Impact of Vertical Stiffness Gradient on the Maximum Divergence Angle

Abstract

Introduction: During vocal fold vibration, the medial surface of both folds forms a convergent shape during opening and a divergent shape during closing. A greater maximum divergence angle is associated with greater closing forces which will increase the closing speed of the glottis. An increased closing speed results in a greater acoustic intensity and greater vocal efficiency. Indentation testing showed that as the strain increases, the inferior aspect of the folds becomes stiffer than the superior aspect, resulting in the vertical stiffness gradient (VSG). We hypothesize that a reduction of the vertical stiffness gradient will reduce the maximum divergence angle.

Methods: Four excised canine larynges were tested. Stress-strain curves of the superior and inferior aspects of the fold in the mid membranous plane of the baseline larynges were taken using the indentation method. Calcium hydroxylapatite (CaHA) crystals were then injected into the superior aspect of the fold. The stress-strain tests were repeated. Particle imaging velocimetry (PIV) of the intraglottal velocity fields was performed in three larynges at different subglottal pressures in the mid coronal plane for the baseline and CaHA-injected larynges.

Results: CaHA injection reduced the inferior-superior stiffness gradient in all larynges. The maximal divergence angle was markedly reduced. In some cases, there was not a divergent angle.

Discussion: Marked reduction of the vertical stiffness gradient significantly reduces the maximum divergence angle. Clinical implications will be discussed.

Level of evidence: NA Laryngoscope, 131:E1934-E1940, 2021.

Keywords: Laryngology; aerodynamics; basic science; vocal fold.

Fig. 1.

Mid-coronal views of the excised larynx model. (a) Normal. (b) Injection of 0.1 to 0.2 mL Calcium Hydroxylapatite (CaHA) per vocal fold along the fold superior edge from the anterior commissure to the vocal process, directly under the epithelium layer to increase stiffness.

Fig. 2.

Experimental setup of the indentation strain testing on the excised larynx model. (a) Axial view of the strain test setup: all the structures above the vocal folds have been removed and the tested vocal fold is adducted. A 3D-printed wedge creates space between the tested fold (right) and the other one for the strain probe to perform indentations testing. (b) Mid-membranous coronal view that defines the indentation location for the superior and inferior edges, and shows the probe’s size relative to the vocal fold. x and y are the coordinate system used in the velocity fields plots.

Fig. 3.

Schematic of the particle image velocimetry (PIV) experimental setup, showing the components of the flow control loop, as well as the larynx placed on the aerodynamic nozzle.

Fig. 4.

Characterization of the average vertical stiffness gradient in four larynges pre-CaHA injection (black) and post-CaHA injection (red). (a) stress–strain curves during unloading by indentation at the inferior edge (solid lines) and the superior edge (dashed lines). (b) Vertical stiffness gradient at maximum strain during unloading, expressed as the ratio of Young’s modulus. Stress–strain curves do not change much qualitatively at the inferior edge post-injection as expected, whereas it takes more stress to achieve the maximum strain after injection of CaHA (red dashed curve is “shifted” upward) at the superior edge. Stiffness gradients are much-reduced post-CaHA injection, as the ratio of Young modulus shows.

Fig. 5.

Contour of velocity magnitude for one larynx. Pink regions represent the glottal wall (vocal folds) and arrows indicate the superior edges on each side of the glottis. The divergence angle (α) is measured in each vignette between the folds’ straight portion of the walls, as indicated in (b) with the dashed arc. Velocity contour plots are shown for (a) baseline at Psg = 17 cmH₂O(b) baseline at Psg = 23 cmH₂O, (c) CaHA at Psg = 15 cmH₂O, (d) CaHA at Psg = 23 cmH₂O.

Fig. 6.

Maximum divergence angle reduction post-stiffening of the superior edge of the glottis. Shown as the average maximum angle for all the baseline cases (black), and the CaHA injected larynges (red).

Fig. 7.

Mapping methodology for circulation strength calculation. The intraglottal flow field is restricted to the vertical region located in the divergent area, between the glottal jet and the vocal fold (blue contours in Fig. 5a). Contours of normal vorticity ω_z are plotted inside the integration area (dashed boundaries), which is used for the calculation of the vortex circulation.

Fig. 8.

Normalized Circulation in the flow-separated area. Shown as the average of the maximum circulation for the baseline cases (black), and the CaHA injected larynges (red).