Observation of perturbations in a lumped-element model of the vocal folds with application to some pathological cases

In this paper a mass-spring model is developed that is a hybrid of the two-mass and the longitudinal string models, proposed by Ishizaka and Flanagan [Bell Sys. Tech. J. 51, 1233-1268 (1972)] and Titze [Phonetica 28, 129-170 (1973)], respectively. The model is used to simulate the vibratory motion of both the normal and asymmetric vocal folds. Mouth-output pressure, lateral tissue displacement, phase plots, and energy diagrams are presented to demonstrate the interaction between vocal fold tissue and the aerodynamic flow between the folds. The results of the study suggest that this interaction is necessary for sustained large amplitude oscillation because the flow supplies the energy lost by the tissue damping. Tissue mass and stiffness were varied locally or uniformly. Decreased stress in the longitudinal string tension produced subharmonic and chaotic vibrations in the displacement, velocity and acceleration phase diagrams. Similar vibratory characteristics also appeared in pathological speech data analyzed using time domain jitter and shimmer measures and a harmonics-to-noise ratio metric. The subharmonics create an effect that has been perceptually described as diplophonia.     

A theoretical model of the pressure field arising from asymmetric intraglottal flows applied to a two-mass model of the vocal folds

A theoretical flow solution is presented for predicting the pressure distribution along the vocal fold walls arising from asymmetric flow that forms during the closing phases of speech. The resultant wall jet was analyzed using boundary layer methods in a non-inertial reference frame attached to the moving wall. A solution for the near-wall velocity profiles on the flow wall was developed based on a Falkner-Skan similarity solution and it was demonstrated that the pressure distribution along the flow wall is imposed by the velocity in the inviscid core of the wall jet. The method was validated with experimental velocity data from 7.5 times life-size vocal fold models, acquired for varying flow rates and glottal divergence angles. The solution for the asymmetric pressures was incorporated into a widely used two-mass model of vocal fold oscillation with a coupled acoustical model of sound propagation. Asymmetric pressure loading was found to facilitate glottal closure, which yielded only slightly higher values of maximum flow declination rate and radiated sound, and a small decrease in the slope of the spectral tilt. While the impact on symmetrically tensioned vocal folds was small, results indicate the effect becomes more significant for asymmetrically tensioned vocal folds.     

An anatomically based, time-domain acoustic model of the subglottal system for speech production

A time-domain model of sound wave propagation in the branching airways of the subglottal system is presented. The model is formulated as an extension to an acoustic transmission-line modeling scheme originally developed for simulating the supraglottal system in the time-domain during speech production [Maeda (1982). Speech Commun. 1, 199-229; Mokhtari et al. (2008). Speech Commun. 50, 179-190]. The approach allows for predictions of time-varying acoustic pressure and volume velocity at any point along the various generations of subglottal airways from trachea to alveoli. In addition, the model can be configured so that its overall structure simulates different geometric forms, including airways that branch in a symmetric or asymmetric pattern. Three subglottal configurations, two symmetric and one asymmetric, were represented based on reported anatomical dimensions of the subglottal airways. Estimates of the acoustic input impedances of these subglottal configurations revealed resonant characteristics similar to those found in the previous studies. Simulations of voiced sound propagation into the subglottal airways, achieved by coupling the subglottal model to a two-mass vocal fold model and a supraglottal tract configured for different vowels, yielded predictions of time-domain sound pressure waveforms below the vocal folds that compare favorably to previous measurements in human subjects.     

