Optimal glottal configuration for ease of phonation

Journal of Voice Vol. 12, No. 2, pp. 151-158 © 1998 Singular PublishingGroup, Inc.

Optimal Glottal Configuration for Ease of Phonation J o r g e C. L u c e r o Departmento de Matem6tica, Universidade de Brasflia, Brazil

S u m m a r y : Recent experimental studies have shown the existence of optimal values of the glottal width and convergence angle, at which the phonation threshold pressure is minimum. These results indicate the existence of an optimal glottal configuration for ease of phonation, not predicted by the previous theory. In this paper, the origin of the optimal configuration is investigated using a low dimensional mathematical model of the vocal fold. Two phenomena of glottal aerodynamics are examined: pressure losses due to air viscosity, and air flow separation from a divergent glottis. The optimal glottal configuration seems to be a consequence of the combined effect of both factors. The results agree with the experimental data, showing that the phonation threshold pressure is minimum when the vocal folds are slightly separated in a near rectangular glottis. Key Words: Glottal configuration--Phonation--Phonation threshold pressure--Vocal fold oscillation--Glottal aerodynamics--Mucosal wave model.

Phonation threshold pressure is defined as the minimum lung pressure required for the onset of the vocal fold oscillation (1-3), and may be considered a measure of "ease of phonation" (4). According to theory (1-3), it is the level of pressure at which the energy transferred from the airflow to the vocal folds is large enough to overcome the energy dissipated in the tissues, so that an oscillation can be initiated and sustained. In recent experimental studies in a physical model of the vocal fold mucosa, the threshold pressure was measured at various glottal shapes, varying the glot-

tal width in a rectangular glottis (4) and the glottal angle of convergence (defined as the angle formed by the medial surface of the vocal fold in a coronal section) (5). Theory predicted that the threshold pressure should monotonically decrease for decreasing glottal width and angle of convergence, i.e., phonation should be easier to achieve as the glottis becomes narrower and more divergent. However, experiments showed the existence of optimal values for the glottal width and angle, at which the threshold pressure is minimum. In the first set of experiments (4), the minimum pressure occurred when the vocal folds were slightly separated, and in the second set (5), it occurred when the glottis was near rectangular. This paper will look for a probable origin of the optimal glottal configuration. Two phenomena of the glottal aerodynamics usually neglected in previous analytical treatments will be considered: pressure losses due to air viscosity and airflow separation (formation of a free jet) from a divergent glottis. A pre-. liminary work (6) showed that viscous losses might

Accepted for publication June 16, 1997. Address: correspondence and reprint requests to Departamento de Matem~itica, Universidade de Brasflia, Brasflia DF 70910-900, Brazil. The results in this manuscript were presented at the Third Joint Meeting of the Acoustical Society of America and the Acoustical Society of Japan, in Honolulu, Hawaii, December 2-6, 1996.

151

152

JORGE C LUCERO

be the cause for the minimum of the threshold pressure in relation with the glottal width. Also, airflow separation might be responsible for the minimum in relation with the glottal angle of convergence, by reducing the area of the vocal folds which interacts with the airflow as the glottis is made more divergent (5). VOCALFOLD MODEL We use Titze's mucosal wave model (1), shown in Fig. I. The vocal fold is composed by two structures: body and cover. The body consists of the muscle and deep layers of the vocal ligament, and is assumed to be stationary. The cover consists of the more superficial layers of the ligament and the epithelium around the body, and propagates a surface wave X(t - z/c) in the direction of the airflow, where X is the lateral

displacement of the cover, z is the distance from the glottal entry (lower edge of the vocal fold) in the direction of the airflow, and c is the wave velocity. In Fig. 1, Tis the thickness of the glottis in the direction of the airflow. The anteroposterior length of the glottis is in the direction normal to the plane of the paper, and will be denoted by L. To derive the equations of motion we follow similar steps as in Titze's work, but including an explicit term for viscous losses and the effect of airflow separation. Assuming a small time delay of the surface wave in moving from the lower to the upper edges of the vocal fold, the lateral displacement of the cover may be approximated by:

ventricle

cover

trachea FIG. 1. Vocalfold model (coronalsection)(I). Journal of Voice, Vol. 12, No. 2, 1998

