Mean intraglottal pressure in vocal fold oscillation

Journal of Phonetics (1986) 14, 359-364

Mean intraglottal pressure in vocal fold oscillation logo R. Titze The University of Iowa, Iowa City, Iowa 52242, and The Denver Center for the Performing Arts, 1245 Champa Street, Denver, Colorado 80204 , U.S.A.

Mean intraglottal pressure is discussed in relation to subglottal pressure, supraglottal pressure and transglottal pressure. In the opening phase, mean intraglottal pressure (the net vocal fold driving pressure) is strongly dependent on transglottal pressure, but in the closing phase it depends almost entirely on supraglottal pressure. Negative mean intraglottal pressure occurs briefly in the glottal cycle, but is not essential for oscillation. The Bernoulli principle is re-examined.

1. Introduction

Flow-induced oscillation of the vocal folds is based on intraglottal pressure variations. Temporal and spatial variations within the glottal cycle are central to understanding the principles of vocal fold vibration. The primary spatial variation occurs vertically along the glottis, from glottal entry to glottal exit. The primary temporal variation occurs with respect to opening, closing and closure of the glottis. During complete or partial closure, contact pressures between opposing vocal folds are combined with aerodynamic pressures. Important as these pressures are in understanding the principles of self-oscillation (Ishizaka & Flanagan, 1972; Baer, 1975; Stevens, 1977; Broad, 1979; Titze, 1980; Conrad, 1980; Titze, 1983), they have never been measured under dynamic (oscillatory) conditions. The task is extremely difficult because the glottis is very small and inaccessible. No available pressure transducers are small enough not to disturb the glottal airflow or the vibratory patterns of the vocal folds, nor are they robust enough to measure both aerodynamic pressures and contact pressures in moist environments, nor can their position in the glottis (or attachment to the tissue) be stabilized without introducing vibration artifacts. A theoretical approach is therefore the only viable option at the present time for studying intraglottal pressures during phonation. Such an approach usually combines limited empirical results with traditionally accepted laws of physics that describe the movement of air and tissue. Thus, the static pressure-flow relationships obtained on physical models of the glottal airway (van den Berg, Zantema & Doornenbal, 1957; Scherer & Titze, 1983; Scherer, Titze & Curtis, 1983; Gauffin, Binh, Ananthapadmanabha, & Fant, 1983) are combined with acoustic pressure calculations above and below the glottis to obtain intraglottal pressures that are assumed to be valid for oscillatory conditions. Such an approach will be used here. 0095-4470/86/030359

+ 06 $03.00/0

© 1986 Academic Press Inc. (London) Ltd.

360

I. R. Titze

The objective of this paper will be to clear up some misconceptions about the so-called Bernoulli Effect and the role oftransglottal pressure in vocal fold vibration. Specifically, we hope to revise the notion that the transglottal pressure, which drives the glottal flow , also drives the vocal fold tissue. In addition , we hope to soften the claim that negative pressure (suction) in the glottis is the primary mechanism by which the vocal folds come together (the Bernoulli Effect). Although these notions are not entirely false, they need clarification. A more fundamental condition for oscillation is that the mean intraglottal pressure (the driving pressure) be in phase with the mean tissue velocity (Titze, 1985a). This velocity-dependence of the driving pressure comes from two sources: (I) a mucosal surface-wave propagating upward in the glottis, and (2) an inertive vocal tract load. We begin the quantitative discussion by drawing upon recent theoretical results obtained by our group (Titze, 1983, 1985a, b) . Some key formulae will be restated for completeness and for later discussion. 2. Summary of derivations for mean intraglottal pressure

Consider the glottal airway as shown in Figure I. (Details of the tissue modeling will be given later.) The mean intraglottal pressure Pg has been derived by Titze (1985b) for general open and closed conditions of the glottis on the basis of a glottal area ratio r, which relates the glottal entry area a 1 to the glottal exit area a2 , and a continuously varying vocal fold contact area ac. Details are beyond the scope of this paper, but it may suffice here to say that the mean intraglottal pressure was obtained by integrating the

a;

Body I

I

Cover /

/

/

-01 /' /' / / /

/

I

I

I I

/

/

' ..... .....

