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Cooperative Regulation of Vocal Fold Morphologyand Stress by the Cricothyroid and ThyroarytenoidMuscles

*Shinji Deguchi,Yuki Kawahara, andSatoshi Takahashi, *Sendai andyOkayama, Japan

Summary:Voice is produced by vibrations of vocal folds that consist of multiple layers. The portion of the vocal fold

tissue that vibrates varies depending primarily on laryngeal muscle activity. The effective depth of tissue vibrationshould significantly influence the vibrational behavior of the tissue and resulting voice quality. However, thus far,

the effect of the activation of individual muscles on the effective depth is not well understood. In this study, a three-

dimensional finite element analysis is performed to investigate the effect of the activation of two major laryngealmuscles, the cricothyroid (CT) and thyroarytenoid (TA) muscles, on vocal fold morphology and stress distribution in

the tissue. Because structures that bear less stress can easily be deformed and involved in vibration, information on

the morphology and stress distribution may provide a useful estimate of the effective depth. The results of the analyses

indicate that the two muscles perform distinct roles, which allow cooperative control of the morphology and stress.

When the CT muscle is activated, the tip region of the vocal folds becomes thinner and curves upward, resulting in

the elevation of the stress magnitude all over the tissue to a certain degree that depends on the stiffness of each layer.

On the other hand, the TA muscle acts to suppress the morphological change and controls the stress magnitude in

a position-dependent manner. Thus, the present analyses demonstrate quantitative relationships between the two mus-cles in their cooperative regulation of vocal fold morphology and stress.

Key Words:Vocal foldLarynxCricothyroid muscleThyroarytenoid muscleFinite element analysis.

INTRODUCTION

Voice is produced at the larynx by structure-fluid interactions

between the vocal folds and respiratory airflow. Mechanical

stress in the vocal folds during these interactions is believed

to cause vocal fold injuries.1 The stress produced by external

loading has been previously studied in finite element

models.27 However, actual vocal folds are complex structures

that experience not only passive stress but also active stress

because of muscle contractions. The vocal folds primarily

consist of membranous, ligamentous, and thyroarytenoid (TA)

muscular layers

8

(hereafter referred to as the membrane, liga-ment, and TA muscle, respectively) and are also externally

stretched inthe anterior-posterior direction by the cricothyroid

(CT) muscle7 in addition to responding to the action of the TA

muscle inside them. Thus, even without airflow forces, thevocal

folds should undergo complicated stresses depending on the

way in which they are acted on by the attached muscles.It has been proposed that vocal folds use muscle-dependent

structural changes to control phonation onset and voice

pitch.811 This theory suggests that tension in the membrane,

regulated by the CT muscle, plays a dominant role in

controlling voice at small-amplitude vibrations, whereas the

TA muscle is involved in large-amplitude vibrations and acts

as an antagonist for the CT muscle.12

Because the mechanical

properties of each layer differ,13 the amplitude of vibration

and parts of the tissue that vibrate are critical. The effective

depth of tissue vibration might be regulated actively by the

attached muscles and passively by the air pressure. Although

such muscle-dependent active stress has been considered in pre-

vious analyses,2,14,15 there has been no study that describes the

relationship between muscle activation and the resultant stress

distribution and morphology, which would be useful as an

estimate of the effective depth. Because the physical depthof

each layer changes in response to the muscle activation,16 in

addition to a longitudinal length change, a three-dimensionalanalysis considering all directions should be used.

The present study evaluates the effect of muscle activation on

the morphology and stress in the prephonatory position using

a three-dimensional finite element analysis. We focus on the

roles of the CT and TA muscles because they are the principal

laryngeal muscles that directly influence vocal fold dynam-

ics.9,17 Static changes in the morphology and stress caused by

the muscle activation are investigated. Tissue thixotropy and

viscoelasticity are not considered in this study. The expected

effective depth is discussed in terms of the stress distribution

in the tissue without analyses of the structure-fluid interaction

because structures that bear less stress could be deformed

relatively easily in response to the driving air pressure.18,19

METHODS

Vocal fold geometry and boundary condition

Finite element calculations are performed using Ansys 11.0

(Cybernet Systems, Tokyo, Japan). A pair of vocal folds is stud-

ied (Figure 1) in which the membrane, ligament, and TA muscle

are included. In the absence of muscle activation, the model is

14-mm long intheZdirection and has the same shape inany cor-

onal (X-Y) plane along theZ-axis. The membrane geometry of

the coronal section is formed by a combination of two sectors,

Accepted for publication November 15, 2010.From the *Department of Biomedical Engineering, Tohoku University, Sendai, Japan;