OPTIMAL GLOTTAL CONFLGURATION FOR EASE OF PHONATION

We consider first the aerodynamics in the portion between the traquea and the point of flow separation in the glottis. Assuming that the subglottal pressure is constant and equal to the lung pressure PL, the pressure drop in this portion can be modeled through the extended energy equation:

where x(t) is the lateral displacement at the midpoint of the glottis (see Fig. 1). Also, lumping the biomechanical properties of the vocal fold at the midpoint, we obtain the equation of motion: M£ + B2 + Kx = Pg

(2)

where M, B, and K are the lumped effective mass, damping, and stiffness, respectively, of the vibrating portion of the vocal folds (roughly, the cover) per unit area of the vocal fold surface L T over which the glottal pressure acts, and Pg is the glottal pressure ( 1). According to recent experimental and theoretical studies on the glottal aerodynamics in a divergent glottis (7,8), the relation between the glottal area at entry a / (minimum glottal area) and the area a s at which airflow separation occurs (Fig. 2) may be assumed constant in a first approximation, with a value: as/a I = 1.1

(3)

This approximation is valid for high Reynolds numbers, within the approximate range Re > 1000. Typical Reynolds numbers for the glottal airflow are in the order of 3000 (7), so the approximation holds. Thus, for the general case of divergent and convergent glottal shapes we assume:

PL -- ~ = APBe,,ou,i + Z~viscous

(4)

A PBernoulli = 2PaU22 s

(6) where p is the air density and u is the air volume flow velocity. Note that contrary to previous works (1-3), we do not consider special coefficients to account for the formation of a vena contracta at the glottal entry (9) and pressure recovery at the glottal exit. Recent experimental studies (7) have found no formation of vena contracta and negligible pressure recovery. Following previous works (6-8), the viscous losses are modeled by:

W L

FIG. 2. Airflow separation in a divergent glottis.

Jodz

d__PP=_ 12pL2u dz a3(z)

as

k

fz, ap z

(7)

where Zs denotes the distance at which airflow separation occurs (see Fig. 2), and dP/dz is given by the Poiseuille equation:

where a 2 is the area at the glottal exit.

al

(5)

where ~ is the pressure at the point of airflow separation. Assuming further that the cross sectional area of the trachea is much larger than the glottal area, the Bernoulli term is:

=

[1.1al, f o r a 2 > 1.1a 1 a s = [ a 2 , f o r a 2 < 1.1a I

153

(8)

where p is the air viscosity and a(z) is the glottal area along the z-axis. We must note that the Poiseuille equation (equation 8) was derived for parabolic velocity profiles in parallel plates. Van den Berg et al. (9) observed a good agreement between this equation and measurements on a physical model of the larynx for steady flow and a rectangular glottal shape. However, in the present case the glottal flow varies in time and the glottis may take convergent and divergent shapes. Equation 8 is then adopted as an approximation, valid for small angles of glottal convergence or divergence and small oscillation amplitudes. Journal of Voice, Vol. 12, No. 2, 1998

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J O R G E C. L U C E R O

Assuming a linear variation of the glottal area along the z-axis (3), we have: a(z) = a/ + a2 - al z T

(9)

and the relation between Zs and a s is: as=al +

a2-- al, T

(10)

Integrating equation 7 and substituting equations 6 and 7 into equation 5, we obtain: PL -- Ps =

pu2 + 6~L2zsU(a! +as) 2a~ ~ 2 aTas

(11)

Below the point of flow separation we assume that the diameter of the air jet is much smaller than the supraglottal diameter, and that no pressure recovery occurs (7). As further simplification, we assume that the supraglottal pressure is equal to the atmospheric pressure (1). Then, the pressure at the point of airflow separation reduces to: Ps = 0

(12)

The glottal pressure Pg acting on the vocal folds is next calculated as the mean air pressure along the z-axis:

1 fZ, Pg = T o P(z)dz

(13)

