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Fluid flow through the larynx channel
Journal of Sound and Vibration (1988) 121(2), 277-290
FLUID FLOW T H R O U G H THE LARYNX C H A N N E L J. A.
I~IILLER,J.
C. PEREIRA AND D. W. TtIOXtAS
Department of Electronics and Computer Science, Unit'ersi O' of Southampton, Southampton S09 5 NH, England (Received 31 October 1985, and in rerixed form 12 Afa)" 1987) The classic two-mass model of the larynx channel is extended by including the false vocal folds and the laryngeal ventricle. Several glottis profiles are postulated to exist which are the result of the forces applied to the mucus membrane due to intraglottal pressure variation. These profiles constrain the air flow which allows the formation of one or two "'venae contractae". The location of these influences the pressure in the glottis and layrngeal ventricle and also gives rise to additional viscous losses as well as losses due to flow enlargement. Sampled waveforms are calculated from the model for volume velocity, glottal area, Reynolds number and fluid forces over the vocal folds for various profiles. Results show that the computed waveforms agree with physiological data [ !, 2] and that it is not necessary to use any empirical constants to match the simulation results. Also, the onset of phonation is shown to be possible either with abduction or adduction of the vocal folds. 1. INTRODUCTION AND BACKGROUND Early models of the h u m a n larynx were based on simplified geometric shapes and air flow conditions [ I-3]. The trend has been to increase model complexity in terms of more elaborate mechanical structures representing the vocal tissue [4-6] with less attention being paid to the source of vocal energy, the air. In this research an attempt has been made to redress this imbalance by focusing on the laryngeal air flow and its interaction with the surrounding elastic tissue. In order to establish the contribution of air patterns to realistic phonatory behaviour, it was decided to adopt a relatively simple mechanical model o f the vocal folds. This consisted of two coupled masses representing each true vocal fold plus a third mass representing the false fold. The air-tissue boundary is represented by a series of straight lines joining the fixed and movable points in the proposed model, as shown in Figure l(c), and approximating the actual profile shown in Figure l(b). Several theories of vocal fold vibration have been proposed in the literature and some of these are summarized as follows. (a) The Aerodynamic-Myoelastic Theory. Helmholtz, in 1863, proposed that the "puffs'" of air originating at the glottis are the main source o f sound. According to Husson's Neuro-Chronaxic Theory [7], these puffs are induced by an impulse sent by the brain to the vocalis muscle via the recurrent nerve so setting the vocal fold into vibration. The Aerodynamic-Myoelastic Theory was proposed by van den Berg [8] to explain p h e n o m e n a which cannot be accounted for by the Neuro-Chronaxic Theory. This theory postulates that the vocal fold oscillation is actuated by the stream of air delivered by the lungs and trachea. The experimental basis for this theory was established by van den Berg et al. in 1957 [9]. Using a cast of a h u m a n larynx, van den Berg et al. measured the air resistance of the larynx and the intra-glottal pressure. The negative pressure measured inside the glottis led them to propose that the Bernoulli effect was responsible for the closure o f t h e glottis. 277 0022-46ox/88/050277 + 14 S03.00/0 O 1988 Academic Press Li. dted
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J.A.
MILLER. J. C. P E R E I R A A N D D. W. THOMAS
They also suggested that other phenomena were occurring in the flow, like the formation of "eddies" in the laryngeal ventricle and contraction of the flow at the entry of the glottis. Based on their measurements, and because of the complexity of the problem, they proposed an empirical formula for the air resistance of the larynx, in which pressure loss at the entry of the glottis and the pressure recovery at the exit were considered as fractions of the kinetic pressure for a wide range of glottal diameters. (b) The Flow-separation Theory [10]. The theory of lshizaka and Matsudaira represents a significant conceptual advance by showing how vocal fold vibration can occur in terms of the laws of mechanics in relation to the interaction between the fluid flow and tissue displacement at the glottis. The flow-pressure equations derived in their analytical study account for the losses due to the vena contracta formation at the entry of the glottis, the sudden expansion at the exit of the glottis and the viscous losses. However, they adopted van der Berg's empirical figure for the loss at the entry of the glottis. Their model represents the tissue by two coupled masses (the simplest structure which can capture the vertical phase difference). They derived coefficients for the pressure variations in terms of aerodynamic reacting pressure and aerodynamic coupling pressure, and showed thata sizeable aerodynamic coupling stiffness exists between the upper and lower masses. (c) The Layered Viscoelastic Tissue Theory [6]. A layered tissue approach for the vocal fold model was proposed by Titze and Talkin. In this model phonation is considered as a wave phenomenon in a layered configuration of viscoelastic tissues, the analysis of which may be accomplished by the solutiori of a boundary value problem. The theory is not only an attempt to reproduce human-like speech, but also, although through an analog, to replicate the physical processes as completely as possible. This approach is classified by the authors as "speech simulation" rather than "speech synthesis". In this context, the model presents a very high level of sophistication of the modelling process. By closely modelling the physiological process, they have obtained results that are a very good match to the actual behaviour of the vocal folds. (d) 77~e Collapsible Tube Theory [11]. This was proposed by Conrad in 1980 and is based on the idea that the vocal folds behave like a flow-controlled non-linear resistor with negative slope resistance. His results on collapsible tubes are used to suggest a simple phenomenological model for the dynamics of the vocal folds; however, we believe the supragiottal orifice area used is physiologically unrealistic [12]. When a realistic value for the supraglottal constriction area of 0.4 sq cm is used an abnormal subglottal pressure must be supplied in order to obtain the negative dynamic resistance (conditiori for oscillation). The flow rate at these conditions is also unrealistic for normal phonation. Some important experimental results have been presented in the literature [ 13-15]. In these studies the consequences of different glottal shapes (divergent, convergent or rectangular) were investigated in order to determine the glottal pressure-flow relationship. Ishizaka [ 16] has incorporated into the pressure-flow equation of his theoretical model the pressure drop due to converging or diverging glottis profiles but has not considered the case of divergent flow within a convergent glottis. In contrast, results from our model suggest the possibility of divergent flow within a convergent glottis profile, or vice versa. Scherer and Titze [14] pointed out that the existing equations may be adequate first approximations to the pressure-flow characteristics of the larynx, but the results of their studies indicated that the equations lack sufficient sensitivity to glottal shape and geometry to be applicable to a wide range of phonation conditions. Although some experiments have included the laryngeal ventricle and false vocal folds, the larynx channel has been kept as a fixed profile and just a few values of the glottis
FLUID FLOW TIlROUGI! TIlE LARYNX CHANNEL
279
dimensions have been permitted to exist. Furthermore, the results have been presented in terms of"translaryngeal pressure d r o p " or entry or exit glottal pressure. The intraglottal pressure profile seems to have been largely ignored. The experimental work carried out by Dejonckere and Lebacq [17] is important as they investigated in vivo patterns of the movements of the glottal area and aerodynamic conditions of the phonatory onset. They showed that the initial movement of the vocal folds can be inward or outward in an unpredictable way, and suggested that they could be set in motion by turbulence developing in the air flow at the glottic level. The false vocal folds are two prominent mucous covered folds, which are soft and somewhat flaccid downstream from the glottis and it seems reasonable to consider that there is some interaction between these folds and the air flow. As we understand the literature there are two main areas that require investigation, as follows. (1) Some larynx models described in the literature are basically o f one type, that is, the fluid flow is restricted at one point only, so a sharp discontinuity is formed at the trailing edge of the vocal folds which act as the restriction. We believe that the inclusion of the laryngeal ventricle and the false vocal folds as an active part of the model is a more reasistic condition of the internal larynx profile. (2) No in-depth investigation has been undertaken into the types offluid flow conditions necessary to allow energy transfer from a fluid to its elastic container, in particular, one would like to know what is the geometric profile o f the containment vessel that is necessary for sustained oscillatory interaction between the flow and its elastic boundary. Figure l(a) shows the model generally used in computer simulations whilst Figure l(b) shows the model we believe should be used. Figure l(c) is the current model simulated. Van den Berg [8] has given some X-ray tomograms which show different shape and dimension for the laryngeal ventricle and false vocal folds. These tomograms suggest a relative movement between false and true vocal folds as well as a " d e f o r m a t i o n " of the false vocal folds. In the model proposed here some fluid flow efiects are considered in order that real conditions are approximated as closely as possible. The pressure on the laryngeal channel is considered in order that real conditions are approximated as closely as possible. The pressure drop at the glottis entry is not considered as a percentage of the kinetic pressure, but as a function of the vena contracta position. In the same way the pressure recovery at the glottal exit is based on the equation for diffuser flow. Six flow types are proposed to cover all possible situations inside the larynx. These flow-types are defined by the profile itself--convergent or divergent (a parallel profile is
t
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~
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Figure I. Model of larynx cross-section. (a) Profile generally used; (b) actual profile; (c) proposed profile.
