## Abstract

The nonlinear viscoelastic passive properties of three canine intrinsic laryngeal muscles, the lateral cricoarytenoid (LCA), the posterior cricoarytenoid (PCA), and the interarytenoid (IA), were fit to the parameters of a modified Kelvin model. These properties were compared with those of the thyroarytenoid (TA) and cricothyroid (CT) muscles, as well as previously unpublished viscoelastic characteristics of the human vocal ligament. Passive parameters of the modified Kelvin model were summarized for the vocal ligament, mucosa, and all five laryngeal muscles. Results suggest that the LCA, PCA, and IA muscles are functionally different from the TA and CT muscles in their load-bearing capacity. Furthermore, the LCA, PCA, and IA have a much larger stress-strain hysteresis effect than has been previously reported for the TA and CT or the vocal ligament. The variation in this effect suggests that the connective tissue within the TA and CT muscles is somehow similar to the vocal ligament but different from the LCA, PCA, or IA muscles. Further demonstrating the potential significance of grouping tissues in the laryngeal system by functional groups in the laryngeal system was the unique finding that, over their working elongation range, the LCA and PCA were nearly as exponentially stiff as the vocal ligament. This paper was written in conjunction with an online technical report (http://www.ncvs.org/ncvs/library/tech) in which comprehensive muscle data and sensitivity analysis, as well as downloadable data files and computer scripts, are made available.

- canine larynx
- lateral cricoarytenoid muscle
- posterior cricoarytenoid muscle
- interarytenoid muscle
- thyroarytenoid muscle
- cricothyroid muscle
- vocal ligament
- vocal posturing
- laryngeal muscles
- Young's Modulus

during the past few decades, significant advances have occurred in our understanding of the structure and function of the human laryngeal system (10, 17, 20, 21, 31, 36, 47), whose main functions are airway protection, respiration, and phonation. With these advances, laryngological surgical procedures have become more precise, emphasizing preservation/protection of voice, ventilation, swallowing, and partial muscle function (30, 41, 66). Other benefits from these recent advances have been the improvement of pedagogical practices and nonsurgical clinical prevention and treatment of voice disorders (57, 58), as well as education of individuals who rely on their voice for performance of their occupational duties (e.g., singers, actors, broadcasters, and teachers; Refs. 62, 65). These improvements have occurred in large part due to enhanced understanding of the laryngeal system in terms of both structure and function.

Nevertheless, there is still much to learn and apply in our understanding; for this to happen, we must accurately define the composition of the laryngeal system (tissue geometry and properties; Fig. 1) and gain the ability to apply this knowledge to realistic laryngeal models using computational simulations and laboratory experiments. Of particular importance are the vocal folds, the primary sound generator of the human voice, as well as a source of airway protection. The vocal folds attach to the thyroid cartilage on one end and the arytenoid cartilage on the other and have been described as having three layers (21, 40): the mucosa, the vocal ligament, and the thyroarytenoid (TA) muscle (see Fig. 1*C*).

The kinetics of the vocal folds can be classified into two parts. The first is vocal fold vibration with its small and relatively fast deformations (at 100–1,000 Hz or more), which occurs when the tissue is driven into oscillation for sound generation. It transpires primarily in the lamina propria of the vocal fold (which includes vocal fold mucosa and vocal ligament) and the most medial portion of the TA muscle. The second is the positioning of the vocal folds (commonly referred to as vocal fold posturing) with its large and relatively slow deformations (at frequencies of 1–10 Hz); vocal fold posturing occurs when the vocal folds are positioned for prephonation, pitch change, inhalation, and airway closure. In further contrast to vocal fold vibration, vocal fold posturing is powered primarily by the laryngeal muscles, with passive contributions by the vocal ligament, laryngeal muscles not in contraction, and surrounding cartilages and joints.

There are two primary means by which vocal fold posturing transpires. The first is vocal fold elongation, which occurs via movement of the cricoid cartilage around the cricothyroid joint and is controlled by the cricothyroid (CT) muscle and the TA muscle (see Fig. 2*A* for a simplified representation of the joint's posturing movement; compare with Fig. 1*B*; Refs. 15, 59, 63). This elongation is the primary source of pitch control wherein the tensile strains in the vocal fold can range from 0 to 40% during high-pitch phonation such as singing (67); such strains are much higher than what is seen in most limb muscles and ligaments but somewhat comparable with ocular muscles (52). Furthermore, during this elongation, the stretching of the vocal ligament (35, 60) and vocal fold mucosa (12, 13), as well as either the CT or TA muscle, contribute a passive resistance to this elongation.