and the yDepartment of Mechanical Engineering, Graduate School of Natural Scienceand Technology, Okayama University, Kita-ku, Okayama, Japan.

Address correspondence and reprint requests to Shinji Deguchi, Department ofBiomedical Engineering, Tohoku University, 6-6-11-1306-2 Aramaki-Aoba, Sendai980-8579, Japan. E-mail:[email protected]

Journal of Voice, Vol. 25, No. 6, pp. e255-e2630892-1997/$36.00 2011 The Voice Foundationdoi:10.1016/j.jvoice.2010.11.006
mailto:[email protected]://dx.doi.org/10.1016/j.jvoice.2010.11.006http://dx.doi.org/10.1016/j.jvoice.2010.11.006mailto:[email protected]

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X 5:52Y 22 22 20 q 0; (1)

X2 Y2 7:72 90 q< 20; (2)

where the origin of theX-Ycoordinates is located at the upper-right corner of the plane, and the angle q is measured from the

X-axis. The shapes of the ligament and TA muscle without

activation are geometrically similar to that of the membrane.

The membrane and ligament are 0.35- and 0.8-mm thick,

respectively, according to the literature.8,11 Although different

mechanical properties are assigned to each of these three

layers, relative displacement at any of the interfaces is not

considered. The geometry of this vocal fold model was

determined on the basis of anatomic images anddrawings1,2022 and other modeling studies.3,23 According to

these anatomic data, the entire vocal fold surface is lined with

the vocal ligament layer, as modeled inFigure 1. However, the

ligament is relatively thick around the tip region (which vibrates

the most), and such a localized ligament, without any lining

between the membrane and the TA muscle, has been used in

some previous modeling studies.2,57,24 Thus, the geometry of

the ligament could be a point of controversy; however, in the

present study, we focus on aspects that are essentially

unaffected by whether the ligament is localized or distributed

along the layer.

It is assumed that the relative positions among the elements

in each of the most anterior (the plane initially at Z 14 mm),posterior (Z 0 mm), or lateral (X 0 mm) flat surfaces,where the thyroid and arytenoid cartilages are located, remain

unchanged at each muscle activation level. This boundary

condition may result in an unnaturally high stress concentration

near those planes. However, in this study, we consider themedial

portion involved in actual vibration, and the other laryngealmuscles responsible for adduction/abduction of the vocal folds

are not included.

Constitutive equation

The vocal fold is assumed to consist of a linear elastic material.

A tensor expression of Hookes law is written as

sij Cijkl3kl i;j; k; l x;y; z; (3)

where sij, 3kl, and Cijkl are the stress, strain, and stiffness ten-

sors, respectively. Because of the symmetry of the tensors, 21

independent constants appear in the stiffness tensor compo-

nents. The vocal fold is assumed to be orthotropic with two or-thogonal planes of symmetry, thereby reducing the number ofindependent constants to nine. As the fiber tissues run in the

Zdirection, the vocal fold is further assumed to be transversely

isotropic, that is, having material properties that are symmetric

about the Z-axis. Equation 3can thus be written explicitly as

whereEx(Ey)andEzare the Youngs moduli in the transverse(X or Y) and longitudinal (Z) directions, respectively;

Gyz (Gzx) is the shear modulus in the longitudinal direction;and nx (ny) and nz are the Poisson ratios in the transverseand longitudinal directions, respectively. The number of inde-

pendent constants is now five, which is consistent with Alipour

et al.