280
J.A. MILLER, J. C. PEREIRA AND D. %V.THOMAS
regarded as a limiting case of a convergent profile)--and the position of the venae contractae which depend upon the constriction heights. 2. PROPOSED MODEL The results of some preliminary research conducted at the University of Southampton [18] suggested that the exact shape o f the tip of the vocal folds is very important in determining if vocal fold oscillation will occur. A c o m p u t e r model representing a vertical section through the larynx has been developed and is a three-mass representation o f the vocal folds. This model is an extension of those o f other researchers in that various air flow conditions are allowed to occur within the glottis and the false vocal folds are included as an active section of the larynx. A schematic diagram of the model is shown in Figure 2. The mass situated at the flow entry side of the model (the subglottal side) is assumed to represent the bulk of the vocalis muscle. The mass situated at the glottal exit (m2) represents the bulk o f the mucosa membrane and the mass situated at the flow output side (the supraglottal side) represents the false folds. The soft mucous-covered false vocal folds are simulated as a homogeneous mass (m3) (m3 is assumed to have the same characteristics as m2) with r~, r+, s~ and s4 representing the folds' natural damping and stiffness parameters; these values are based on their equivalents in the glottis. The parameters shown in Figure 2 are based on the work of Ishizaka and Kaneko [ 19], and lshizaka and Flanagan [ ! ]. Table I shows the parameter range available to our model and the published physiological values for normal phonation. ---'r--'x
"//////////////////////////~///
~
S3
9
1.,
L~,o,
~[~ - - L,,
r3
N
--,+ Lr~to~
Figure 2. Proposed schematic diagram of the three-mass model. (a) Glottis; (b) laryngeal ventricle; (e) false glottis. (~) denotes the station number.
The air flow conditions to be discussed result from changes in the glottis profile as the mucous membrane responds to the varying intraglottal pressures. Air flow conditions in the larynx are classified into six types, see Figure 3: type 0 to type 5. Type 0 flow occurs in the ease of a closed glottis (i.e., no air flow). There are three possible configurations the glottis may assume when in a closed state: the glottis is sealed at entry, at exit or at both. The forces acting on masses 1, 2 or 3 depend on the particular configuration giving rise to this flow type. The remaining flow types represent different glottis profiles controlling the formation o f venae contractae in the vicinity of the larynx air channel. Type I has a divergent profile with a vena contracta inside the glottis. Type 2 is also divergent but has a vena contracta in the laryngeal ventricle. The other types are all convergent. Type 3 has two venae contractae (one inside the glottis and other in the laryngeal ventricle), while types 4 and 5 have only one. Type 4 has the vena contracta controlled by the entry angle of the glottis and in the type 5 flow the vena contracta position is controlled by the glottal exit angle.
FLUID FLOW THROUGH TIlE LARYNX CHANNEL
281
TABLE I
Physiological parameters for tlle mode/ Parameter
Normal phonation Range
Ps~ m=
0-50 --
m~
m3 sl s2 s,s3 s4 rt r2 r3
r4 x t X2 X3 X4
values 10 g/cm ~ 0.10g 0"05 g 0.15 g 60 g/cm I0 g/cm 20 g/cm 10 g/cm 10 g/cm 0.1 0.6 0.6 0.6 --t --1" --'P 1"43 cm
--
-0-500 0-100 0-100 0-100 0-100 0-1 0-1 0-1 0-1 0-0-2 0-0.2 0" 1-0"35 1"13-1"78
1"Time variables within the range.
T h e basis for c a l c u l a t i n g the position o f the vena c o n t r a c t a , Zc a n d the c o n t r a c t i o n coefficient Cc are given in A p p e n d i x A, where, for our m o d e l ,
Z,. = (X, x 1.26) x (!
-Xt/R).
(l)
Here Z~ is the position of the vena contracta downstream relative to the area discontinuity, X~ is the height of the glottis at the discontinuity, R is the radius of the trachea, and C~ = 1 - 4 . 7 x 10-Sx entry angle.
(2)
(A list o f s y m b o l s is given in A p p e n d i x B.) Type 0
Type t
Type 2
= . r
Type 3
Type4
.
.
.
. ~
Type 5
-4..-- - - - .
---~
I.~
_ _ _ .
I
Figure 3. Vocal fold profiles that give rise to the various flow types as defined by the model.
__..~
282
J. A. M I L L E R , J. C. P E R E I R A A N D
D. ~,V. T I I O M A S
Figure 3 shows how various glottis profiles in the model give rise to the defined flow types. These various flow types, which are described above, represent particular geometric formations of the vocal folds as they respond to the flowing air. Although this simple model has limitations in terms of having a very angular profile the results from the model compare favourably with published data. 3. MODELLING ASSUMPTIONS The model presented has overall energy and mass balances and the losses are due to viscosity and to an enlargement in the flow (a diffuser). Several assumptions are made since it is extremely difficult to describe a complicated system like that of the laryngeal dynamics in a model. These assumptions are as follows. (i) The flow is one-dimensional and the glottal flow is assumed to be quasi-steady. Flanagan [20] has shown that the inertia of the air in the glottis is sufficiently low so that static modelling o f dynamic flow appears reasonable: the results support this conclusion. (ii) The flow is incompressible. Figures furnished by Plint and Boswirth [21] show that the reduction in density for air when accelerated from rest to 60 m/s is less than 2%. On the other hand, laryngeal air velocities measured by Dejonckere and Lebacq [17], at the onset o f phonation, were of the order of seven metres per second; therefore, an incompressibility assumption results in insignificant errors. (iii) Gravity effects are neglected as a significant force on the air. (iv) There is no real data available for the false vocal folds. The parameters used in the simulation are based on tomograms reported in the literature and the folds are considered to consist of a fleshy tissue with a few muscle fibres. Upon adopting the assumptions above and by involving conservation o f mass, one finds from the general momentum equation that the losses are given as follows. First, the viscous losses are L~.(,_j, = {K,.IIL/2gWX3m} U,
(3)
where L,.u_~) is the viscous loss between stations i and j, L is the distance between the sections to be considered, W is the width o f t h e duct, X,,,i, is the minimum height between stations i and j, K,. is the loss coefficient, U is the volume velocity, and p. is the dynamic viscosity. Second, the loss due to enlargement of the flow is Ld(,-n = (p/2gAy)[ l - A / A,] 2 U 2,
(4)
where La(~-n is the loss due to flow enlargement between stations i and j, A~ is the cross-sectional area o f the initial station, Ai is the cross-sectional area of the final station, p is the fluid density, and U is the volume velocity. From the equation of conservation o f energy and the losses given above, the volume velocity can be found and pressures at several stations can be calculated. The equation used to calculate the pressure at a station n is given as
P"=P'g
2gA2.,
,=, 2gtVgm, X3m,, j-i+l
,., 2-"-~
1 A,J
j=i+l
Aj> A,
The stations at which the pressure is calculated are the entry and exit of the glottis, the entry of the false glottis and the venae contractae positions, which are not fixed. According to the types of profile it is possible to have one or two venae contractae, which implies that the number of stations is variable as well. The second term in the right side of equation (5) is the kinetic pressure at station n. The third term accounts for the
FLUID
FLOW THROUGII
TIlE
LARYNX CHANNEL
283
viscous losses from the entry of the glottis up to section n and the last term accounts for the losses due to an enlargement of the flow. This term is calculated when the area at station j (A~) is larger than the area at station i (A~). As an example, the type 3 flow profile defined in Figure 3 has the following set o f equations: P, = P , ~ - Pk ~
Pc1 = Ps~ - Pkcl -- L~.tl-cl), P2 = Psg - Pk2 - L t . o - 2 ~ - Ld~cl-2>, P*2 = P,, - Pk~2- L,,(,-~2)-
La(c,-2),
P3 = P,~ - P k 3 - L,.(i-3) = L d ( c l - 2 ) - Ld(c2-3), P~=Pa,m,
(6)
where Po,,, is the atmospheric reference pressure. Losses incurred in converging regions have been neglected in this model. In general, such regions result in minimal losses and provide an efficient conversion of potential to kinetic flow energy. Pressure variations between two adjacent stations are considered linear and the forces due to fluid flow acting on masses 1, 2 and 3 are calculated as a mean pressure over the vocal folds surfaces. The differential equation linking the motion of the masses of the true vocal fold (masses m~ and m2) to the forces of fiuid flow (the average pressure under the masses times Wgto, x Lgto,/2) are the same as those used by Ishizaka and Flanagan [1]. For the false vocal fold (mass m3) the equation is also the same, but the terms for collision and coupling are made zero. 4. RESULTS Figure 4 shows a computer generated profile of the tip o f the vocal folds as defined by the model. The sequence o f frames shows the motion of the vocal folds as it responds to the air flow 109 ms from the start of simulation. The fold profiles are generated at 12 instants within a cycle (from 109 ms to about 117 ms). These frames compare favourably with the sketches made by Hirano [22], which show various phases of vocal fold vibration. The related graphs of the laryngeal channel area and flow volume velocity are given in Figure 5, where it is noted that the vocal fold profiles applicable during normal phonation are types 0, 1 and 3. The subglottal pressure used for the simulation results shown in Figure 5 increases (as 1 - e x p ( - a t ) ) from zero to 8.0 cm of H20 in 60 ms. The initial profile is parallel with an initial area of about 0.07 cm 2. It can be seen from this figure that the vocal folds are pushed apart as the subglottal pressure increases and then fluctuate sinusoidally until the first "closure plateau" is reached. The number of cycles before the "plateau" is consistent with the results published by Dejonckere and Lebacq [17]. The graphs in Figure 6 show that vocal fold oscillation can commence either with vocal fold abduction or vocal fold adduction. Both results have a parallel initial profile. In Figure 6(a) the flow velocity is not sufficient to Cause an adduction because of the high losses which occur in the small glottal area. As a result, the force applied to mass 1 causes an outward movement (abduction). In Figure 6(b) the initial area of the glottis is relatively large and the high flow velocity causes a negative pressure to occur in the glottal channel; therefore an inward ~novement (adduction) results. This result arises because the glottis profile is not fixed as in the other models reported in the literature.