Parameter estimation of an asymmetric vocal-fold system from glottal area time series using chaos synchronization

In this paper, we apply an iterative parameter adaption scheme based on chaos synchronization to estimate system parameters of the asymmetric vocal folds from glottal area time series. The original asymmetric vocal-fold system associated with recurrent laryngeal paralysis shows chaotic vibrations with positive Lyapunov exponents. Aperiodic glottal area time series from the original system will be applied as the feedback variable coupling the simulative and the original vocal-fold systems. The parameter adaption technique based on chaos synchronization is employed to manipulate the simulative system parameters. The chaotic vibrations, system parameters, and the bifurcation diagram of the original vocal-fold system can be exactly reproduced in the simulative system, and the two chaotic systems can be synchronized. Furthermore, the effects of noise, sampling rate, and equation difference due to nonlinear spring terms on vocal-fold parameter estimations are investigated. Despite large noise perturbations, large equation differences, and low sampling rate, the parameter adaption scheme can effectively estimate the original vocal-fold system parameters. This study provides a theoretical base to apply chaos synchronization to estimate the vocal-fold system parameters from the glottal area data and show its potential application in laryngeal physiology.     

Two-mass model of the vocal folds: negative differential resistance oscillation

The vocal folds and glottis are analyzed as a single system rather than as two separate but interacting systems, i.e., an aerodynamic one (the glottis) and a mechanical one (the vocal folds). Simplified steady flow calculations based on the two-mass model, and similar to those of Ishizaka and Matsudaira [SCRL Monograph No. 8, Santa Barbara, CA (1972)], are made except that flexible walls are assumed for both dc and ac flows. A negative differential resistance is found for steady flow when the coupling spring is weak compared to that of the lower mass. Dynamic transverse motion of the masses is represented by two transverse series resonant circuits in parallel within the glottis. The vocal tract is represented by a lumped resistance and inertance in series. Sustained, self-excited, small-amplitude oscillations can be obtained when the magnitude of the negative differential resistance is equal to the real part of the impedance of the rest of the circuit. The oscillation frequency depends only on the elasticity and mass of the vocal folds. The present analysis differs from Ishizaka and Matsudaira's analysis because their oscillation frequency decreases as dc volume velocity increases.     

Comparison of biomechanical modeling of register transitions and voice instabilities with excised larynx experiments

Voice instabilities were studied using excised human larynx experiments and biomechanical modeling. With a controlled elongation of the vocal folds, the experiments showed registers with chest-like and falsetto-like vibrations. Observed instabilities included abrupt jumps between the two registers exhibiting hysteresis, aphonic episodes, subharmonics, and chaos near the register transitions. In order to model these phenomena, a three-mass model was constructed by adding a third mass on top of the simplified two-mass model. Simulation studies showed that the three-mass model can vibrate in both chest-like and falsetto-like patterns. Variation of tension parameters which mimic activities of laryngeal muscles could induce transitions between both registers. For reduced prephonatory areas and damping constants, extended coexistence of chest and falsetto registers was found, in agreement with experimental data. Subharmonics and deterministic chaos were observed close to transitions between the registers. It is concluded that the abrupt changes between chest and falsetto registers can be understood as shifts in dominance of eigenmodes of the vocal folds.     

Phonation thresholds as a function of laryngeal size in a two-mass model of the vocal folds

This letter analyzes the oscillation onset-offset conditions of the vocal folds as a function of laryngeal size. A version of the two-mass model of the vocal folds is used, coupled to a two-tube approximation of the vocal tract in configuration for the vowel /a/. The standard male configurations of the laryngeal and vocal tract models are used as reference, and their dimensions are scaled using a single factor. Simulations of the vocal fold oscillation and oral output are produced for varying values of the scaling factor. The results show that the oscillation threshold conditions become more restricted for smaller laryngeal sizes, such as those appropriate for females and children.     

Glottal flow through a two-mass model: comparison of Navier-Stokes solutions with simplified models

A new numerical model of the vocal folds is presented based on the well-known two-mass models of the vocal folds. The two-mass model is coupled to a model of glottal airflow based on the incompressible Navier-Stokes equations. Glottal waves are produced using different initial glottal gaps and different subglottal pressures. Fundamental frequency, glottal peak flow, and closed phase of the glottal waves have been compared with values known from the literature. The phonation threshold pressure was determined for different initial glottal gaps. The phonation threshold pressure obtained using the flow model with Navier-Stokes equations corresponds better to values determined in normal phonation than the phonation threshold pressure obtained using the flow model based on the Bernoulli equation. Using the Navier-Stokes equations, an increase of the subglottal pressure causes the fundamental frequency and the glottal peak flow to increase, whereas the fundamental frequency in the Bernoulli-based model does not change with increasing pressure.     