_o,2(1 - o: )+fz.,, aT(z)

z -~z "

(16)

a2

(17)

=

2L(xo2 + x + "Ci')

where Xol and Xo2 are the prephonatory displacements at the lower and upper edges of the glottis, respectively, and "c= T/(2c) is a time delay. PHONATION T H R E S H O L D PRESSURE No viscous losses and no airflow separation (previous theory) The phonation threshold pressure may be obtained from the condition for instability of the equilibrium position of the vocal folds. Let us consider first the simple case of p = 0 (no viscous losses) and zs = T (no airflow separation). This is the case studied in previous works (1,3). Equations 11 and 15 reduce to the simple form: Pg=PL 1

_ a2 ) -~-l

(18)

The equilibrium position is determined by setting the time derivatives in the equations 2, 13 and 14 to zero, and solving for x, which yields the result: £ = - x0----t~-+ 2

PL(xOI-

Xo2)

K

(14)

P~(x,X)=ed£,O)+~ (£O)x+ ~ (£0)x In the right side of the above equation 14, the first term is the Bernoulli term and the second represents viscous losses. Using also the Poiseuille equation (equation 8) for the integral, we obtain: pU2Zs P g - 2a2T

(1 -- a s aI

+ 6 ~ L2z2u

(15)

(20)

and setting the total damping to zero, i.e., the sum of all terms proportional to the fold velocity: B - aP~ (£,0) = 0

(21)

Ta l a 2s

Finally, using equation 1, we may express the glottal areas by:

Journal ofVoice, Vol. 12, No. 2, 1998

(19)

where £ denotes the equilibrium position. The instability condition of the equilibrium position is next obtained linearizing Pg around the equilibrium position:

where P(z) is given by:

P(z) - 2a2

a/ = 2L(Xol + x + ~c)

(not e that the damping term due to the glottal pressure is negative, because the glottal pressure appears

O P T I M A L GLOTTAL C O N F I G U R A T I O N FOR EASE O F P H O N A T I O N

at the right side of equation 2). Calculating the derivative and solving for Pc, we finally obtain the oscillation threshold pressure Pth: Prh - B(x° + kx° / 2 + .r )2 2Z(xo + ~c)

(22)

where x 0 = (x0t + x02)/2 is the prephonatory glottal half-width at the midpoint, and Ar 0 = x0, - x02 is a measure of the glottal convergence. Equation 22 may be further simplified considering that for most conditions 5cis negligible in comparison with x 0 ( 1), which yields the approximate expression: Pth = B(x° + Ax°/2)2 2zx o

(23)

We can clearly see in equation 23 that the threshold pressure should monotonically decrease with the glottal convergence and also with the glottal width. Further, in a rectangular glottis (Ax0 = 0) the threshold pressure should linearly decrease with the glottal width x 0, and should be zero for x 0 = 0. General case

For the general case, the threshold pressure was calculated by modifying equation 21 to the difference equation: (24) f(pL)=B

-

2~

with 8.~ = 0.01 cm/s, and numerically searching (e.g., using a bisection algorithm) the value of Pc = Pth to satisfyf(P,,) = 0. Note that at each step, the value of the equilibrium position .~ for a given Pc is required to compute equation 24. The equilibrium position was calculated by setting to zero all the derivatives in equation 2, which yields: Kx = Pg (x,0)

(25)

and numerically searching (as with equation 24) the value of x = .7cto satisfy equation 25. Figs. 3 to 6 show the threshold pressure calculated at various glottal configurations, using the values M = 4.76 kg/m e, B = 1000 N s/m 3, K = 2000000 N/m 3,