284
J. A. MILLER, J. C. PEREIRA A N D D. W. THOMAS
0.3 0.2 0.1
i
ool Frame no. 363 Vol. vet G-area Time F-type
Frame no. 388
= 0 . 0 cm3/s = 0 . 0 0 cmz = 109"0 ms =0
VoL vel. G-area Time F-type
= O 0 cm3/s = 0 . 0 0 cm 2 = 109.9 ms =0
Frame no, 368 VOl. vel. G-area Time F-type
= 0 . 0 cm3/s = 0 . 0 2 Cm2 = 1t0.5 ms =0
0.2 0-1
0"0[
Frame no. ,370 Vol. vel. G-area Time F-type
Frame no. 372 Vol. vel. G-area Time F-type
= 108.7 cm3/s = 0 - 0 7 cmz =111"I ms =3
= 4 2 0 . 9 cm3/s = 0 . 1 3 cmz = 111-7 ms =3
Frame no. 37,5
VOI. vet. G-area Time F-type
= 7 8 2 . 5 cm3/s = 0.19 cm E = 112.6 ms =3
Vol. vel. G-area Time F-type
Frame no. 381 --- 598-5 cm3/s = 0 . 1 5 cmz = 114-4 ms = 3
0-5 0.2 0.1 e,-
0.0 Frame no. :377 Vol. vel. = 868-3 cm3/s G-area = 0 . 2 0 cmz Time =113-2 ms F-type =3
L
---Frame no. 379 VoL vel. =801-7 cm3/s G-area =0"19 cm 2 J Time =113-8 ms J F-type = 3
/
~" 0 . 3 0-2 0-1 0-0 --Vol. vet. G- area Time F-type
Frame no. 383 = 250.7 cm3/s = 0 - 0 9 cm 2 = 115'0 m.s = 1
Vol. vel. G-area Time F-type
Frame no. 386 = 0 . 0 cm3/s = 0 0 1 cmz = 115-9 ms = 0
Frame no. Vol. vel. G - area Time F-type
= 0 " 0 cm3/s
= 0 . 0 0 cmz =116",'5 ms =0
Figure 4. Computer generated vocal fold profiles. Only one vocal fold is shown as medial symmetry is assumed in this model.
5. TRANSLARYNGEAL PRESSURE DROP
In order tO compare the model presented in this paper with experimental and theoretical results reported in the literature, especially the comparisons given by Scherer, Titze and Curtis [ 15], a fixed profile model was adopted to allow measurement of the translaryngeal pressure drop. The profile used to obtain the results presented in Figure 7 had a 0.104 cm diameter by 1.4 cm width parallel glottis, with a false glottis diameter of 0.4 cm by 1.4 cm width.
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Time(ms) Figu[e 5. Onset of glottal oscillations (breathy attack). Figure 7 shows the pressure drop between stations as a function of volume velocity. The first graph shows the pressure drop between trachea and glottis entry (as no losses are considered in this region the drop is the kinetic pressure at the glottis entry). The second graph considers the pressure drop between trachea and glottis exit (station 2 in the model); this is the transglottal pressure drop. Because of the small dimensions of the glottis, the losses are larger than in the other sections. The translaryngeal pressure drop (P,R - 'Dr) is presented in the third graph. This pressure drop is calculated betwecn trachea and false vocal folds. It can be seen from Figure 7 that negative pressure is developed at the glottis exit for high volume velocity (P,g - Pr graph). The pressure recovery, which occurs in the laryngeal ventricle, can be evaluated from the second and third graphs. If the translaryngeal pressure drop curve (Figure 7, P , ~ - P c ) is fitted to the results given by Schercr et aL, then it shows that our model compares very favourably, as the results lie between those of Van den Berg and close to that o f Ishizaka and Matsudaira.
6. CONCLUSIONS The response of the model described here to various input parameters, such as subglottal pressure, vocal fold tension and phonation neutral area, indicate a behaviour that is consistent with observations of human vocal folds. C o m p a r i s o n of this model with the two-mass formulation of lshizaka and Flanagan also reveals many similarities. However, the present model offers greater flexibility in its behaviour. We believe more realistic mechanical behaviour has resulted, because more elaborate fluid flow situations can be accounted for and the false vocal folds have been included as an active section of the glottal source. Several profile types have been presented and the interaction between the flow and the laryngeal channel has been considered. It has been shown that oscillation can start either
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Figure 6. These figures show that abduction and adduction can occur with different parameters values as is shown in the in t i t o experiments of Dejonckere and Lebacq. (a) a vocal fold abduction at the beginning of oscillation; (b) a vocal fold adduetion at the beginning o f oscillation.
as an abduction or adduction. The results presented compare favourably with the experimental results and conclusions of Dejonckere and Lebacq. The mechanical complexity o f the model is not sufficient to reproduce the exact undulating behaviour of the mucosa membrane reported by Hirano: however, the gross movement of the true vocal folds throughout a cycle can be modelled.
FLUID FLOW THROUGH TIlE LARYNX CHANNEL
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40
u
24
Q." 0
ZOO
400
600
800
1000
Volume velocity (crn3/s)
a 401.
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:~32t-
/
40 32 u
0
200
400
600
800
24
O/ .
.
1000 0 200 Volume velocity (cm3/s)
.
400
.
.
600
,
800 1000
Figure 7. Pressure drop profiles between several stations of the model. The pressure axis can be represented in terms of dynes/cm2 by noting that I dyne/era ~= 1-019/1000cm of 1120. P,~, P0, P2, and Pr, respectively, are the subglottal pressure, the pressures at the glottis entry and exit, and the pressure in the false glottis.
S i m u l a t e d values o f pressure drop b e t w e e n several stations through the laryngeal c h a n n e l having a parallel profile were o b t a i n e d . C o m p a r i s o n s o f the t r a n s l a r y n g e a l pressure d r o p with e x p e r i m e n t a l data [ 15] a n d theoretical values show the model b e h a v i n g in a realistic m a n n e r . In a d d i t i o n , due to the flexibility o f the m o d e l , i n s t a n t a n e o u s values o f pressure drop a n d v o l u m e velocity as well as profiles of the laryngeal c h a n n e l can be obtained. A l t h o u g h static c o n c e p t s have been a p p l i e d to a d y n a m i c s i m u l a t i o n the results have s h o w n that the model is c a p a b l e of p r o d u c i n g results that are in line with the in vivo e x p e r i m e n t s c o n d u c t e d by D e j o n c k e r e a n d Lebacq.
