pressure, and kinematic viscosity. The equations are solved on a Cartesian mesh. A sharp-interface immersed boundary method is used to treat complex and moving boundaries. The details of the solver, numerical schemes, and discretization can be found in Reference [20]. Appl. Sci. 2021, 11, 1221 3 of 17

Linearized perturbed compressible equations (LPCE) based on the hydrodynamic/acoustic splitting method [21] were used to simulate vocal tract acoustics:

∂v0 ∂v0V 0 i + i i + 1 ∂p = 0 ∂t ∂xj ρ ∂xj (2) 0 0 ∂v0 ∂p + V ∂p + γP i + v0 ∂P = − dP ∂t i ∂xi ∂xi i ∂xi dt

0 where vi and p0 are the perturbed compressible velocity components and pressure, and γ is the ratio of specific heats. The equations are also solved on a Cartesian mesh with the im- mersed boundary method to treat the boundaries. In the computation, the incompressible flow is computed first, and the obtained flow solution is used to compute the perturbed compressible variables. The final total velocity and pressure variables are computed as follows: v = v0 + V i i i (3) p = p0 + P where v and p are the total flow velocity and pressure. This method was successfully verified in simulating acoustics in [22]. Further details of the LPCE equations can be found in Reference [21]. The governing equations of the vocal fold dynamics are Navier equations, as shown be- low: 2 ∂ d ∂σij i = + ρ 2 ρ fi (4) ∂t ∂xj where ρ, d, σ, and f are tissue density, displacement, stress, and body force per unit , respectively. The finite element method was used to solve Equation (4) [23]. The above three solvers are explicitly coupled through a Lagrangian interface, which is represented by triangular surface meshes. The incompressible flow solver is marched by one step with the existing deformed shape and velocities of the solid tissue as the boundary conditions. The perturbed compressible solver is then marched with the updated incom- pressible flow field, as well as the existing deformed shape and velocities of the solid tissue as the boundary conditions. The forces at the vocal fold surface are then calculated with the new incompressible flow pressure and the perturbed pressure. At last, the solid solver is marched by one step with the updated surface traction. The deformation and velocities on the solid grid are then transferred to the vocal fold surface, so that the fluid/solid interface can be updated. The details of the coupling can be found in Zheng et al. [23] and Jiang et al. [24].

2.2. Geometric Models Figure1a shows the geometric models of the vocal folds, SGS, supraglottal tract, and subglottal tract, as well as the dimension of the computational domain. The configuration of the supraglottal airway was generated based on a neutral area function reported in Story [25], which is the mean area function of 12 vowels of a male speaker. According to another study by Story [26], the first and second natural frequencies (F1 and F2) of the mean area function are similar to those from a uniform tube, but higher frequencies are more subject-specific. The overall length of the supraglottal airway centerline is 17.4 cm. The geometries of the ventricle, false vocal folds, and subglottic tract were superimposed to the tract based on CT scans of a male subject larynx [27]. The geometry of the vocal fold (Figure1b) was generated by using a mathematical model proposed by Titze and Talkin [28] that matches physiologically realistic key pa- rameters and features (e.g., shape of medial surface, anterior–posterior shape variation, and the subglottal angle). The model assumes a three-layer structure of the vocal fold including the cover, ligament, and body, as shown in Figure1b. The thickness of the cover and ligament layer was set as 0.5 and 1.1 mm, respectively, based on the average values of human measurement [29]. The vocal fold tissues were modeled as viscoelastic transversely isotropic materials. The material properties of each layer are listed in Table1. These values Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 18

The geometry of the vocal fold (Figure 1b) was generated by using a mathematical model proposed by Titze and Talkin [28] that matches physiologically realistic key pa- rameters and features (e.g., shape of medial surface, anterior–posterior shape variation, and the subglottal angle). The model assumes a three-layer structure of the vocal fold in- cluding the cover, ligament, and body, as shown in Figure 1b. The thickness of the cover and ligament layer was set as 0.5 and 1.1 mm, respectively, based on the average values of human measurement [29]. The vocal fold tissues were modeled as viscoelastic transversely isotropic materials. The material properties of each layer are listed in Table 1. These values were adopted from previous studies which showed how to reproduce typical vocal fold dy- namics [24,30,31]. The subglottic tract was extended downward by 6.6 cm to incorporate the SGS. An idealized geometry of the SGS was created by using a cosine function (Figure 2a). Specif- ically, for a given surface point (𝑥,𝑦,𝑧) at the SGS region where 𝑥, 𝑦, and 𝑧 are the original coordinates of the point, the new coordinates (𝑥,𝑦,𝑧) of the point after adding the SGS were determined by using the equation below:

𝑎 𝑦 𝑥 =𝑥 −𝑥 × ×(1−𝑐𝑜𝑠2𝜋 ) 2 𝐿 𝑎 𝑦 𝑧 =𝑧 −𝑧 × ×(1−𝑐𝑜𝑠2𝜋 ) (5) 2 𝐿 Appl. Sci. 2021, 11, 1221 4 of 17 𝑦 =𝑦 where 𝐿 is the SGS length and was set as 1.0 cm, which is the most common SGS length [32]; 𝑦 is the distance of the lower bound of the SGS to the 𝑦 component of the point (𝑦); wereand 𝑎 adopted is the percentage from previous reduction studies of whichthe hydraulic showed diameter how to reproduce of the airway typical cross-sec- vocal fold tion dynamicsdue to the [SGS.24,30 ,31].

(b)

(a)

FigureFigure 1. (a) 1.The(a) computational The computational domain domain and geometry and geometry of the of larynx, the larynx, subglottic subglottic stenosis, stenosis, vocal vocal fold, fold, and vocaland vocal tract. tract.(b) The (b) inner The inner layers’ layers’ structure structure of the of vocal the vocal fold. fold.

Table 1. Material properties of the inner layers of the vocal fold. Ep is the transversal Young’s modulus, Epz is the longitudinal Young’s modulus, Gpz is the longitudinal shear modulus, and η is the damping ratio.

Ep (kPa) Epz (kPa) Gpz (kPa) η (Pa·s) Cover 2.014 40 10 0.15 Ligament 3.306 66 40 0.225 Body 3.99 80 20 0.375

The subglottic tract was extended downward by 6.6 cm to incorporate the SGS. An ide- alized geometry of the SGS was created by using a cosine function (Figure2a). Specifically, for a given surface point (x1, y1, z1) at the SGS region where x1, y1, and z1 are the original coordinates of the point, the new coordinates (xs, ys, zs) of the point after adding the SGS were determined by using the equation below:

a yc  xs = x1 − x1 × 2 × 1 − cos 2π L

a yc  (5) zs = z1 − z1 × 2 × 1 − cos 2π L ys = y1

where L is the SGS length and was set as 1.0 cm, which is the most common SGS length [32]; yc is the distance of the lower bound of the SGS to the y component of the point (y1); and a is the percentage reduction of the hydraulic diameter of the airway cross-section due to the SGS. In this study, the severity and location of SGS were both varied. The SGS severity was quantified by the percentage of area reduction due to SGS. Seven values were used, including 0% (baseline case), 25%, 50%, 75%, 90%, 96%, and 99%, by varying “a” in Equation (5) from 0 to 0.9 (Figure2b). The SGS location was defined as the distance between the narrowest section of the SGS and the superior surface of the vocal folds Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 18

Table 1. Material properties of the inner layers of the vocal fold. Ep is the transversal Young’s modulus, Epz is the longitudinal Young’s modulus, Gpz is the longitudinal shear modulus, and η is the damping ratio.

