Development and Evaluation of Ligament Phantoms Targeted for Shear Wave Tensiometry

Abstract

Developing a shear wave tensiometer capable of non-invasively measuring ligament tension holds promise for enhancing research and clinical assessments of ligament function. Such development would benefit from tunable test specimens fabricated from well-characterized and consistent materials. Although previous work found that yarn can replicate the mechanical behavior of collateral ligaments, it is not obvious whether yarn-based phantoms would be suitable for development of a shear wave tensiometer for measuring ligament tension. Accordingly, the primary objective of this study was to characterize the mechanical properties and shear wave speed – stress relationships of ligament phantoms fabricated from yarn and silicone, and compare these results to published data from biological ligaments. We measured the mechanical properties and shear wave speeds during axial loading in nine phantoms with systematically varied material properties. We performed a simple linear regression between shear wave speed squared and the axial stress to determine the shear wave speed – stress relationship for each phantom. We found comparable elastic moduli, hysteresis, and shear wave speed squared – stress regression parameters between the phantoms and collateral ligaments. For example, the ranges of the coefficients of determination (R²) and slopes across the nine phantoms were 0.84 to 0.95, and 0.78 to 1.27 kPa/m²/s², respectively, which overlapped with the ranges found in a prior study in porcine collateral ligaments (0.84 to 0.996 and 0.34 to 1.18 kPa/m²/s², respectively). Additionally, the shear wave speed squared – stress regression parameters varied predictably with the density of the phantom and the shear modulus of the silicone. In summary, we found that yarn-based phantoms serve as mechanical analogs for ligaments (i.e., are ligament mimicking), and thus should prove beneficial in the development of a shear wave tensiometer for measuring ligament tension as well as investigations into ligament structure-function relationships.

Graphical Abstract

1. Introduction

Shear wave tensiometry has recently emerged as a method to non-invasively and directly measure tension in ligaments (Blank et al., 2020b) and other load-bearing soft tissues (Martin et al., 2018). A shear wave tensiometer is composed of an actuator (e.g., a piezoelectric tapper) that excites shear waves in the tissue of interest, and downstream sensors (e.g., laser Doppler vibrometers, miniature accelerometers) that measure the shear wave propagation speed (Martin et al., 2018). Since shear wave speed squared increases linearly with axial stress, measured shear wave speeds can be used to determine axial stress (Martin et al., 2018). However, accurately determining axial stress from shear wave speed is complicated by the dependence of shear wave speed on tensiometer design and use factors, such as the frequency of the signal that excites shear waves in the tissue (Brum et al., 2014; Cortes et al., 2015) and the application force of the tensiometer against the tissue (Blank et al., 2020a). Optimizing design factors and standardizing the measurement process are necessary steps toward reliable use of a shear wave tensiometer tuned for ligaments in both research and clinical settings. To date, these efforts have been conducted with biological tissues (Martin et al., 2019) or with computational models (Blank et al., 2020a), but each of these approaches have limitations for thorough device optimization.

One approach to overcome the limitations of using biological tissues or computational models is to use ligament phantoms, which are test specimens that replicate the mechanical behavior of ligaments, but are fabricated from engineering materials. While biological ligaments degrade over time due to insufficient hydration (Thornton et al., 2001), prolonged freezing (Lansdown et al., 2017), and/or multiple freeze-thaw cycles (Huang et al., 2011), engineering materials are typically inert. Furthermore, ligament properties vary in the population due to natural variability (Smeets et al., 2017), ligament damage and healing (Woo et al., 2006), and pathology (e.g., osteoarthritis (Fishkin et al., 2002)), while repeatable fabrication processes can minimize differences between ligament phantoms. Thus, controlled, well-characterized ligament phantoms are advantageous for developing a shear wave tensiometer because variability in measured shear wave speeds can be attributed to tensiometer design and use factors rather than specimen factors. Finally, although computational models facilitate exploration of a large design space, a physical ligament phantom is needed to test tensiometer design iterations, acquire feedback from surgeons, and establish repeatable measurement protocols.

Recently, Pineda Guzman and Kersh demonstrated that Caron 4-plied acrylic spun yarn has a similar load-displacement relationship to a cadaveric medial collateral ligament (Pineda Guzman and Kersh, 2021). Despite the analogous structural properties between the yarn and collateral ligament, a yarn-based ligament phantom must satisfy other design requirements in order to be suitable for shear wave tensiometry. First, the phantom must have similar mechanical properties (e.g., elastic modulus and hysteresis) and shear wave speed – stress relationships to ligaments, because designing a tensiometer tuned for ligaments requires a ligament-mimicking specimen. Furthermore, integrating phantoms into biomechanically realistic joint models requires similar mechanical properties to achieve accurate haptics (Ruiz and Dhaher, 2021). It is also important that these phantoms have repeatable fabrication processes to minimize specimen-to-specimen differences. Finally, as repeated loading is an expected loading regime for these phantoms during tensiometer optimization experiments, it is important to characterize the fatigue behavior of a yarn-based ligament phantom.

Along with axial stress and tensiometer design and use factors, previous work has revealed that measured shear wave speeds are dependent on the material properties of a specimen, including its fiber alignment (Blank et al., 2021), shear moduli (Blank et al., 2021), and effective density (Martin et al., 2019). An advantage of using ligament phantoms that are composite structures is the opportunity to tune the quantity, type, and arrangement of its components to achieve a range of desired properties. Therefore, in addition to tensiometer optimization, ligament phantoms offer a unique opportunity to explore the effects of the material properties of a specimen on measured shear wave speeds using a controlled experimental model. Such preliminary evaluations should reveal the feasibility of using shear wave tensiometry to investigate ligament structure-function relationships.