2

Because the data available on Poisson ratio are few, forsimplicity, it is assumed that nx nz (n). Consequently, fourindependent parameters (Ex, Ez, n, and Gyz) are required for

each layer.

Muscle activation

Contraction of the CT muscle shortens the distance between the

anterior parts of the cricoid and thyroid cartilages, resulting in

an elongation of the vocal fold and accumulation of tensile

strain in its fiber direction (Figure 2A). In the present study,

activation of the CT muscle is approximated by a stretching

of the entire vocal fold in the Z direction. To evaluate the

FIGURE 1. Vocal fold model.

26666664

sxx

syy

szz

syz

szx

sxy

37777775

2666666664

Ex1vz1vz

12v2zvx1vxExvxv2z

12v2zvx1vxExvz

12v2zvx0 0 0

Exvxv2z12v2zvx1vx

Ex1vz1vz

12v2zvx1vxExvz

12v2zvx0 0 0

Ezvz12v2zvxEzvz12v2zvx

Ez1vx12v2zvx 0 0 0

0 0 0 Gyz 0 00 0 0 0 Gyz 00 0 0 0 0 Ex

21vx

3777777775

2

666664

3xx

3yy

3zz

3yz

3zx

3xy

3

777775; (4)

Journal of Voice, Vol. 25, No. 6, 2011e256

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activity of the CT muscle, a parameterAct is introduced (follow-

ing previous analytical studies

9

) that ranges from 0%(no stretch) to 100% (maximum stretch). In the previous study,

maximum CT muscle activation with no TA muscle activation

was reported to yield a tensile strain of approximately 0.3 in the

vocal fold length direction. Here, the maximum stretch is

assumed to be 4 mm, corresponding to a strain of 0.29 (roughly

4 mm divided by the original or nonstress length L0of 14 mm).

Consequently, an enforced displacement DLct(04 mm) of the

following form is given to the entire vocal fold in the positive

Zdirection:

DLct 4Act=100 mm: (5)

To assess the effect of the TA muscle activation, the muscle

contraction is modeled as a shortening of the reference (or non-

stress) length in theZdirection (ie, muscle fiber direction) from

L0toL00(Figure 2B). If a muscle contracts without a constraint

that prevents a free displacement, the active shortening results

in no appearance of strain or tension even with substantive de-

crease in length. A tensile strain (3active >0) is generated if there

are constraints, that is, the thyroid and arytenoid cartilages. Inthis manner, the real TA muscle will achieve an isometric con-

traction in which a certain tension is present while keeping the

apparent length unchanged. To quantify the activity of the TA

muscle, another parameter, Ata, is introduced that ranges from

0% (no contraction) to 100% (maximum contraction). Accord-

ing to previous studies of the TA muscle,25,26 considerable

variability exists between samples in terms of active strain

that appears when the muscle fully contracts. However,

a strain of 0.20.3 seems to be reasonably typical as the

maximum value. Here, the maximum strain is assumed to be

0.3; more specifically, an active tensile strain of 0.3 (because

of internal sliding of the muscle fibers to achieve the

reference length L00 in the Z direction) is produced atAta 100% in the Z direction by the TA muscle while theoriginal length L0 in the Z direction is kept unchanged (ie,

isometric contraction). Thus,

Ata 3active100

0:3

L0 L00

L00$

100

0:3%: (6)

A resultant strain in the TA muscle, resulting from the

activities of the two muscles, is the sum of the active strain

caused by the TA muscle and passive strain caused by the

CT muscle (Figure 2B). Both strains are included in the

calculation of the strain tensor for the TA muscle. In separate

analyses, we confirmed that the resultant force caused by theactions of the two different muscles were actually a linear