Ep (kPa) Epz (kPa) Gpz (kPa) η (Pa·s) Cover 2.014 40 10 0.15 Ligament 3.306 66 40 0.225 Body 3.99 80 20 0.375

In this study, the severity and location of SGS were both varied. The SGS severity Appl. Sci. 2021, 11, 1221 was quantified by the percentage of area reduction due to SGS. Seven values were used,5 of 17 including 0% (baseline case), 25%, 50%, 75%, 90%, 96%, and 99%, by varying “𝑎” in Equa- tion (5) from 0 to 0.9 (Figure 2b). The SGS location was defined as the distance between the narrowest section of the SGS and the superior surface of the vocal folds (Figure 2c). Two(Figure values2c). Twowere values used, were2.1 and used, 3.13 2.1 cm, and which 3.13 cm,respectively which respectively correspond correspond to the minimum to the andminimum average and values average of 92 values SGS patients of 92 SGS reported patients in reported the past instudies the past [32,33]. studies [32,33].

(a) (b) (c)

Figure 2. ((a)) Schematics Schematics of of the the cross-section cross-section of an idealized subglotti subglotticc stenosis (SGS) geometry an andd the parameters used in Equation (5), toto createcreate thethe SGS.SGS. (b(b)) The The different different values values of of SGS SGS severity, severity, which which is is defined defined as as the the percentage percentage of areaof area reduction reduc- tion due to SGS. (c) The SGS location, defined as the distance between the narrowest section of the SGS and superior due to SGS. (c) The SGS location, defined as the distance between the narrowest section of the SGS and superior surface of surface of the vocal folds. the vocal folds. 2.3. Computational Domain and Boundary Conditions 2.3. ComputationalThe entire geometric Domain model and Boundary was immersed Conditions into a 2.3 cm × 17.8 cm × 9.9 cm rectangu- lar computationalThe entire geometric domain. model The domain was immersed was discretized into a 2.3 cm by ×a 17.864 × cm256× × 9.992 cmCartesian rectangular grid incomputational the x, y, and domain.z directions, The respectively. domain was A discretized 0.8 kPa pressure by a 64 drop× 256 was× 92 applied Cartesian between grid thein the inlet x, and y, and outlet z directions, to drive the respectively. flow. The tota A 0.8l reflection kPa pressure boundary drop condition was applied for acoustics between wasthe inletapplied and at outlet the tooutlet. drive To the eliminate flow. The the total ac reflectionoustic reflection boundary at the condition inlet boundary, for acoustics an anechoicwas applied zone at was the outlet.enforced To in eliminate the LPCE the solver acoustic to buffer reflection the acoustics at the inlet [34]. boundary, Vocal tract an wallsanechoic were zone treated was enforced as no-slip in theboundaries LPCE solver in tothe buffer incompressible the acoustics solver [34]. Vocaland tracthard wallswall boundarywere treated condition as no-slip in the boundaries LPCE solver in the [35]. incompressible A traction boundary solver and condition hard wall was boundary applied atcondition the fluid–structure in the LPCE interaction solver [35 interface,]. A traction while boundary the lateral, condition anterior wasand appliedposterior at sur- the facesfluid–structure of the vocal interaction folds were interface, fixed in while the solid the lateral,solver. anteriorA penalty-method-based and posterior surfaces contact of modelthe vocal was folds applied were to fixedprevent in thethe solidpenetration solver. of A the penalty-method-based vocal folds. An artificial contact minimum model gapwas of applied 0.2 mm to was prevent enforced the penetrationbetween the of vocal the vocalfolds, folds.which Anis necessary artificial minimumfor the success gap of the0.2 mmflow was solver. enforced The betweenvocal fold the was vocal discreti folds,zed, which using is necessary 28,997 tetrahedral for the success elements. of the flowThe time-stepssolver. The of vocal 1.149 fold × was 10 discretized, s and 0.05 using × 1.149 28,997 × 10 tetrahedral s were used elements. in the The incompressible time-steps of 1.149 × 10−6 s and 0.05 × 1.149 × 10−6 s were used in the incompressible and LPCE solvers, respectively, to satisfy their Courant–Friedrichs–Lewy condition of below 1. 256 processors were used to simulate 4.6 × 10−2 s on the XSEDE COMET cluster (Intel Xeon E5-2680v3 processors) which resulted in the computational cost of 161,000 CPU hours.

3. Results and Discussion In this section, the results of the baseline case, including the dynamics of the glot- tal flow, vocal fold vibration, and acoustics, are first presented to demonstrate that the simulation represents typical human phonatory dynamics. Next, the changes of the dynam- ics with the varied SGS severity (area reduction) and location in the cases are quantified. The relationship between the quantities and SGS severity and locations is discussed. Finally, the underlying mechanism responsible for the relationships is analyzed. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 18

and LPCE solvers, respectively, to satisfy their Courant–Friedrichs–Lewy condition of be- low 1. 256 processors were used to simulate 4.6 × 10s on the XSEDE COMET cluster (Intel Xeon E5-2680v3 processors) which resulted in the computational cost of 161,000 CPU hours.

3. Results and Discussion In this section, the results of the baseline case, including the dynamics of the glottal flow, vocal fold vibration, and acoustics, are first presented to demonstrate that the simu- lation represents typical human phonatory dynamics. Next, the changes of the dynamics with the varied SGS severity (area reduction) and location in the cases are quantified. The Appl. Sci. 2021, 11, 1221 relationship between the quantities and SGS severity and locations is discussed. Finally,6 of 17 the underlying mechanism responsible for the relationships is analyzed.

3.1. Baseline Case Dynamics 3.1. Baseline Case Dynamics 3.1.1. Glottal Flow Rate Waveform 3.1.1. Glottal Flow Rate Waveform The simulation was successfully carried out for eight vocal fold vibration cycles. Self- The simulation was successfully carried out for eight vocal fold vibration cycles. sustained vibration was established after the second cycle. Figure 3 shows the phase-av- Self-sustained vibration was established after the second cycle. Figure3 shows the phase- eraged volumetric flow rate at the glottic exit based on the steady cycles. Phase-averaging averaged volumetric flow rate at the glottic exit based on the steady cycles. Phase-averaging isis donedone byby dividingdividing eacheach steadysteady cyclecycle intointo 10001000 phasesphases andand calculatingcalculating thethe averageaverage valuevalue at eacheach phasephase overover allall thethe steadysteady cycles.cycles. The flowflow raterate showsshows aa typicaltypical glottalglottal flowflow raterate waveformwaveform ofof humanhuman phonationphonation whichwhich is characterizedcharacterized by a slower rising, faster dropping, and a lasting lasting period period of of glottic glottic closure. closure. Notice Notice that thatthe flow the flowrate did rate not did drop not to drop zero to during zero duringglottic closure glottic closuredue to the due artificial to the artificial minimum minimum glottic gap glottic enforced gap enforced in the simulation. in the simulation. Several Severaltypical voice typical quality voice related quality quantities related quantities were computed were computed based on basedthe waveform, on the waveform, including the f0; the skewness quotient (𝜏 ), defined as the ratio of the duration of flow acceleration including the f 0; the skewness quotient (τs), defined as the ratio of the duration of flow accelerationto flow deceleration; to flow deceleration;the peak glottal the flow peak ra glottalte; the mean flow rate;glottal the flow mean rate; glottal and the flow MFDR. rate; andThe thevalues MFDR. are found The values within are the found typical within phys theiological typical ranges physiological of human ranges phonation of human [31] phonation(Table 2). [31] (Table2).

Figure 3.3. The phase-averaged flowflow raterate overover oneone vibrationvibration cyclecycle inin thethe baselinebaseline case.case.