Accordingly, there were four objectives in this study. The first and primary objective was to characterize the mechanical properties and shear wave speed – stress relationships of ligament phantoms constructed from yarn and silicone and compare them to biological ligaments. We hypothesized that these yarn-silicone phantoms would have similar mechanical properties and shear wave speed – stress relationships to ligaments because of their similar composite, hierarchical structures. The second objective was to determine whether the mechanical properties and shear wave speed – stress relationships differed between phantoms fabricated using the same process. The third objective was to determine the effect of repeated loading on the mechanical properties and shear wave speed – stress relationships of the phantoms. Finally, the fourth objective was to determine the effect of phantom material properties on the shear wave speed – stress relationship.

2. Materials and Methods

2.1. Phantom Fabrication

We fabricated ligament phantoms as a composite of acrylic spun yarn embedded in a silicone. We used Caron 4-plied acrylic spun yarn with a linear density of 590 tex because this yarn has a similar load-displacement relationship to knee collateral ligaments (Pineda Guzman and Kersh, 2021). The yarn served as a load-bearing component analogous to collagen fibers in a biological ligament. We aligned the yarn in parallel strands along the longitudinal axis of the phantom using a custom 3D-printed mold (Figure 1). We used a tensioning fixture to apply a pretension of 10 N to the yarn throughout the molding process to remove slack and maintain alignment of the yarn strands. We used an isotropic silicone rubber (Smooth-On) to simulate the ground substance of a ligament. We prepared the silicone following the manufacturer datasheet, which involved first vacuum degassing the silicone for two minutes to remove entrapped air. We then poured the degassed silicone into the mold to embed the yarn in silicone. After curing, we removed the phantom from the mold and yarn-tensioning fixture.

Figure 1:

The flowchart shows the methods for fabrication and characterization of ligament phantoms. We embedded longitudinally aligned, pretensioned yarn in silicone to fabricate two sets of ligament phantoms (§ 2.1). To assess their mechanical properties, we cyclically loaded the phantoms at 1 Hz (§ 2.2.1). We computed the elastic modulus and hysteresis using the steady-state stress-strain curve. To evaluate the shear wave speed – stress relationship of the phantoms, we measured shear wave speeds in the phantoms during ramp loading (§ 2.2.2). We performed a linear regression between shear wave speed squared and axial stress to determine the slope, y-intercept, and coefficient of determination (R²) to describe the shear wave speed – stress relationship. The testing protocol was repeated once for each of the nine phantoms in Set 1, and once every other day for 15 days for each of the five phantoms in Set 2. We analyzed the phantoms in Set 1 for objectives 1 and 4, and the phantoms in Set 2 for objectives 2 and 3.

We fabricated two sets of phantoms following the fabrication procedure described above (Figure 1). We used the first set of phantoms to characterize the mechanical properties and shear wave speed – stress relationships of the phantoms (objective 1), compare the phantoms to ligaments, and determine the effect of phantom material properties on the shear wave speed – stress relationship (objective 4). This first set included nine phantoms that varied in the type of silicone (Dragon Skin 10 Fast, Dragon Skin FX-Pro, or Dragon Skin 20) and number of yarn strands (8, 10, 12) used in fabrication following a two-factor, three-level, full factorial experimental design (Table 1). The three silicones varied in their elastic and shear moduli (Table 2). We experimentally measured the shear moduli of the silicones as described in Supplement A. We chose to vary properties of the silicone and the number of yarn strands to model changes in the ground substance and collagen content of ligaments, respectively, that may result from ligament pathology or healing (Woo et al., 2006). We fabricated one phantom of each type.

Table 1:

We fabricated nine phantoms following a two-factor, three-level, full factorial experimental design.

Phantom	Dragon Skin Silicone Type	Number of Yarn Strands
1	10 Fast	8
2	FX-Pro	8
3	20	8
4	10 Fast	10
5	FX-Pro	10
6	20	10
7	10 Fast	12
8	FX-Pro	12
9	20	12

Table 2:

The three silicones varied in their elastic and shear moduli. Elastic moduli are based on the manufacturer data sheets, and we experimentally measured the shear moduli using the methods in Supplement A.

Dragon Skin Silicone Type	Elastic Modulus [kPa]	Shear Modulus [kPa]
10 Fast	151.7	59.0
FX-Pro	260.6	98.6
20	337.8	121.3

We used a second set of phantoms to evaluate the consistency of phantom fabrication (objective 2) and measure the change in mechanical properties and shear wave speed – stress relationships of the phantoms with repeated loading (objective 3). The second set included five phantoms, each with 12 strands of yarn and Dragon Skin 20 silicone. We chose this combination of fabrication parameters based on preliminary findings that the phantom with this combination (Phantom #9, Table 1) had an elastic modulus closest to ligaments and a high R² of the shear wave speed squared – stress relationship.

2.2. Phantom Characterization

A single operator measured the mechanical properties and shear wave speed – stress relationships of each of the nine phantoms in Set 1. For each of the five phantoms in Set 2 (denoted phantoms R1, R2, R3, R4, and R5), the same operator measured the mechanical properties and shear wave speed – stress relationships every other day for 15 days, for a total of eight testing sessions. We chose this loading regimen to replicate the expected use case of the phantoms in tensiometer development studies. The protocols for measuring the mechanical properties (§ 2.2.1) and shear wave speed – stress relationships (§ 2.2.2) are described below.