'

.....

'

'''

\

I I

Mean intraglottal pressure

361

pressure vertically along the glottis. The result was Pg

Where

r

= =

[P;

+

+

t(P;

(P, - P;)(l - r- ke)/ k 1 ](I - ae / LT)

+

P,)ac/LT.

a2 j a

(I) (2)

is the glottal convergence ratio, P; is the vocal tract input (supraglottal) pressure, P, is the subglottal pressure, ke is the exit pressure coefficient, k, is the transglottal pressure coefficient, L is the vocal fold length, and Tis the vocal fold thickness. The term in the first line of Equation (I) represents the mean aerodynamic pressure on the tissue surface and the term in the second line represents the mean hydrostatic pressure transmitted through the portions of the tissue in contact. Note that, as the contact area ac approaches the medial surface area LT (complete contact), the aerodynamic pressure is gradually turned off and the mean contact pressure is fully applied. The contact area is defined as (3) where the choice of 0 or I depends on whether there is contact at the differential surface dy dz along the medial plane. This depends on the pre-phonatory shape and the amplitude (and mode) of vibration (Titze, 1984, 1985b). The pressure coefficients k, and ke measure the fraction of the exit kinetic pressure 2 tQu ja~ dropped across the entire glottis (k,) or gained at exit (kc), where Q is the tissue density and u is the flow. The exit pressure coefficients used by Ishizaka & Flanagan (1972) was (4) where A; is the vocal tract input (supraglottal) area. This coefficient is generally less than 0.2. The transglottal pressure coefficient k, is based on results obtained by Scherer ( 1981 ), Scherer & Titze (1983), Scherer et al. ( 1983) and Gauffin et al. ( 1983) and is approximated here by the formula (5) The value ranges between 1.0 for a highly convergent glottis and 2.0 for a highly divergent glottis. Justification for the formula is given by Titze (1985b) . Note that the mean intraglottal pressure in Equation (I) has components that depend on supraglottal pressure P; , transglottal pressure P, - P; , and the average between sub glottal and supraglottal pressure (P, + P;)/ 2. Which of these components dominates depends on the portion of the glottal cycle. We will discuss them in terms of simulated waveshapes obtained by computer.

3. Time-variation of mean intraglottal pressure Figure 2 shows simulated waveforms obtained from the body-cover model of Figure I. The model is a self-oscillating version of the model used for parameterizing glottographic signals (Titze, 1984). It represents a hybrid between lumped-parameter and distributedparameter models, where the body is modeled by a single mass-spring-damper and the cover is a distributed tissue layer that can propagate a surface wave (Titze, 1985b).

I. R . Titze

362 Length

1.60

FV0 XOI

XD2

VM

Pres

0.00 0.05 0 .05 150.00 8.00

A B C

Figure 2. Outline of the vocal tract (above) and (below) waveform simulation: see text for detailed description.

Shown from bottom to top in Figure 2 are 100 ms traces of contact area (ac) , glottal area (ag), glottal flow (ug), subglottal pressure (P,), mean intraglottal pressure (Pg), vocal tract input pressure (PJ, transglottal pressure (P, - PJ, and the mean pressure [(P, + PJ /2] used for contact. All pressure waveforms have been normalized with respect to their maximum value to clarify shape. Relative magnitudes can therefore not be compared. Above the waveforms is an outline of the vocal tract for a non-nasalized [a] vowel, showing trachea on the left, the glottis in the pre-phonatory state, an oral tract below the midline and the nasal tract above the midline. Input parameters used in the simulation are echoed above the outline. (Vocal fold length is 1.6 em, active vocalis force is zero, upper and lower glottal half-widths are 0.05 em in the prephonatory state, mucosal wave velocity is 150 cmf s at this length, and lung pressure is 8.0cm H 2 0.) In order to examine details of the waveforms, one cycle has been marked with vertical lines to separate the opening phase, the closing phase, and the closed phase. These are labeled A, B, C, respectively. Consider first the opening phase. Here the glottis is known to be convergent (bottom leads top, as in Figure 1). Just prior to opening, only the upper margins of the folds made contact. Full subglottal pressure was applied to the entire thickness of the folds , which cause Pg to reach its peak value just prior to opening (note peak in the Pg waveform in Figure 2, phase A). Since Equation (l) was used in the simulation, one should be able