TableTable 2.2. Voice-quality-relatedVoice-quality-related quantities quantities computed computed from from the the baseline baseline model model and and their their typical typical physio- physiological ranges; f0 is the fundamental frequency; 𝜏 is the skewness quotient; 𝜏 is the open logical ranges; f0 is the fundamental frequency; τs is the skewness quotient; τo is the open quotient; quotient; 𝑞 and 𝑞 are the mean and peak glottal flow rates, respectively; MFDR is the qpeak and qmean are the mean and peak glottal flow rates, respectively; MFDR is the maximum flow maximum flow declination rate. declination rate. Computed Value Typical Range [31] Computed Value Typical Range [31] f0 (Hz) 201 65–260 f𝜏0 (Hz)1.45 2011.1–3.4 65–260 τs 1.45 1.1–3.4 𝜏 0.67 0.4–0.7 τo 0.67 0.4–0.7 𝑞 (mL/s) 220 200–580 qpeak (mL/s) 220 200–580 qmean (mL/s) 113 110–220 MFDR (L/s2) 240 -

3.1.2. Vocal Fold Dynamics Glottal area waveform, which measures the minimum glottal cross-section area versus time during sustained vibrations, typically obtained from videostroboscopy images, is an important clinical measure of vocal fold dynamics. Different from the glottal flow rate waveform, which includes the effect of both vocal folds’ kinematics and air inertial effect, glottal area waveform is determined only by the vocal fold kinematics; therefore, it is widely used to quantify the vocal fold opening amplitude, speed, and time. Figure4 shows the phase-averaged glottal area over one vibration cycle in the baseline case. Due to the artificially enforced minimum glottal gap, the glottal area did not drop to zero during glottic closure. The open quotient (τo) of the vibration, which is defined as the ratio of the Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 18

𝑞 (mL/s) 113 110–220 MFDR (L/s2) 240 -

3.1.2. Vocal Fold Dynamics Glottal area waveform, which measures the minimum glottal cross-section area ver- sus time during sustained vibrations, typically obtained from videostroboscopy images, is an important clinical measure of vocal fold dynamics. Different from the glottal flow rate waveform, which includes the effect of both vocal folds’ kinematics and air inertial effect, glottal area waveform is determined only by the vocal fold kinematics; therefore, it is widely used to quantify the vocal fold opening amplitude, speed, and time. Figure 4 Appl. Sci. 2021, 11, 1221 shows the phase-averaged glottal area over one vibration cycle in the baseline case.7 ofDue 17 to the artificially enforced minimum glottal gap, the glottal area did not drop to zero dur- ing glottic closure. The open quotient (𝜏) of the vibration, which is defined as the ratio of the duration of open glottis to the duration of the vibration cycle, was computed and duration of open glottis to the duration of the vibration cycle, was computed and found to found to be 0.67. Note that the closed glottis was defined when the time rate of change of be 0.67. Note that the closed glottis was defined when the time rate of change of the glottal areathe glottal was less area than was 2% less of than the maximum 2% of the timemaxi ratemum of time change rate of of the change glottal of area.the glottal Moreover, area. Moreover, 𝜏 is another laryngeal dynamics quantity of important implication to voice τo is another laryngeal dynamics quantity of important implication to voice quality. This parameterquality. This typically parameter ranges typically between ranges 0.4 and between 0.7 for normal 0.4 and phonation. 0.7 for normal A value phonation. lower than A 0.4value indicates lower than a “pressed” 0.4 indicates voice, a and“pressed” a value voice, greater and than a value 0.7 indicates greater than a “lax” 0.7 indicates voice [36 ].a The“lax” value voice of [36]. 0.67 The in thevalue baseline of 0.67 casein the is baseline within thecase normal is within range the normal but implies range a but lightly im- laxplies voice. a lightly lax voice.

FigureFigure 4.4. TheThe phase-averagedphase-averaged glottalglottal areaarea waveformwaveform ofof thethe baselinebaseline case.case.

ToTo showshow the detailed detailed vibration vibration pattern pattern of of the the vocal vocal folds, folds, Figure Figure 5 plots5 plots the the mid-cor- mid- coronalonal profile profile of the of the vocal vocal folds folds at four at four different different time time instants instants during during one one vibration vibration cycle. cycle. At At0.17 0.17 T, when T, when the vocal the vocal folds folds were wereopening, opening, the glottis the glottis formed formed a convergent a convergent shape. At shape. 0.36 AtT, 0.36the glottis T, the glottisformed formed a parallel a parallel channel. channel. At 0.55 At T, 0.55 when T, when the vocal the vocal folds folds were were closing, closing, the the glottis formed a divergent shape. Finally, at 0.74 T, the vocal folds fully closed and glottis formed a divergent shape. Finally, at 0.74 T, the vocal folds fully closed and re- remained closed for the rest of the cycle. This alternative convergent–divergent shape mained closed for the rest of the cycle. This alternative convergent–divergent shape change of the glottis clearly indicates the formation and propagation of a mucosal wave change of the glottis clearly indicates the formation and propagation of a mucosal wave on the vocal fold medial surface. This mucosal wave propagation is a hallmark of vocal on the vocal fold medial surface. This mucosal wave propagation is a hallmark of vocal Appl. Sci. 2021, 11, x FOR PEER REVIEWfold dynamics, found to be essential for the sustained energy transfer from flow to8 vocalof 18 fold dynamics, found to be essential for the sustained energy transfer from flow to vocal folds [36]. Both the results of the glottal area waveform and vocal fold vibration pattern folds [36]. Both the results of the glottal area waveform and vocal fold vibration pattern indicate that our simulation reproduced typical vocal fold vibratory dynamics. indicate that our simulation reproduced typical vocal fold vibratory dynamics.

Figure 5. Figure 5. The vibration pattern ofof thethe vocalvocal foldsfolds inin thethe baseline baseline case, case, shown shown by by the the mid-coronal mid-coronal profile profile of of the the vocal vocal folds folds at atfour four different different time time instants instan overts over one one vibration vibration cycle. cycle. Vocal fold vibration is known to be the result of a dynamic entrainment process of Vocal fold vibration is known to be the result of a dynamic entrainment process of a a number of vibration modes [37,38]. To reveal this dynamic process, we applied the number of vibration modes [37,38]. To reveal this dynamic process, we applied the proper proper orthogonal decomposition (POD) method to obtain the primary vibration modes of orthogonal decomposition (POD) method to obtain the primary vibration modes of the the vocal fold vibration. The POD method is a common method to use low-dimensional vocal fold vibration. The POD method is a common method to use low-dimensional quan- quantities to express complex dynamics. It computes the most energetic vibratory modes tities to express complex dynamics. It computes the most energetic vibratory modes and their modal coefficients through the proper orthogonal decomposition of vocal fold kine- matics. Figure 6a shows the 3D shape, as well as the mid-coronal profile of the first and second POD modes of the vocal fold vibration, which captured 65% and 27% of the total vibration energy, respectively. In the 2D plot, the solid line illustrates the undeformed shape of the vocal fold, and the dashed and dashed-dot lines illustrate the two-extreme states of these two modes. It can be seen that the first and second modes primarily repre- sent a lateral motion and a convergent–divergent type motion, respectively. Figure 6b shows the time history of the modal coefficient of the two modes. It reveals that the two modes vibrated at the same frequency, with a 0.38 T phase lag.

(a)

(b) Figure 6. The proper orthogonal decomposition (POD) analysis of the vocal fold vibration in the baseline case. (a) The mid-coronal profile and three-dimensional shape of the first and second POD modes of the vocal fold vibration. In the 2D plot, the solid line illustrates the undeformed shape of the vocal fold, and the dashed and dashed-dot lines illustrate the two-extreme states of these two modes. (b) Time history of the modal coefficients of the two modes.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 18

Figure 5. The vibration pattern of the vocal folds in the baseline case, shown by the mid-coronal profile of the vocal folds at four different time instants over one vibration cycle.