2.2.1. Mechanical Properties

To measure the mechanical properties of the phantoms, we loaded each phantom in an electrodynamic testing machine (Acumen 3, MTS, Eden Prairie, MN) (Figure 1). Because spun yarn is viscoelastic (Manich et al., 2000), we preconditioned the phantom to elicit a consistent force-displacement relationship. This preconditioning included 500 cycles of sinusoidal cyclic loading from 0 to 300 N at 1 Hz. Our preliminary experiments revealed that this preconditioning was sufficient to achieve consistency in the force-displacement relationship of each phantom (Supplement B). To compute the axial stress, we divided the force measured by the MTS load cell (661.18SE-02, MTS, reported hysteresis/non-linearity = 0.08% full scale) by the unloaded cross-sectional area of each phantom measured with digital calipers (VINCA DCLA-0605, Clockwise Tools, Valencia, CA, reported accuracy: ±0.03 mm). We computed the strain by dividing the MTS axial displacement by the unloaded gauge length of each phantom measured with digital calipers. We set a gauge length of 51.5 ± 0.4 mm in all mechanical tests.

We computed the mechanical properties of each phantom for each cycle of cyclic loading. First, we shifted the stress-strain data along the strain axis to set the initial, unloaded strain to zero for each cycle. The toe and linear regions of the shifted, loading portion of the stress-strain data were fit simultaneously using exponential and linear functions, respectively (Tanaka et al., 2011). We defined the slope of the linear region as the elastic modulus of each phantom. We also computed the hysteresis (expressed as a percentage) as the area between the loading and unloading portions of the curve divided by the area under the loading portion of the curve. The hysteresis represents the energy dissipation per cycle and is attributed to viscoelastic behavior. We excluded data from the 500^th cycle from these analyses due to a controller delay that caused a small lag between loading and unloading segments and introduced errors in the computed hysteresis. However, because there was a negligible change in the force-displacement relationship between the 450^th and 499^th cycle (0.63% average change in maximum displacement from 450^th to 499^th cycle, Supplement B), we expect a small and negligible change between the 499^th and 500^th cycles as well. Therefore, we considered the mechanical properties computed from the 499^th cycle of preconditioning to be the steady-state properties of each phantom (Figure 1).

2.2.2. Shear Wave Speed – Stress Relationship

To characterize the shear wave speed – stress relationships of each of the phantoms, we measured shear wave speeds in the phantoms during three ramp loading trials (Figure 1). In each trial, we loaded the phantom using a ramp from 0 to 300 N at 20 N/s. During ramp loading, a piezoelectric actuator (PK4JQP1, Thorlabs Inc., Newton, NJ) encased in a custom housing excited shear waves in each phantom by delivering 20 μm taps at a frequency of 10 Hz. Lasers from two laser Doppler vibrometers (PDV-100, Polytec Inc., Irvine, CA) spaced 5.5 mm apart on each phantom measured the transverse velocity of the phantom’s surface. To increase the curvature of the laser signals for improved cross-correlation performance, we differentiated the laser signals to compute the transverse acceleration. Next, we filtered the differentiated laser signals using a 2^nd order Butterworth bandpass filter with cut-off frequencies of 150 and 1,500 Hz. We then cross-correlated the transient portions of the two processed laser signals to find the time delay in wave arrival at the two laser locations. We used a 3-point cosine fit of the normalized cross correlation values to identify the peak correlation with sub-sample resolution (Cespedes et al., 1995). We then computed the shear wave speed by dividing the spacing of the lasers by the time delay identified from the peak correlation. Finally, we low pass filtered the computed shear wave speeds and load signals with a cut-off frequency of 4.99 Hz. To compute a single load per measured shear wave speed, we averaged the 5,000 load measurements acquired within each tap event. We computed the axial stress as previously described (§ 2.2.1).

To characterize the shear wave speed – stress relationship, we performed a simple linear regression between shear wave speed squared (independent variable) and axial stress (dependent variable) (Figure 1). For the phantoms in Set 1, we performed the linear regressions between data pooled across the three trials, while for the phantoms in Set 2, we performed separate linear regressions for each trial. From each linear regression, we extracted the slope, y-intercept, and coefficient of determination (R²).

2.3. Phantom – Ligament Comparison

To determine whether the phantoms are indeed ligament mimicking, we compared the range of mechanical properties and shear wave speed squared – stress regression parameters between the phantoms in Set 1 and collateral ligaments from prior studies. We chose collateral ligaments for comparison because collateral ligament tensioning is a key step during total knee arthroplasty (Babazadeh et al., 2009), making it an important target for clinical translation of shear wave tensiometry. We performed a literature search to assemble elastic modulus and hysteresis values for the medial collateral ligament (MCL) and lateral collateral ligament (LCL). We restricted our search to human and porcine ligaments. We also assembled the range of slopes, y-intercepts, and R² values for the linear regression between shear wave speed squared and axial stress for porcine MCL and LCL specimens (Blank et al., 2020b). We compared the published literature values for collateral ligaments to the range of values measured in the nine phantoms with varying properties.

2.4. Consistency of Phantom Fabrication and Effect of Repeated Loading

We fit linear mixed models to the data acquired using phantom Set 2 to determine the consistency of phantom fabrication and the effect of repeated loading on the mechanical properties and shear wave speed – stress relationships. We used the mechanical properties and shear wave speed squared – stress regression parameters as the dependent variables in the linear mixed models. The mechanical property dependent variables were the elastic moduli and hysteresis, which we computed for each phantom specimen per testing session. The linear mixed models for the mechanical property dependent variables included two factors, “testing session” and “specimen”, which were both modeled as fixed effects. We defined “testing session” as a continuous factor, with eight levels corresponding to the numbered testing session (1, 2, …, 8). We defined “specimen” as a categorical factor, with five levels corresponding to the phantom specimen (R1, R2, …, R5). The shear wave speed dependent variables were the slope, y-intercept, and R² of the shear wave speed squared – stress relationship, which we computed for each phantom specimen per trial per testing session. The linear mixed models for the shear wave speed squared – stress regression parameters included the fixed effects “testing session” and “specimen”, as well as a random effect for the trial number. This “trial” factor corresponded to the ramp loading trial number (1, 2, or 3). Along with the linear regression coefficient table, we computed an analysis of variance (ANOVA) table for each linear mixed model (“fitlme” and “anova” functions, MATLAB R2019b, Natick, MA).

