26. The simulation of the electroglottographic signal with the two-mass model

The EGG device produces a signal which contains information about the contact area of the vocal folds (see section 8). The two-mass model of vocal fold vibration provides a description of the glottal area instead. The transition between glottal area and contact area was introduced by Childers et al. (1986) and involves a simplified three-dimensional model of the folds.

As Childers et al. (1986:1311) put it: "A limitation of this two-mass model is that the projected glottal area is always rectangular. The folds are either in contact or separated; there is no gradation of closure or opening. The progression of the contact area is stair-stepped; first of it is a minimum (open glottis), next there is complete closure, partial contact (lower mass in contact), and finally there is full contact (lower and upper masses both in contact). This approximation does not adequately model the vibratory motion of true vocal folds during phonation."

The better approximation of vocal fold motion is the one in section 25. In the two-mass model the folds are observed from above the glottis. An additional cross-sectional description of the movement is needed to obtain a more realistic description of vocal folds contact. The vocal fold does not move like a piston. A better approximation of its motion can be achieved by representing it as a flexible ribbon fixed at both ends, with the upper and lower margins moving differently and out of phase. The movements of the flexible folds are depicted in Fig. 45.

In three-dimensional space, the opening has the shape of a tetrahedron, whereas the contact area is trapezoidal in shape (Fig. 46). The sequence depicted in Fig.45 corresponds directly to the rolling and zippering action modelled in the proposed description of the EGG (contact area) waveform. High-speed film observations of vocal fold vibration have shown that there is a time delay in the closing and the opening of the folds along their lengths. That is an additional delay which is different to that of the lower and upper margin movement which also out of phase (as displayed in Fig.45). This effect is sometimes called a mucosal wave.

Figure 45. Three-dimensional stylized view of vocal fold movement (according to Childers et al.,1986). During the first phase of motion the folds are fully closed and the contact area is maximal. Then the folds begin to separate, usually starting posterior lower margins and ending at the anterior upper margins. When the lower margins are fully open, the upper margins continue to open, which leads to the formation of a contact area in the shape of tetrahedron. When the folds are fully open, no contact occurs.

During closure the contact between the folds first occurs between small portions along their length and continues, zipper-like. As the folds come into contact at the instant of closure, a zipper is pulled along the full length of the fold margins from anterior to posterior. During opening the reverse process occurs and the folds separate. Physiological observations of the folds also suggest that the angles of closing and opening can be different (Childers et al., 1986:1312). As far as the cross-section of the glottis is concerned, the time delay between the motion of the lower and upper margins causes a non-simultaneous contact across the whole width of the vocal folds. The contact area has a rather trapeziodal form: the lower margin precedes the upper margin in both opening and closing movements (which is typical for modal voice), thus the contact is more advanced along the lower margin than along thue upper margin. In the model it is assumed that the contact occurs along a straight line, connecting the most advanced contact points at the lower and upper margins, as is schematically depicted in Fig. 46.

Figure 46. The model of contact area calculation from the superior and sagittal views (after Childers et al., 1986:1312).

The modelling of the EGG waveform can be included in the main loop of the two-mass model calculation, using the differential approximation of the equations (36)-(39). The modelling of the contact area can then be described as follows (after Childers et al., 1986):

• the displacements x1 and x2 of the lower and upper mass in relation to the midsagittal line of the glottis are computed and used to derive the glottis areas at the lower and upper margins Ag1, Ag2
• in the three-dimensional space the opening has the form of a tetrahedron. Thus the contact area is trapezoidal in shape. The representation in Fig.46 corresponds directly to the rolling and zippering action modelled in the proposed description of the EGG (contact area) waveform. As stated before, there is a time delay caused by both the the motion of the upper and lower margins and the zippering action along them.
• in order to include the initial positions of the folds, the positions at the posterior ends of the vocal folds (see Fig.46) are computed as:

  

• the glottal areas of the upper and lower margins are computed using the triangular area configuration (not the projection of Ag1, Ag2) and depend on the closing and opening angles theta_c, theta_o, as follows from Fig. 46,
• the contact area is:
  1. nonexistent, if the folds are not in contact
  2. triangular in form, if the lower margins of the folds are in contact but the upper margins are not
  3. also triangular in form, if the upper margins are closed but the lower margins are not
  4. trapezoidal in form, if both upper and lower folds are in contact but not in phase (the contact edge along one of the margins is shorter than one along the other margin)

These leads to the following model of the electroglottographic waveform:

where EGG(i) denotes the i-th sample of the EGG waveform, Ac(i) is the contact area, C is a constant proportional to the shunt impedance (impedance of the larynx structures and of the tissue surrounding the larynx) and k is a scaling constant.

The influence of the parameters theta_c, theta_o and of the time delay in the motion of both masses was described in detail by Childers et al. (1986). They came to the important conclusion that even for small delays the maximum of the EGG derivative does not correctly mark the instant of closure, because the glottal area becomes zero prior to the time at which the maximum contact occurs. The folds come into the contact along the lower margins and block off the glottal airflow, although the upper margins are not completely closed. The effect of varying opening and closing angles is generally reflected in the shape of the waveform, but when the values are within the reasonable ranges the influence is rather limited. The proposed ranges are 0.01º..0.5º for the closing angle and 0.5º..2º for the opening angle.

In this study the model will be used to simulate an EGG waveform for different laryngeal settings. It is the main goal of the simulation to relate the experimental evidence about stress and intonation to the modelled physiological constraints. The simulations conducted in this study are limited to the modelling of modal, non-pathological voices. Nevertheless, a simulation of pathological (disordered) voices is possible within this model. It was indeed performed by Childers et al. (1986) for various laryngeal configurations. However, a realistic simulation requires an estimation of many physiological parameters, tissue properties, etc. which are hard to obtain. Additionally, such a model will be valid for a very small group of cases only, as pathologies are usually individual cases. Thus, it was decided to use the model to verify hypotheses regarding stress and intonation patterns.