Vocal fold vibration is known to be the result of a dynamic entrainment process of a Appl. Sci. 2021, 11,number 1221 of vibration modes [37,38]. To reveal this dynamic process, we applied the proper 8 of 17 orthogonal decomposition (POD) method to obtain the primary vibration modes of the vocal fold vibration. The POD method is a common method to use low-dimensional quan- tities to express complex dynamics. It computes the most energetic vibratory modes and their modal coefficientsand their through modal coefficients the proper throughorthogonal the decomposition proper orthogonal of vocal decomposition fold kine- of vocal fold matics. Figure kinematics.6a shows the Figure 3D shape,6a shows as well the as 3D the shape, mid-coronal as wellas profile the mid-coronal of the first and profile of the first second POD modesand second of the vocal POD modesfold vibration, of the vocal which fold captured vibration, 65% which and 27% captured of the 65%total and 27% of the vibration energy,total respectively. vibration energy, In the respectively. 2D plot, the In solid the 2D line plot, illustrates the solid the line undeformed illustrates the undeformed shape of the vocalshape fold, of theand vocal the dashed fold, and and the dashed-dot dashed and lines dashed-dot illustrate linesthe two-extreme illustrate the two-extreme states of these twostates modes. of these It twocan be modes. seen Itth canat the be first seen and that second the first modes and second primarily modes repre- primarily represent sent a lateral motiona lateral and motion a convergent–diver and a convergent–divergentgent type motion, type motion,respectively. respectively. Figure 6b Figure 6b shows shows the timethe history time of history the modal of the coefficient modal coefficient of the two of themodes. two modes.It reveals It revealsthat the that two the two modes modes vibratedvibrated at the same at the frequency, same frequency, with a 0.38 with T a phase 0.38 T lag. phase lag.

(a)

(b)

Figure 6. TheFigure proper 6. The orthogonal proper orthogonal decomposition decomposition (POD) analysis (POD) of an thealysis vocal of the fold vocal vibration fold vibration in the baseline in the case. (a) The mid-coronalbaseline profile andcase. three-dimensional (a) The mid-coronal shape profile of the and first three-dime and secondnsional POD shape modes of ofthe the first vocal and fold second vibration. In the 2D POD modes of the vocal fold vibration. In the 2D plot, the solid line illustrates the undeformed plot, the solid line illustrates the undeformed shape of the vocal fold, and the dashed and dashed-dot lines illustrate the shape of the vocal fold, and the dashed and dashed-dot lines illustrate the two-extreme states of two-extreme states of these two modes. (b) Time history of the modal coefficients of the two modes. these two modes. (b) Time history of the modal coefficients of the two modes.

3.1.3. Acoustics Figure7a plots the phase-averaged sound pressure monitored at a hypothetical point at a distance of 5 cm outside of the tract exit (mouth). The sound pressure was calculated by assuming a 3D spherical sound wave propagation, starting from the mouth and using Equation (6): ρ dQ p = (6) s 4πr dt

where ps is the sound pressure, ρ is the air density, r is the distance of the location to the source (mouth), and dQ/dt is the time derivative of the flow rate at the mouth. It can be seen that the sound pressure oscillates with a frequency different from vibration f 0 (201 Hz), suggesting that the highest acoustic energy was not on f 0. Figure7b shows the frequency spectrum of the sound pressure. The total time length of the pressure wave is ∆t = 0.0249s, based on which the frequency resolution is 1/∆t = 40.2 Hz. The figure shows the first three of the sound pressure wave at 602, 1365, and 2449 Hz. Note that, by assuming the tract as a uniform open-closed tube, its natural frequencies can be estimated c by using Fn = (2n − 1) 4L [36], where n = 1, 2, 3, ..., c is the speed of sound and L is the length of the tube. By employing c = 351.88 m/s and L = 17.4 cm, the first three natural frequencies of the tract are 506, 1517, and 2528 Hz, consistent with the formants. Several voice-quality-related quantities were computed. The signal-to-noise ratio (SNR) is Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 18

3.1.3. Acoustics Figure 7a plots the phase-averaged sound pressure monitored at a hypothetical point at a distance of 5 cm outside of the tract exit (mouth). The sound pressure was calculated by assuming a 3D spherical sound wave propagation, starting from the mouth and using Equation (6): 𝜌 𝑑𝑄 𝑝 = (6) 4𝜋𝑟 𝑑𝑡

where 𝑝 is the sound pressure, 𝜌 is the air density, 𝑟 is the distance of the location to the source (mouth), and 𝑑𝑄⁄ 𝑑𝑡 is the time derivative of the flow rate at the mouth. It can be seen that the sound pressure oscillates with a frequency different from vibration f0 (201 Hz), suggesting that the highest acoustic energy was not on f0. Figure 7b shows the fre- quency spectrum of the sound pressure. The total time length of the pressure wave is ∆𝑡 = 0.0249𝑠, based on which the frequency resolution is 1∆𝑡⁄ =40.2 Hz. The figure shows the first three formants of the sound pressure wave at 602, 1365, and 2449 Hz. Note that, by assuming the tract as a uniform open-closed tube, its natural frequencies can be estimated by using 𝐹 =(2𝑛−1) [36], where 𝑛 = 1, 2, 3, …, 𝑐 is the speed of sound and 𝐿 is the Appl. Sci. 2021, 11, 1221 9 of 17 length of the tube. By employing 𝑐 = 351.88 m/s and 𝐿 = 17.4 ,cm the first three natural frequencies of the tract are 506, 1517, and 2528 Hz, consistent with the formants. Several voice-quality-related quantities were computed. The signal-to-noise ratio (SNR) is 11.45 2 I −12 ∆∆ps 11.45dB. Sound dB. Sound intensity, intensity, calculated calculated as 10𝑙𝑜𝑔 as 10log10 inI inwhich which 𝐼 I=100 = 10 andand 𝐼=I = 2ρ c(∆𝑝(∆ piss 0 is the amplitude of the pressure waveform, ρ is air density, and c is the employed sound the amplitude of the pressure waveform, 𝜌 is air density, and 𝑐 is the employed sound 2 (SPL(−120)/10)⁄ speed),speed), isis 89.3189.31 dB.dB. VocalVocalefficiency, efficiency, calculated calculated as as4 π4𝜋𝑟r 1010 /P𝑃pulU𝑈g (P𝑃pul isis thethe −5 pulmonarypulmonary pressure,pressure, and andU g𝑈is theis the mean mean glottal glottal flow flow rate) rate) [36], [36], is 4.74 is ×4.7410 ×. 10 All of. All these of quantitiesthese quantities are within are within or close or to close the typicalto the typical range ofrange human of human phonation phonation [36,39– [36,39–42].42].

(a) (b)

FigureFigure 7.7. ((aa)) TheThe phase-averagedphase-averaged soundsound pressurepressure waveformwaveform atat aa distancedistance ofof 55 cmcm outsideoutside ofof thethe mouthmouth inin thethe baseline case. (b(b)) Frequency Frequency spectrum spectrum of of the the sound sound pressure. pressure.

3.2. Effects of SGS Severity on Glottal Flow Dynamics, Vocal Fold Vibration, and Acoustics 3.2. EffectsIn this of section, SGS Severity we investigate on Glottal the Flow effects Dynamics, of SGS Vocal severity Fold Vibration, on voice andproduction Acoustics for the area Inreductions this section, of 25%, we investigate50%, 75%, 90%, the effects96%, and of SGS99% severityand SGS on distance voice productionof 3.13 cm. Im- for theportant area quantities reductions of of the 25%, dynamics 50%, 75%, of 90%,glotta 96%,l flow, and vocal 99% fold and vibrations, SGS distance and of acoustics 3.13 cm. Importantwere computed quantities for each of the case. dynamics The relationship of glottals between flow, vocal these fold quantities vibrations, and and SGS acoustics severity wereare discussed. computed for each case. The relationships between these quantities and SGS severity are discussed. 3.2.1. Glottal Flow Rate Waveform For all of the cases, self-sustained vibration was reached after the second cycle, and eight steady vibration cycles were generated. Following the method in the baseline case, the phase-averaged glottal flow rate waveform over one vibration cycle was plotted for each case and shown in Figure8a. It is found that there are no noticeable changes (less than 1%) in the peak glottal flow rate up to 75% area reduction, which is similar to the results in Smith and Thomson [18]. Beyond 75% area reduction, the flow rate drops rapidly with the area reduction. It is also interesting to notice that, when the area reduction is beyond 75%, the glottal flow rate waveform shows a delayed glottic closure. Quantitatively, the glottic closure occurs at the phase of 0.71 T, 0.71 T, 0.72 T, and 0.76 T, corresponding to 75%, 90%, 96%, and 99% area reduction, respectively. With the phase of the peak flow rate remaining unchanged in all of the cases, the delayed glottic closure means a prolonged duration of flow deceleration. As a result, the skewness quotient decreases much quicker when the area reduction is beyond 90% (Figure8b). The combined effect of the reduced peak flow rate and prolonged flow deceleration time also results in corresponding decreases of MFDR (Figure8c). A lower MFDR leads to reduced sound intensity. Therefore, our results indicate that both the flow rate and sound intensity start to show noticeable reduction when the area reduction is above 75% and quicker reduction when the area reduction is above 90%; below 75% area reduction, they are nearly not affected. Interestingly, the 75% threshold corresponds well to the clinical observation that area reduction below 70% is categorized into Grade I and Grade II SGS, for which no treatment is normally needed. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 18