From each linear mixed model, we extracted the p-value for the “specimen” fixed effect from the ANOVA table to determine whether any of the five phantom specimens differed significantly from the others with respect to the output of interest. We also extracted the p-value and model coefficient for the “testing session” fixed effect to ascertain whether and how, respectively, each output of interest varied across testing sessions, and thus across repeated loading. We used a significance level (α) of 0.05.

Finally, we calculated the intermediate precision standard deviations (S_I) for the factors “testing session,” “specimen,” and “trial” to quantify and compare the variability introduced by testing sessions, phantom fabrication, and the measurement method, respectively (ISO, 1994). We used the following equations (Equations 1–3) to calculate the intermediate precision standard deviations for each of the shear wave speed squared – stress regression parameters (y):

$$s_{I(\mathit{\text{session}})}=\sqrt{\frac{1}{tn(d-1)}\sum _{i=1}^n\sum _{j=1}^t\sum _{k=1}^d{\left(y_{ijk}-{\overline{y}}_{ij}.\right)}^2}$$ (1)

$$s_{I(\mathit{\text{specimen}})}=\sqrt{\frac{1}{td(n-1)}\sum _{j=1}^t\sum _{k=1}^d\sum _{i=1}^n{\left(y_{ijk}-{\overline{y}}_{\cdot jk}\right)}^2}$$ (2)

$$s_{I(\mathit{\text{trial}})}=\sqrt{\frac{1}{nd(t-1)}\sum _{i=1}^n\sum _{k=1}^d\sum _{j=1}^t{\left(y_{ijk}-{\overline{y}}_{i\cdot k}\right)}^2}$$ (3)

where n is the number of phantom specimens (5), d is the number of testing sessions (8), and t is the number of trials per specimen per testing session (3). Since only one trial of mechanical property measurements was performed on each specimen each testing session, the intermediate precision standard deviations for the mechanical properties were calculated using Equations (1) and (2) with t=1.

2.5. Effect of Phantom Properties on the Shear Wave Speed – Stress Relationship

To investigate how the material properties of the phantoms affect the relationship between shear wave speed and axial stress, we performed simple linear regressions between the properties of interest (independent variables; i.e., density and shear modulus) and the shear wave speed squared – stress regression parameters (dependent variables; i.e., slope and y-intercept) using a significance level (α) of 0.05. According to the analytical relationship between shear wave speed (SWS) and axial stress (σ) derived from a tensioned beam model (Equation 4) (Martin et al., 2018), the slope of this relationship is the effective density of the specimen (ρ), and the y-intercept is proportional to the shear modulus (μ) of the specimen:

$$\sigma =\rho {(SWS)}^2-k^{\prime }\mu$$ (4)

where k′ is a shear correction factor (0.822, based on phantom geometry (Dong et al., 2010)). Thus, we chose to perform simple linear regressions between (1) the mass density of the phantoms and the slope of the shear wave speed squared – stress relationship, and (2) the shear modulus of the silicone and the y-intercept of the shear wave speed squared – stress relationship. In ex vivo experiments on a specimen that is isolated from surrounding tissues and not immersed in fluid, the mass density of the specimen is comparable to its effective density (Martin et al., 2019). The mass density of each phantom was computed by dividing the mass of the phantom measured using a digital scale (GH-202, A&D, San Jose, CA, reported readability: 0.1 mg) by the volume calculated from the dimensions of the phantom (measured using VINCA DCLA-0605 digital calipers, reported accuracy: ±0.03 mm), assuming a rectangular prism geometry. We used the shear modulus of the silicone in our linear regression because the phantoms are transversely isotropic, and thus the isotropic ground substance (i.e., silicone) is the main contributor to the transverse shear modulus of the phantoms (Gardiner and Weiss, 2001). We compared the linear model coefficients from the linear regressions to those predicted by the analytical tensioned beam model.

3. Results

The stress-strain relationship for each of the ligament phantoms was nonlinear and had strain-stiffening behavior (Figure 2a). During preconditioning, the elastic moduli and hysteresis tended to increase and decrease, respectively, as a function of the number of cyclic loading cycles (Figure 2b). The elastic moduli calculated from the linear region of the steady-state cycle ranged from 104.7 to 176.3 MPa across the nine phantoms fabricated with varying numbers of yarn strands and silicone types (Phantom Set 1) (Table 3). Although the phantom elastic moduli were on the order of magnitude of collateral ligament elastic moduli, the phantoms tended to be more compliant (Table 3). The hysteresis, or energy dissipation per cycle, ranged from 19.3 to 29.7% across the nine phantoms, which spans the range of published hysteresis values for the MCL and LCL (Table 3). Phantom #9 (Dragon Skin 20 with 12 strands of yarn, Table 1), with an elastic modulus of 176.3 MPa and hysteresis of 19.3% (Figure 2), had the most ligament-like mechanical properties of all tested phantoms (Table 3).