The second means of posturing is vocal fold abduction (opening) and adduction (closing), caused by movement of the arytenoid cartilage around the cricoarytenoid joint (compare Fig. 2*B* with 2*C*). Adduction, originating by pivoting the cricoarytenoid joint, is controlled by the lateral cricoarytenoid (LCA), the inner arytenoid (IA), and the TA muscles; abduction, on the other hand, is controlled by the combined effort of the two bellies of the posterior cricoarytenoid (PCA) muscle, the oblique PCA (PCA_{O}), and the vertical PCA (PCA_{V}; Refs. 10, 11, 16, 36). The abductor/adductor muscles (working in concert as an agonist/antagonist function group) also affect changes in glottal width and, thereby, glottal flow resistance (an essential component of respiration and airway protection). These muscles have been shown to be different histologically, with the adductors having more fast fibers and the abductor PCA having more of a fast-slow mix (50).

In summary, these five laryngeal muscles (i.e., CT, TA, LCA, IA, and PCA), along with the passive contribution of the vocal ligament and vocal fold mucosa, are key to vocal fold posturing. Therefore, they are also key to voicing (through vocal onset, self-sustained oscillation, intensity, and pitch; Refs. 14, 22, 24, 37, 54, 64) and overall health (through ventilation, swallowing, and effort closure of the airway; Refs. 33, 34). Generally, these specialty muscles are described as less developed for force as they are for speed and precise coordination of movements (30), which is helpful for both airway protection and vocalization.

Because interest has been traditionally focused on phonation and pitch control, research has focused on the related tissues, including the contractile (active) and noncontractile (passive stress-strain or normalized force-elongation) characteristics of the canine^{1} CT and TA muscles (4, 6–8, 15), as well as the passive characteristics of the human mucosa and vocal ligament (7, 12, 13, 35, 60). However, to truly understand the laryngeal system as a whole and vocal fold adduction/abduction specifically, the remaining three laryngeal muscles (the IA, LCA, and PCA) must be studied in more detail. Thus far, these three muscles have been examined in a few canine studies (5, 16), but these studies do not provide a rigorous quantization of their passive properties (i.e., tissue elasticity and tissue viscosity) that are important to posturing.

The purposes of this report were to measure more accurately and quantify in greater detail the passive properties of the canine LCA, IA, PCA_{O}, and the PCA_{V} (36) by using Titze's (55) adaptation of the Kelvin model. New details of the hysteresis of the vocal ligament's stress-strain curve were also presented. A summary of all five intrinsic laryngeal muscles (using canine tissues), in addition to the human vocal ligament and mucosa, was given in terms of the tissue model parameters. This paper then provides the mathematical model and parameters to simulate the dynamic longitudinal stress-strain response of the five laryngeal muscles, vocal ligament, and vocal fold mucosa. Finally, this paper was written in conjunction with an online technical report (25) that provides the raw tissue data used in the analysis below, a sensitivity analysis^{2} of the model parameters, and downloadable data files and computer scripts implementing the model.

## MATERIALS AND METHODS

In this section we review the methodology used to collect the stress-strain data, construct Titze's (55) modified Kelvin model for one-dimensional elongation, and optimize the model parameters to fit the tissue data.

### Review of Methodology for Previously Collected Tissue Data

The tissue data used in the current study were collected as part of several previous experiments on human and canine laryngeal tissue conducted at the National Center for Voice and Speech at the University of Iowa (5, 35); the method of dissection and measurement was similar in all studies and is summarized below. Before dissection, the average length of each sample was measured in situ with a caliper (±0.1 mm accuracy, although human uncertainty was on the order of 0.33 mm; Ref. 26). Then the tissue samples were dissected with the tissue continually submerged in the Krebs-Ringer solution. For all tissue samples, portions of the cartilage attachments were left intact.