155

L = 1.4 cm, T= 0.3 cm, c = 1 m/s, p = 1.15 kg/m3,/.1 = 1.86 10-5 kg/(s m) (1). Fig. 3 shows the phonation threshold pressure vs. the glottal half-width, as predicted by the theories and when viscous losses are included. At the left end, the curves stop at the point where glottal closure occurs, except for the case of viscous losses in a divergent glottis (full lines 1 and 2) which stop at a point where no equilibrium position could be found. The previous theories show in all cases a monotonic decrease of the threshold pressure when the glottal width is reduced. Viscous losses in general cause an increase of the threshold pressure at low glottal widths, and a minimum appears in the rectangular or divergent cases. However, in the rectangular case, after reaching a maximum the threshold pressure also tends to zero at lower glottal widths, in contradiction with the experimental evidence. A more accurate modeling of the viscous losses than the simple Poiseuille formulation (equation 8) might be required at this range of smaller glottal widths. Fig. 4 shows the phonation threshold pressure vs. the glottal half-width when the airflow separation is also included. For reference, the curves for no airflow separation are also shown. Only the case of a divergent glottis (Ax0 < 0) is considered, since there is no airflow separation in a rectangular or convergent glottis. Note that the airflow separation also causes a much larger increase in the threshold pressure than the increase produced by the viscous losses. All the curves have a minimum, which is located at the point where airflow separation within the glottis starts. This value of the glottal width depends on the glottal divergence, and is larger at steeper glottal divergences. Fig. 5 shows the threshold pressure vs. the glottal convergence, as predicted by the previous theory and when viscous losses are included. The viscous losses cause an increase of the threshold pressure, and a minimum appears in a divergent glottis. As the glottal angle is increased in absolute value keeping the glottal width at midpoint constant, the glottal area at entry (in the divergent case) decreases, which causes an increase in the viscous losses. Fig. 6 shows the threshold pressure vs. the glottal convergence when airflow separation is included. For reference, the curves for no airflow separation are also shown. As in Fig. 3, the increase of the threshold

Journal of Voice, Vol. 12, No. 2, 1998

156

JORGE C. LUCERO

450 400

~350

i

~300

~

250

~

200

/ ~

.~

/

~a. 150

~

~

.~

//'"

'

~

'

/

! 1t

'

~

100

50

s. ~ ,~,

0 ~" 0

I

0.01

~

~t I

i

0.02

0.03

/

/

-----r

~

I

I

0.04 0.05 0.06 glottal half-width (em)

I

I

I

0.07

0.08

0.09

0.1

FIG. 3. Phonation threshold pressure (P,,) vs. glottal half-width (x0), for glottal convergences Axo = - 0 . 0 6 cm (1), - 0 . 0 2 cm (2), 0 (3), 0.02 cm (4), and 0.06 cm (5). Broken lines: previous theory. Full line: with viscous losses.

2000 1800

1600 m-~1400 ~ 1200

2 lOOO 8oo 0

600

/

400 200 I

0

0.05

I

I

0.1 0.15 glottal half-width (cm)

I

0.2

0.25

FIG. 4. Phonation threshold pressure (Pth) vs. glottal half-width (x0), for glottal convergences Ax0 = - 0 . 0 0 5 cm (1), -0.01 cm (2), - 0 . 0 2 (3). Broken lines: with viscous losses. Full line: with viscous losses and airflow separation.

Journal of Voice, Vol. 12, No. 2, 1998

OPTIMAL GLOTTAL CONFIGURATION FOR EASE OF PHONATION 500

I

i

i

i

157

I

450 400

~ 350 ~ 300 "o ~ 250

..J "

..//

777

. i I"1

7

// I/

i

200

/

"~150

"1

I-

100

II/

2

I

/I

/ II

ii

1/11

I ""

I /

50 0

-0.06

-0.04

-0.02 0 0.02 glottal convergence (cm)

0.04

0.06

FIG. 5. Phonation threshold pressure (~,) vs. glottal convergence Axo, for glottal half-widths x o = 0.02 cm (I), 0.05 cm (2), and 0.1 cm (3). Broken lines: previous theory. Full line: with viscous losses.

2500

.

.

.

.2ooo,= I

.

g

~ 1500 e-

7 1000 500

~

......... 0

-0.06

I

-0.04

-

:I

I

I

-0.02 0 0.02 glottal convergence (cm)

I

0.04

0.06

FIG. 6. Phonation threshold pressure (Pth) vs. glottal convergence Ax0, for glottal half-widths x o = 0.02 cm (l), 0.05 cm (2), and 0.1 cm (3). Broken lines: with viscous losses. Full line: with viscous losses and airflow separation.