3.2.1. Glottal Flow Rate Waveform For all of the cases, self-sustained vibration was reached after the second cycle, and eight steady vibration cycles were generated. Following the method in the baseline case, the phase-averaged glottal flow rate waveform over one vibration cycle was plotted for each case and shown in Figure 8a. It is found that there are no noticeable changes (less than 1%) in the peak glottal flow rate up to 75% area reduction, which is similar to the results in Smith and Thomson [18]. Beyond 75% area reduction, the flow rate drops rap- idly with the area reduction. It is also interesting to notice that, when the area reduction is beyond 75%, the glottal flow rate waveform shows a delayed glottic closure. Quantita- tively, the glottic closure occurs at the phase of 0.71 T, 0.71 T, 0.72 T, and 0.76 T, corre- sponding to 75%, 90%, 96%, and 99% area reduction, respectively. With the phase of the peak flow rate remaining unchanged in all of the cases, the delayed glottic closure means a prolonged duration of flow deceleration. As a result, the skewness quotient decreases much quicker when the area reduction is beyond 90% (Figure 8b). The combined effect of the reduced peak flow rate and prolonged flow deceleration time also results in corre- sponding decreases of MFDR (Figure 8c). A lower MFDR leads to reduced sound inten- sity. Therefore, our results indicate that both the flow rate and sound intensity start to show noticeable reduction when the area reduction is above 75% and quicker reduction when the area reduction is above 90%; below 75% area reduction, they are nearly not af- Appl. Sci. 2021, 11, 1221 fected. Interestingly, the 75% threshold corresponds well to the clinical observation10 of that 17 area reduction below 70% is categorized into Grade I and Grade II SGS, for which no treatment is normally needed.

(b)

(a) (c)

FigureFigure 8. 8.( a()a) The The phase-averaged phase-averaged flow flow rate rate waveform waveform in in each each case case with with the the varied varied SGS SGS severity. severity. (b (b) The) The skewness skewness quotient quotient versusversus SGS SGS severity. severity. (c ()c The) The maximum maximum flow flow deceleration deceleration rate rate (MFDR) (MFDR) versus versus SGS SGS severity. severity.

3.2.2. Vocal Fold Dynamics 3.2.2. VocalFigure Fold 9a plots Dynamics the phase-averaged glottal area waveform for different SGS severities. SimilarFigure to the9a plotsglottal the flow phase-averaged rate, the glottal glottal area is area reduced waveform beyond for a threshold different SGS severity. severi- ties.However, Similar different to the glottal from the flow glottal rate, flow the glottalrate, the area threshold is reduced value beyond is at the a SGS threshold area reduc- SGS severity.tion of 90%. However, Specifically, different with from the the area glottal reduction flow rate, of 90%, the threshold96%, and value99%, the is at maximum the SGS areaglottal reduction area, respectively, of 90%. Specifically, drops 1.2%, with 7%, the and area 30%; reduction the average of 90%, speed 96%, of glottic and 99%, opening, the maximumrespectively, glottal drops area, 1.7%, respectively, 15%, and drops49%; and 1.2%, th 7%,e average and 30%; speed the of average glottic speed closing, of glotticrespec- opening, respectively, drops 1.7%, 15%, and 49%; and the average speed of glottic closing, Appl. Sci. 2021, 11, x FOR PEER REVIEWtively, drops 1.5%, 10%, and 57% relative to the baseline case. Figure 9b plots the11 openof 18 respectively,quotient versus drops the 1.5%, SGS severity 10%, and for 57% different relative cases. to the The baseline variation case. of Figurethe open9b plotsquotient the is opennegligible quotient up versusto 90% the severity SGS severity and increases for different rapidly cases. beyond The variationthat. Our of results the open suggest quotient that, is negligible up to 90% severity and increases rapidly beyond that. Our results suggest that, whilewhile SGS SGS tends tends to to reducereduce vocal vocal fold fold vibration, vibration, the the effect effect is is noticeable noticeable only only when when the the area area reductionreduction is is beyond beyond 90% 90% (Grades (Grades III III and and IV IV SGS). SGS).

(b)

(a)

FigureFigure 9.9. (a) The phase-averaged phase-averaged glottal glottal area area waveform waveform for for different different SGS SGS severities. severities. (b) (Theb) The open open quotient quotient versus versus SGS SGSseverity. severity.

The POD analysis was performed to investigate the effect of SGS severity on vocal fold vibration modes and their dynamic entrainment. Figure 10a shows the energy per- centage of the two most energetic vibration modes in each case. The total energy of the two modes is also plotted. For all of the cases, the two modes captured more than 90% of the total vibration energy. It can be observed that there are no noticeable changes in the modal energy percentage up to 90% of area reduction. Beyond that, at a 96% and 99% area reduction, the energy of the first mode respectively increases by 8.5% and 7.6%; the energy of the second mode respectively reduces by 7% and 1.6%; and the total energy respectively increases by 1.5% and 6%. To evaluate if any changes occurred in the modal shape, the similarity of the shape of the corresponding modes in the different cases is quantified by computing the dot product of the modal shape vectors. The result is plotted in Figure 10b. A value of 1 of the dot product represents two identical modes, and a value of 0 represents two completely orthogonal modes. It is observed that the similarity of the corresponding modes between different SGS severity cases and baseline case remains as high as 98% up to 96% area reduction. A significant drop in the modal similarity on the second mode is observed when the area reduction is 99%. To show the details, Figure 10c plots the extreme states of the first and second modes of the baseline case and those of the case with 99% area reduction. As expected, the shape of the first mode remains highly similar between the two cases. However, in the second mode, a large difference is observed on the superior surface of the vocal fold that the waves on the superior surface appear to have opposite phases between the two cases. It indicates that increasing the SGS area reduction to 99% results in different dynamic entrainment between the waves on the medial surface and superior surface. Figure 10d shows the relationship between vibration frequency and SGS severity. Consistent with vocal fold dynamics, f0 is unaffected up to a 90% area reduction. At 90%, the severity f0 was slightly reduced to 199 Hz, and then it was reduced further to 195 and 183 Hz at 96% and 99% severities, respectively. In sum, our results show that the SGS starts to show noticeable effects on vocal fold vibrations when the area reduction reaches 90%, including reducing the vibration ampli- tude, increasing the glottal open quotient, and reducing the vibration frequency. The re- sults also suggest that the changes in the vocal fold dynamics are associated with the var- iation in the energy percentage of the vibration modes or the generation of different vi- bration modes. It is interesting to note that the threshold of SGS area reduction for notice- able effect on vocal fold dynamics is 90%, while on flow rate, it is 75%, suggesting that the Appl. Sci. 2021, 11, 1221 11 of 17

