Assessing sensitivity of predictive shoulder simulations to uncertain soft tissue properties

Abstract

Accurate modeling of glenohumeral joint mechanics requires reliable representations of passive structures such as ligaments and articular cartilage, yet these tissue properties are un-available in vivo on a subject-specific basis. This study evaluated how variability in ligament and cartilage properties propagates to joint-level outcomes. Using a validated musculoskeletal shoulder model and two Monte Carlo simulation frameworks, we independently varied glenohumeral ligament (stiffness and slack length) and cartilage (elastic modulus and thickness) parameters to predict secondary kinematics and joint contact mechanics during scaption. Regression analysis revealed that ligament slack length, particularly for the anterior and middle glenohumeral ligaments (A-IGHL and MGHL), had the most significant influence across outcomes. Articular cartilage properties also contributed significantly to contact mechanics, reinforcing their role as key contributors to load distribution. These findings are especially relevant for modeling pathological conditions such as osteoarthritis and surgical intervention such as joint replacement and ligament reconstructions, where cartilage and ligament structure and properties are altered. This work underscores the importance of accounting for soft tissue variability in subject-specific shoulder simulations and highlights targets for advancing in vivo characterization methods.

1. Introduction

Musculoskeletal models are powerful tools for investigating joint mechanics and have supported exploration across various domains including (1) general biomechanical principles, (2) effects of task conditions, (3) injury or surgical intervention, and (4) inter-individual morphological differences (Bolsterlee et al., 2013). Specifically, upper limb models have been applied to study shoulder biomechanics, ergonomics (McFarland et al., 2019), tendon transfers (Saul et al., 2003), rotator cuff injuries (Vidt et al., 2018), and wheelchair propulsion (Bregman et al., 2009), among other applications (Bolsterlee et al., 2013).

Musculoskeletal modeling inherently requires choices to be made that balance model fidelity versus computational demand and complexity. Model simplifications are often necessary due to two primary constraints: the lack of comprehensive, high-quality anatomical or physiological data, and the need to manage computational cost and stability. As a result, many models rely on generalized assumptions or estimated parameters that may not fully capture individual variability or complex biomechanical behavior. For example, few models have attempted to characterize soft tissue loading within the shoulder joint due to limitations addressed above—a stark contrast to more extensively studied joints like the knee (Bei and Fregly, 2004; Lenhart et al., 2015; Smith et al., 2018). Although a few six degree-of-freedom (DOF) shoulder models have been proposed (Khandare and Vidt, 2023; Quental et al., 2016), current limitations include the simplification of articulating surfaces using basic geometries like spheres or radial curves, lack of key stabilizing structures such as the joint capsule and ligaments, and generalized property assignment to passive structures. To predict internal tissue loads, detailed material properties for soft tissues—particularly ligaments and articular cartilage—are essential. Unfortunately, these properties are not easily or reliably measured in vivo. This introduces parametric uncertainty into model predictions, particularly for dynamic movements where soft tissues play a stabilizing and load-bearing role. To determine whether this uncertainty materially impacts joint mechanics, it is critical to assess how sensitive model outcomes are to variations in soft tissue properties. If these passive structures and their properties do exert a meaningful effect, this indicates the need for further characterization of the soft tissues of the shoulder for accurate or personalized models. To address this gap, our objective was to use a probabilistic modeling approach to investigate how variability in ligament and cartilage material properties propagates through shoulder simulations. Specifically, we investigated parameters with limited or incomplete characterization in literature: ligament slack length, which remains unquantified; ligament stiffness, reported in cadaveric studies (Bigliani et al., 1992; Boardman et al., 1996; McMahon et al., 1998; Smeets et al., 2017; Ticker et al., 1996); articular cartilage thickness, characterized in limited MRI and cadaveric studies (Carlson et al., 2023; Schleich et al., 2017; Yeh et al., 1998; Zumstein et al., 2014); and articular cartilage modulus, characterized in one cadaveric study (Huang et al., 2005). We examined the influence of these parameters on joint kinematics and internal tissue loading during scaption—a functional and clinically relevant movement (Timmons et al., 2013). Our hypotheses are: (1) ligament parameters will have primary influence on secondary kinematics due to their primary function as limiters of humeral head dislocations, (2) articular cartilage parameters will have primary influence on contact outcomes due to their primary function as the cushioning between bony surfaces.

