Three-dimensional biomechanical properties of human vocal folds: Parameter optimization of a numerical model to match in vitro dynamics
(Color online) (a) Schematic representation of the 3DM. Every vocal fold consists of five mass elements on each transverse plane. Each mass element is elastically connected to a rigid body by using an anchor spring. Also the mass elements are connected to their adjacent masses through springs in vertical and longitudinal directions.21 (b) Exemplary results of the optimized trajectories (dotted lines) of the mass elements m 3,3 located in the median cross section at the middle transverse plane on the right side of the model compared to the hemilarynx trajectories (solid lines) at the corresponding suture point. The longitudinal component is not shown here because the longitudinal displacement is relatively small ( ± 0.1 mm). The fundamental frequency is 120 Hz. The global normalized error (Γ) and correlation (k) are 0.09% and 88%, respectively. The local are (0.02, 99%) for lateral component and (0.11, 99%) for vertical component.
(a) and (b) Charts for general distributions of local effective mass and local lateral stiffness of the vocal fold. They were averaged over all optimization results on the 24 experimental data sets. The black points are the positions of different mass elements distributed at vocal fold surface. Different gray levels correspond to different values of local mass/stiffness. Dark gray level to light gray level denotes small value to large value. The dotted lines are iso-lines.
(a) and (b) Charts for general distributions of local vertical stiffness and local longitudinal stiffness of vocal fold, respectively. The same descriptions are shown in the caption of Fig. 2.
(a) to (d) Charts for comparisons of mass and stiffnesses between different horizontal planes s. , , , and denote the grand average values of mass, lateral stiffness, vertical stiffness, and longitudinal stiffness after applications of optimization procedure over all 24 hemilarynx experimental data sets, respectively. The dotted lines indicate the corresponding regressions for mass and different stiffness.
(a) to (d) Charts for the comparisons of mass and stiffness among different crosssections i. The descriptions are the same as Fig. 4.
(a) to (d) Charts for distributions of P-values of change of mass, lateral stiffness, vertical stiffness, and longitudinal stiffness of the vocal fold for increased airflow. They are on the basis of the optimization procedure to the 24 hemilarynx data sets. The black points are positions of different mass elements distributed across the vocal fold. Different gray levels correspond to different P-values of mass/stiffness. Dark gray level to light gray level denotes small value to large value. The black solid bold line indicates contour of P-value 0.95 (significant increase: P > 0.95), while the white solid bold line denotes contour of P-value 0.05 (significant decrease: P < 0.05).
(a) to (d) Charts for distributions of P-values of changing of mass, lateral stiffness, vertical stiffness, and longitudinal stiffness of the vocal fold for increased muscular process traction. Corresponding descriptions are shown in the caption of Fig. 6.
(a) to (d) Charts for distributions of P-values of changing of mass, lateral stiffness, vertical stiffness, and longitudinal stiffness of the vocal fold for increased thyroid traction. Corresponding descriptions are shown in the caption of Fig. 6.
Optimization results for 24 hemilarynx experimental data sets: Objective function Γ and correlation coefficient k are optimization measures exhibiting the quality of the results. Corresponding experimental conditions include applied airflows Ug [ml/s], muscular process weights Mp [g], and thyroid cartilage weights Tc [g]. Psub [cmH2O] is the subglottal pressure applied in the experiments, while is the optimized value within the 3DM. Additionally, [Hz] is the computed fundamental frequency, [cm] is the computed length of vocal fold. The indices of recordings are defined in the hemilarynx experiments.17
Maximum and minimum of the optimized local mass mi , s , lateral stiffness , vertical stiffness , and longitudinal stiffness . These values are based on the applications of optimization procedure to 24 hemilarynx data sets. The unit of mass is [g]. The unit of stiffness is [N/m]. i = 1,…, 5 denote the dorsal cross section to the ventral cross section along the longi- tudinal direction. s = 1,…, 5 denote the inferior plane to the superior plane along the vertical direction.
Statistically relevant changes of mass and stiffnesses (stif.) under influences of increased airflow Ug , muscular process Mp , and thyroid simulation Tc after statistical left-tail and right-tail t-tests. Within each cell, the three numbers (#-#-#) denote significant decrease (Pl < 0.05) - nonsignificant change (0.05 ≤ Pl ≤ 0.95) − significant increase (Pl > 0.95, due to Pr = 1 − Pl ). The sums of the three numbers represent the amounts of mass (25), lateral stif. (25), vertical stif. (20), longitudinal stif. (20).
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