REFERENCES I. K. ISHIZAKA and J. L. FLANAGAN 1972 Bell System TechnicaIJounla151, 1233-1268. Synthesis of voiced sounds from a two-mass model of the vocal cords. 2. R. L. WEGEL 1930 Bell System Technical Journal 9, 207-227. Theory of vibration of the larynx. 3. J. L. FLANAGAN and L. L. LANDGRAF 1968 lnstintte of Electrical and Electronic Engineers, Transactions on Audio and Electroacoustics 16, 57-64. Self-oscillating source for vocal-tract synthesizers. 4. I. R. TITZE 1973 Phonetica 28, 129-170. The human vocal cords: a mathematical model. Part I. 5. I. R. TITZE 1974 Phonedca 29, 1-21. The human vocal cords: a mathematical model. Part II. 6. I. R. TITZE and D. T. TALKIN 1979 Journal of the Acoustical Society of America 66, 60-74. A theoretical study of the effects of various laryngeal configurations on the acoustics of phonation. 7. R. HUSSON 1953 Annals ofOtology, Rhinology and larynology 69, 124-137. Sur la physiologic voeale. 8. J. VAN DEN BERG 1958 Journal of Speech and Hearing Research I, 227-244. Myoelasticaerodynamic theory of voice production. 9. J. VAN DEN BERG, J. T. ZANTEMA and P. DOORNENBAAL JR 1957 Journal of the Acoustical Society of America 29, 626-631. On the air resistance and the Bernouli effect ofthe human larynx. 10. K. ISIIIZAKA and M. t~,'iATSUI)AIRA 1972 Speech Communications Research Laboratory-Monograph No. 8. Fluid mechanical considerations of vocal cord vibration. I I. W. A. CONRAD 1980 Medical Research Engineering 13 7-10. A new model of the vocal cords based on a collapsible tube analogy.
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AND
D. XV. T H O M A S
12. W. A. CONRAD 1983 in lliomecltanics, Acoustics and Phonator)' Control. (1. R. Titze and R. C. Schcrer, (ed.). The Denver Center for the Performing Arts, 328-348. Collapsible tube model of the larynx. 13. J. GAUFFIN, N. BINH, T. V. ANANTIIAPADMANABHAand G. ]:ANT 1983 in 2rid Vocal Fold Physiology Conference. (D. M. Bless and J. H. Abbs Editors), Vocal Fold Physiology--College Hill Press, 194-201. Glottal Geometry and volume velocity waveform. 14. R. C. SCllERER and I. R. TI-rZE 1983 in 2nd Vocal Fold Physiology Conference (D. M. Bless and J. H. Abbs editors), Vocal Fold Physiology--College Hill Press, 179-193. Pressure-flow relationships in a model of the laryngeal airway with a diverging glottis. 15. R.C. SCHERER, 1. R. TITZE and J. F. CURTIS 1983 Journal of the Acoustical Society of America 73, 668-676. Pressure-flow relationships in two models of the larynx having rectangular glottal shapes. 16. K. ISIIIZAKA 1983 in Vocal Fold Physiology: Biomechanics, Acoustics and Phonatory Control. (I. R. Titze and R. C. Scherer, editors), The Denver Center for the Performing Arts, 414-424. Air resistance and intraglottal pressure in a model of the larynx. 17. P. DEJONCKERE and J. LEBACQ 1981 Archi~'es internationales de Ph)'siologie et de Biochimie 89, 127-136. Mechanism of initiation of oscillatory motion in human glottis. 18. J. A. MILLER 1982 M.Sc. Thesis, UniversiO' of Southampton. System identification in speech. 19. K. ISHIZAKA and T. KANEKO 1968 Journal of the Acoustical Society of Japan 24, 312-313. On equivalent mechanical constants of the vocal cords. 20. J. L. FLANA(3AN 1959 Journal of Speech and Hearing Research, 2, 168-172. Estimates of intraglottal pressure during phonation. 21. M.A. PLINT AND L. IIOSWlRTH 1978 Fhdd Alechanics: A Laboratory Course. London: Charles Griffin. 22. M. HIRANO 1977 in Dynamic Aspects of Speech Production. M. Sawashima and F. S. Cooper, editors), Tokyo: University of Tokyo Press, 13-30. Structure and vibratory behavior of the vocal folds. 23. R. D. GROSE" 1985 Transactions of the American Society of Mechanical Engineers, Journal of Fhdds Engineering 107, 36-43. Orifice contraction coefficient for inviscid incompressible flow. 24. B. S. MASSEY 1979 Mechanics of Fhdds. New York: Van Nostrand Reinhold. 25. J. SZEKELY 1979 Fluid Flow Phenonzena in Metals Processing. New York: Academic Press.
APPENDIX A: VENA CONTRACTA A.I. C O E F F I C I [ - . N T O F C O N T R A C T I O N The coefficient o f contraction as a Function o f the entry angle has been reported by Grose [23]. Based on the Navier-Stokes equation and using a control volume with an elliptical surface at the entry o f the orifice he has presented a Formulation for the contraction coefficient. The coefficient derived for an entry angle o f 90 ~ i's given by Cc = &2 _ [~b4 _ ~b2] i/2
(AI)
where Cc is the coefficient o f contraction and ~b is the potential surface area ratio, i.e., the ratio o f the elliptical surface area to the orifice o p e n i n g area, and is given by = [2/(1 + K)] '/2,
(A2)
where K is the orifice to upstream channel height ratio. By applying this Formulation to the t r a c h e a - v o c a l Folds junction the contraction coefficient can be determined for the limiting values of the glottis heights (XI) during phonation. As s h o w n in Table l, the b o u n d a r y values o f XI during phonation are zero and 0.2 cm. For an average value for the area o f trachea o f 5 cm 2. (R = i . 2 6 c m ) , the contraction coefficients are Found to be 0.5858 and 0.6065, respectively. The figures presented in the preceding paragraph are calculated for a 90 ~ entry angle. By choosing an a p p r o p r i a t e value for r which is related to the entry angle ( a ) , G r o s e has derived the coefficient o r c o n t r a c t i o n as a Function o f t h i s angle (curve l in Figure A l ) .
FLUID
FLOW TIIROUGH
THE
LARYNX
CIIANNEL
289
In order to find the region of validity for our particular case an extrapolation of the curve for a higher value of K was made ( K = 0-159), as the orifice opening for the glottis ranges between 0.0 and 0.4 cm. The linear approximation assumed for this is shown in Figure A1.
o8~. 1.O
'
i
'
i
'
i
'
i
I
,
!
i0"4 0
40
80
120
160
e
Figure At. lnviscid incompressible flow contraction coefficient as a function of the entry angle. (l) Elliptical theory curve (K = 0) estimated by Grose; (2) extrapolated curve (K = 0-159); (3) linear approximation curve.
A.2.
POSITION
OF THE
VENA
CONTRACTA
Massey [24] states that for a sharp edged orifice in a vessel wail, a vena contracta forms downstream at about a distance equal to the orifice radius. The position of the vena contracta is nearly independent of the rate of flow and is a function of the orifice-to-pipe diameter ratio K, moving upstream as the value of K becomes larger. U p o n assuming a linear approximation for the position o f the vena contracta from Szekely's plot of correlation [.25] for the discharge coefficient, then Zc is given by Zc = (r x 1.26) x (1 - r / R ) ,
(A3)
where Zc is the position of the vena contracta relative to the area discontinuity, r is the radius o f the pipe downstream from the contraction, and R is the radius of the pipe upstream from the contraction. It should be noted here that the relationship for Zc has been derived for axisymmetric flow and at the trachea-glottis junction the flow changes from axisymmetrical flow in the trachea to two-dimensional flow in the glottis. APPENDIX B: LIST OF SYMBOLS
tglot Lt~,
t.lglo U
A, A~
x,
length of the glottis (0.4 cm) length of laryngeal ventricle (0.3 cm) length of the false glottis (0.8 cm) width of the glottis ( 1.4 cm) volume" velocity (cm3/s) final area of flow enlargement (cm 2) initial area of flow enlargement (cm 2) height of the laryngeal channel at station i (cm)
290
P~ Kv
P, P~, Pc P~ Pat,,I LL,ti-j) Ldti-~) g p tx
J. A. M I L L E R , J. C. P E R E I R A A N D
D. W. T H O M A S
minimum laryngeal flow height (cm) normalized subglottal pressure (g/cm 2) constant for viscous loss at glottis normalized piezometric pressure at station i (g/cm 2) normalized kinetic pressure at station i (g/cm 2) normalized piezometric pressure at vena contracta (g/cm 2) normalized kinetic pressure at vena contracta (g/cm 2) atmospheric pressure viscous loss (between stations i and j) (g/cm 2) loss due to an enlargement (between stations i and j) (g/cm 2) acceleration due to gravity (981 cm/s 2) air density (1.489 x 10-3 gm/cm 3) air viscosity (1.878 x 10-4 g/cms)
FLUID FLOW T H R O U G H THE LARYNX C H A N N E L J. A.
I~IILLER,J.