2. Methods

Model.

An existing 50th percentile male upper extremity musculoskeletal model (Saul et al., 2015) was used as a foundation. The original model included 7 rotational degrees of freedom for the shoulder, elbow, forearm, and wrist (Wu et al., 2005), as well as 50 Hill-type muscle actuators spanning these joints (properties defined in (Saul et al., 2015)). Clavicular and scapular movement is defined by regression equations relating rotations to humeral posture (de Groot and Brand, 2001). In this study, the base model was augmented in OpenSim (Delp et al., 2007) to include additional translational degrees of freedom (DOFs), force actuators representing ligaments and other passive structures, and contact between the humeral head and glenoid fossa. The 3 translational DOFs allow humeral head translation relative to the glenoid center in the superior/inferior (S/I), anterior/posterior (A/P), and medial/lateral (M/L) directions.

Articular cartilage surfaces for the humeral head and glenoid were represented by triangulated meshes obtained by duplicating and reducing the bone surfaces to the articulating surface (Meshmixer v3.5, Autodesk, Inc., 2017 San Rafael, CA). Surfaces were enlarged relative to each underlying bone to account for cartilage thickness. Cartilage thickness for each surface was set to 2 mm, based on reported values for humeral head and glenoid cartilage derived from four published studies (range of reported means: 1.07 – 2.14 mm) (Carlson et al., 2023; Schleich et al., 2017; Yeh et al., 1998; Zumstein et al., 2014). The glenoid and humeral head surfaces were meshed using 9,836 and 1,942 triangle elements for a mesh density of 2.5 triangle/mm², respectively, based on the convergence study performed for a knee model (Smith et al., 2018). Glenoid and humeral head articular cartilage were assigned elastic moduli of 2.2 and 4.2 MPa, respectively, based on average measurements reported in a cadaveric indentation study (Huang et al., 2005) and a Poisson’s ratio of 0.45 (Dixit et al., 2020; Lenhart et al., 2015). The contact pressure ($p$) on an individual triangle in each cartilage mesh is calculated according to elastic foundation theory developed for articular cartilage (Eq. (1)) as a function of the depth of penetration between the contacting cartilage surfaces:

$$p=-E\frac{(1-\nu )}{(1+\nu )(1-2\nu )}ln\left(1-\frac{d}{h}\right)$$ (1)

where $E$ is cartilage elastic modulus, $\nu$ is cartilage Poisson’s ratio, $d$ is local overlap depth and $h$ is local cartilage thickness (Bei and Fregly, 2004; Blankevoort and Huiskes, 1991; Smith et al., 2018).

Ligament actuators representing the coracohumeral ligament (CGHL) and superior (SGHL), medial (MGHL), and anterior (A-IGHL) and posterior (P-IGHL) components of the inferior glenohumeral ligaments were included. Ligaments were placed at mean attachment points previously reported based on computed tomography of cadaveric specimens (Yang et al., 2010). Each glenohumeral ligament was represented by one nonlinear passive actuator and the combined structure of the glenohumeral capsule and labrum was represented by 13 actuators distributed around the glenohumeral joint. Wrapping objects were used to model ligament wrapping over bony geometries. Passive actuators were modeled using a nonlinear passive spring-damper formulation originally introduced by Blankevoort and Huiskes. This model captures the characteristic force-strain behavior of ligaments, with a quadratic “toe” region at low strains and a linear region at higher strains (Blankevoort and Huiskes, 1991). Linear ligament stiffnesses, defined as the slope of load versus strain, were determined based on cadaveric tensile data reporting yield or ultimate force and corresponding strain. These average values were used to estimate stiffness for each ligament and ensure stable simulation performance (Bigliani et al., 1992; Boardman et al., 1996; McMahon et al., 2001; Smeets et al., 2017; Ticker et al., 1996) (Table 1). Currently, no experimental data are available to inform glenohumeral ligament slack lengths. Therefore, the initial slack length – the length at which the ligament begins to produce a force – was chosen such that it would occur at 30° abduction, where the shoulder passive moment is closest to zero (Kodek and Marko Munih, 2003).