Vibration in a self-oscillating vocal fold model with left-right asymmetry in body-layer stiffness

This study compares the phonatory behavior of an asymmetric vocal fold model to that of each individual vocal fold model in a hemi-configuration. Although phonation frequencies of the two folds in hemi-configurations had a ratio close to 1:3, a subharmonic synchronization between the two folds was not observed in the asymmetric model. Instead, the vibratory behavior was dominated by the dynamics of one fold only, and the other fold was enslaved to vibrate at the same frequency. Increasing subglottal pressure induced a shift in relative dominance between the two folds, leading to abrupt changes in both vibratory pattern and frequency.     

Model-based classification of nonstationary vocal fold vibrations

Classification of vocal fold vibrations is an essential task of the objective assessment of voice disorders. For historical reasons, the conventional clinical examination of vocal fold vibrations is done during stationary, sustained phonation. However, the conclusions drawn from a stationary phonation are restricted to the observed steady-state vocal fold vibrations and cannot be generalized to voice mechanisms during running speech. This study addresses the approach of classifying real-time recordings of vocal fold oscillations during a nonstationary phonation paradigm in the form of a pitch raise. The classification is based on asymmetry measures derived from a time-dependent biomechanical two-mass model of the vocal folds which is adapted to observed vocal fold motion curves with an optimization procedure. After verification of the algorithm performance the method was applied to clinical problems. Recordings of ten subjects with normal voice and ten dysphonic subjects have been evaluated during stationary as well as nonstationary phonation. In the case of nonstationary phonation the model-based classification into "normal" and "dysphonic" succeeds in all cases, while it fails in the case of sustained phonation. The nonstationary vocal fold vibrations contain additional information about vocal fold irregularities, which are needed for an objective interpretation and classification of voice disorders.     

Irregular vocal-fold vibration--high-speed observation and modeling

Direct observations of nonstationary asymmetric vocal-fold oscillations are reported. Complex time series of the left and the right vocal-fold vibrations are extracted from digital high-speed image sequences separately. The dynamics of the corresponding high-speed glottograms reveals transitions between low-dimensional attractors such as subharmonic and quasiperiodic oscillations. The spectral components of either oscillation are given by positive linear combinations of two fundamental frequencies. Their ratio is determined from the high-speed sequences and is used as a parameter of laryngeal asymmetry in model calculations. The parameters of a simplified asymmetric two-mass model of the larynx are preset by using experimental data. Its bifurcation structure is explored in order to fit simulations to the observed time series. Appropriate parameter settings allow the reproduction of time series and differentiated amplitude contours with quantitative agreement. In particular, several phase-locked episodes ranging from 4:5 to 2:3 rhythms are generated realistically with the model.     

Bifurcations and chaos in register transitions of excised larynx experiments

Experimental data from an excised larynx are analyzed in the light of nonlinear dynamics. The excised larynx provides an experimental framework that enables artificial control and direct observation of the vocal fold vibrations. Of particular interest in this experiment is the coexistence of two distinct vibration patterns, which closely resemble chest and falsetto registers of the human voice. Abrupt transitions between the two registers are typically accompanied by irregular vibrations. Two approaches are presented for the modeling of the excised larynx experiment; one is the nonlinear predictive modeling of the experimental time series and the other is the biomechanical modeling (three-mass model) that takes into account basic mechanisms of the vocal fold vibrations. The two approaches show that the chest and falsetto vibrations correspond to two coexisting limit cycles, which jump to each other with a change in the bifurcation parameter. Irregular vibrations observed at the register jumps are due to chaos that exists near the two limit cycles. This provides an alternative mechanism to generate chaotic vibrations in excised larynx experiment, which is different from the conventionally known mechanisms such as strong asymmetry between the left and right vocal folds or excessively high subglottal pressure.     