C. PEREIRA AND D. W. TtIOXtAS
Department of Electronics and Computer Science, Unit'ersi O' of Southampton, Southampton S09 5 NH, England (Received 31 October 1985, and in rerixed form 12 Afa)" 1987) The classic two-mass model of the larynx channel is extended by including the false vocal folds and the laryngeal ventricle. Several glottis profiles are postulated to exist which are the result of the forces applied to the mucus membrane due to intraglottal pressure variation. These profiles constrain the air flow which allows the formation of one or two "'venae contractae". The location of these influences the pressure in the glottis and layrngeal ventricle and also gives rise to additional viscous losses as well as losses due to flow enlargement. Sampled waveforms are calculated from the model for volume velocity, glottal area, Reynolds number and fluid forces over the vocal folds for various profiles. Results show that the computed waveforms agree with physiological data [ !, 2] and that it is not necessary to use any empirical constants to match the simulation results. Also, the onset of phonation is shown to be possible either with abduction or adduction of the vocal folds. 1. INTRODUCTION AND BACKGROUND Early models of the h u m a n larynx were based on simplified geometric shapes and air flow conditions [ I-3]. The trend has been to increase model complexity in terms of more elaborate mechanical structures representing the vocal tissue [4-6] with less attention being paid to the source of vocal energy, the air. In this research an attempt has been made to redress this imbalance by focusing on the laryngeal air flow and its interaction with the surrounding elastic tissue. In order to establish the contribution of air patterns to realistic phonatory behaviour, it was decided to adopt a relatively simple mechanical model o f the vocal folds. This consisted of two coupled masses representing each true vocal fold plus a third mass representing the false fold. The air-tissue boundary is represented by a series of straight lines joining the fixed and movable points in the proposed model, as shown in Figure l(c), and approximating the actual profile shown in Figure l(b). Several theories of vocal fold vibration have been proposed in the literature and some of these are summarized as follows. (a) The Aerodynamic-Myoelastic Theory. Helmholtz, in 1863, proposed that the "puffs'" of air originating at the glottis are the main source o f sound. According to Husson's Neuro-Chronaxic Theory [7], these puffs are induced by an impulse sent by the brain to the vocalis muscle via the recurrent nerve so setting the vocal fold into vibration. The Aerodynamic-Myoelastic Theory was proposed by van den Berg [8] to explain p h e n o m e n a which cannot be accounted for by the Neuro-Chronaxic Theory. This theory postulates that the vocal fold oscillation is actuated by the stream of air delivered by the lungs and trachea. The experimental basis for this theory was established by van den Berg et al. in 1957 [9]. Using a cast of a h u m a n larynx, van den Berg et al. measured the air resistance of the larynx and the intra-glottal pressure. The negative pressure measured inside the glottis led them to propose that the Bernoulli effect was responsible for the closure o f t h e glottis. 277 0022-46ox/88/050277 + 14 S03.00/0 O 1988 Academic Press Li. dted
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MILLER. J. C. P E R E I R A A N D D. W. THOMAS
They also suggested that other phenomena were occurring in the flow, like the formation of "eddies" in the laryngeal ventricle and contraction of the flow at the entry of the glottis. Based on their measurements, and because of the complexity of the problem, they proposed an empirical formula for the air resistance of the larynx, in which pressure loss at the entry of the glottis and the pressure recovery at the exit were considered as fractions of the kinetic pressure for a wide range of glottal diameters. (b) The Flow-separation Theory [10]. The theory of lshizaka and Matsudaira represents a significant conceptual advance by showing how vocal fold vibration can occur in terms of the laws of mechanics in relation to the interaction between the fluid flow and tissue displacement at the glottis. The flow-pressure equations derived in their analytical study account for the losses due to the vena contracta formation at the entry of the glottis, the sudden expansion at the exit of the glottis and the viscous losses. However, they adopted van der Berg's empirical figure for the loss at the entry of the glottis. Their model represents the tissue by two coupled masses (the simplest structure which can capture the vertical phase difference). They derived coefficients for the pressure variations in terms of aerodynamic reacting pressure and aerodynamic coupling pressure, and showed thata sizeable aerodynamic coupling stiffness exists between the upper and lower masses. (c) The Layered Viscoelastic Tissue Theory [6]. A layered tissue approach for the vocal fold model was proposed by Titze and Talkin. In this model phonation is considered as a wave phenomenon in a layered configuration of viscoelastic tissues, the analysis of which may be accomplished by the solutiori of a boundary value problem. The theory is not only an attempt to reproduce human-like speech, but also, although through an analog, to replicate the physical processes as completely as possible. This approach is classified by the authors as "speech simulation" rather than "speech synthesis". In this context, the model presents a very high level of sophistication of the modelling process. By closely modelling the physiological process, they have obtained results that are a very good match to the actual behaviour of the vocal folds. (d) 77~e Collapsible Tube Theory [11]. This was proposed by Conrad in 1980 and is based on the idea that the vocal folds behave like a flow-controlled non-linear resistor with negative slope resistance. His results on collapsible tubes are used to suggest a simple phenomenological model for the dynamics of the vocal folds; however, we believe the supragiottal orifice area used is physiologically unrealistic [12]. When a realistic value for the supraglottal constriction area of 0.4 sq cm is used an abnormal subglottal pressure must be supplied in order to obtain the negative dynamic resistance (conditiori for oscillation). The flow rate at these conditions is also unrealistic for normal phonation. Some important experimental results have been presented in the literature [ 13-15]. In these studies the consequences of different glottal shapes (divergent, convergent or rectangular) were investigated in order to determine the glottal pressure-flow relationship. Ishizaka [ 16] has incorporated into the pressure-flow equation of his theoretical model the pressure drop due to converging or diverging glottis profiles but has not considered the case of divergent flow within a convergent glottis. In contrast, results from our model suggest the possibility of divergent flow within a convergent glottis profile, or vice versa. Scherer and Titze [14] pointed out that the existing equations may be adequate first approximations to the pressure-flow characteristics of the larynx, but the results of their studies indicated that the equations lack sufficient sensitivity to glottal shape and geometry to be applicable to a wide range of phonation conditions. Although some experiments have included the laryngeal ventricle and false vocal folds, the larynx channel has been kept as a fixed profile and just a few values of the glottis
FLUID FLOW TIlROUGI! TIlE LARYNX CHANNEL
279
dimensions have been permitted to exist. Furthermore, the results have been presented in terms of"translaryngeal pressure d r o p " or entry or exit glottal pressure. The intraglottal pressure profile seems to have been largely ignored. The experimental work carried out by Dejonckere and Lebacq [17] is important as they investigated in vivo patterns of the movements of the glottal area and aerodynamic conditions of the phonatory onset. They showed that the initial movement of the vocal folds can be inward or outward in an unpredictable way, and suggested that they could be set in motion by turbulence developing in the air flow at the glottic level. The false vocal folds are two prominent mucous covered folds, which are soft and somewhat flaccid downstream from the glottis and it seems reasonable to consider that there is some interaction between these folds and the air flow. As we understand the literature there are two main areas that require investigation, as follows. (1) Some larynx models described in the literature are basically o f one type, that is, the fluid flow is restricted at one point only, so a sharp discontinuity is formed at the trailing edge of the vocal folds which act as the restriction. We believe that the inclusion of the laryngeal ventricle and the false vocal folds as an active part of the model is a more reasistic condition of the internal larynx profile. (2) No in-depth investigation has been undertaken into the types offluid flow conditions necessary to allow energy transfer from a fluid to its elastic container, in particular, one would like to know what is the geometric profile o f the containment vessel that is necessary for sustained oscillatory interaction between the flow and its elastic boundary. Figure l(a) shows the model generally used in computer simulations whilst Figure l(b) shows the model we believe should be used. Figure l(c) is the current model simulated. Van den Berg [8] has given some X-ray tomograms which show different shape and dimension for the laryngeal ventricle and false vocal folds. These tomograms suggest a relative movement between false and true vocal folds as well as a " d e f o r m a t i o n " of the false vocal folds. In the model proposed here some fluid flow efiects are considered in order that real conditions are approximated as closely as possible. The pressure on the laryngeal channel is considered in order that real conditions are approximated as closely as possible. The pressure drop at the glottis entry is not considered as a percentage of the kinetic pressure, but as a function of the vena contracta position. In the same way the pressure recovery at the glottal exit is based on the equation for diffuser flow. Six flow types are proposed to cover all possible situations inside the larynx. These flow-types are defined by the profile itself--convergent or divergent (a parallel profile is
t
~Vo~
To moufh
oI L_..._...~ ttoct]
FoSse oi foTd ~ \ ryngeo! n Ir icte "~l~___. ' Vocol fold
~
To
g
r OCheo-~.--.~-
Figure I. Model of larynx cross-section. (a) Profile generally used; (b) actual profile; (c) proposed profile.