Table 1.

Literature and Nominal Glenohumeral Ligament Properties.

Glenohumeral Ligament (GHL)	Literature	Nominal Model Values
	Approx. Average Linear Stiffness (N/ strain)	Linear Stiffness (N/strain)	Slack Length (mm)
Superior (SGHL)	5004	600	35.5
Medial (MGHL)	–	600	26.3
Anterior Inferior (A-IGHL)	8001 28002 20003	1000	19.7
Posterior Inferior (P-IGHL)	–	1000	25.3
Coracohumeral (CGHL)	10004	1500	44.0

¹(Bigliani et al., 1992).

²(McMahon et al., 1998).

³(Ticker et al., 1996).

⁴(Boardman et al., 1996).

Experimental Protocol.

Experimental data from a previously recorded healthy male subject (age:31 years; height:180.2 cm; weight:155.2 lbs.; right hand dominant) performing scaption was used to inform computational sensitivity analyses evaluated here (McFarland et al., 2020). The participant had: (1) no history of upper limb injury or pathology, (2) no neuromuscular impairments, and (3) no physical impediments to performing the required exertions. The subject provided written informed consent in accordance with NC State University Institutional Review Board. These data were used to anthropometrically scale the model (using OpenSim ScaleTool) and perform inverse kinematics to define the joint angles during scaption (IKTool).

Sensitivity Analyses.

To evaluate the influence of ligament and articular cartilage properties on loading and kinematic outcomes, two Monte Carlo approaches were used. Based on a power analysis with α = 0.05 and power = 0.80, a minimum of 196 simulations were required to detect a small effect size (f² = 0.05). To account for an anticipated 20 % simulation failure rate, a total of 250 simulations were conducted (GPower v.3.1.9.7). 250 parameter sets of ligament properties (slack length and linear stiffness) were randomly selected from independent normal distributions centered around the model’s nominal values. Standard deviations were set to ± 10 % of the nominal value for slack length and ± 30 % of the nominal value for linear stiffness, which represents approximately 1.5–2 standard deviations and is consistent with the approach used by Smith et al. (Smith et al., 2016) (Fig. 1). Separately, 250 parameter sets of articular cartilage properties (thickness and elastic modulus) were randomly selected from independent normal distributions centered around the model’s nominal values. Standard deviations were set to ± 30 % of the nominal value for both parameters, which represents approximately 1.5–2 standard deviations (Carlson et al., 2023; Huang et al., 2005; Schleich et al., 2017; Yeh et al., 1998; Zumstein et al., 2014)(Fig. 1).

Fig. 1.

Parameter selection for investigating soft tissue property variability within the musculoskeletal model. Top: A total of 250 parameter sets were randomly generated for ligament slack length and linear stiffness using independent normal distributions centered on the model’s nominal values, with standard deviations of 10 % and 30 %, respectively. Bottom: Likewise, 250 sets of cartilage property parameters—thickness and elastic modulus—were sampled from independent distributions with standard deviations of 30 %.

Each model was used in a simulation applying the Concurrent Optimization of Muscle Activations and Kinematics (COMAK) algorithm (Smith, 2017; Smith et al., 2018) over 31°-130° of scaption (as recorded from the participant). For each simulation, a settling period was first used to allow initial glenohumeral contact to reach equilibrium; in this period the initial posture was held constant for 0.02 s seconds (20 simulation steps). COMAK solves for the muscle activation and secondary kinematics (humeral head translations) necessary to generate the desired primary kinematics (joint rotations). When solving for muscle activations and secondary kinematics, COMAK minimizes an objective function proportional to the sum of weighted muscle volumes and activations squared while satisfying constraints. These constraints include muscle activations between 0 and 1, simulated accelerations equivalent to those defined from Inverse Kinematics, and equilibrium such that the secondary coordinate accelerations approach zero.