Vocal fold and ventricular fold vibration in period-doubling phonation: physiological description and aerodynamic modeling

Occurrences of period-doubling are found in human phonation, in particular for pathological and some singing phonations such as Sardinian A Tenore Bassu vocal performance. The combined vibration of the vocal folds and the ventricular folds has been observed during the production of such low pitch bass-type sound. The present study aims to characterize the physiological correlates of this acoustical production and to provide a better understanding of the physical interaction between ventricular fold vibration and vocal fold self-sustained oscillation. The vibratory properties of the vocal folds and the ventricular folds during phonation produced by a professional singer are analyzed by means of acoustical and electroglottographic signals and by synchronized glottal images obtained by high-speed cinematography. The periodic variation in glottal cycle duration and the effect of ventricular fold closing on glottal closing time are demonstrated. Using the detected glottal and ventricular areas, the aerodynamic behavior of the laryngeal system is simulated using a simplified physical modeling previously validated in vitro using a larynx replica. An estimate of the ventricular aperture extracted from the in vivo data allows a theoretical prediction of the glottal aperture. The in vivo measurements of the glottal aperture are then compared to the simulated estimations.     

Pressure-flow relationships during phonation in the canine larynx

Measurements of air pressure and flow were made using an in vivo canine model of the larynx. Subglottic pressures at varying flow rates were taken during phonation induced by laryngeal nerve stimulation. Results showed that during constant vocal fold stiffness, subglottic pressure rose slightly with increased air flow. The larynx in the in vivo canine model exhibited a flow-dependent decrease in laryngeal airway resistance. Increasing flow rate was associated with an increase in frequency of phonation and open quotient, as measured glottographically. Results from this experiment were compared with a theoretical two-mass model of the larynx and other theoretical models of phonation. The influence of aerodynamic forces on glottal vibration is explained by increased lateral excursion of the vocal folds during the open interval and shortening of the closed interval during the glottal cycle.     

Influence of supraglottal structures on the glottal jet exiting a two-layer synthetic, self-oscillating vocal fold model

A synthetic two-layer, self-oscillating, life-size vocal fold model was used to study the influence of the vocal tract and false folds on the glottal jet. The model vibrated at frequencies, pressures, flow rates, and amplitudes consistent with human phonation, although some differences in behavior between the model and the human vocal folds are noted. High-speed images of model motion and flow visualization were acquired. Phase-locked ensemble-averaged glottal jet velocity measurements using particle image velocimetry (PIV) were acquired with and without an idealized vocal tract, with and without false folds. PIV data were obtained with varying degrees of lateral asymmetric model positioning. Glottal jet velocity magnitudes were consistent with those measured using excised larynges. A starting vortex was observed in all test cases. The false folds interfered with the starting vortex, and in some cases vortex shedding from the false folds was observed. In asymmetric cases without false folds, the glottal jet tended to skew toward the nearest wall; with the false folds, the opposite trend was observed. rms velocity calculations showed the jet shear layer and laminar core. The rms velocities were higher in the vocal tract cases compared to the open jet and false fold cases.     

Comments on "Two-mass models of the vocal cords for natural sounding voice synthesis"

Koizumi et al. [J. Acoust. Soc. Am. 82, 1179-1192 (1987)] have proposed a way to incorporate mucosal waves into previous two-mass mechanical models of the vocal folds. This was accomplished by allowing the mass of the masses to vary with time. The equations of motion Koizumi et al. used to mathematically describe this model neglected terms from the time rate of change of momentum of Newton's second law. In this letter, approximations of the magnitude of this term indicate that it must not be neglected.     

Nonlinear dynamics of the voice: Signal analysis and biomechanical modeling

Irregularities in voiced speech are often observed as a consequence of vocal fold lesions, paralyses, and other pathological conditions. Many of these instabilities are related to the intrinsic nonlinearities in the vibrations of the vocal folds. In this paper, bifurcations in voice signals are analyzed using narrow-band spectrograms. We study sustained phonation of patients with laryngeal paralysis and data from an excised larynx experiment. These spectrograms are compared with computer simulations of an asymmetric 2-mass model of the vocal folds. (c) 1995 American Institute of Physics.