#### Canine muscle preparation.

Multiple samples of each of the three abductor/adductor laryngeal muscles (i.e., IA, LCA, PCA) were extracted from five fresh canine larynges (Table 1). [Due to the difficulty in obtaining viable human laryngeal tissues, the canine has been traditionally used as a human vocal model (1–8, 15–17, 31, 39, 46, 47) because of its availability and comparable size.] Extraction occurred under a surgical microscope, and tissues were immediately hydrated in an oxygenated Krebs-Ringer solution at 37°C with a pH of 7.4. Only a portion of the entire muscle (i.e., a few muscle bundles) was used for the experiment. LCA and IA samples were taken from their whole muscle and trimmed to widths of ∼4–5 mm. Due to the large fanning out of the PCA muscle on the cricoid cartilage, samples from this muscle were taken from either the vertical portion (PCA_{V}) or the oblique portion (PCA_{O}). To retain a piece of the arytenoid cartilage on the end of both muscles, the LCA samples were taken from the side opposite from which the PCA sample was taken. The cross-sectional area of each sample was calculated from the mass and length after removal from the hydration bath and after excess liquid was removed (26). LCA, PCA_{V}, and PCA_{O} muscle samples ranged between 12 and 18 mm, whereas IA muscle samples ranged from 6 to 10 mm; all muscle samples were trimmed to ∼4–7 mm wide and ∼3–5 mm thick [Ref. 5; see Table 1 for full list of sample in situ lengths and average cross-sectional areas; for a more in-depth look at laryngeal muscle lengths and full muscle area, see Cox et al. (17) and Mineck et al. (36)].

#### Human ligament preparation.

Eight human vocal fold ligaments were obtained from surgery (4 hemilarynges from 3 surgical patients, 2 men and 1 woman; mean age of 65 yr; range 61–67) and autopsy (4 hemilarynges from 2 autopsy cases, 1 man and 1 woman; age of 30 and 46, in no particular order; Ref. 35). The ligaments from surgery were taken from larynges resected during total laryngectomy, with at least one of the two vocal folds being free of disease; the time interval from laryngeal resection to the elongation experiment ranged between 4 and 22 h. For the autopsy cases, the time interval from death to experiment ranged between 13 and 20 h (discounting time for 2 samples that were frozen). The absence of disease on all vocal folds used was confirmed under a dissecting microscope at the time of sample preparation.

Under a dissecting microscope, the vocal ligament was isolated by resecting the epithelium, superficial lamina propria, and muscular layers while the tissue was kept moist with Krebs-Ringer solution. The lateral border of the ligament was dissected up to the lateral edge of the vocal process of the arytenoid cartilage. The medioinferior aspect of the vocal ligament was dissected down to the inferior border of the tip of the vocal process of the arytenoid cartilage. The epithelial and superficial layers of the lamina propria (i.e., the mucosa) were resected by dissecting at the plane of the superficial layer, which was loosely attached to the underlying intermediate layer.

#### Force elongation data.

For a given tissue sample, a Tevdek polyester suture was inserted through each end piece of cartilage and tied off. With these sutures, one cartilage portion was tied to the ergometer arm and the other one to the bottom of the hydration bath (Fig. 3). The tension of the suture was adjusted until the sample was mounted with the length closest to its in situ length (defined as the tissue reference length), typically ∼1–2 g of force (0.01–0.02 N). The displacement of the ergometer arm and the force exerted by the tissue were measured electronically with a Dual Servo ergometer (Cambridge Technology, Cambridge, MA). The ergometer had a force resolution of 0.0005 N, displacement accuracy of 0.02 mm, and rise time of 6 ms. The analog signals of displacement and force of the ergometer were sent to an analog-to-digital converter (DI-410, DATAQ Instruments, Akron, OH). From these signals, conventional longitudinal stress (σ) and strain (*ε*) are derived from an axial force using the initial (unstretched) cross-sectional area of the sample, the reference length (in vivo length), and the experimental length (the current elongated length). By defining the change in length as the difference between the experimental and reference length, the stress is defined as the ratio of the axial force and initial cross-sectional area while the strain^{3} is defined as the ratio of the difference between lengths and the reference length (σ=*F*/*A*_{0} and ε=Δ*L*/*L*_{0}). This stress calculation assumed that the cross-sectional area essentially did not change significantly during a stretch cycle.

It should be noted that the above definition of reference length as in vivo length is different from the common muscle physiology definition, the optimal sarcomere length for maximum active contractile stress of a muscle. The common reference length for the vocal ligament, the CT, and the TA was defined as in vivo length because the ligament does not have an “optimal sarcomere length” and *1*) the three tissues were first studied as a group because of their combined importance in pitch control (precedence) and *2*) the TA and vocal ligament are parallel and, thus, experience related strains. This convention has been used in all reported laryngeal muscle studies where reference length was needed (4, 6–8, 26, 35, 42–44).