We augmented COMAK to include an additional glenohumeral stability criterion in which the resultant contact force is required to stay directed within the glenoid boundary (McFarland et al., 2020). This was incorporated into COMAK as a penalty term added to the cost function proportional to the amount of force directed outside of the glenoid boundary. Each simulation was used to predict humeral head location and glenohumeral contact force, pressure, and area. Simulation success was evaluated by convergence, which was defined by whether the optimization algorithm found a solution at each postural increment (reported by COMAK) across the simulated range of motion. Simulations were considered successful if > 95 % of steps converged to a solution.

Using multiple linear regression, unstandardized and standardized regression coefficients were calculated to quantify the magnitude and relative importance of each predictor (p < 0.05). Model fit was evaluated using coefficient of determination (adjusted R²) and root mean square error (RMSE). High adjusted R² values alongside low RMSE indicated strong model performance and accurate outcome prediction based on ligament and articular cartilage parameters. Coefficient of variation (CV) was computed for each outcome to assess variability across simulations and calculated as the ratio of the standard deviation to the mean of the outcome variable. CV analysis elucidates consistency and reliability of the biomechanical responses to changes in ligament slack length and linear stiffness, articular cartilage thickness and elastic modulus.

3. Results

Convergence.

Out of 250 total simulations evaluating ligament parameter effects, the COMAK simulation algorithm successfully completed 193 simulations (77.2 %). The remaining 57 simulations either failed to complete or did not reach 95 % convergence and were excluded from further analysis. Similarly, for the 250 simulations assessing influence of articular cartilage parameters, 200 simulations (80.0 %) were considered successful. When simulation success was examined relative to input parameters (Fig. 2), only the A-IGHL ligament demonstrated a possible relationship with convergence: unsuccessful simulations tended to occur at combinations of shorter and stiffer or longer and more compliant ligament properties.

Fig. 2.

Simulation success outcomes for the two Monte Carlo sensitivity analyses targeting (1) ligament properties (left panel) and (2) articular cartilage properties (right panel). Left: Ligament predictor space is shown with linear stiffness (N) on the y-axis and ligament length (mm) on the x-axis. Right: Cartilage predictor space is plotted with thickness on the y-axis both glenoid (top) and humeral head (bottom) cartilage and elastic modulus on the x-axis for both glenoid (left) and humeral head (right) cartilage. In all panels, incomplete simulations are shown in red, those with less than 95 % convergence of steps in purple, and fully completed simulations with greater than 95 % convergence in yellow. Among all predictors, only the A-IGHL ligament exhibited a potential relationship with simulation success—where failures clustered at combinations of shorter and stiffer or longer and more compliant ligament properties.

The standard error of the mean (SEM) changed by < 1 % after including more than 181 simulations for ligament properties and 194 simulations for articular cartilage (Fig. 3); the SEM for 193 ligament simulations and 200 articular cartilage simulations was < 5 % and 7 % of the range of mean values, respectively (Table 2). These results provide confidence that the predicted outcome means are both stable and representative.

Fig. 3.

Example plots of the standard error of the mean (SEM) for humeral head location as additional simulations were completed. Left panel: SEM results for ligament parameter variations, with final SEM values of 0.02 mm (superior/inferior location) and 0.04 mm (anterior/posterior location). Right panel: SEM results for articular cartilage parameter variations, with final SEM values of 2.97e-3 mm (superior/inferior location) and 0.01 mm (anterior/posterior location). These values confirm the simulations produced stable estimates for humeral head positioning.

Table 2.

Standard Error of Mean (SEM) for Ligament and Cartilage Property Changes.