sum of the active and passive components (Supplementary

Figure S1). Similarly, in previous numerical studiesthat con-

sidered the effect of active stress/strain,14,15,2730 a single

term for active strain (or stress) was linearly added to

another strain (or stress) caused by external loading. Thus,

the current model essentially follows the same approach as

in previous studies.The muscle under contraction thickens,24 and this lateral

expansion or bulging because of longitudinal shortening can

be mimicked by artificially setting new reference lengths in

the X and Y directions, which are longer than the original

lengths. The cover and ligament that line the bulging TA mus-

cle are stretched in the X-Yplane, whereas the TA muscle is

compressed in the same plane because the cover and ligament

function as a constraint that prevents the free lateral expansion

of the muscle inside. At the same time, as long as tension is

generated in the Zdirection because of longitudinal muscle

contraction, the muscle also tends to shrink within the trans-

verse plane to a certain degree, depending on the value of

the Poisson ratio. Therefore, performing a realistic simulation

of the bulging effect requires a reasonable setting of the refer-

ence lengths in theXand Ydirections to yield an expansion in

A

B

FIGURE 2. Active and passive strains. A. Tensile strain appears

when material is stretched (from i to ii). B. Free active shortening with-

out a constraint (from iii to iv) does not yield strain. If there is a con-

straint at the two ends, strain is produced in the material and is thus

under isometric contraction (v). Here, the reference length is reduced

from the original length L0 to L00, which results in the appearance of

the strain. If the material with the active strain is further stretched

externally or passively, the resulting total strain becomes the sum of

the active and passive strains (vi).

Shinji Deguchi, et al Muscle Cooperative Regulation on VF Morphology and Stress e257

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the transverse plane that exceeds the lateral shrinkage caused

by the nonzero Poisson ratio. The fact that the degree of

expansion is related to the action of the TA muscle, because

the expansion is caused by the muscle contraction, leads to

further complications. Because the data available on the

degree of expansion intrinsic to the TA muscle are few, the

lateral expansion is not included here to avoid incorporating

an arbitrary assumption. Later, at the end of the Effect onShapesection, the effect of muscle expansion is investigated

separately.

Material properties

Unless otherwise specified, the material properties given in this

section are used in the finite element models. The longitudinal

elastic modulusEzis 41.9 kPa for the membrane and 20.7 kPa

for the TA muscle, based on data from canine vocal folds in

which a linear approximation of stress-strain curves within

a strain range 00.15 was used.13 However, stress-strain curves

for human vocal ligaments have been found to be significantly

nonlinear.

31

AnEzvalue of 30 kPa is chosen for the ligament onthe basis of data approximated linearly within a strain range

00.15. We have found a quantitative measurement of the

transverse elastic modulus Ex reported by Kakita et al,32 but

their value (40 kPa) in canine cadavers seems high when com-

pared with the above longitudinal value, possibly because of

postrigor stiffening. As the transverse modulus should be lower

than the longitudinal modulus in each layer, we simply multi-

ply the value ofEzby 0.4 to approximateEx. Thus, the values of

Exused are 16 kPa for the membrane, 12 kPa for the ligament,

and 8 kPa for the TA muscle. The Poisson ratio n is assumed to

be 0.3 for all the layers, following the literature,6 and its effect

is investigated separately later on at the end of the Effect on

Shapesection. The longitudinal shear modulus Gyzis assumedto be 10 kPa for the membrane, 40 kPa for the ligament, and

12 kPa for the TA muscle, according to the literature.2 The

tension in the TA muscle of excised feline larynges was

reported to increase by at most 260% based on tetanic stimula-

tions.26 Indeed, a 260% increase in tension in the Zdirection is

achieved in the present model using Ata 100%.

Mesh sensitivity

The mesh sensitivity was examined to ensure that the results

were independent of the size of the computational mesh.

Three mesh sizes were used: the coarse, medium, and fine

meshes consisting of 1049, 2306, and 3025 nodes, respec-tively, distributed over 1719, 3619, and 4724 tetrahedron

elements in the membrane; 872, 2857, and 5180 nodesdistributed over 1371, 3494, and 5700 tetrahedron elements

in the ligament; and 746, 905, and 1784 nodes distributed

over 828, 1033, and 1828 tetrahedron elements in the TA

muscle. The three meshes gave similar patterns of results.

At all muscle activation levels, the maximum difference

among the three computational meshes was 0.5% in thick-ness (defined below in Figure 3A). Therefore, all the results

presented in the remainder of this article are obtained with

the medium mesh.