To obtain the passive properties of these muscles, samples were stretched and released by applying a 1-Hz sinusoidal signal (approximating the rate of vocal fold posturing and similar to other comparable studies; Ref. 67). Muscle samples were stretched and released with a median maximum elongation of 34% (range: 25–48%); tissue samples were not exposed to a set maximum strain but a set displacement as each tissue sample had a different reference length and the elongation apparatus was set to a predetermined displacement cycle. The cyclic elongation lasted for ∼20 s. With each stretch-release cycle, the muscle samples showed stress relaxation (Fig. 4*A*), which stabilized after about 10 cycles. On Fig. 4*A*, *points b* and *c* mark the beginning and end of a cycle after this stabilization; a stress-strain curve from this portion of the data is shown in Fig. 4*B*. Vocal ligament samples were treated similarly but with an elongation of ∼54% (median: 53%, range: 45–66%).

### Description of Titze's Modified Kelvin Model for One-Dimensional Elongation

Fibrous tissues intrinsic to the vocal folds contain both elastin and collagen fibers in their extracellular matrix. These fibers are aligned in a mostly parallel orientation (origin-insertion), which we shall call the axial direction. The fibers have a nonlinear relationship between axial stress (σ) and axial strain^{3} (ε), defined here as a dynamic Young's Modulus (*E*), or a ratio between longitudinal stress and strain. A simple mathematical quantification of this nonlinear relationship, used on both ligaments (35) and vocal fold muscles (7), is (1) This simple mathematical representation is useful in quantifying the nonlinear stretch response of the vocal fold fibrous tissues. However, it cannot be used to describe the viscous or strain-rate dependency of the tissue, which presents as stress hysteresis in repeated stretch-relaxation cycles (i.e., there is less stress during the relaxation phase than the stretch phase for the same strain).

Thus a more advanced mathematical equation has been developed. Titze (55) adapted the classic Kelvin model (28; Fig. 5) to include nonlinear components. Elements were a parallel stiffness *k*_{p}, series stiffness *k*_{s}, and internal viscous damping *d*. Although other models of tissue passive properties have been developed and may better serve to describe the structure of laryngeal tissues (e.g., Refs. 51, 67), the advantages of Titze's model are that it *1*) requires little computation time, lending itself to larger biomechanical models of complex laryngeal system mechanics (27, 61); *2*) incorporates active properties (not discussed here); and *3*) has parameters that all have direct definitional relations to characteristics that can be seen in cyclic stress-strain tissue fiber data.

Titze used the classic Kelvin model to define a differential equation that represents a fibrous tissue with both an active (contractile) and a passive component. The dynamic differential relationship between ε and total modeled axial stress σ_{m} is: (2) where *t*_{s} is a series time constant, *t*_{p} is a parallel time constant, σ_{p} is the passive stress, and ε̇ is the axial strain rate. Contractile (or internal or active) stress is represented by σ_{i}, which is zero for passive tissue; this stress has its own submodel (27, 55) that is not discussed in this paper. Work cycle losses in *Eq. 2* are represented by the σ̇ and ε̇ term, which include time constants defined in terms of the stiffness and damping components of the Kelvin model (*Eqs. 3* and *4*, respectively), (3) (4) Furthermore, *Eq. 4* can be rewritten in terms of a ratio of the spring constants: (5) In other words, the nearer *t*_{p}/*t*_{s} is to one, the more *k*_{p} is dominant and the less damping (or viscous loss) there is. The larger *t*_{p}/*t*_{s}, the more viscous loss is seen in elongating the tissue.

Titze's most significant adaptation of the Kelvin model is the nonlinear axial passive fiber stress σ_{p} for one-dimensional deformation, defined as the following piecewise stress-strain relationship: (6) where the upper relation in *Eq. 6* is a linear term, which is augmented by the exponential term in the lower relation. The function is continuous, as is its first derivative. In the relation above, σ_{0} is the stress at zero strain (representing the prestrained state of the laryngeal fibers), σ_{2} is a scale factor for the exponential function, *B* is an exponential strain constant, ε_{1} is the strain at zero stress (a negative number), ε_{2} is the strain at which the nonlinear exponential function begins, and ε is the axial strain (independent variable). The nonlinear tangent Young's Modulus is the derivative of *Eq. 6* with respect to strain, (7) In summary, using the equations above with seven parameters (i.e., σ_{0}, σ_{2}, ε_{1}, ε_{2}, *B*, *t*_{p}, and *t*_{s}), this mathematical model predicts an output stress σ_{m} for any input strain ε and strain rate ε̇.