Figure 2:

Representative plots (Phantom #9) depict (a) the steady-state stress-strain curve, and (b) the change in mechanical properties throughout preconditioning. (a) The phantoms exhibit nonlinear strain-stiffening behavior, with a toe and linear region at low and high strains, respectively. (b) The elastic modulus and hysteresis tended to increase and decrease, respectively, as a function of the number of cyclic loading cycles. In all results figures, we used colors accessible to those with color-vision deficiencies (Crameri et al., 2020).

Table 3:

Mechanical properties and shear wave speed squared – stress regression parameters for nine phantoms fabricated with varying numbers of yarn strands and silicone types (Phantom Set 1), along with values from human and porcine LCL and MCL specimens from the literature for comparison. Data are presented as ranges (indicated by brackets), means, or means ± standard deviation.

	Mechanical properties		Shear wave speed squared - stress regression parameters
	Elastic modulus [MPa]	Hysteresis [%]	Slope [kPa/m2/s2]	Y-intercept [kPa]	R2
Phantom range	[104.7, 176.3]	[19.3, 29.7]	[0.78, 1.27]	[−978, −64]	[0.84, 0.95]
LCL	493.9 (H)a 289.0 ± 159.7 (H)b 183.5 ± 110.7 (H)c	28.4 (H)e	[0.34, 0.79] (P)h	[−1558, −271] (P)h	[0.95, 0.996] (P)h
MCL	326.8 (H)a 441.8 ± 117.2 (H)b 332.2 ± 58.3 (H)d	19.6 ± 3.1 (P)f 18.8 ± 0.9 (H)g	[0.39, 1.18] (P)h	[−2385, 187] (P)h	[0.84, 0.99] (P)h

(P) porcine tissue; (H) human tissue

^aCho and Kwak, 2020;

^bSmeets et al., 2017;

^cLaPrade et al., 2005;

^dQuapp and Weiss, 1998;

^evan Dommelen et al., 2006 computed from load-displacement curve (Figure 4a, “cycles” curve);

^fHenninger et al., 2013;

^gLujan et al., 2007;

^h Blank et al., 2020b

The axial stress increased linearly with shear wave speed squared for each of the nine phantoms in Set 1 (Figure 3). The coefficient of determination (R²) was high for all phantoms (range: 0.84 to 0.95, Table 3), indicating that the linear relationship between shear wave speed squared and axial stress was strong for all combinations of phantom fabrication parameters used in this study. The phantom shear wave speed data also tended to lie within the range of data collected on porcine collateral ligaments (Figure 3, Table 3). Concerning the regression parameters, the y-intercept for the phantoms varied between −978 and −64 kPa, with values in the range of y-intercepts from all porcine MCLs and most porcine LCLs (Table 3). The range of slopes for the phantoms was 0.78 to 1.27 kPa/m²/s², with nearly all values falling within the range of porcine MCL slopes (Table 3). The range of phantom and porcine LCL slopes overlapped, but the phantom slopes tended to be higher (Table 3). Phantom #7 (Dragon Skin 10 Fast with 12 strands of yarn, Table 1), with a slope of 0.78 kPa/m²/s² and y-intercept of −341 kPa (Figure 3), had the most ligament-like shear wave speed squared – stress relationship of all tested phantoms (Table 3).

Figure 3:

Representative shear wave speed squared – stress relationship (Phantom #7) pooled across three trials, with a linear regression fit line that depicts a strong, positive linear relationship. The phantom data also lies within the range of data from nine porcine MCL (left) and LCL (right) specimens over the stress range measured in the phantoms (Blank et al., 2020b).

We detected statistically significant differences in the mechanical properties of the phantoms in Set 2 between specimens and across testing sessions (Table 4, Figure 4). Both the elastic modulus and hysteresis differed significantly in at least one of the five fabricated phantoms (p<0.001 for both). The elastic modulus significantly decreased across the eight testing sessions at an average rate of −1.57 MPa/session. Hysteresis significantly increased across testing sessions at an average rate of 0.38 %/session. The standard deviations indicate that the elastic moduli of the phantoms tended to vary more between specimens than across testing sessions, while the variability in hysteresis between specimens was comparable to that across sessions (Table 4).

Table 4:

Statistics quantifying the change in mechanical properties and shear wave speed squared – stress regression parameters for the phantoms in Set 2 across specimens, testing sessions, and trials. All properties and parameters except R² differed significantly across specimens. All properties and parameters except the y-intercept differed significantly across testing sessions.

	Specimen		Testing Session			Trial
	p-value	Standard deviation	Linear model coefficient	p-value	Standard deviation	Standard deviation
Mechanical Properties
Elastic modulus	<0.001	10.27 MPa	−1.57 MPa/session	<0.001	5.70 MPa	-
Hysteresis	<0.001	0.94 %	0.38 %/session	<0.001	1.01 %	-
Shear wave speed squared - stress regression parameters
Slope	<0.001	0.18 kPa/m2/s2	0.02 kPa/m2/s2/session	0.002	0.17 kPa/m2/s2	0.12 kPa/m2/s2
Y-intercept	0.014	291.09 kPa	−7.15 kPa/session	0.537	297.52 kPa	285.47 kPa
R2	0.095	0.04	−0.008 /session	<0.001	0.05	0.04

Figure 4:

Variability in mechanical properties between five phantoms fabricated using the same process (Phantom Set 2: R1, R2, R3, R4, and R5) across eight testing sessions.