RESULTS AND DISCUSSION

Effect on shape

In this section, we investigate the effect of muscle activation on

vocal fold morphology and normal stress in the Zdirection

(Figure 4). With an increase in Act, which stretches the entire

vocal fold in the Zdirection, tensional stress is generated in

all the layers. In addition, the middle coronal section shrinks

FIGURE 3. Definitions of parameters. A. Thickness and radius ofcurvature, used in Figures 6 and 9. B. Abscissas used in Figures

5 and 10.


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because of the nonzero Poisson ratio because a stretched mate-

rial tends to shrink in the directions transverse to the direction

of stretching. Meanwhile, with an increase in Ata, which con-

tracts the TA muscle and then produces tension in the Zdirec-

tion, a remarkable increase in stress appears only in the TAmuscle. The stress in the cover/ligament is reduced because lat-

eral shrinkage because of muscle contraction and the nonzero

Poisson ratio compresses the cover/ligament lying alongside

the muscle in the X-Y plane. The compression within the X-Y

plane causes the cover/ligament to expand in the Zdirection

to partially retain the volume; however, as the straight-line dis-

tance between the most anterior and posterior boundaries is

kept unchanged at each Act, the longitudinal tissue expansion

locally relaxes the magnitude of the tensile stress in theZdirec-

tion and may even produce compressive stress as discussed later

in theEffect on stress distributionsection andFigure 5. Thus,

even if a certain set value of longitudinal strain is given accord-

ing to the combination ofAta andAct (Figure 2), the actual stress

observed is different from the set value, particularly around the

peripheral region of the vocal folds containing the cover and

ligament (Figure 4).As muscle contraction proceeds, the tip of the vocal fold

(where vibration primarily occurs in real phonation) becomes

thinner and curves upward, exactly as seen in the detailed obser-

vations of human vocal fold geometry.33 To quantify these

changes, two parameters, thickness and radius of curvature

(Figure 3A), are measured. The thickness is defined as the ver-

tical length of the vocal fold at the leftmost edge of the TA mus-

cle, and the radius of curvature is defined as the radius of

curvature of an imaginary circle tangent to the lowermost por-

tion of the upper line defining the vocal fold. Both Actand Atadecrease the thickness (Figure 6A), and the radius of curvature

FIGURE 4. Vocal fold morphology and normal stress in Zdirection as a function of muscle activation characterized by parameters Actand Ata.

Because the deformation is symmetric with respect to the middle coronal section, only the rear half of the vocal fold is shown. Red represents a stress

magnitude of more than 8 kPa.


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(Figure 6B) indicating that the tip region curves upward. The

decrease in the thickness is caused by the nonzero Poisson ratiobecause a tensed elastic material shrinks compared with the

nonstress state in the directions transverse to the direction of

tension. The decrease in the radius of curvature has two distinct

causes. Shrinkage deformation in the coronal section is in-

hibited at the rightmost surface of the model. This makes the

upper portion of the vocal fold curved, leading to a reduction

in the radius of curvature. In addition, a characteristic stress

distribution because of geometric asymmetry contributes to

the decrease in the radius of curvature. This fact is particularly

explained inFigure 7, which shows the normal stresses in the

Xand Ydirections in the midcoronal section under the maxi-

mum values ofActand Ata. The upper portion of the cover/lig-

ament is compressed in theXdirection because of shrinkage in

the X-Yplane (as shown by the blue area in Figure 7A). The

lower portion of the same layers running obliquely in the planeis also compressed to a similar extent. On the other hand, in the

Ydirection, only the lower tip portion is compressed. The upper

portion is not compressed in the Ydirection (as shown by the

green area in Figure 7B) because it runs perpendicular to

the Y direction. Because of this geometric asymmetry, only

the lower edge resists deformation in the Ydirection. Conse-

quently, a concave curve appears in the upper portion where

there is less resistance to deformation in the Ydirection.

