Dynamic vibration mode decomposition of auto-oscillating vocal fold replicas without and with vertical tilting
Introduction
Unilateral vocal fold paralysis (UVFP) is a common vocal fold (VF) pathology characterised by an air escape due to left–right VF asymmetries of the VF’s shape, tension or/and positioning [1], [2], [3], [4], [5]. The glottic insufficiency associated with UVFP is reported to lead to dysphonia as air leakage is often associated with breathy voice or vocal fatigue [1], [6].
Each aspect of the fluid–structure (FS) interaction underlying normal or abnormal vocal fold vibration can be studied systematically in controlled and known conditions from experiments using deformable mechanical VF replicas [7], [8], [9], [10], [11]. In the current work, three moulded deformable multi-layer silicone VF replicas (labelled M5, MRI and EPI) are considered. The replicas are depicted in Fig. 1(a) [12]. They differ with respect to their shape and layer composition so that they mimic the multi-layer structure and geometry of the human VF’s with different degrees of complexity. Each VF replica consists of a left and right VF placed face-to-face in the transverse plane. The moulded VF replicas are briefly detailed in Section 2. In previous studies [12], [13], these three replicas were used to study the effect of vertical tilting of a single VF on their auto-oscillation. Concretely, the right VF was kept in place whereas the posterior edge of the left VF was tilted in the medio-sagittal plane towards the superior direction as shown in Fig. 1(b) and in Fig. 1(c). The resulting vertical tilting is parameterised by angular asymmetry angle . Imposing angular asymmetry in the range from 0° up to 25° results in a vertical level difference up to a few mm [12], which is of the same order of magnitude as observed in patients suffering from UVFP [2], [3]. Furthermore, vertical tilting of a VF causes left–right VF positioning asymmetry whereas left–right VF tension and shape symmetry is maintained.
In [12], the upstream pressure driving VF auto-oscillation was analysed for different imposed angular asymmetry angles using the setup schematically illustrated in Fig. 1(c). It was found that both the oscillation onset characteristics as well as common voice quality measures are affected. For instance, a decrease of the oscillation frequency and an increase of the oscillation onset threshold pressure () with increasing was observed. The gradual loss of VF’s contact with , inducing increased glottal air leakage, was pointed out to catalyse these tendencies.
In [13], high-speed (HS) imaging of the VF’s for different was performed in order to directly observe the vibrating VF’s. Local image features, exploiting HS videokymographic (VK) line-scans [14], were quantified during steady state oscillation (Fig. 1(d)). Left–right vibration asymmetry parameters showed that the normal VF (right VF in Fig. 1(b)) entrains the movement of the tilted VF (left VF in Fig. 1(b)). This left–right vibration asymmetry caused the mucosal wave velocity in the tilted VF to be lower than the one in the normal VF.
Local VK line-scan analysis [13] showed the influence of angular asymmetry on VF vibration. In this work, it is sought to further investigate sustained steady state VF vibration without () and with () angular asymmetry by analysing high-speed (HS) image sequences of the vibrating VF’s. Instead of focusing on local spatial features or line scans [13], [15], [16], [17], [18] it is sought to assess simultaneously temporal as well as spatial information of the global VF’s auto-oscillation. A dynamic vibration mode decomposition (DMD) of the observed vibration snapshots is applied as other VF vibration mode decomposition methods such as the empirical eigenfunction analysis [19] do not inform on the temporal mode dynamics. It is aimed to evaluate how the effect of angular asymmetry on the VF’s auto-oscillation affects the spatial and temporal vibration mode features.
Section snippets
Deformable silicone VF replicas
The M5, MRI and EPI replicas illustrated in Fig. 1(a) are moulded so that their elasticity is constant as detailed in [12]. They consist of respectively two (M5), three (MRI) or four (EPI) silicone-based moulding layers and a backing layer attaching it to a rigid support, i.e. region outside of the top view frames shown in Fig. 1(a). The M5 replica is a two-layer (muscle and superficial layer) reference model following the so called M5 geometrical VF model [20]. The MRI replica has a more
High-speed imaging of VF auto-oscillation
The experimental setup and flow supply used to generate the FS interaction underlying the VF auto-oscillation is schematically depicted in Fig. 1(c). The setup is similar to the one described in [12], [13]. Airflow, characterised by volume flow rate (sampling frequency 0.5 kHz), is applied along the inferior–superior direction as depicted in Fig. 1(c) and in Fig. 2. The pressure difference driving the VF’s auto-oscillation corresponds to the measured upstream pressure (sampling frequency
Dynamic mode decomposition
The temporal and spatial decomposition of the global VF auto-oscillation is assessed by a dynamic mode decomposition (DMD) of the gathered image sequences [23], [24].
The DMD framework is model free so that the system dynamics is estimated from each measured image sequence only. Assuming a locally linear system dynamics the solution is given as where and denote the eigenvectors and eigenvalues of the system matrix and where the coefficients are the
Temporal vibration analysis: DMD eigenvalue spectra
The temporal behaviour of the vocal folds vibration for different VF replicas and imposed angular asymmetry angles is analysed considering the DMD eigenvalue spectra. Spectra of stable vibration modes for all three replicas and top view angle are plotted in Fig. 3.
All spectral branches originate with a non-oscillation mode (0 Hz, ) associated with the zero eigenvalue at the origin ( and ). This non-oscillation mode expresses the influence of the mean flow along the
Spatial vibration analysis: DMD modes
In this section, stable vibration modes along the main spectral branches, indicated by filled symbols in Fig. 3, Fig. 4, are arranged () according to increasing frequency.
Conclusion
Dynamic vibration mode decomposition in space and time of high-speed images of auto-oscillating silicone vocal folds replicas without and with vertical tilting is assessed. Temporal properties obtained from the eigenvalue spectra show that imposing vertical tilting affects mode decay in a way that for the assessed replicas is hypothesised to be associated with the presence or absence of a stiffer epithelium-like surface layer. The effect is observed most clearly for decaying higher order
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was partly supported by Full3DTalkingHead, France project (ANR-20-CE23-0008-03) and JSPS, Japan International Research Fellowship (SP18205). Authors thank Dr. K. Ishimura (Ritsumeikan Univ., Japan) for experimental support.
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