We also detected statistically significant differences between specimens and across testing sessions in some parameters from the linear regression between shear wave speed squared and stress (Table 4, Figure 5). The slope and y-intercept differed significantly in at least one of the five fabricated phantoms in Set 2, but we did not detect any specimen-dependent differences in R² values. The slope of the shear wave speed squared – stress regression significantly increased across testing sessions at an average rate of 0.02 kPa/m²/s²/session, while there was no significant change in the y-intercept across testing sessions. R² tended to significantly decrease across testing sessions at a rate of −0.008/session. Based on the computed standard deviations, the variability in the slope, y-intercept, and R² tended to be smaller between trials than between testing sessions or specimens (Table 4).

Figure 5:

Variability in shear wave speed squared – stress linear regression parameters between five phantoms fabricated using the same process (Phantom Set 2: R1, R2, R3, R4, and R5) across eight testing sessions. Each point represents the mean value across three trials.

For the phantoms in Set 1, we detected a significant positive relationship between the phantom density and the slope of the shear wave speed squared – stress relationship (linear model coefficient = 10.65, p-value < 0.001). We also detected a significant negative relationship between the shear modulus of the silicone and the y-intercept of the shear wave speed squared – stress relationship (linear model coefficient = −8.05, p-value = 0.038) (Figure 6). The directions of these trends are consistent with the analytical relationship between shear wave speed and axial stress (Equation 4). However, the analytical relationship tends to over- and under-predict the slope of the shear wave speed squared – stress relationship at low and high phantom densities, respectively (Figure 6a). Additionally, the analytical model tends to over-predict the y-intercept of the shear wave speed squared – stress relationship over the full range of silicone shear moduli (Figure 6b). Furthermore, the constants of proportion between the phantom density and slope of the shear wave speed squared – stress relationship (10.65) and between the silicone shear modulus and y-intercept of the shear wave speed squared – stress relationship (−8.05) are higher in magnitude than those predicted by the analytical model (1 and −0.822, respectively, Equation 4).

Figure 6:

Scatterplots with linear fits show (a) the positive relationship between the phantom density and slope of the shear wave speed squared – stress relationship, and (b) the negative relationship between the silicone shear modulus and y-intercept of the shear wave speed squared – stress relationship, which both agree in direction with the analytical tensioned beam model (Equation 4). However, the analytical model predicts a one-to-one relationship between density and slope, and a negative relationship between silicone shear modulus and y-intercept, with the shear correction factor (0.822) as the constant of proportion. Data in these scatterplots were acquired from the phantoms in Set 1.

4. Discussion

The objectives of this study were to: (1) characterize the mechanical properties and shear wave speed – stress relationships of ligament phantoms fabricated from yarn and silicone; (2) determine whether the mechanical properties and shear wave speed – stress relationships differed between phantoms fabricated using the same process; (3) determine the effect of repeated loading on the mechanical properties and shear wave speed – stress relationships of the phantoms (4) determine the effect of phantom material properties on the shear wave speed – stress relationships. Related to these objectives, there were four key findings regarding the mechanical behavior of ligament phantoms fabricated from yarn and silicone. First, these phantoms have comparable mechanical properties and shear wave speed squared – stress relationships to collateral ligaments (Figures 2 and 3, Table 3). Second, although some properties varied significantly between specimens, the standard deviations were small, indicating that our fabrication process is repeatable (Figures 4 and 5, Table 4). Third, the changes in mechanical properties and shear wave speed squared – stress regression parameters across testing sessions were likely due to fatigue damage (Figures 4, 5, and C.1; Table 4). Finally, the shear wave speed squared – stress regression parameters varied predictably with the density of the phantom and the shear modulus of the silicone (Figure 6).

Our first key finding was that the phantoms have comparable mechanical properties to collateral ligaments. The nonlinear, strain-stiffening mechanical behavior exhibited by the phantoms has been documented in both yarn-silicone phantoms (Pineda Guzman and Kersh, 2021) and collateral ligaments (Cho and Kwak, 2020; Quapp and Weiss, 1998; Smeets et al., 2017; Woo et al., 2006). Pineda Guzman and Kersh found that spun yarn serves as an appropriate foundation for constructs that replicate the mechanical behavior of ligaments (Pineda Guzman and Kersh, 2021). They attributed this similar behavior to structural similarities between collagen and yarn, including their rope-like hierarchical structures and viscoelastic properties. We extend upon their work by finding that the intrinsic mechanical properties of yarn-silicone phantoms, particularly the elastic modulus and hysteresis, are also comparable to those of collateral ligaments (Table 3). Specifically, phantom #9 had the most ligament-like mechanical properties, likely due to its high ratio of yarn to silicone and high silicone shear modulus. Furthermore, the increase in elastic modulus and decrease in hysteresis with low-to-moderate amounts of cyclic loading (Figure 2) have been documented in rabbit collateral ligaments (Thornton et al., 2003), and rat patellar tendons (Andarawis-Puri et al., 2012; Fung et al., 2010). Along with comparable mechanical properties, the composite structure of our ligament phantoms achieves a ligament-like macrostructure, including an isotropic ground substance and a fibrous load-bearing component.

We also identified similar shear wave speed – stress relationships between the phantoms and collateral ligaments. Similar to collateral ligaments (Blank et al., 2020b) and tendons (Martin et al., 2018), the squared shear wave speeds in each of the ligament phantoms increased linearly with increasing axial stress. The range of R² was high for all phantoms (0.84 to 0.95) and comparable to the range found previously in the LCL (0.95 to 0.996) and MCL (0.84 to 0.99) from porcine knees (Blank et al., 2020b) (Table 3). Like ligaments, the phantoms have a relatively small shear modulus due to their highly aligned yarn strands that provide minimal resistance to transverse shear (Weiss et al., 2002). Thus, as axial load increases, the axial stress term becomes much larger than the shear modulus term in the analytical tensioned beam model (Equation 4) (Martin et al., 2018), resulting in linearly increasing squared shear wave speeds with increasing axial stress.