### Optimization of Model Parameters to Data

To adjust a model to fit a given data set, the model parameters must be systematically optimized to laboratory data. For the current study, MATLAB technical computing software scripts (25) were written to solve *Eq. 2* with its dependent equations. These scripts calculated modeled stress σ_{m} from the differential equation given a time-varying strain by using the fourth-order Runge-Kutta method. For a numerical solution, an initial strain rate was needed, which represented the history of deformation; therefore, to ensure steady state, the equation solver was always run for 0.15 s of data before any final calculations were made (Fig. 4*A*, *point a* to *point b*).

Optimization of the parameters to the measured data was completed using the Nelder and Mead Simplex method (NMSM), a direct-search method. The NMSM routine was also written as a MATLAB script modified from a version of an algorithm described by Barton and Ivey (9) and provided in MATLAB. The NMSM method of fitting parameter values was chosen for the following reasons: *1*) its handling of functions with multiple parameters; *2*) its performance on both deterministic objective functions (39) and stochastic functions (18); and *3*) its lack of sensitivity to starting values, relying neither on initial derivatives nor on the continuity of the objective function.

The NMSM routine controlled the differential equation solution (*Eq. 2*) by regulating the parameters σ_{0}, σ_{2}, ε_{1}, ε_{2}, *B*, *t*_{p}, and *t*_{s} for a given a strain ε. A summed square error, (8) of the point-by-point difference between the measured stress σ_{d} and the modeled (predicted) stress was minimized by the search method.

Seed values, or first-guess values, were given to the search method to expedite the optimization. These values were based on rudimentary handpicked matches of the stress-strain curves for each muscle. Table 2 lists the initial starting point for these parameters, as well as upper and lower range of the parameter search.^{4} For a given set of initial parameter values, an initial modeled stress value σ_{m} was calculated at which point the NMSM routine systematically adjusted parameter values from which an updated stress value was calculated. This process was repeated until the NMSM routine minimized the error function. Optimization was concluded when the NMSM routine's two types of tolerance values (set at 1%^{5} ) were met: *1*) the difference in each parameter from attempt to attempt and *2*) the value of the error function.

## RESULTS

Stress-strain data from 15 canine laryngeal muscle samples and eight human vocal ligaments were used in the optimization. Figure 6*A* shows a typical fit, whereas Fig. 6*B* shows an optimal fit. Within the MATLAB environment and using the scripts discussed above, optimizing a set of parameters to a single stress-strain curve took on the order of 2,500 iterations (∼5 min of computation time on a 2-GHz processor). The resulting sets of parameters are displayed as follows: Table 3 contains the IA and LCA muscles and, under the heading Adductor, their combined values; Table 4 contains the PCA_{O} and PCA_{V} muscles and, under the heading Abductor, their combined values; and Table 5 contains the vocal ligament and the combined values of the abductor/adductor muscles.

### Obtaining Mean Parameters for a Tissue

Mean stress parameters of a tissue type were obtained in two ways. First, a set of mean parameters was calculated from each sample's optimized parameters (Tables 3 and 4, Mean of Parameters); a stress curve was then predicted from these mean parameters using a sinusoidal strain (0–35% peak-to-peak) as the input to the tissue model (Fig. 7*A*, dotted line). Upper and lower standard deviation (SD) stress curves were also created by taking the extreme (±SD) of each parameter (mean parameter + SD then mean parameter − SD).

This method of averaging parameters in a mathematical model is commonly used and is helpful in describing an average and range of individual parameters. Nevertheless, in nonlinear response curves the averages of the parameters, when taken as a set, may be misleading. For example, one parameter set of values for σ_{0}, σ_{2}, ε_{1}, ε_{2}, *B*, *t*_{p}, and *t*_{s} may predict a particular stress curve and another very different set of values might predict a similar curve (within a reasonable measure of accuracy); however, the averages of these two sets of parameters would not likely produce a curve anything like the original two.