Journal of Voice, Vol. 12, No. 2, 1998

158

JORGE C. LUCERO

pressure caused by the airflow separation is much larger than the increase due to viscous losses. A minimum appears in all the curves at the glottal divergence where airflow separation within the glottis begins. Note that this value of the glottal divergence is small, so that the minimum threshold pressure occurs at a near rectangular glottis, in agreement with the experimental evidence. DISCUSSION The observed minimum value of the oscillation threshold in relation with the prephonatory glottal width and glottal angle of convergence seems to be consequence of the combined effect of pressure viscous losses and airflow separation at the glottis. In a convergent or near rectangular glottis, decreasing the glottal angle of convergence facilitates the oscillation. The transfer of energy from the airflow to the oscillating vocal folds depends on the asymmetry of the glottal pressure in the opening and closing cycles of the oscillation (1), which is harder to achieve at larger convergence angles. However, as the glottis becomes more divergent, airflow separation within the glottis begins, reducing the vocal fold area to interact with the airflow. Thus, the glottis must stay near rectangular (more precisely, slightly divergent and at the angle at which air flow separation within the glottis begins) for its optimal shape. A second way to facilitate the oscillation is to decrease the glottal width, which increases the glottal pressure asymmetry. However, as the glottal width becomes smaller, pressure losses due to air viscosity increase, reducing the amount of energy that can be transferred to the vocal folds and thus making the oscillation more difficult. The result is an optimal glottal width for the oscillation. We also computed the optimal glottal configuration applying a simplex algorithm to the above equations, with both the glottal half-width and glottal convergence as variables. The optimal values found were

Journal of Voice, VoL 12, No. 2, 1998

x 0 = 0.0182 cm and Ax0 = -0.009 cm ( - 1.72°), respectively, which is in a general agreement with experimental data. The above description should be tested using more accurate models of the glottal aerodynamics, especially for the pressure viscous losses at very small glottal widths where the present study failed to reproduce the experimental results.

Acknowledgment: This research was done while I was at the Speech Perception and Production Laboratory of the Department of Psychology, Queen's University (Canada), and was funded by NIH grant # DC-00594 from the National Institute of Deafness and other Communications Disorders, and NSERC. I am grateful to Dr. I. R. Titze for his discussions on the optimal glottal configuration, and to Dr. K. G. Munhall for his interest and support.

REFERENCES 1. Titze IR. The physics of small-amplitude oscillation of the vocal folds. J Acoust Soc Am 1988;83:1536-52. 2. Titze IR. Phonation threshold pressure: A missing link in glottal aerodynamics. J Acoust Soc Am 1992;91:2926-35. 3. Lucero JC. The minimum lung pressure to sustain vocal fold oscillation. J Acoust Soc Am 1995;98:779-84. 4. Titze IR, Schmidt SS, Titze MR. Phonation threshold pressure in a physical model of the vocal fold mucosa. JAcoust S o c A m 1995;97:3080--4. 5. Chan RW, Titze IR, Titze MR. Glottal geometry and phonation threshold pressure in a vocal fold physical model. JAcoust Soc Am 1996;99:2471 (A). 6. Lucero JC. Relation between the phonation threshold pressure and the prephonatory glottal width in a rectangular glottis. J Acoust Soc Am 1996; 100:255 I--4. 7. Pelorson X, Hirschberg A, van Hassel RR, Wijnands APJ, Auregan Y. Theoretical and experimental study of quasisteadyflow separation within the glottis during phonation. Application to a modified two-mass model. J Acoust Soc Am 1994; 96:341 6-31. 8. Pelorson X, Hirschberg A, Wijnands APJ, Bailliet H. Description of the flow through in-vitro models of the glottis during phonation. Acta Acustica 1995;3:191-202. 9. van den Berg J, Zantema JT, Doomenbal J. On the air resistance and the Bernoulli effect of the human larynx. J Acoust Soc Am ! 957;29:626-31.