An interesting question is which ofActandAtahas a stronger

effect on inducing the characteristic morphological change? As

mentioned at the end of the Muscle Activation section, lateral

expansion of the TA muscle because of longitudinal contraction

was not included in the deformation shown inFigure 4. A sep-arate analysis demonstrates that the lateral expansion could

suppress the morphological change caused by longitudinal con-

traction withAta 100% (Figure 8). Here, 3exrepresents an ac-tive expansion strain applied to the TA muscle in bothXand Y

directions by setting new reference lengths for the two direc-

tions that are longer than the original reference lengths. At

3ex 0.25, the decrease in the volume of the TA muscle becauseof longitudinal contraction is approximately equal to the in-

crease in volume because of lateral expansion. Therefore, if

the actual TA muscle contracts while maintaining its volume,a case where 3ex 0.25 would simulate a realistic deformation.A comparison of the cases for 3ex 0.25 and Ata 100%(Figure 8) and 3ex 0 and Ata 0% (shown in the upper leftinFigure 4) shows that the stress magnitude is elevated only

in the muscle in the former case, and the shape of the vocal

fold remains almost unchanged. Taken together, the results de-

scribed so far show that CT muscle activation makes the tip re-

gion thinner and upward oriented. In contrast, a similarmorphological shrinkage caused by the TA muscle would be

suppressed in reality because of the lateral expansion discussed

above.

The Poisson ratio was set to 0.3 throughout the calculations

shown here, as mentioned in the Material Properties section;

however, we confirmed the effect of Poisson ratio on thickness

in separate analyses (Figure 9). As the Poisson ratio increases,the effect ofAtaon thickness increases because a low Poisson

ratio does not significantly affect the lateral deformation (or

thickness measured within theX-Yplane) even with the longi-

tudinal stretching caused byAta. As expected, the cross section

around the middle coronal section under longitudinal tension

shrinks as the Poisson ratio increases.

Effect on stress distribution

Next, we investigate the normal stress in the Z direction

(Figure 5) in the element immediately beneath the vocal fold

profile, as defined inFigure 3B. AtAct 0%,Atahas virtually

FIGURE 5. Stress distribution in cover as a function of muscle

activation.

FIGURE 6. Effect of muscle activation on thickness (A)and radius

of curvature (B). Note that the valuesof radius of curvature atAct 0%

andAta 0% are infinite in magnitude and are hence omitted from the

graph.


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no influence on the stress magnitude around the tip region,

where significant vibration occurs in real phonation (at about

8 mm along the abscissa in Figure 5), because only a minor

stress induced by Ata in the TA muscle is transmitted to the

cover. In the upper (10 mm, subglottis) regions, Ata induces considerable com-pression in the cover within theX-Yplane (Figure 7). This trans-

verse compression also causes a local relaxation of the tension

in theZdirection because the longitudinal straight-line distance

between the most anterior and posterior boundaries is kept con-

stant at eachAct. With an increase in Act, the stress magnitude is

elevated over the vocal fold, and then the compression isreplaced by positive tensional stress. Thus, although Act and

Ata show distinct effects, they cooperatively allow regulation

of tension in the cover.

To assess how the cooperation betweenActandAtaaffects the

expected effective tissue depth, the distribution of normal stress

in the Zdirection across a line from the medial border to the

lateral margin (Figure 3B) is investigated (Figure 10). An in-

crease in Act with a constant value ofAta (33%) increases the

stress magnitude almost equally in all layers (Figure 10A).

The relative magnitude in the layers is determined essentially

by the longitudinal Youngs modulus and Ata. The effect of

Ata on the stress along the same line is examined with

Act 100% (Figure 10B). As TA muscle activation proceeds,tensile stress builds in the muscle, whereas the magnitude of

the stress in the ligament decreases. Therefore, vibration may

be limited to the surface region at high values ofAta.