Although the regression parameters for the phantoms were similar to or in range of those found in porcine MCLs and LCLs, the shear wave speeds were generally lower in the phantoms than in the collateral ligaments (Figure 3). Two likely sources of this discrepancy are differences in boundary conditions and shear behavior between the phantoms and ligaments. Concerning the boundary conditions, we tested the phantoms in rigid metal grips, while the collateral ligaments were tested with their bony attachments intact. Although an effort was made to achieve uniform tension across the width of the porcine collateral ligaments (Blank et al., 2020b), ligaments undergo non-uniform loading at many knee postures (Willinger et al., 2020). Thus, the measure of stress in the porcine collateral ligament study should be considered an average stress that may underestimate the local stress where shear wave speed measurements were made. In contrast, because manipulating the boundary conditions of an isolated phantom is more straightforward, we are confident that we achieved near-uniform loading across the width of the phantom, and thus the average stress is more representative of the local stress in the yarn at the shear wave speed measurement location. The well-controlled boundary conditions of the phantoms will prove beneficial in future studies examining the effect of boundary conditions on measured shear wave speeds.

A second possible explanation for why the shear wave speeds were generally lower in the phantoms than in the collateral ligaments is the greater resistance to transverse shear exhibited by ligaments. While the phantoms were fabricated with all yarn strands aligned as longitudinally as possible, ligaments have a distribution of fiber alignments, including some fibers that are unaligned with the longitudinal axis (Provenzano and Vanderby, 2006; Stender et al., 2018). These unaligned fibers have been shown to increase the shear modulus and thus shear wave speeds at a given stress level (Blank et al., 2021). Furthermore, tendon and ligament have fiber-fiber and fiber-matrix interactions that are believed to further elevate the transverse shear modulus (Gardiner and Weiss, 2001; Guerin and Elliott, 2004). Since the y-intercept of the shear wave speed squared – stress relationship is proportional to the shear modulus (Equation 4), the higher magnitude of y-intercepts in the collateral ligaments than the phantoms (Table 3) is consistent with the greater shear resistance of ligaments.

Regarding the second key finding, the inter-specimen standard deviations in mechanical properties and shear wave speed squared – stress regression parameters are small compared to biological ligaments (Table 4). For example, the inter-specimen standard deviation in elastic modulus is 11.4 and 15.6 times smaller for the phantoms than it is for biological MCLs and LCLs, respectively (Smeets et al., 2017). Thus, using these ligament phantoms instead of biological ligaments decreases specimen-to-specimen differences because of the repeatable fabrication of the phantoms.

Regarding the third key finding, the statistically significant, but small, change in mechanical properties and shear wave speed squared – stress regression parameters across testing sessions is likely due to fatigue damage caused by repeated loading. Across the eight sessions of testing, each phantom was subject to more than 4,000 cycles of cyclic loading. Our supplemental fatigue tests found that the elastic modulus begins to decline after 1,934 consecutive cycles, while the hysteresis begins to increase after 3,630 consecutive cycles (Supplement C). Although both types of testing caused fatigue-induced changes in the mechanical properties of the phantoms, consecutive cycling (Supplement C, Figure C.1) had a greater effect on properties than intermittent loading across testing days (Figure 4). A reduction in elastic modulus and increase in hysteresis with high-cycle loading has also been documented in rat tendons and ligaments (Fung et al., 2010; Fung et al., 2009) and rabbit MCLs (Thornton et al., 2007), and has been attributed to the rupture of collagen fibrils and fibers. Similarly, we speculate that the rupture of yarn plies is responsible for the fatigue-induced decline in elastic moduli and increased energy dissipation across testing sessions. To maintain the elastic modulus at 97% of the maximum value, we recommend limiting the number of consecutive cycles to ~4,000 for loading up to ~5 MPa (Figure C.1), although smaller and intermittent loading may extend the lifetime of the phantoms by allowing for more total cycles before the elastic modulus declines. Given our findings that the mechanical properties of the phantoms change with repeated loading (Figures 4 and C.1), and that the shear wave speed – stress relationship is dependent on phantom material properties (Figure 6), the statistically significant increase in slope and decrease in R² across testing sessions is likely due to fatigue damage as well.

Concerning the trial-to-trial variability, tensiometer measurement error is likely a key contributor to the standard deviations in regression parameters across trials (Table 4). Minimizing this error through optimization of the tensiometer design is a subject of ongoing work with the phantoms. Of note, although we only evaluated the variability in mechanical properties and shear wave speed squared – stress regression parameters for phantom #9, we expect similar variability across specimens, testing sessions, and trials for the other types of phantoms (phantoms #1–8, Table 1). This is because these phantoms are also susceptible to fabrication inconsistences, yarn ply rupture resulting in fatigue, and tensiometer measurement errors.

Regarding the fourth key finding, our results indicate that the properties of the phantoms can be tuned to modulate the slope and y-intercept of the shear wave speed squared – stress relationship, with trends that are consistent in direction with the analytical tensioned beam model (Equation 4). First, the positive relationship between phantom density and slope (Figure 6a) is consistent with the analytical tensioned beam model, which predicts that the slope of the shear wave speed squared (independent variable) – stress (dependent variable) relationship should increase with the density of the specimen. In contrast to the analytical model, we did not detect a one-to-one relationship between density and slope in our phantoms. Many shear wave tensiometry studies on biological tissues have also documented a large discrepancy between the model-predicted slope (i.e., the density of the specimen) and the computed slope of the shear wave speed squared – stress relationship (Blank et al., 2020b; Martin et al., 2018). This discrepancy is a subject of ongoing work using the phantoms and computational models.