Outcome	Ligament		Cartilage
	SEM (N = 203)	% Range of Mean	SEM (N = 208)	% Range of Mean
Superior/ Inferior Location (mm)	0.02	0.77 %	2.97e-3	0.13 %
Anterior/ Posterior Location (mm)	0.04	2.62 %	0.01	0.65 %
Total Contact Area (mm2)	0.54	0.64 %	2.76	2.62 %
Resultant Contact Force (N)	20.22	2.35 %	8.27	0.82 %
Compressive Contact Force (N)	20.11	2.38 %	8.29	0.83 %
Shear Contact Force (N)	2.40	0.93 %	0.48	0.18 %
Peak Contact Pressure (MPa)	0.14	2.88 %	0.37	5.40 %
Average Contact Pressure (MPa)	0.04	4.95 %	0.07	6.44 %

Coefficient of Variation.

S/I humeral head location exhibited greatest variability in response to altered ligament parameters, with CV ranging from 6.85 % to 109.25 % across the motion (Fig. 4, Table S1). A/P location and shear force also demonstrated moderate variability, ranging from 2.36 % to 24.67 % and 5.99 % to 25.96 %, respectively. Notably, variability was highest at lower angles of scaption and progressively decreased as the arm elevated. This pattern suggests that ligament contributions to joint stability are more variable—and potentially more influential— in postures near the onset of ligament engagement.

Fig. 4.

Coefficient of variation ( %) for predicted outcomes across the simulated motion cycle. Higher variability was observed at lower arm angles and decreased with arm elevation. Superior/inferior location exhibited high variability, while anterior/posterior location and shear force exhibited moderate variability. All other outcomes remained below 12 %.

Multilinear Regression Models.

The multiple linear regression models demonstrated strong predictive ability, with adjusted R² values ranging from 0.82 to 0.99. Ligament slack length (SL) had a greater unstandardized effect (B) on predicted outcomes compared to linear stiffness (SF) (Fig. 5B, Table S2). This finding is consistent with the mechanical behavior of ligaments, which typically operate within the toe region of their force–elongation curve as they begin to engage (Fig. 6). In this region, small variations in slack length can determine whether a ligament is engaged at all, thus strongly influencing joint mechanics. Specifically, the middle (B = 0.06 mm/mm, β = 0.70, p < 0.0001) and superior (B = 0.02 mm/mm, β = 0.28, p < 0.0001) glenohumeral ligament slack lengths had a positive effect on S/I location, where an increase in slack length resulted in a more superior location. Whereas the anterior-inferior glenohumeral ligament (B = −0.03 mm/mm, β = −0.21, p < 0.0001) has a negative effect on S/I location. With regard to A/P location, the anterior-inferior glenohumeral ligament (B = 0.19 mm/mm, β = 0.60, p < 0.0001) had a positive effect, while the coracohumeral (B = −0.07 mm/mm, β = −0.54, p < 0.0001) and superior glenohumeral (−0.05 mm/mm, β = −0.26, p < 0.0001) ligaments have a negative effect. In contrast, linear stiffness becomes more influential once the ligament is under tension, a relationship reflected in the standardized coefficients where the relative importance of SL and SF is more balanced (Fig. 5C, Table S3).

Fig. 5.

Multilinear regression analysis of ligament parameter influence on predicted outcomes. Panel A shows the mean and standard deviation (orange) of superior/inferior humeral head location (top), anterior/posterior location (middle), and shear force (bottom). Panel B presents the unstandardized regression coefficients (B), illustrating that ligament slack length (SLACK – pink) has a greater effect on predicted outcomes than linear stiffness (STIFF – solid purple), particularly due to its role in early-phase ligament engagement. Panel C displays the standardized coefficients (β), indicating that the relative importance of slack length and stiffness varies across outcomes. The models yielded high adjusted R² values (0.85-0.91), indicating that the ligament parameters explained a substantial portion of the variability in these predicted outcomes.