As the Youngs modulus of the cover is higher than that of the

ligament in this analysis, significant increases in depth becauseof TA muscle contraction appear almost entirely in the liga-

ment, and hence, the magnitude of the stress in the cover

remains relatively unchanged. In real vocal fold tissues, nonlin-

earity ofthe stress-strain curve is most profound for the vocal

ligament,31 and hence, the incremental elastic modulus of the

ligament would in fact increase as the Act-induced stretch

FIGURE 7. Stresses at Act 100% andAta 100%.A. Stress in Xdirection.B. Stress in Ydirection.

FIGURE 8. Effect of lateral expansion of TA muscle because of longitudinal contraction.


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proceeds. Therefore, when the TA muscle is not activated, most

of the longitudinal stress at high strains (ie, for large values of

Act) would be borne by the ligament. This will cause the cover

to loosen, thereby facilitating vibrations localized near thesurface. Taken together, the analyses suggest that the TA mus-

cle regulates how deep the vibration reaches into the tissue. The

CT muscle would also regulate the effective depth, but it takes

advantage of different nonlinear stress-strain relationships for

each layer because the principal stress-bearing component

could be replaced by stretching the vocal fold.

CONCLUSIONS

In previous studies, changes in longitudinal vocal fold length in

response to muscle activation were investigated using simple

analytical models,1,9,10,17 and the expected effective depth

of tissue vibration and its role in regulating the voicefundamental frequency were discussed. The present study

provides not only results consistent with those of previous

studies but also more detailed explanations and new findings.

In particular, given that the three-dimensional vocal fold mor-

phology and stress distributionwhich the previous analytical

studies had no means to obtainmust be one of the major

determinants of the effective depth, their close relationship

with muscle activation revealed here would be important in

evaluating the effect of muscles on the vibrational behavior

and resultant voice quality. Moreover, active and passive

stresses, which were produced in the vocal folds by the TA

and CT muscles, respectively, were considered separately inthe present analysis. This distinction between active and passive

contributions has not typically been established in previous

modeling studies, even when elongation in the longitudinal

vocal fold length was incorporated (eg, study by Tao and

Jiang7). The present study, thus, provides a likely mechanism

regarding the two muscles cooperative regulation of the effec-

tive depth of tissue vibration.The present results show that the activity of both muscles

makes the tip region of the vocal fold thinner and curve upward.

However, lateral expansion of the TA muscle because of its

longitudinal contraction suppresses the morphological change

induced by the TA muscle. These analyses suggest that the char-

acteristic morphology, which seems suitable for efficient vibra-

tion,34 is realized primarily by the CT muscle in actual vocal

folds. The CT muscle activation increases the stress magnitude

in each layer to an equal degree because only linear analyses are

performed in this study. If nonlinear stress-strain relationships

were incorporated instead, the ligament that has the most

FIGURE 9. Effect of Poisson ratio on thickness (A) and radius of

curvature (B).

FIGURE 10. Stress distribution in depth direction of vocal fold

model.A. Effect ofActat Ata 33%.B. Effect ofAtaat Act 33%.


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profound nonlinearityand is firmly attached to themuscle would

become the principal stress-bearingcomponent. Because the CT

muscle regulates the vocal fold, the principal stress-bearing

component and resulting effective depth of tissue vibration

vary according to the incremental elastic moduli of the layers.

In contrast, TA muscle activation produces a stress barrier

at the interface between the muscle and the ligament. This bar-

rier prevents the enlargement of movable portions of the vocalfold. When the TA muscle is relaxed, a relatively flat stress

distribution is obtained over the tissue. In this case, tissue

motions occur even in deep layers including the muscle, and

the regulation of the effective depth by the CT muscle described

above would dominate. When the TA muscle is sufficiently

activated and becomes the principal stress-bearing component,

tissue motion is limited to the surface region. Thus, the present

analyses demonstrate the relationship between the CT and

TA muscles in their cooperative regulation of the vocal fold

morphology, stress, and possibly effective depth as well. The

results obtained in this study should be useful in future stress

estimations in various phonation situations.

Acknowledgments

This work was supported by the Japan Science and Technology

Agency and the Ono Acoustics Research Fund.

Supplementary data

Supplementary data associated with this article can be found, in

the online version, at doi:10.1016/j.jvoice.2010.11.006.

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