The POD analysis was performed to investigate the effect of SGS severity on vocal fold vibration modes and their dynamic entrainment. Figure 10a shows the energy percentage of the two most energetic vibration modes in each case. The total energy of the two modes is also plotted. For all of the cases, the two modes captured more than 90% of the total vibration energy. It can be observed that there are no noticeable changes in the modal energy percentage up to 90% of area reduction. Beyond that, at a 96% and 99% area reduction, the energy of the first mode respectively increases by 8.5% and 7.6%; the energy of the second mode respectively reduces by 7% and 1.6%; and the total energy respectively increases by 1.5% and 6%. To evaluate if any changes occurred in the modal shape, the similarity of the shape of the corresponding modes in the different cases is quantified by computing the dot product of the modal shape vectors. The result is plotted in Figure 10b. A value of 1 of the dot product represents two identical modes, and a value of 0 represents two completely orthogonal modes. It is observed that the similarity of the corresponding modes between different SGS severity cases and baseline case remains as high as 98% up to 96% area reduction. A significant drop in the modal similarity on the second mode is observed when the area reduction is 99%. To show the details, Figure 10c plots the extreme states of the first and second modes of the baseline case and those of the case with 99% area reduction. As expected, the shape of the first mode remains highly similar between the two cases. However, in the second mode, a large difference is observed on the superior surface of the vocal fold that the waves on the superior surface appear to have opposite phases between the two cases. It indicates that increasing the SGS area reduction to 99% results in different dynamic entrainment between the waves on the medial surface and superior surface. Figure 10d shows the relationship between vibration frequency and SGS severity. Consistent with vocal fold dynamics, f 0 is unaffected up to a 90% area reduction. At 90%, the severity f 0 was slightly reduced to 199 Hz, and then it was reduced further to 195 and 183 Hz at 96% and 99% severities, respectively. In sum, our results show that the SGS starts to show noticeable effects on vocal fold vibrations when the area reduction reaches 90%, including reducing the vibration amplitude, increasing the glottal open quotient, and reducing the vibration frequency. The results also suggest that the changes in the vocal fold dynamics are associated with the variation in the energy percentage of the vibration modes or the generation of different vibration modes. It is interesting to note that the threshold of SGS area reduction for noticeable effect on vocal fold dynamics is 90%, while on flow rate, it is 75%, suggesting that the change of the flow rate associated with the SGS area reduction from 75% to 90% is likely due to the flow viscous loss across the SGS itself.

3.2.3. Acoustics The signal-to-noise ratio, sound intensity, and vocal efficiency for each SGS severity case were computed based on the sound pressure and spectrum. These quantities are plotted against the severity level in Figure 11. These figures show, again, negligible differences of these quantities up to a 75% SGS severity. Beyond this threshold, SNR decreases by 0.6, 3.5, and 7.1 dB, and sound intensity decreases by 0.93, 3.34, and 15.1 dB at 90%, 96%, and 99% SGS severity, respectively. Most significantly, the vocal efficiency decreases by 38%, 82%, and 99.9% at the SGS severity of 90%, 96%, and 99%, respectively, suggesting a significant increase of vocal effort when the area reduction is extremely large. Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 18

Appl. Sci. 2021, 11, 1221 12 of 17 change of the flow rate associated with the SGS area reduction from 75% to 90% is likely due to the flow viscous loss across the SGS itself.

(a)

(c)

(d) (b)

Figure 10.Figure(a) The10. (a energy) The energy percentage percentage of the of two the dominanttwo dominant proper proper orthogonal orthogonal decomposition decomposition (POD) (POD) modes modes ofof thethe vocal fold fold vibration versus SGS severity. (b) The similarity of the corresponding POD modes of the vocal fold vibration versus vibration versus SGS severity. (b) The similarity of the corresponding POD modes of the vocal fold vibration versus SGS SGS severity. (c) The mid-coronal profile of the vocal fold in the two dominant POD modes in the baseline case (solid severity.lines) (c) The and mid-coronalthe 99% SGS severity profile ofcase the (das vocalhed foldline). in The the two two subfigures dominant of PODeach mo modesde show in the the baselinetwo extreme case phases (solid of lines) the and the 99%mode. SGS severity(d) The fundamental case (dashed frequency line). The of vocal two subfiguresfold vibration of versus each modeSGS severity. show the two extreme phases of the mode. (d) Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 18 The fundamental frequency of vocal fold vibration versus SGS severity. 3.2.3. Acoustics The signal-to-noise ratio, sound intensity, and vocal efficiency for each SGS severity case were computed based on the sound pressure and spectrum. These quantities are plot- ted against the severity level in Figure 11. These figures show, again, negligible differences of these quantities up to a 75% SGS severity. Beyond this threshold, SNR decreases by 0.6, 3.5, and 7.1 dB, and sound intensity decreases by 0.93, 3.34, and 15.1 dB at 90%, 96%, and 99% SGS severity, respectively. Most significantly, the vocal efficiency decreases by 38%, 82%, and 99.9% at the SGS severity of 90%, 96%, and 99%, respectively, suggesting a sig- nificant increase of vocal effort when the area reduction is extremely large.

FigureFigure 11. 11.The The signal-to-noise signal-to-noise ratio, ratio, sound sound intensity, intensity, and and vocal vocal efficiency efficiency for for different different SGS SGS severities. severi- ties.

3.3. Effects of SGS Location on Glottal Flow Dynamics, Vocal Fold Vibration and Acoustics In this paragraph, we investigate the effects of SGS location on voice production. Since the above results show that SGS only shows noticeable effects on voice production when the area reduction is above 90%, the effects of SGS location are only studied on the cases with SGS area reduction of 90% and 96%. Two SGS distances of 3.13 and 2.1 cm were employed in the simulations with the varied SGS severity. Figure 12 compares the glottal flow rate waveform, glottal area waveform, f0, sound intensity, and the similarity of the vibration modes with those of the baseline case. The overall observation is that the SGS location does not change its effects. The changes in the peak flow rate, maximum glottal area, and other quantities in Figure 12, due to the change of the SGS location, are all within 2%. We have also computed other quantities including the skewness and open quotients, MFDR, mode energies, SNR, and vocal efficiency. Overall, we do not observe noticeable effects of SGS location on them. This clearly indicates that the voice outcome is insensitive to the location of SGS.

Appl. Sci. 2021, 11, 1221 13 of 17

3.3. Effects of SGS Location on Glottal Flow Dynamics, Vocal Fold Vibration and Acoustics In this paragraph, we investigate the effects of SGS location on voice production. Since the above results show that SGS only shows noticeable effects on voice production when the area reduction is above 90%, the effects of SGS location are only studied on the cases with SGS area reduction of 90% and 96%. Two SGS distances of 3.13 and 2.1 cm were employed in the simulations with the varied SGS severity. Figure 12 compares the glottal flow rate waveform, glottal area waveform, f 0, sound intensity, and the similarity of the vibration modes with those of the baseline case. The overall observation is that the SGS location does not change its effects. The changes in the peak flow rate, maximum glottal area, and other quantities in Figure 12, due to the change of the SGS location, are all within 2%. We have also computed other quantities including the skewness and open quotients, Appl. Sci.MFDR, 2021, 11, x FOR mode PEERenergies, REVIEW SNR, and vocal efficiency. Overall, we do not observe noticeable14 of 18

effects of SGS location on them. This clearly indicates that the voice outcome is insensitive to the location of SGS.