M ean in traglot tal pressure

363

to identify this peak pressure directly from the formula. This is possible if we note that for a 2 = 0 and a 1 > 0 (upper margins just touching), Equations (2), (4) and (5) yield r = 0, ke = 0 and k, = 1. Also, since there is no significant contact area, ac = 0, everything drops out of Equation (1) except P, the peak pressure. As the glottis now opens, a downward concave trend is noticed in Pg. This follows the transglottal pressure P, - P; and the subglottal pressure P, in shape (Figure 2, phase A). Since P, is decreasing slightly and P; is increasing markedly, the transglottal pressure is reduced sharply (by about 80%) in the opening phase. The driving pressure Pg does not reduce as sharply, however, because only a fraction of the transglottal pressure is applied to the folds. The greater the convergence angle (i.e. the smaller r is), the greater the transglottal pressure component of the driving pressure will be, as seen from Equation(!). Consider now the closing phase B. Here it is evident from Figure 2 that Pg follows P; in shape almost completely. The transglottal pressure varies in the opposite direction. This brings us to one of the more important points to be made in this discussion. Large transglottal pressures can occur in the presence of small mean intraglottal (driving) pressures, and vice versa. Just prior to closure, for example, Pg is maximally negative where P, - P; is maximally positive (end of phase B). This departure of the mean driving pressure from the transglottal pressure during closing is important for self-oscillation. It is facilitated by reduced convergence (or increased divergence) of the glottis. By selecting a mode of vibration that alternates between greater and lesser degrees of convergence on alternate half-cycles, the vocal folds are able to "dodge" the large positive transglottal pressures during closing. Consider the source of this large transglottal pressure. The inertia of the air in the vocal tract above and below the glottis tends to maintain the forward glottal flow during closing. As less and less air is allowed to flow through the glottis, however, a compression is created subglottally and a rarefaction is created supraglottally. This raises P, and lowers P; , resulting in a large P, - P; difference. Equation (1) can be used to quantify P, for rectangular or divergent shapes. For r = I (rectangular glottis), only the fracti on - kc/k of the transglottal pressure (which is usually less than 10% in magnitude) affects the driving pressure, and the sign is opposite the P; component. This, together with the decreasing trend in the supraglottal pressure, drives the mean intraglottal pressure toward the negative value in Figure 2. As the glottis diverges (r > 1), negative Pg values are even more likely. This brings us back to the Bernoulli Effect, which is often used to make a strong point about these negative (sucking) pressures. In Figure 2, the negative portions of Pg and P; occur very briefly and are not exceedingly large. It is not important that Pg ever be negative, but that it be lower (on the average) in the closing phase than in the opening phase. This asy mmetry of the driving force with respect to peak opening reduces to a net component in phase with the mean tissue velocity over the whole cycle. As the folds move outward, a strong positive (outward) pressure exists. As they move inward, a lesser pressure opposes them. This allows energy to be transferred to the tissue by the airstream. A driving force that exhibits such a net component in phase with velocity directly opposes the normal frictional energy losses, and thereby can induce oscillation by reducing the energy losses per cycle to zero (Titze, 1983, 1985b). Another way to highlight this asymmetry is to observe the slope of Pg in the open portion of the cycle. It is the only wave shape in Figure 2 that continuously decreases from opening to closing. This is always the case if self-oscillation occurs. If one were to