280
J.A. MILLER, J. C. PEREIRA AND D. %V.THOMAS
regarded as a limiting case of a convergent profile)--and the position of the venae contractae which depend upon the constriction heights. 2. PROPOSED MODEL The results of some preliminary research conducted at the University of Southampton [18] suggested that the exact shape o f the tip of the vocal folds is very important in determining if vocal fold oscillation will occur. A c o m p u t e r model representing a vertical section through the larynx has been developed and is a three-mass representation o f the vocal folds. This model is an extension of those o f other researchers in that various air flow conditions are allowed to occur within the glottis and the false vocal folds are included as an active section of the larynx. A schematic diagram of the model is shown in Figure 2. The mass situated at the flow entry side of the model (the subglottal side) is assumed to represent the bulk of the vocalis muscle. The mass situated at the glottal exit (m2) represents the bulk o f the mucosa membrane and the mass situated at the flow output side (the supraglottal side) represents the false folds. The soft mucous-covered false vocal folds are simulated as a homogeneous mass (m3) (m3 is assumed to have the same characteristics as m2) with r~, r+, s~ and s4 representing the folds' natural damping and stiffness parameters; these values are based on their equivalents in the glottis. The parameters shown in Figure 2 are based on the work of Ishizaka and Kaneko [ 19], and lshizaka and Flanagan [ ! ]. Table I shows the parameter range available to our model and the published physiological values for normal phonation. ---'r--'x
"//////////////////////////~///
~
S3
9
1.,
L~,o,
~[~ - - L,,
r3
N
--,+ Lr~to~
Figure 2. Proposed schematic diagram of the three-mass model. (a) Glottis; (b) laryngeal ventricle; (e) false glottis. (~) denotes the station number.
The air flow conditions to be discussed result from changes in the glottis profile as the mucous membrane responds to the varying intraglottal pressures. Air flow conditions in the larynx are classified into six types, see Figure 3: type 0 to type 5. Type 0 flow occurs in the ease of a closed glottis (i.e., no air flow). There are three possible configurations the glottis may assume when in a closed state: the glottis is sealed at entry, at exit or at both. The forces acting on masses 1, 2 or 3 depend on the particular configuration giving rise to this flow type. The remaining flow types represent different glottis profiles controlling the formation o f venae contractae in the vicinity of the larynx air channel. Type I has a divergent profile with a vena contracta inside the glottis. Type 2 is also divergent but has a vena contracta in the laryngeal ventricle. The other types are all convergent. Type 3 has two venae contractae (one inside the glottis and other in the laryngeal ventricle), while types 4 and 5 have only one. Type 4 has the vena contracta controlled by the entry angle of the glottis and in the type 5 flow the vena contracta position is controlled by the glottal exit angle.
FLUID FLOW THROUGH TIlE LARYNX CHANNEL
281
TABLE I
Physiological parameters for tlle mode/ Parameter
Normal phonation Range
Ps~ m=
0-50 --
m~
m3 sl s2 s,s3 s4 rt r2 r3
r4 x t X2 X3 X4
values 10 g/cm ~ 0.10g 0"05 g 0.15 g 60 g/cm I0 g/cm 20 g/cm 10 g/cm 10 g/cm 0.1 0.6 0.6 0.6 --t --1" --'P 1"43 cm
--
-0-500 0-100 0-100 0-100 0-100 0-1 0-1 0-1 0-1 0-0-2 0-0.2 0" 1-0"35 1"13-1"78
1"Time variables within the range.
T h e basis for c a l c u l a t i n g the position o f the vena c o n t r a c t a , Zc a n d the c o n t r a c t i o n coefficient Cc are given in A p p e n d i x A, where, for our m o d e l ,
Z,. = (X, x 1.26) x (!
-Xt/R).
(l)
Here Z~ is the position of the vena contracta downstream relative to the area discontinuity, X~ is the height of the glottis at the discontinuity, R is the radius of the trachea, and C~ = 1 - 4 . 7 x 10-Sx entry angle.
(2)
(A list o f s y m b o l s is given in A p p e n d i x B.) Type 0
Type t
Type 2
= . r
Type 3
Type4
.
.
.
. ~
Type 5
-4..-- - - - .
---~
I.~
_ _ _ .
I
Figure 3. Vocal fold profiles that give rise to the various flow types as defined by the model.
__..~
282
J. A. M I L L E R , J. C. P E R E I R A A N D
D. ~,V. T I I O M A S
Figure 3 shows how various glottis profiles in the model give rise to the defined flow types. These various flow types, which are described above, represent particular geometric formations of the vocal folds as they respond to the flowing air. Although this simple model has limitations in terms of having a very angular profile the results from the model compare favourably with published data. 3. MODELLING ASSUMPTIONS The model presented has overall energy and mass balances and the losses are due to viscosity and to an enlargement in the flow (a diffuser). Several assumptions are made since it is extremely difficult to describe a complicated system like that of the laryngeal dynamics in a model. These assumptions are as follows. (i) The flow is one-dimensional and the glottal flow is assumed to be quasi-steady. Flanagan [20] has shown that the inertia of the air in the glottis is sufficiently low so that static modelling o f dynamic flow appears reasonable: the results support this conclusion. (ii) The flow is incompressible. Figures furnished by Plint and Boswirth [21] show that the reduction in density for air when accelerated from rest to 60 m/s is less than 2%. On the other hand, laryngeal air velocities measured by Dejonckere and Lebacq [17], at the onset o f phonation, were of the order of seven metres per second; therefore, an incompressibility assumption results in insignificant errors. (iii) Gravity effects are neglected as a significant force on the air. (iv) There is no real data available for the false vocal folds. The parameters used in the simulation are based on tomograms reported in the literature and the folds are considered to consist of a fleshy tissue with a few muscle fibres. Upon adopting the assumptions above and by involving conservation o f mass, one finds from the general momentum equation that the losses are given as follows. First, the viscous losses are L~.(,_j, = {K,.IIL/2gWX3m} U,
(3)
where L,.u_~) is the viscous loss between stations i and j, L is the distance between the sections to be considered, W is the width o f t h e duct, X,,,i, is the minimum height between stations i and j, K,. is the loss coefficient, U is the volume velocity, and p. is the dynamic viscosity. Second, the loss due to enlargement of the flow is Ld(,-n = (p/2gAy)[ l - A / A,] 2 U 2,
(4)
where La(~-n is the loss due to flow enlargement between stations i and j, A~ is the cross-sectional area o f the initial station, Ai is the cross-sectional area of the final station, p is the fluid density, and U is the volume velocity. From the equation of conservation o f energy and the losses given above, the volume velocity can be found and pressures at several stations can be calculated. The equation used to calculate the pressure at a station n is given as
P"=P'g
2gA2.,
,=, 2gtVgm, X3m,, j-i+l
,., 2-"-~
1 A,J
j=i+l
Aj> A,
The stations at which the pressure is calculated are the entry and exit of the glottis, the entry of the false glottis and the venae contractae positions, which are not fixed. According to the types of profile it is possible to have one or two venae contractae, which implies that the number of stations is variable as well. The second term in the right side of equation (5) is the kinetic pressure at station n. The third term accounts for the
FLUID
FLOW THROUGII
TIlE
LARYNX CHANNEL
283
viscous losses from the entry of the glottis up to section n and the last term accounts for the losses due to an enlargement of the flow. This term is calculated when the area at station j (A~) is larger than the area at station i (A~). As an example, the type 3 flow profile defined in Figure 3 has the following set o f equations: P, = P , ~ - Pk ~
Pc1 = Ps~ - Pkcl -- L~.tl-cl), P2 = Psg - Pk2 - L t . o - 2 ~ - Ld~cl-2>, P*2 = P,, - Pk~2- L,,(,-~2)-
La(c,-2),
P3 = P,~ - P k 3 - L,.(i-3) = L d ( c l - 2 ) - Ld(c2-3), P~=Pa,m,
(6)
where Po,,, is the atmospheric reference pressure. Losses incurred in converging regions have been neglected in this model. In general, such regions result in minimal losses and provide an efficient conversion of potential to kinetic flow energy. Pressure variations between two adjacent stations are considered linear and the forces due to fluid flow acting on masses 1, 2 and 3 are calculated as a mean pressure over the vocal folds surfaces. The differential equation linking the motion of the masses of the true vocal fold (masses m~ and m2) to the forces of fiuid flow (the average pressure under the masses times Wgto, x Lgto,/2) are the same as those used by Ishizaka and Flanagan [1]. For the false vocal fold (mass m3) the equation is also the same, but the terms for collision and coupling are made zero. 4. RESULTS Figure 4 shows a computer generated profile of the tip o f the vocal folds as defined by the model. The sequence o f frames shows the motion of the vocal folds as it responds to the air flow 109 ms from the start of simulation. The fold profiles are generated at 12 instants within a cycle (from 109 ms to about 117 ms). These frames compare favourably with the sketches made by Hirano [22], which show various phases of vocal fold vibration. The related graphs of the laryngeal channel area and flow volume velocity are given in Figure 5, where it is noted that the vocal fold profiles applicable during normal phonation are types 0, 1 and 3. The subglottal pressure used for the simulation results shown in Figure 5 increases (as 1 - e x p ( - a t ) ) from zero to 8.0 cm of H20 in 60 ms. The initial profile is parallel with an initial area of about 0.07 cm 2. It can be seen from this figure that the vocal folds are pushed apart as the subglottal pressure increases and then fluctuate sinusoidally until the first "closure plateau" is reached. The number of cycles before the "plateau" is consistent with the results published by Dejonckere and Lebacq [17]. The graphs in Figure 6 show that vocal fold oscillation can commence either with vocal fold abduction or vocal fold adduction. Both results have a parallel initial profile. In Figure 6(a) the flow velocity is not sufficient to Cause an adduction because of the high losses which occur in the small glottal area. As a result, the force applied to mass 1 causes an outward movement (abduction). In Figure 6(b) the initial area of the glottis is relatively large and the high flow velocity causes a negative pressure to occur in the glottal channel; therefore an inward ~novement (adduction) results. This result arises because the glottis profile is not fixed as in the other models reported in the literature.
