An Optimal Contact-Mechanically Consistent and Flow-Separation Adapted Modeling of Vocal Fold Dynamics

Abstract: Single mass-spring-damper models of vocal folds have been effective in simulating vocal fold vibrations without added complexity. However, single-degree-of-freedom models cannot sustain oscillation in the presence of structural damping unless source-tract interaction is considered. Moreover, existing lumped models struggle to accurately simulate vocal fold closure during phonation. This study aims to develop a reliable and simplified single-degree-of-freedom model of phonation that can simulate sustained oscillation in a damped system without incorporating a vocal tract model. Additionally, the proposed model maintains vocal fold closure in a manner consistent with the physics of phonation, addressing a longstanding challenge in existing lumped models. High-speed videoendoscopy (HSV) data from four normophonic subjects producing sustained vowel /i/ were used to extract glottal area waveforms (GAWs) via deep learning-based image segmentation for particle swarm optimization of the model parameters. An additional resistance force was incorporated to compensate for flow separation and generate the force imbalance required for sustained oscillation. An external structural force was also added during closure to sustain the closed phase. The 4th-order Runge-Kutta method was used to solve the governing equations with enhanced numerical stability and accuracy. The model parameters were optimized for individual subjects, resulting in normalized errors below 3% between experimental and simulated GAWs. The proposed model accurately reproduced subject-specific vocal fold vibrations and vocal fold closure in agreement with experimental data. Overall, the proposed model provides a computationally efficient framework for simulating sustained phonation without requiring complex source-tract coupling while capturing the key biomechanical and aerodynamic mechanisms of phonation.

• The paper presents a single mass–spring–damper model enhanced with dynamic flow separation and contact mechanics for sustained oscillation.
• It employs a displacement-dependent nonlinear damping and reaction force to achieve physiologically consistent glottal closure.
• Parameter optimization using particle swarm on high-speed endoscopy data yields normalized errors as low as 1.9%, validating the model's accuracy.

Optimal Flow-Separation-Adapted Contact-Mechanical Lumped Modeling of Vocal Fold Dynamics

Introduction and Motivation

The paper "An Optimal Contact-Mechanically Consistent and Flow-Separation Adapted Modeling of Vocal Fold Dynamics" (2606.29071) presents a computational and mechanistic framework for modeling the vibratory dynamics of vocal folds using a single-degree-of-freedom, lumped element approach. Traditionally, accurate vocal fold models have required higher-order, computationally expensive implementations (multi-mass or finite element methods), especially when capturing flow separation, contact mechanics at closure, and self-sustained oscillation under damping. The single-mass models, despite their utility, have historically failed to capture critical phenomena such as sustained oscillation in the presence of tissue damping and physiologically plausible closure. This work aims to close that gap by technical advancements in the treatment of flow separation and mechanical contact within a minimal model.

Model Formulation and Innovations

The authors’ framework is based on a single mass–spring–damper system, augmented with two essential innovations. First, the aerodynamic force formulation incorporates a dynamic flow separation model, adapted to represent force asymmetries during the closing phase, counterbalancing the energy loss due to damping and enabling sustained self-oscillation without explicit source-tract coupling. Second, during glottal closure, contact mechanics are modeled through a reaction force derived from conservation of linear momentum, combined with a cancellation of elastic forces to sustain closure over physiologically correct durations. The damping coefficient itself is modeled as a nonlinear function of displacement using a clipped hyperbolic tangent relationship, reflecting the progressive tissue compression and nonlinearity observed in vivo.

Aerodynamic forces are computed under a one-dimensional Bernoulli principle with viscous correction, and the flow rate is calculated from experimental subglottal pressure data, further improving the physical realism of the airflow–tissue interaction. The governing ODEs are integrated with a fourth-order Runge–Kutta scheme, favoring both numerical stability and computational efficiency.

Parameter Optimization and Experimental Validation

High-speed videoendoscopy (HSV) data were collected from four normophonic human subjects (two male, two female), providing gold-standard ground truth for glottal area waveforms (GAW). The authors leverage a deep neural network (U-Net) for image segmentation of the glottal area in each HSV frame, enabling precise extraction of experimental GAWs. The lumped-model parameters for each individual (spring constant, nonlinear damping factor, scaling, closure duration, etc.) are then subject-specifically optimized via a global particle swarm optimization (PSO) routine, targeting the minimization of normalized error between simulated and experimentally observed GAWs.

The model achieves normalized errors below 3% across all subjects, with error rates as low as 1.9% in the best-case, demonstrating that the minimal model, when appropriately regularized and physically adapted, is capable of replicating physiologically relevant vibratory motion and glottal closure.

Numerical Analysis: Impact of Model Components

Multiple numerical experiments are reported to dissect the individual roles of flow separation, elastic stiffness, and nonlinear damping in vocal fold dynamics:

• Flow separation force: Its removal leads to rapid decay in oscillation amplitude, highlighting its importance for energy balance in the presence of damping. The inclusion of a displacement-dependent separation point is shown to be critical for closure and maintained amplitude.
• Elastic stiffness ($K$): Increasing $K$ increases oscillation frequency; values above the subject-specific optimum inhibit full closure and rapidly dampen oscillation.
• Damping ($C_f$): Over- or under-estimating results in loss of closure or growth in oscillation amplitude, respectively, confirming the necessity of subject-adaptive, nonlinear damping for physiologically plausible cycles.

Scaling factors are found to linearly modulate amplitude but do not affect the phase or frequency, as expected.

Theoretical and Practical Implications

This work demonstrates that a single-degree-of-freedom system, appropriately treated for flow separation and contact mechanics, can accurately and efficiently reproduce subject-specific vocal fold dynamics without reliance on source-tract coupling or anatomically detailed multi-mass models. The implications are significant for real-time simulation and individualized clinical modeling, since parameter optimization can be performed directly from kinematic data, and the minimal state representation substantially reduces computational costs and complexity.

The method also provides a principled framework for incorporating more advanced rheological models (e.g., fractional viscoelasticity, power-law behaviors) as future extensions, aiming to further reconcile model simplicity with biological realism. Such directions are motivated in the discussion, referencing recent advances in nonlocal, fractional, and memory-dependent material modeling ([i28]-[i32]), and suggesting the modular extension of the current paradigm to model cycle-to-cycle irregularities, asymmetric pathologies, and AI-driven lumped-element models that can bridge to higher-level systems modeling (Ikram et al., 15 Jun 2026).

Conclusion

The presented model achieves a technically rigorous synthesis of physical consistency and computational parsimony in lumped-element modeling of vocal fold dynamics. By explicitly capturing flow separation-induced force asymmetry and closure-phase contact mechanics within a single mass–spring–damper architecture, it delivers robust simulation of sustained phonation and closure in normophonic subjects, with subject-specific fidelity validated against high-speed endoscopy data. The results clarify the core mechanisms required for sustained self-oscillation in single-mass vocal fold models and provide a foundation for both theoretical exploration and practical deployment in biomechanical voice analysis, clinical simulation, and AI-enhanced modeling of vocal biomechanics.