364

I. R. Titze

differentiate the ag or ug waveforms (the closest correlates here to mean tissue displacement) to obtain the tissue velocity, a similar monotonic decrease would be observed, indicating the "in-phase" relationship between driving pressure and tissue velocity. In summary, we have reasoned that the transglottal pressure, which drives the glottal airflow, does not generally constitute the driving pressure of the folds. The mean intraglottal pressure (defined here as the driving pressure) does have a component proportional to transglottal pressure, but it varies directly with the degree of convergence of the glottis. Thus, during glottal opening, where convergence is large, a significant portion of the driving pressure is related to the transglottal pressure. During glottal closing, however, the folds are driven primarily by the supraglottal pressure. This asymmetry in the driving pressure is facilitated by vertical phase differences in tissue motion. A one-mass system would be driven almost entirely by the supraglottal pressure, making the source- vocal tract coupling very strong. In either case, negative driving pressures can exist, but are not essential for sustained oscillation as long as the driving force asymmetry is maintained and net energy loss per cycle is zero. This work was supported by a grant from the National Institutes of Health, RO 1-NS 16320-05, for which we are grateful.

References Baer, T. (1975) Investigation of phonation using excised larynxes, Ph.D. Dissertation, M.l.T., Cambridge, MA, U.S.A. Broad, D. (1979) The new theories of vocal fold vibration. In Speech and language: advances in basic research and practice, Vol. 2 (N. Lass, editor), pp. 203-256. London: Academic Press. Conrad, W. (1980) A new model of the vocal cords based on the collapsible tube analogy, Medical Research Engineering, 13 (2), 7- 10. Fant, G. (1983) Preliminaries to analysis of the human voice source, Quarterly Progress and States Report STL-QPSR4, Speech Transmission Laboratory, Royal Institute of Technology, Stockholm, Sweden, l- 27. Gauffin , J. , Binh, N. , Ananthapadmanabha, T.V. & Fant, G. (1983) Glottal geometry and volume velocity waveform. In Vocal fold physiology: contemporary research and clinical issues (D. M. Bless, & J. H . Abbs, editors), pp. 194-201. San Diego, CA: College Hill Press. Ishizaka, K. & Flanagan, J. L. (1972) Synthesis of voiced sounds from a two-mass model of the vocal cords, Bell System Technical Journal, 51 , 1233- 1268. Scherer, R. (1981) Laryngeal fluid mechanics: steady flow considerations using static models. Ph.D. Dissertation , University of Iowa. Scherer, R. & Titze, I. R. (1983) Press ure- flow relationships in a model of the laryngeal airway with diverging glottis. In Vocal fold physiology: con temporary research and clinical issues (D. M. Bless & J. H. Abbs, editors). San Diego, CA: College Hill Press. Scherer, R. C., Titze, I. R. & Curtis, J. F. (1983) Pressure- flow relationships in two models of the larynx having rectangular glottal shapes, Journal of the Acoustical Society of America, 73 (2) , 668- 676. Stevens, K . N. (1977) Physics of laryngeal behavior and larynx modes, Phonetica, 34, 264-279. Titze, I. R. (1980) Comments on the myoelastic-aerodynamic theory of phonation, Journal of Speech and Hearing Research, 23(3), 495- 510. Titze, I. R. ( 1983) Mechanisms of sustained oscillation of the vocal folds. In Vocal fold physiology: biomechanics, acoustics and phonatory control (l.R. Titze & R. C. Scherer, editors). Denver: The Denver Center for the Performing Arts, Inc. Titze, I. R. (1984) Parameterization of glottal area , glottal flow , and vocal fold contact area, Journal of the Acoustical Society of America, 75, 570--580. Titze, I. R. (1985a) The physics of flow-induced oscillation of the vocal folds. Part I. Small-amplitude oscillations. Research Report DCPA-RRJ, The Denver Center for the Performing Arts. Titze, I. R. (1985b) The physics of flow-induced oscillation of the vocal folds. Part II. Limit cycles. Research Report DCPA-RR&, The Denver Center for the Performing Arts. van den Berg, J., Zantema, J. & Doornenbal, P. (1957) On the air resistance and Bernoulli effect of the human larynx, Journal of the Acoustical Society of America, 29, 626-631.