Therefore, a second method of predicting mean parameters that could produce an average stress for each muscle was necessary, herein referred to as the mean stress curve parameters. In this method, each individual set of parameter values from the multiple samples of a muscle was used to model a stress curve with the same given strain, a sinusoidal strain ranging from 0 to 35% peak to peak. The predicted stresses from each sample's individual parameter values were averaged and the SD of the stresses for a given strain was computed (Fig. 7*B*), resulting in a mean stress cycle from the modeled stresses for a given muscle type (Fig. 7*C*, dotted line). Mean stress curve parameters were fit to this mean stress cycle, using the optimization method described in *Optimization of Model Parameters to Data* above. These values were included in the Tables 3 and 4 under the column Mean Stress Parameters. Figure 7*D* compares the results of the two methods of calculating mean stress; as can be seen, the mean parameter stress curve is significantly greater than the mean stress curve parameters. Figure 8 illustrates the corresponding results for the human vocal ligament with the values listed in Table 5.

Once the model parameter values were obtained, an isometric stress-strain curve was plotted using *Eq. 6*, which does not include viscous losses (ε̇=0; see Figs. 9 and 10). A curve of the isometric Young's Modulus was then obtained (*Eq. 7*). To provide a comparison to previous studies, which used the simpler exponential function (*Eq. 1*), resulting stress from each muscle's respective elongation and the simulated mean stress for a type of muscle was fit to the simpler mathematical representation (*A*_{1} and *B*_{1}) from *Eq. 1* (bottom of Tables 3–5).

The small size of laryngeal tissues makes their tissue properties, and thus any resultant model parameter values, susceptible to errors in laboratory measurement, particularly errors in length and cross-sectional area (26). Therefore, it is important to know the potential effect of these errors on the model. The online technical report written in conjunction with this manuscript (25) showed that some of the parameters were more sensitive to error (i.e., σ_{2}, ε_{1}, *t*_{p}, *t*_{s}) than others. The discussion and final conclusions will focus on those parameters that are less sensitive to measurement errors (i.e., σ_{0}, ε_{2,} *B*, ratio *t*_{p}/*t*_{s}) because their value results can be compared from one sample to another with more confidence as any uncertainty from the laboratory experiments would have little effect on them.

## DISCUSSION

In previous studies of the passive properties of the vocal ligament, mucosa, TA, and CT, the nonlinear tissue elasticity has been of great interest because its exponential-like nature is directly relevant to pitch control (56). In this study, investigations were expanded to include the passive properties (i.e., exponential tissue elasticity and viscosity) of the canine abductor/adductor muscles (i.e., IA, LCA, and PCA) via a modified Kelvin model; because viable (pre-rigor mortis) human laryngeal tissues are difficult to obtain, canine tissues have been traditionally used as a common model for the human laryngeal system (1–6, 8, 15–17, 38, 48, 49). These two effects (exponential elasticity and viscous loss) were captured by a model parameter *B* and the ratio of the time constants *t*_{p}/*t*_{s}; these two quantities were not sensitive to laboratory errors (25) and, thus, are used below to compare the passive tissue properties from several tissues.

The three abductor/adductor muscles (i.e., LCA, IA, and PCA) were very similar in their ratio of time constants (*t*_{p/}*t*_{s} between 1.7 and 2.0). As a group, the muscle samples' mean ratio was ∼1.8 (Table 5 under Abductor/Adductor Combined). Reflecting back to the description of the passive model where the time constants are defined from two parameters *k*_{s} and *k*_{p} (Fig. 5 via *Eq. 5*), a ratio of 1.8 implies that *k*_{s} is ∼80% of *k*_{p}. With one spring constant being 80% of the other, the damper in the model plays a large role, demonstrated by the large difference between the elongation stress and the relaxation stress of the stretch-release cycle (Fig. 7). However, the freshness of the tissue (which can be questioned in any in vitro study) would likely impact all parameters, particularly the viscous (damping) parameters.

In comparing the IA, LCA, and PCA muscle parameters, summarized in Tables 3 and 4, the IA seemed to have the smallest exponential factor (*B* = 7–8 compared with *B* = 14–17 for LCA and PCA). This result can be seen in Fig. 8*A*, where the exponential growth of the PCA and LCA is significantly more than that of the IA. This finding correlates with a previous study's finding that the IA muscle is histologically different from the PCA muscle (Ref. 53; LCA was not discussed in the study). The disparity may be the result of the distinct functional differences between the IA and the other two muscles. The LCA and PCA muscles are nearly an agonist/antagonist pair, rotating the arytenoids through a rocking motion and thereby positioning the vocal process. Thus a strong exponential factor in both muscles stabilizes movement through the resistance of the opposing muscle contraction; more important, in the case of laryngeal paralysis, a stable vocal fold in cadaveric position would be less likely to obstruct the airway. On the other hand, the IA's active role is to close the posterior glottis by the medial sliding of the arytenoids on the CA joint, playing its role as a vocal fold positioner associated with extended phonations and during forceful abduction tasks (38). If the IA had a strong exponential resistance, a quick inhalation may be more difficult as there is not a direct antagonist to counter the motion caused by IA contraction.