(a)

(c)

(b)

FigureFigure 12. The 12. comparisonThe comparison of the phase-averaged of the phase-averaged (a) flow rate waveform (a) flowand (b) rate glottal waveform area waveform and in (theb) simulation glottal area cases with the varied SGS location and severity. (c) The fundamental frequency, sound intensity, and similarity of the correspondingwaveform modes in the in simulation the cases with cases the varied with SGS the location varied and SGS severity. location and severity. (c) The fundamental frequency, sound intensity, and similarity of the corresponding modes in the cases with the varied 3.4. Underlying Mechanism—Correlation to the Flow Resistance SGS location and severity. Same as Smith and Thomson [18], we hypothesized that the above-observed effects 3.4. Underlying Mechanism—Correlationof SGS on voice production to are the primarily Flow Resistance due to its effect on the flow resistance. To exam- ine this hypothesis, we computed the flow resistances across the SGS (trans-stenosis re- Same as Smithsistance) and Thomson and glottis [ 18(trans-glottis], we hypothesized resistance) for that all theof the above-observed SGS severity cases. effects The flow of |∆| SGS on voice productionresistance are is defined primarily as due, where to its∆𝑃 effect is the mean on the pressure flow drop resistance. across the To SGS examine or glottis, this hypothesis, weand computed 𝑄 is the corresponding the flow resistances incompressible across glottal the flow SGS rate. (trans-stenosis The total flow resistance) resistance is and glottis (trans-glottisdefined as resistance) the sum of the for two. all ofFigure the 13a SGS shows severity the cycle-averaged cases. The flow flow resistances resistance ver- | | is defined as ∆P ,sus where the SGS∆P severityis the meanin which pressure the SGS location drop acrossis 3.13 cm. the SGS or glottis, and Q is Q It is noticed that, up to the SGS area reduction of 75%, the trans-stenosis resistance the correspondingstays incompressible nearly zero, and glottal as a result, flow the rate. total The flow total resistance flow is resistance the same as isthe defined trans-glottis as the sum of the two.flow Figure resistance. 13a Moreover, shows the the cycle-averaged change of the SGS area flow reduction resistances up to versus75% does the not SGSaffect severity in which thethe trans-glottis SGS location flow is resistance. 3.13 cm. It indicates that, when the SGS area reduction is below 75%, the SGS does not generate additional flow resistance nor affect the trans-glottis flow resistance. When the SGS area reduction is 75%, a very slight increase in the trans-stenosis flow resistance and total flow resistance can be seen. Beyond 75% SGS severity, the trans- stenosis resistance starts to increase rapidly with the increasing SGS severity. In the mean- while, the trans-glottis resistance slightly drops. Because the increase of the trans-stenosis Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 18

resistance is much more than the drop of the trans-glottis resistance, the total flow re- sistance also rapidly increases with the SGS severity. When the SGS area reduction reaches 99%, the trans-stenosis resistance has become comparable to the trans-glottis resistance; as a result, the total flow resistance is about twice its values in the baseline case. These observations indicate that, when the SGS area reduction is beyond 75%, the SGS starts to generate additional flow resistance, which increased rapidly with the SGS area reduction. The effect of the SGS on the flow resistance is found to be consistent with its effect on the flow rate and vocal fold vibration. The SGS is found to start to lower the flow rate at 75% area reduction and lower the vocal fold vibration amplitude at 90% area reduction. These are consistent with its threshold to increase the flow resistance. That the threshold of SGS severity to affect vocal fold vibration appears higher than that to affect the flow rate is likely because the flow rate is more sensitive to the changes in flow resistance. Fur- thermore, Titze and Talkin [28] reported that a decrease in trans-glottis pressure drop could result in a slight decrease in the f0. Our results show that increasing the SGS severity beyond 75% decreases the flow rate and trans-glottis flow resistance, suggesting that the trans-glottis pressure would also be decreased. It thus explains the drop of the f0 with the Appl. Sci. 2021, 11, 1221 SGS severity beyond 75%. 14 of 17 To investigate why the threshold SGS severity to increase the flow resistance occurs at 75% area reduction, we computed the area ratio between the glottis and SGS and plot its relationship with SGS severity in Figure 13b. The area ratio is defined as the cycle- averagedIt is noticed glottal that, area to up the to SGS the area. SGS The area fi reductiongure shows ofa similar 75%, therelationship trans-stenosis between resistance the staysarea nearly ratio and zero, SGS and severity as a result, to that the of the total trans-stenosis flow resistance flow resistance: is the same The as area the ratio trans-glottis re- flowmains resistance. nearly zero Moreover, until the SGS the severity change reaches of the 75%; SGS after area that, reduction it rapidly up increases to 75% with does not affectthe theSGStrans-glottis severity. Therefore, flow resistance.the effect of SGS It indicates on the area that, ratio when correlates the SGSwell with area its reduction effect is belowon the 75%, trans-stenosis the SGS does flow not resistance, generate suggesting additional that flow the resistance area ratio noris a key affect parameter, the trans-glottis to flowdetermine resistance. the Wheneffect of the SGS SGS on areathe flow reduction resistance. is75%, Specifically, a very when slight the increase glottal inarea the is trans- stenosissignificantly flow resistance smaller than and the total SGS area flow (SGS resistance severity canbelow be 75%), seen. the Beyond SGS generates 75% SGS nearly severity, theno trans-stenosis additional flow resistance resistance, startsand the to vocal increase fold dynamics rapidly withare little the affected. increasing By increasing SGS severity. In thethe meanwhile,SGS severity beyond the trans-glottis 75%, the area resistance ratio starts slightly to increase drops. rapidly, Because and the the trans-steno- increase of the sis flow resistance increases accordingly. It is noticed that, when the SGS severity is 96%, trans-stenosis resistance is much more than the drop of the trans-glottis resistance, the the area ratio reaches 0.62, meaning that the SGS area is only less than two times the cycle- total flow resistance also rapidly increases with the SGS severity. When the SGS area averaged glottal area, but the trans-stenosis resistance is only 0.14 of the trans-glottis re- reductionsistance. reaches When the 99%, SGS the severity trans-stenosis reaches 99%, resistance the area has ratio become becomes comparable 2.14, meaning to thethat trans- glottisthe SGS resistance; area is only as a half result, of the the cycle-averaged total flow resistance glottal area, is about but the twice trans-stenosis its values resistance in the baseline case.is about These the observations same as the trans-glottis indicate that, resistance when. the The SGS reason area is that reduction very high is beyondtrans-glottis 75%, the SGSflow starts resistance to generate is generated additional during flow the early resistance, glottal opening which increasedand late glottal rapidly closing with when the SGS areathe reduction. glottal area is very small.

(a) (b)

FigureFigure 13. (a )13. The (a) cycle-averaged The cycle-averaged trans-glottis, trans-glottis, trans-stenosis, trans-stenosis, and and total total flowflow resistanceresistance versus versus SGS SGS severity. severity. (b) ( bThe) The ratio ratio of of the cycle-averaged glottal area to the SGS area versus SGS severity. the cycle-averaged glottal area to the SGS area versus SGS severity.

The effect of the SGS on the flow resistance is found to be consistent with its effect on the flow rate and vocal fold vibration. The SGS is found to start to lower the flow rate at 75% area reduction and lower the vocal fold vibration amplitude at 90% area reduction. These are consistent with its threshold to increase the flow resistance. That the threshold of SGS severity to affect vocal fold vibration appears higher than that to affect the flow rate is likely because the flow rate is more sensitive to the changes in flow resistance. Furthermore, Titze and Talkin [28] reported that a decrease in trans-glottis pressure drop could result in a slight decrease in the f 0. Our results show that increasing the SGS severity beyond 75% decreases the flow rate and trans-glottis flow resistance, suggesting that the trans-glottis pressure would also be decreased. It thus explains the drop of the f 0 with the SGS severity beyond 75%. To investigate why the threshold SGS severity to increase the flow resistance occurs at 75% area reduction, we computed the area ratio between the glottis and SGS and plot its relationship with SGS severity in Figure 13b. The area ratio is defined as the cycle-averaged glottal area to the SGS area. The figure shows a similar relationship between the area ratio and SGS severity to that of the trans-stenosis flow resistance: The area ratio remains nearly zero until the SGS severity reaches 75%; after that, it rapidly increases with the SGS severity. Therefore, the effect of SGS on the area ratio correlates well with its effect on the trans-stenosis flow resistance, suggesting that the area ratio is a key parameter, to determine the effect of SGS on the flow resistance. Specifically, when the glottal area is significantly smaller than the SGS area (SGS severity below 75%), the SGS generates nearly no additional flow resistance, and the vocal fold dynamics are little affected. By increasing Appl. Sci. 2021, 11, 1221 15 of 17

the SGS severity beyond 75%, the area ratio starts to increase rapidly, and the trans-stenosis flow resistance increases accordingly. It is noticed that, when the SGS severity is 96%, the area ratio reaches 0.62, meaning that the SGS area is only less than two times the cycle-averaged glottal area, but the trans-stenosis resistance is only 0.14 of the trans-glottis resistance. When the SGS severity reaches 99%, the area ratio becomes 2.14, meaning that