Fig. 6.

Ligament force-strain curve showing the early-phase quadratic “toe-region” (pink) and the linear region (purple).

Coefficient of Variation.

When examining simulations with varying articular cartilage properties, the greatest variability in predicted outcomes was observed early in the simulated motion. Peak contact pressure and average contact pressure exhibited the most consistently high variability across the whole motion in response to altered cartilage parameters (CV: 10.14 % – 78.26 %) (Fig. 7, Table S4). Thus, the regression analyses for cartilage parameters focused on pressure-related outcomes.

Fig. 7.

Coefficient of variation ( %) for predicted outcomes across the simulated motion with varying articular cartilage properties. The highest variability occurs early in the motion, likely reflecting the model’s effort to establish a stable contact interface between articulating surfaces. After this initial settling phase, variability becomes more consistent across the range of motion. Among the outcomes, peak and average contact pressure exhibited the most consistent high variability across the motion (18.93 % – 78.26 % and 10.14 % – 52.41 %, respectively). All other outcomes remained below 14 % outside of the initial posture (31.96°).

Multilinear Regression Models.

The multiple linear regression models demonstrated strong predictive ability, with adjusted R² values ranging from 0.70 to 0.97 (excluding shear force R² = 0.46) (Tables S5 and S6). The multiple linear regression analysis revealed that cartilage thickness (TH) consistently emerged as the most influential predictor of both mean (β = −0.74 and −0.45, p < 0.0001) and peak contact pressures (β = −0.65 and −0.67, p < 0.0001) (Fig. 8). For instance, a 1 mm increase in humeral cartilage thickness (H_TH) and glenoid cartilage thickness (G_TH) was associated with a 6.59 MPa and 5.92 reduction in peak pressure, respectively. Similarly, 1 mm increase in H_TH and G_TH resulted in 1.19 MPa and 0.77 MPa reduction in average pressure. In contrast, elastic modulus (M), while statistically significant, demonstrated smaller effect sizes for peak pressure (H_M: B = 0.75 MPa/MPa, β = 0.17, p < 0.0001; G_M: B = −2.11 MPa/MPa, β = −0.24, p < 0.0001) and average pressure (H_M: B = 0.26 MPa/MPa, β = 0.34, p < 0.0001; G_M: B = 0.64 MPa/MPa, β = 0.42, p < 0.0001).

Fig. 8.

Multilinear regression analysis of articular cartilage parameter influence on predicted outcomes. Panel A shows the mean and standard deviation of peak contact pressure (top) and average contact pressure (bottom). Panel B presents the unstandardized regression coefficients (B), demonstrating that cartilage thickness (THICK – pink) has a greater effect on predicted outcomes than cartilage elastic modulus (MOD – solid purple). Panel C displays the standardized coefficients (β), reinforcing the greater relative importance of thickness over modulus. However, all parameters were statistically significant in their contributions. The models yielded high adjusted R² values (0.84–0.92), indicating that the articular cartilage parameters explained a substantial portion of the variability in these predicted outcomes.

4. Discussion

Our primary objective was to investigate the effects of soft tissue properties—specifically, glenohumeral ligament slack length and stiffness, and glenoid and humeral cartilage thickness and elastic modulus—on shoulder biomechanics. To accomplish this, we employed a Monte Carlo simulation framework integrated with a co-simulation musculoskeletal model to predict secondary kinematics, muscle forces, and joint contact mechanics during arm elevation in the scapular plane.

Among all parameters, ligament slack length emerged as the most influential on simulation outcomes. This is likely because ligaments primarily operate within the quadratic “toe region” of their force–strain curve, meaning that the posture at which each ligament engages has a particularly strong influence on the predicted outcomes. In this study, the nominal ligament slack length was defined such that engagement occurred at 30° of abduction, the posture where the passive shoulder moment is closest to zero, to reflect a physiologically reasonable reference configuration in the absence of experimental data (Kodek and Marko Munih, 2003). Because we systematically varied ligament slack lengths, we implicitly altered the posture at which each ligament begins to engage. This is evident in the variability in model predictions at low scaption postures as ligaments begin to engage differently across simulations, but performance stabilizes quickly even a few degrees past engagement. Improved data to inform ligament tension behavior near and below 30° of abduction will be critical for reducing variability associated with ligament engagement and improving the numerical stability of future simulations.