Looking specifically at the two bellies of the PCA, the *B* value of PCA_{O} was 11–12, whereas the value of PCA_{V} was ∼17. This difference is associated with previous assertions that the two bellies have unique functions in vocal fold posturing function (10, 11, 16, 36, 46), nerve innervation (45, 47), and histology (fiber types; Ref. 10). Nevertheless, additional investigation is recommended given the small sample size and overlap of SDs.

In contrast to the PCA, LCA, and IA, the vocal ligament had a time constant ratio of 1.35 (Table 5). Thus *k*_{s} is only 34% of *k*_{p} for the ligament, which suggests that the ligament is much less viscous during an axial stretch cycle than the abductor/adductor muscles. This is illustrated in the dynamic stress-strain curves of the ligament (as compared with the muscles above) as the elongation stress and the relaxation stress in the stretch-release cycle are similar (compare Figs. 7 and 8).

Previous discussions reported that the vocal ligament is stiffer than laryngeal muscles; however, these postulations have been based only on measurements of two of the five intrinsic laryngeal muscles, the TA and CT. Preliminary results from the current study suggest, however, that the LCA and PCA have a significantly greater exponential factor, *B*, than the vocal ligament over their respective test ranges.

All of the differences discussed above between the abductor/adductor muscles (i.e., IA, LCA, PCA) and the vocal ligament are even more interesting when expanding the comparison to other laryngeal tissues. Table 6 contains Titze's parameters for the TA, CT, and mucosa in the Kelvin model (56). Of the tissue types, the TA and CT are very similar (perhaps because they are an agonist/antagonist pair) and are slightly stiffer than the mucosa when comparing *B* values. Furthermore, the vocal ligament has an even higher *B* value than the CT muscle, TA muscle, and mucosa; the vocal ligament's role in pitch range makes this stiffness useful as stiffness is directly related to fundamental frequency and an exponential stiffness curve increases the range of frequencies possible without much length change.

However, it is important to note the TA, CT, mucosa, and vocal ligament (Figs. 9*C* and 10*C*) can be regarded as a group distinct from the abductor/adductor muscles. They are less exponentially stiff than the abductor/adductor muscles; furthermore, their *t*_{p}/*t*_{s} values are all between 1.1 and 1.3 (mean of 1.2). Thus these four tissues have similar viscous losses in a stretch cycle but are significantly less viscous than the abductor/adductor muscles.

The grouping of abductor/adductor muscles vs. pitch control muscles is in general agreement with the histology and muscle fiber type found in the laryngeal muscles where abductor and adductor muscles are described as “allotypically different” and “kinetically faster than the fastest limb myosin heavy chain” (53). In addition, the TA is known to be different from the PCA (23). While muscle fiber type is usually associated with muscle contraction speed and fatigue, it is likely that different muscle fiber types would have different tensile stress characteristics (hysteresis, elasticity).

In conclusion, in this study, the passive properties of the canine abductor/adductor muscles (i.e., the LCA, IA, and PCA), along with the human vocal ligament, were quantified with the passive portion of Titze's modified Kelvin model. The study addressed not only the traditional isometric stretch response curve, but also a viscous component under a 1-Hz strain rate. Details of curve fitting were outlined, as was the methodology for obtaining the parameters from laboratory data. The mathematical model could accurately reproduce a dynamic stress-strain cycle with hysteresis. A supplement to this paper is an online technical note that contains computer scripts and instructions on usage, as well as extra details on the muscle data (25).

Previous studies of the laryngeal system have extended the passive characteristics of the CT and TA to the LCA, IA, and PCA. Nevertheless, this study suggests that the passive properties of the LCA, IA, and PCA are quite different from those of the CT and TA. In retrospect, this should not be surprising given their distinct functions. Further demonstrating the potential significance of functional groups in the laryngeal system, the four tissues related to pitch control (CT, TA, ligament, and mucosa) were less viscous during an axial stretch cycle than the LCA, IA, and PCA, the three muscles related to arytenoid positioning. These results correlate with the differences found in histological studies of the laryngeal muscles (50, 53).