284
J. A. MILLER, J. C. PEREIRA A N D D. W. THOMAS
0.3 0.2 0.1
i
ool Frame no. 363 Vol. vet G-area Time F-type
Frame no. 388
= 0 . 0 cm3/s = 0 . 0 0 cmz = 109"0 ms =0
VoL vel. G-area Time F-type
= O 0 cm3/s = 0 . 0 0 cm 2 = 109.9 ms =0
Frame no, 368 VOl. vel. G-area Time F-type
= 0 . 0 cm3/s = 0 . 0 2 Cm2 = 1t0.5 ms =0
0.2 0-1
0"0[
Frame no. ,370 Vol. vel. G-area Time F-type
Frame no. 372 Vol. vel. G-area Time F-type
= 108.7 cm3/s = 0 - 0 7 cmz =111"I ms =3
= 4 2 0 . 9 cm3/s = 0 . 1 3 cmz = 111-7 ms =3
Frame no. 37,5
VOI. vet. G-area Time F-type
= 7 8 2 . 5 cm3/s = 0.19 cm E = 112.6 ms =3
Vol. vel. G-area Time F-type
Frame no. 381 --- 598-5 cm3/s = 0 . 1 5 cmz = 114-4 ms = 3
0-5 0.2 0.1 e,-
0.0 Frame no. :377 Vol. vel. = 868-3 cm3/s G-area = 0 . 2 0 cmz Time =113-2 ms F-type =3
L
---Frame no. 379 VoL vel. =801-7 cm3/s G-area =0"19 cm 2 J Time =113-8 ms J F-type = 3
/
~" 0 . 3 0-2 0-1 0-0 --Vol. vet. G- area Time F-type
Frame no. 383 = 250.7 cm3/s = 0 - 0 9 cm 2 = 115'0 m.s = 1
Vol. vel. G-area Time F-type
Frame no. 386 = 0 . 0 cm3/s = 0 0 1 cmz = 115-9 ms = 0
Frame no. Vol. vel. G - area Time F-type
= 0 " 0 cm3/s
= 0 . 0 0 cmz =116",'5 ms =0
Figure 4. Computer generated vocal fold profiles. Only one vocal fold is shown as medial symmetry is assumed in this model.
5. TRANSLARYNGEAL PRESSURE DROP
In order tO compare the model presented in this paper with experimental and theoretical results reported in the literature, especially the comparisons given by Scherer, Titze and Curtis [ 15], a fixed profile model was adopted to allow measurement of the translaryngeal pressure drop. The profile used to obtain the results presented in Figure 7 had a 0.104 cm diameter by 1.4 cm width parallel glottis, with a false glottis diameter of 0.4 cm by 1.4 cm width.
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FLOW
THROUGII
TIlE
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CIIANNEL
~9
~ X
~ E >, 1500 I ~ ' B 12001=.'~ 9 0 0 "
600
-=
3o
~ 8~176176
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B 2000 tr
5
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.
,
,
I
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. . . .
I
30
. . . .
I
45
60
t5
90
105
120
135
150
Time(ms) Figu[e 5. Onset of glottal oscillations (breathy attack). Figure 7 shows the pressure drop between stations as a function of volume velocity. The first graph shows the pressure drop between trachea and glottis entry (as no losses are considered in this region the drop is the kinetic pressure at the glottis entry). The second graph considers the pressure drop between trachea and glottis exit (station 2 in the model); this is the transglottal pressure drop. Because of the small dimensions of the glottis, the losses are larger than in the other sections. The translaryngeal pressure drop (P,R - 'Dr) is presented in the third graph. This pressure drop is calculated betwecn trachea and false vocal folds. It can be seen from Figure 7 that negative pressure is developed at the glottis exit for high volume velocity (P,g - Pr graph). The pressure recovery, which occurs in the laryngeal ventricle, can be evaluated from the second and third graphs. If the translaryngeal pressure drop curve (Figure 7, P , ~ - P c ) is fitted to the results given by Schercr et aL, then it shows that our model compares very favourably, as the results lie between those of Van den Berg and close to that o f Ishizaka and Matsudaira.
6. CONCLUSIONS The response of the model described here to various input parameters, such as subglottal pressure, vocal fold tension and phonation neutral area, indicate a behaviour that is consistent with observations of human vocal folds. C o m p a r i s o n of this model with the two-mass formulation of lshizaka and Flanagan also reveals many similarities. However, the present model offers greater flexibility in its behaviour. We believe more realistic mechanical behaviour has resulted, because more elaborate fluid flow situations can be accounted for and the false vocal folds have been included as an active section of the glottal source. Several profile types have been presented and the interaction between the flow and the laryngeal channel has been considered. It has been shown that oscillation can start either
286
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60
75 Time (ms) (b)
90
10.5
120
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1.~3
Figure 6. These figures show that abduction and adduction can occur with different parameters values as is shown in the in t i t o experiments of Dejonckere and Lebacq. (a) a vocal fold abduction at the beginning of oscillation; (b) a vocal fold adduetion at the beginning o f oscillation.
as an abduction or adduction. The results presented compare favourably with the experimental results and conclusions of Dejonckere and Lebacq. The mechanical complexity o f the model is not sufficient to reproduce the exact undulating behaviour of the mucosa membrane reported by Hirano: however, the gross movement of the true vocal folds throughout a cycle can be modelled.
FLUID FLOW THROUGH TIlE LARYNX CHANNEL
287
40
u
24
Q." 0
ZOO
400
600
800
1000
Volume velocity (crn3/s)
a 401.
I
/
:~32t-
/
40 32 u
0
200
400
600
800
24
O/ .
.
1000 0 200 Volume velocity (cm3/s)
.
400
.
.
600
,
800 1000
Figure 7. Pressure drop profiles between several stations of the model. The pressure axis can be represented in terms of dynes/cm2 by noting that I dyne/era ~= 1-019/1000cm of 1120. P,~, P0, P2, and Pr, respectively, are the subglottal pressure, the pressures at the glottis entry and exit, and the pressure in the false glottis.
S i m u l a t e d values o f pressure drop b e t w e e n several stations through the laryngeal c h a n n e l having a parallel profile were o b t a i n e d . C o m p a r i s o n s o f the t r a n s l a r y n g e a l pressure d r o p with e x p e r i m e n t a l data [ 15] a n d theoretical values show the model b e h a v i n g in a realistic m a n n e r . In a d d i t i o n , due to the flexibility o f the m o d e l , i n s t a n t a n e o u s values o f pressure drop a n d v o l u m e velocity as well as profiles of the laryngeal c h a n n e l can be obtained. A l t h o u g h static c o n c e p t s have been a p p l i e d to a d y n a m i c s i m u l a t i o n the results have s h o w n that the model is c a p a b l e of p r o d u c i n g results that are in line with the in vivo e x p e r i m e n t s c o n d u c t e d by D e j o n c k e r e a n d Lebacq.