When examining individual ligament contributions during scaption, the anterior-inferior and middle glenohumeral ligament properties consistently emerged as strong predictors of superior–inferior, anterior–posterior translation, and shear force. This finding aligns with their primary mechanical role in constraining anterior translation of the humeral head—the direction in which the shoulder is most susceptible to dislocation. Indeed, anterior instability represents the most common type of shoulder dislocation, accounting for approximately 90 % of all cases (Goss, 1988). However, as the direction and complexity of tasks differ from scaption, such as during multiplane or functional movements, the influence of other ligaments is expected to increase as they begin to engage in different loading directions. Furthermore, under conditions involving active exertion or interaction with external forces, higher joint contact pressures and forces would likely occur, though experimental data to validate these effects remain limited.

The standardized multilinear regression results highlighted that linear stiffness (∣β∣ up to 0.90) can be just as important as slack length (∣β∣ up to 0.82) (Table S3.3), supporting the need to consider both properties when modeling shoulder ligaments. Taken together, our findings underscore the need for accurate characterization of ligament slack length to capture ligament engagement behavior and highlight the importance of linear stiffness in later phases of loading. This work provides an important foundation for future investigation of how variations in ligament properties and loading patterns contribute to the onset of shoulder pathologies such as tearing and avulsion of the labroligamentous complex. These insights are particularly relevant in the context of clinical interventions where ligament or capsular properties may be surgically altered. For example, in anterior cruciate ligament (ACL) reconstruction, tendon grafts such as the hamstring are commonly used despite being substantially less stiff than the native ACL (Biuk et al., 2015). Although grafting is less common in shoulder ligament reconstruction, most repair procedures involve reattachment of the labrum, capsule, or glenohumeral ligaments following injuries such as Bankart or SLAP tears. Additionally, selective release of portions of the capsule or ligaments is a recognized treatment for addressing shoulder stiffness after shoulder replacement (Wagner et al., 2021) and for frozen shoulder (Galasso et al., 2023; Liem et al., 2008).

Despite their importance, ligament properties are not readily measurable in vivo. Although implantable sensors capable of directly assessing ligament strain have been developed for the human knee (Zhang et al., 2021), they remain invasive, limited in duration, and require adequate implantation space—factors that restrict their clinical utility. Alternative approaches, such as numerical optimization methods with direct manipulation of cadavers (laxity tests), have been used to estimate tendon slack lengths (Cox et al., 2019; Manal and Buchanan, 2004) and knee ligaments (Kia et al., 2016; Razu et al., 2023; Zaylor et al., 2019), but have not been applied to shoulder ligaments. In recent years, shear wave elastography has emerged as a promising tool for quantifying the stiffness of tendons (Breda et al., 2020; Pasurka et al., 2024) and ligaments (Alzouhayli et al., 2024; Wei et al., 2024) of the knee. However, its application in the shoulder remains underexplored. Further research should prioritize developing robust methods for estimating subject-specific glenohumeral ligament slack lengths and stiffness, which would enhance accuracy and clinical relevance of musculoskeletal models, particularly in scenarios where ligament properties are altered or reconstructed.