Appl. Sci. 2021, 11, x FOR PEER REVIEWthe SGS area is only half of the cycle-averaged glottal area, but the trans-stenosis resistance16 of 18 is about the same as the trans-glottis resistance. The reason is that very high trans-glottis flow resistance is generated during the early glottal opening and late glottal closing when the glottal area is very small. Finally,Finally, wewe comparecompare thethe flowflow resistancesresistances inin thethe casescases withwith differentdifferent SGSSGS locations.locations. AsAs shownshown inin FigureFigure 1414,, thethe SGSSGS locationlocation hashas negligiblenegligible effectseffects onon flowflow resistance.resistance. TheThe 2.12.1 andand 3.133.13 cmcm casescases havehave nearlynearly thethe samesame flowflow resistance,resistance, withwith reductionsreductions ofof 0.8%0.8% ofof thethe totaltotal flowflow resistanceresistance andand 1.2%1.2% ofof thethe trans-glottistrans-glottis resistanceresistance forfor thethe 90%90% areaarea reductionreduction casecase andand 0.4%0.4% ofof thethe totaltotal flowflow resistanceresistance andand 0.9%0.9% ofof thethe trans-glottistrans-glottis resistanceresistance forfor thethe 96%96% areaarea reductionreduction case.case. ThisThis alsoalso nicelynicely correspondscorresponds toto thethe observationobservation thatthat thethe SGSSGS locationlocation hashas nearlynearly nono effecteffect onon voicevoice productionproduction withinwithin thatthat stenosis-glottisstenosis-glottis distancedistance interval.interval.

FigureFigure 14.14. The cycle-averaged trans-glottis, trans-stenos trans-stenosis,is, and and tract tract flow flow resistance resistance of of the the 2.1 2.1 and and 3.133.13 cmcm SGSSGS distancesdistances forfor thethe 90%90% andand 96%96% SGSSGS severities.severities.

4.4. Conclusions InIn thisthis study,study, aa 3D3D laryngeallaryngeal modelmodel waswas employedemployed toto studystudy thethe effectseffects ofof subglotticsubglottic stenosisstenosis on dynamics dynamics of of glottal glottal flow, flow, vocal vocal fold fold vibration, vibration, and and acoustics acoustics during during voice voice pro- production.duction. The The results results show show that that the the SGS SGS starts starts to to have have a a noticeable noticeable effect effect on glottal flowflow raterate onlyonly whenwhen thethe areaarea reductionreduction isis beyondbeyond 75%75% andand hashas anan effecteffect onon vocalvocal foldfold vibrationvibration whenwhen thethe areaarea reduction reduction is is beyond beyond 90%. 90%. Specifically, Specifically, when when the the SGS SGS area area reduction reduction is below is be- thislow threshold,this threshold, varying varying the SGS the areaSGS area reduction reduction had littlehad little effect effect on glottal on glottal flow, flow, vocal vocal fold vibration,fold vibration, and acoustics; and acoustics; when itwhen is beyond it is beyond this threshold, this threshold, increasing increasing the SGS area the reductionSGS area quicklyreduction decreases quickly thedecreases flow rate, the flow vocal rate, fold vocal vibration fold vibration amplitude, amplitude, and vocal and efficiency; vocal effi- it slightlyciency; it decreases slightly thedecreases skewness the quotient,skewness vibrationquotient, frequency,vibration frequency, signal-to-noise signal-to-noise ratio, and vocalratio, intensity;and vocal and intensity; it slightly and increases it slightly the increases open quotient. the open Furthermore, quotient. Furthermore, our results show our thatresults large show SGS that area large reduction SGS area can reduction also affect can vocal also foldaffect dynamics vocal fold by dynamics affecting by the affecting energy percentagethe energy ofpercentage the vibration of the modes vibration or generating modes differentor generating vibration different modes. vibration Changing modes. SGS locationChanging shows SGS location no effect shows on the no dynamics. effect on the dynamics. TheThe detaileddetailed analysis analysis on on the the trans-stenosis trans-stenosis flow flow resistance, resistance, trans-glottis trans-glottis flow resistance, flow re- andsistance, total and flow total resistance flow resistance shows that shows the effects that th ofe effects SGS are of related SGS are to related its effect to onits theeffect total on flowthe total resistance, flow resistance, which is which further is related further to related the ratio to the of theratio cycle-averaged of the cycle-averaged glottal area glottal to SGS area. When the SGS area reduction is below 75%, the area ratio was nearly zero, area to SGS area. When the SGS area reduction is below 75%, the area ratio was nearly meaning that the glottis opening is much smaller than the SGS opening. In these cases, the zero, meaning that the glottis opening is much smaller than the SGS opening. In these trans-stenosis resistance is nearly zero, and the total flow resistance is almost the same as cases, the trans-stenosis resistance is nearly zero, and the total flow resistance is almost the trans-glottis flow resistance, suggesting that the SGS does not generate additional flow the same as the trans-glottis flow resistance, suggesting that the SGS does not generate additional flow resistance. As a result, the SGS shows nearly no effect on the dynamics. By increasing the SGS severity beyond 75%, the area ratio starts to increase rapidly with the SGS severity, and the trans-stenosis flow resistance is observed to increase rapidly too, resulting in the quick increase of the total flow resistance. As a result, the flow rate and vocal fold vibration reduce quickly. In the meantime, increasing SGS severity also causes a decrease in the trans-glottis resistance and trans-glottis pressure drop, which further causes a drop in the fundamental frequency. Therefore, our results indicate that the effect of SGS on phonatory dynamics is largely determined by the area ratio between the glottis Appl. Sci. 2021, 11, 1221 16 of 17

resistance. As a result, the SGS shows nearly no effect on the dynamics. By increasing the SGS severity beyond 75%, the area ratio starts to increase rapidly with the SGS severity, and the trans-stenosis flow resistance is observed to increase rapidly too, resulting in the quick increase of the total flow resistance. As a result, the flow rate and vocal fold vibration reduce quickly. In the meantime, increasing SGS severity also causes a decrease in the trans-glottis resistance and trans-glottis pressure drop, which further causes a drop in the fundamental frequency. Therefore, our results indicate that the effect of SGS on phonatory dynamics is largely determined by the area ratio between the glottis and SGS. A threshold area ratio exists, below which the SGS has nearly no effect on the dynamics, and, beyond which, the effect of SGS grows rapidly with its severity. In our results, the threshold area ratio is around 0.1, corresponding to an SGS area reduction of 75%. The results are found to be consistent with clinical observations that Grades I and II SGS (area reduction below 70%) had very minor effects on both voice production, and respiration and normally no treatment is needed; only at Grades III and IV SGS, effects on breathing and voice production become significant and treatments are needed.

Author Contributions: Conceptualization, X.Z. and S.T.; methodology, D.B., Q.X., and X.Z.; formal analysis, D.B., Q.X., and X.Z.; investigation, D.B., Q.X., and X.Z.; writing—original draft preparation, D.B., Q.X., and X.Z.; writing—review and editing, Q.X., X.Z., and S.T.; supervision, Q.X. and X.Z.; project administration, X.Z. and S.T.; funding acquisition, S.T. All authors have read and agreed to the published version of the manuscript. Funding: This research was partially funded by the National Institute on Deafness and Other Communication Disorders (NIDCD), Grant No. 5R01DC009616. The numerical simulation used the Extreme Science and Engineering Discovery Environment (XSEDE) (allocation award number TG-BIO150055). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available upon request. Conflicts of Interest: The authors declare no conflict of interest.

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