The negative relationship between silicone shear modulus and y-intercept (Figure 6b) is consistent with the structural composition of the phantoms and the analytical tensioned beam model. In transversely isotropic materials with highly aligned fibers, such as ligaments and the phantoms, the unloaded, transverse shear response is dominated by the isotropic ground substance (Gardiner and Weiss, 2001). Thus, a higher silicone shear modulus translates to a higher phantom shear modulus, which decreases the y-intercept according to the analytical tensioned beam model (Equation 4). The analytical model may have under-predicted the magnitude of the y-intercept because it does not account for the strain-stiffening effect commonly observed in transversely isotropic tissues such as the phantoms and ligaments (Weiss et al., 2002). Because the analytical model does not accurately predict changes in the shear wave speed squared – stress regression parameters with changes in phantom properties, there is a limitation in applying this model to the phantoms. Therefore, we recommend using the relationships between phantom properties and regression parameters from this study (Figure 6) when choosing a phantom composition to achieve a particular shear wave speed squared – stress relationship.

The high strength of the linear relationship between shear wave speed squared and axial stress in all phantoms in Set 1 demonstrates the versatility of shear wave tensiometry. Although varying the phantom fabrication parameters varied the mechanical properties of the phantoms in Set 1 (elastic modulus range: 104.7 to 176.3 MPa, hysteresis range: 19.3 to 29.7%), each phantom still demonstrated a strong, linear relationship between shear wave speed squared and axial stress (R² range: 0.84 to 0.95) (Table 3). This finding suggests that shear wave tensiometry, which operates based on the linear relationship between shear wave speed squared and axial stress, can be employed in biological or synthetic transversely isotropic materials with varying mechanical properties. Because properties of ligaments are known to vary with ligament damage, pathology, and age (Woo et al., 2006), this finding is important for clinical translation of shear wave tensiometry.

Although we found our phantoms to be ligament mimicking, there are three limitations in our phantoms and testing methods to consider when interpreting and applying our findings. First, while an effort was made to fabricate phantoms with diverse properties (Table 3), we did not capture the wide range of ligament properties that exists in healthy (e.g., due to age, activity level) nor pathologic (Woo et al., 2006) populations. To obtain these diverse properties, we recommend tuning the type, quantity, and arrangement of the phantom constituents. For example, the phantoms tended to have lower elastic moduli than collateral ligaments, particularly the MCLs. If higher elastic moduli are of interest for a specific application, then the ratio of yarn to silicone in the cross section can be increased or a different fibrous material with a higher elastic modulus can be incorporated into the composite. Second, although we axially loaded the phantoms to force levels exceeding those imposed on porcine collateral ligaments (Blank et al., 2020b), the phantoms had larger cross-sectional areas, and therefore achieved lower peak stresses than those studied in collateral ligaments. However, collateral ligament loads in common knee poses and clinical laxity assessments are small (estimated to be <2 MPa using average tensions (Höher et al., 1998; Schafer et al., 2016) and cross-sectional areas (Wilson et al., 2012)). Thus, the average peak stress of 5.3 MPa imposed on the phantoms in our study should capture the stresses in ligaments during many daily activities and clinical assessments. Third, we measured shear wave speeds with the phantoms clamped into rigid metal grips that provided consistent, controlled boundary conditions, but do not replicate the softer bony boundary conditions nor the non-uniform loading across the width of collateral ligaments in vivo (Willinger et al., 2020). Ongoing work in our group is focused on building upon this study to explore the effects that more physiological boundary conditions have on measured shear wave speeds, which should facilitate the translation of our tensiometer in more realistic loading conditions.

The tunable properties of the phantoms and their predictable effects on the shear wave speed squared – stress relationship make the phantoms extendable to a variety of applications. For example, to investigate the application of shear wave tensiometry to the diagnosis of ligament pathologies, the properties of the phantoms could be modified to simulate diseased or damaged ligaments. Furthermore, these ligament phantoms could facilitate ongoing investigations into implementing shear wave elastography on transversely isotropic tissues (Aristizabal et al., 2014; Brum et al., 2014). Beyond shear wave tensiometry and elastography, there have been growing efforts to use synthetic materials to model biomechanically realistic tissues (Pineda Guzman and Kersh, 2021; Sparks et al., 2015) and joints (Cui et al., 2021; Ruiz and Dhaher, 2021) for use in medical education, preoperative planning, and research investigations. Along with the phantom’s tunable properties and simple fabrication process, the phantoms are fabricated from low cost, accessible (i.e., commercially available), and easy-to-handle materials, making them attractive options for fabricating biomechanically realistic fibrous tissues or for incorporating into joint models.

In summary, we found that shear wave speeds measured on ligament phantoms are sensitive to differences in tension, as well as their material properties. Despite the widespread use of phantoms for shear wave elastography, we report the first known use of phantoms targeted for shear wave tensiometry. The similar structure, mechanical properties, and shear wave speed – stress relationships between the phantoms and collateral ligaments should make our ligament phantoms useful for enhancing ligament tensiometers to improve surgical interventions and diagnose ligament pathologies, as well as for investigating the structure-function relationships of ligaments.

Highlights.

• Yarn-silicone phantoms have similar mechanical properties to ligaments.

• The phantoms have similar shear wave speed – stress relationships to ligaments.

• The phantoms undergo fatigue-induced changes in properties with repeated loading.

• The shear wave speed – stress relationship is dependent on phantom properties.

• Yarn-silicone phantoms are viable specimens for shear wave tensiometer development.