In addition, our analysis demonstrated that articular cartilage thicknesses were more impactful and influential than elastic moduli on model outcomes. These results are consistent with elastic foundation theory, which describes how compliant materials such as cartilage distribute load (Equation (1). Specifically, cartilage thickness (h) has a logarithmic relationship with contact pressure, meaning that increases in thickness lead to a more even distribution of stress across the contact surface, thereby reducing localized pressure. However, both properties were statistically significant (p < 0.05) for their contributions to predicted outcomes. This work highlights that the uncertainty and variability of parameters such as articular cartilage thickness and elastic modulus have a meaningful effect on model predictions, emphasizing that accurately characterizing the physiological range associated with these parameters is essential for improving fidelity of subject-specific musculoskeletal models. In the current model, cartilage thickness was assumed uniform at 2 mm, reflecting average values reported across MRI (Schleich et al., 2017) and cadaveric studies (Carlson et al., 2023; Yeh et al., 1998; Zumstein et al., 2014). However, cartilage thickness varies across joint surfaces, and incorporating full segmentations and spatially varying thickness would likely yield more accurate predictions of joint pressure. Future modeling that implements variable cartilage thickness may reduce localized stress concentrations that could contribute to numerical instability or unrealistic joint translations, particularly during high-demand tasks where increased load could result in higher contact loading within the shoulder. Cartilage morphology and thickness can be captured non-invasively using high-resolution, soft tissue-weighted MRI (Graichen et al., 2003; Kladny et al., 1996; Schleich et al., 2017), offering a pathway toward more personalized modeling.

In contrast, cartilage elastic modulus is traditionally measured ex vivo, often via nanoindentation (Armstrong and Mow, 1982; Kabir et al., 2021; Maeda et al., 2024; Mieloch et al., 2019). Recent innovations, such as dualMRI under loading (Chan et al., 2016) and shear wave elastography in the knee (Bedewi et al., 2020; Deng et al., 2022), have shown promise for estimating modulus in vivo. However, comparable methods and similar efforts to quantify cartilage properties in the shoulder remain limited (Huang et al., 2005). This is especially relevant for modeling cartilage degeneration in osteoarthritis (Maeda et al., 2024), where both thickness and modulus evolve with disease.

To properly interpret the results of this study, several methodological limitations must be acknowledged. The parameter ranges used here were selected to capture biologically relevant variability based on values reported in the literature (Bigliani et al., 1992; Boardman et al., 1996; Huang et al., 2005; McMahon et al., 2001; Smeets et al., 2017; Ticker et al., 1996) and precedent from similar sensitivity analyses in the knee joint (Smith et al., 2016). These ranges were informed by cadaveric data, which, while widely used and valuable for understanding structural properties, may not fully represent the biomechanical behavior of living tissues due to changes in tissue hydration, loading history, and biological remodeling. This motivates future work that integrates subject-specific measurements to more accurately reflect physiological conditions.

In addition, muscle geometries were based on a generic upper limb musculoskeletal model (Saul et al., 2015), which may not reflect individual anatomical variability. Ligament attachment sites were defined relative to anatomical landmarks and literature-reported locations and were not varied across simulations. Additionally, each ligament was modeled using a single line of action, which simplifies the complex, multi-fiber structure of ligaments. Modeling ligaments as bundles with multiple strands spanning their full attachment footprints could improve anatomical fidelity and reduce error in force transmission predictions.

Lastly, the capsule and labrum complex were represented in the model as a series of passive structures surrounding the glenohumeral joint. Initially, the labrum was explicitly included as a contact surface, but this approach resulted in substantially increased computational cost with inconsistent simulation convergence. This simplification may underrepresent the passive constraint and stabilizing effects of the capsulolabral complex, highlighting an area for future model refinement.

5. Conclusion

This study is the first to evaluate how variability in passive soft tissue properties—namely ligament slack length and stiffness, and cartilage thickness and modulus—impacts secondary shoulder kinematics and joint loading estimates in a musculoskeletal model. The findings underscore the importance of incorporating parameter variability and sensitivity into musculoskeletal simulations, especially for subject-specific applications. These results highlight the critical role of soft tissue mechanics in model accuracy. Key opportunities supported by these simulations are a need for in vivo characterization of glenohumeral ligament slack length and of glenohumeral ligament and shoulder articular cartilage material property, possibly using methods such as shear wave elastography which has been applied to other soft tissues.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jbiomech.2025.113114.