REFERENCES I. K. ISHIZAKA and J. L. FLANAGAN 1972 Bell System TechnicaIJounla151, 1233-1268. Synthesis of voiced sounds from a two-mass model of the vocal cords. 2. R. L. WEGEL 1930 Bell System Technical Journal 9, 207-227. Theory of vibration of the larynx. 3. J. L. FLANAGAN and L. L. LANDGRAF 1968 lnstintte of Electrical and Electronic Engineers, Transactions on Audio and Electroacoustics 16, 57-64. Self-oscillating source for vocal-tract synthesizers. 4. I. R. TITZE 1973 Phonetica 28, 129-170. The human vocal cords: a mathematical model. Part I. 5. I. R. TITZE 1974 Phonedca 29, 1-21. The human vocal cords: a mathematical model. Part II. 6. I. R. TITZE and D. T. TALKIN 1979 Journal of the Acoustical Society of America 66, 60-74. A theoretical study of the effects of various laryngeal configurations on the acoustics of phonation. 7. R. HUSSON 1953 Annals ofOtology, Rhinology and larynology 69, 124-137. Sur la physiologic voeale. 8. J. VAN DEN BERG 1958 Journal of Speech and Hearing Research I, 227-244. Myoelasticaerodynamic theory of voice production. 9. J. VAN DEN BERG, J. T. ZANTEMA and P. DOORNENBAAL JR 1957 Journal of the Acoustical Society of America 29, 626-631. On the air resistance and the Bernouli effect ofthe human larynx. 10. K. ISIIIZAKA and M. t~,'iATSUI)AIRA 1972 Speech Communications Research Laboratory-Monograph No. 8. Fluid mechanical considerations of vocal cord vibration. I I. W. A. CONRAD 1980 Medical Research Engineering 13 7-10. A new model of the vocal cords based on a collapsible tube analogy.
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J.A.
MILLER,
J. C. P E R E I R A
AND
D. XV. T H O M A S
12. W. A. CONRAD 1983 in lliomecltanics, Acoustics and Phonator)' Control. (1. R. Titze and R. C. Schcrer, (ed.). The Denver Center for the Performing Arts, 328-348. Collapsible tube model of the larynx. 13. J. GAUFFIN, N. BINH, T. V. ANANTIIAPADMANABHAand G. ]:ANT 1983 in 2rid Vocal Fold Physiology Conference. (D. M. Bless and J. H. Abbs Editors), Vocal Fold Physiology--College Hill Press, 194-201. Glottal Geometry and volume velocity waveform. 14. R. C. SCllERER and I. R. TI-rZE 1983 in 2nd Vocal Fold Physiology Conference (D. M. Bless and J. H. Abbs editors), Vocal Fold Physiology--College Hill Press, 179-193. Pressure-flow relationships in a model of the laryngeal airway with a diverging glottis. 15. R.C. SCHERER, 1. R. TITZE and J. F. CURTIS 1983 Journal of the Acoustical Society of America 73, 668-676. Pressure-flow relationships in two models of the larynx having rectangular glottal shapes. 16. K. ISIIIZAKA 1983 in Vocal Fold Physiology: Biomechanics, Acoustics and Phonatory Control. (I. R. Titze and R. C. Scherer, editors), The Denver Center for the Performing Arts, 414-424. Air resistance and intraglottal pressure in a model of the larynx. 17. P. DEJONCKERE and J. LEBACQ 1981 Archi~'es internationales de Ph)'siologie et de Biochimie 89, 127-136. Mechanism of initiation of oscillatory motion in human glottis. 18. J. A. MILLER 1982 M.Sc. Thesis, UniversiO' of Southampton. System identification in speech. 19. K. ISHIZAKA and T. KANEKO 1968 Journal of the Acoustical Society of Japan 24, 312-313. On equivalent mechanical constants of the vocal cords. 20. J. L. FLANA(3AN 1959 Journal of Speech and Hearing Research, 2, 168-172. Estimates of intraglottal pressure during phonation. 21. M.A. PLINT AND L. IIOSWlRTH 1978 Fhdd Alechanics: A Laboratory Course. London: Charles Griffin. 22. M. HIRANO 1977 in Dynamic Aspects of Speech Production. M. Sawashima and F. S. Cooper, editors), Tokyo: University of Tokyo Press, 13-30. Structure and vibratory behavior of the vocal folds. 23. R. D. GROSE" 1985 Transactions of the American Society of Mechanical Engineers, Journal of Fhdds Engineering 107, 36-43. Orifice contraction coefficient for inviscid incompressible flow. 24. B. S. MASSEY 1979 Mechanics of Fhdds. New York: Van Nostrand Reinhold. 25. J. SZEKELY 1979 Fluid Flow Phenonzena in Metals Processing. New York: Academic Press.
APPENDIX A: VENA CONTRACTA A.I. C O E F F I C I [ - . N T O F C O N T R A C T I O N The coefficient o f contraction as a Function o f the entry angle has been reported by Grose [23]. Based on the Navier-Stokes equation and using a control volume with an elliptical surface at the entry o f the orifice he has presented a Formulation for the contraction coefficient. The coefficient derived for an entry angle o f 90 ~ i's given by Cc = &2 _ [~b4 _ ~b2] i/2
(AI)
where Cc is the coefficient o f contraction and ~b is the potential surface area ratio, i.e., the ratio o f the elliptical surface area to the orifice o p e n i n g area, and is given by = [2/(1 + K)] '/2,
(A2)
where K is the orifice to upstream channel height ratio. By applying this Formulation to the t r a c h e a - v o c a l Folds junction the contraction coefficient can be determined for the limiting values of the glottis heights (XI) during phonation. As s h o w n in Table l, the b o u n d a r y values o f XI during phonation are zero and 0.2 cm. For an average value for the area o f trachea o f 5 cm 2. (R = i . 2 6 c m ) , the contraction coefficients are Found to be 0.5858 and 0.6065, respectively. The figures presented in the preceding paragraph are calculated for a 90 ~ entry angle. By choosing an a p p r o p r i a t e value for r which is related to the entry angle ( a ) , G r o s e has derived the coefficient o r c o n t r a c t i o n as a Function o f t h i s angle (curve l in Figure A l ) .
FLUID
FLOW TIIROUGH
THE
LARYNX
CIIANNEL
289
In order to find the region of validity for our particular case an extrapolation of the curve for a higher value of K was made ( K = 0-159), as the orifice opening for the glottis ranges between 0.0 and 0.4 cm. The linear approximation assumed for this is shown in Figure A1.
o8~. 1.O
'
i
'
i
'
i
'
i
I
,
!
i0"4 0
40
80
120
160
e
Figure At. lnviscid incompressible flow contraction coefficient as a function of the entry angle. (l) Elliptical theory curve (K = 0) estimated by Grose; (2) extrapolated curve (K = 0-159); (3) linear approximation curve.
A.2.
POSITION
OF THE
VENA
CONTRACTA
Massey [24] states that for a sharp edged orifice in a vessel wail, a vena contracta forms downstream at about a distance equal to the orifice radius. The position of the vena contracta is nearly independent of the rate of flow and is a function of the orifice-to-pipe diameter ratio K, moving upstream as the value of K becomes larger. U p o n assuming a linear approximation for the position o f the vena contracta from Szekely's plot of correlation [.25] for the discharge coefficient, then Zc is given by Zc = (r x 1.26) x (1 - r / R ) ,
(A3)
where Zc is the position of the vena contracta relative to the area discontinuity, r is the radius o f the pipe downstream from the contraction, and R is the radius of the pipe upstream from the contraction. It should be noted here that the relationship for Zc has been derived for axisymmetric flow and at the trachea-glottis junction the flow changes from axisymmetrical flow in the trachea to two-dimensional flow in the glottis. APPENDIX B: LIST OF SYMBOLS
tglot Lt~,
t.lglo U
A, A~
x,
length of the glottis (0.4 cm) length of laryngeal ventricle (0.3 cm) length of the false glottis (0.8 cm) width of the glottis ( 1.4 cm) volume" velocity (cm3/s) final area of flow enlargement (cm 2) initial area of flow enlargement (cm 2) height of the laryngeal channel at station i (cm)
290
P~ Kv
P, P~, Pc P~ Pat,,I LL,ti-j) Ldti-~) g p tx
J. A. M I L L E R , J. C. P E R E I R A A N D
D. W. T H O M A S
minimum laryngeal flow height (cm) normalized subglottal pressure (g/cm 2) constant for viscous loss at glottis normalized piezometric pressure at station i (g/cm 2) normalized kinetic pressure at station i (g/cm 2) normalized piezometric pressure at vena contracta (g/cm 2) normalized kinetic pressure at vena contracta (g/cm 2) atmospheric pressure viscous loss (between stations i and j) (g/cm 2) loss due to an enlargement (between stations i and j) (g/cm 2) acceleration due to gravity (981 cm/s 2) air density (1.489 x 10-3 gm/cm 3) air viscosity (1.878 x 10-4 g/cms)