The primary weaknesses of this study were the small number of tissue samples (reducing the power of statistical significance) and the always questionable viability of any in vitro tissue samples. Another weakness was the use of both canine and human laryngeal tissue. Because of the extreme difficulty in obtaining viable human laryngeal muscles, the canine larynx has been used for many years as a model for the human larynx (17). Nevertheless, there are some anatomical differences in general morphology (19, 31), the canine larynx's lack of a vocal ligament being the primary difference. Therefore, although it is possible that not all of the results of this study may be directly relevant to the human, some information from a reasonable laryngeal model (canine) may still provide useful insight into the vocal system, as noted above. In addition, there seems to be a histological similarity between laryngeal tissues across many species (23) allowing for some interpretation of human laryngeal tissue using canine tissue.

As stated previously, the freshness of the tissue would likely have a large impact on all parameters, especially the viscous parameters; therefore, the exact comparison of the viscous properties across all tissues discussed may not be useful using the current data set. Furthermore, because the TA and CT data were collected at different times under different laboratory circumstances and the parameter fit was accomplished using a different method than used here, it is possible that some difference was methodological. The amount of this possible difference is not known as the raw data from the CT and TA experiments were not available; nevertheless, as *B* is a robust parameter, comparing the exponential factor should still be valid even with any additional uncertainties.

Future studies should be conducted to investigate any other physiological differences from those discussed above that would account for the measured mechanical response disparities in the TA and CT compared with the IA, LCA, and PCA. Additional studies should include the contractile properties of human laryngeal muscles, although procurement of such tissue is difficult and would take a great deal of coordination between an adequate in vitro muscle laboratory and an otolaryngologist at a nearby surgical suite (ideally in the same building). Finally, stretch studies should be expanded to include transverse and biaxial loading (29). With these quantifications, biomechanical modeling of vocal posturing (27, 32, 61) will allow for predictive mechanics of cartilage and joint motion in the larynx.

Finally, while previous studies of laryngeal muscle (canine) active properties have been accomplished (4–6, 8, 15, 16), none have presented the results in such a way to be able to directly contribute to the active parameters of the Titze model (the raw data are needed); thus an all encompassing set of parameters for the Titze model is not presented. While the contractile data from Alipour et al. (5) are available to the authors, it was not presented in terms of the Titze parameters as additional work is to be done. The contractile part of the Titze model is its own submodel (55). The entire model (both active and contractile) has been used in a larger biomechanical model of laryngeal posturing (27) but the muscle parameters for all the muscles was not known at that time. The data from any future report detailing canine muscle contractility in terms of the Titze model could be used to construct a complete integrated set of Titze parameters and the combined results of all model parameters could be discussed.

## GRANTS

Muscle data and previous analysis were collected with support from DC-004347 [(NIDCD)] and with special help from Drs. Fariborz Alipour, Douglas Montequin, Roger Chan, Niro Tayama, and Young Min.

The analysis and modeling presented here were conducted as part of a research program entitled, “A Computational Tool for Simulation of Phonosurgical Procedures,” also supported by the NIDCD (R03-DC-006801).

## Acknowledgments

The authors thank the research team at the National Center for Voice and Speech with many supporting roles in this work, particularly Farihorz Alipour.

Thank you to Laura M. Hunter for her technical writing skills.

## Footnotes

↵1 It is difficult to obtain viable (pre rigor mortis) human laryngeal muscles from surgery intact and in a timely manner. Therefore, human laryngeal muscles have not yet been used in a laryngeal study. Animal models have thus been used in the past, including the murine, lapine, porcine, bovine, ovine and canine; the canine has been the most popular human vocal model (1–8, 15–17, 31, 38, 48, 49) because of its availability and comparable size.

↵2 That is, sensitivity of the model parameters to experimental uncertainty and parameter fits to individual samples.

↵3 In engineering, large strains are usually defined in terms of principal stretches

*λ*. For uniaxial loading of a tissue assumed to be anisotropic, the axial principal stress can be assumed to be*λ*= 1 + ε.↵4 The boundary implementation scheme was based on a script donated by John D'Errico as found on MATLAB.com's Matlab Central.

↵5 This tolerance was chosen because the change in optimized parameters when tolerance was set from 1.0 to 0.1% was on the order of 0.0005%.

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- Copyright © 2007 the American Physiological Society