Introduction:: Soft biological tissues rely on complex and resilient extracellular matrices for support during large in vivo deformations. Many of these tissues are thin and nearly membranous [1], [2], making planar biaxial testing an attractive loading modality for characterizing their passive mechanical behavior [3]. Advances in digital image correlation (DIC) [4], [5], boundary force acquisition [6], and inverse approaches describing the sample’s mechanical properties [7]–[9] have helped overcome limitations of planar biaxial testing, offering researchers access to unprecedented information regarding the mechanical function of soft tissues. The objective of this study was to improve quantifications of soft tissue mechanics produced from our novel biaxial testing protocols [6]. To do so, we introduced a generally orthotropic constraint to our Generalized Anisotropic Inverse Mechanics (GAIM) method requiring positive-definite stiffness tensors and strain energy functions [10]–[12]. Here, we validate our updated method using simulated and experimental samples that assess GAIM’s ability to quantify mechanical stiffness, anisotropy, and the spatial heterogeneity of these properties [13]–[15].
Materials and Methods:: GAIM assumes a linear relationship between Green strains and second Piola-Kirchoff stresses at static equilibrium, allowing us to quantify sample stiffness (first Kelvin modulus, K1) and the degree (R) and direction (φ) of relative mechanical anisotropy [12]. We introduced a constraint to GAIM limiting the neo-Hookean stiffness tensor to a generally orthotropic form using techniques from Li and Barbič to promote numerical stability and positive-definite strain energy functions [16]. To validate the updated form of GAIM, we computationally simulated biaxial testing and experimentally biaxially tested thin PDMS gels (1 mm thickness) [13], thick PDMS gels (1 – 5 mm thickness, increasing from left to right), and TissueMend[14], [15], a mechanically anisotropic patch used to repair tendons. Finite element simulations of our unique biaxial extensions were constructed in ANSYS R2021, then nodal displacements and boundary forces were exported as inputs for GAIM analyses. Cruciform geometries were created and assigned elastic constants representing idealized mechanical behavior for PDMS (linear isotropy; E = 1.32 MPa, ν = 0.49) or TissueMend (linear orthotropy; E1= 12 MPa, E2= 2 MPa, μ12= 1.64 MPa, and ν12= 1.2). Planar biaxial testing was also conducted. All samples were preloaded and preconditioned [6], [17], then subjected to each of our fifteen unique biaxial extensions [6], [11], [12]. Full-field displacements, determined via DIC, and normal and shear boundary forces were inputs for the experimental GAIM analyses. Simulated and experimental results were then compared and related to literature where possible.
Results, Conclusions, and Discussions:: Results:There was excellent agreement between simulations and experiments. Thin PDMS gels (Sim. K1 = 2.59 N/mm; Exp. K1 = 2.50 N/mm) exhibited stiffnesses in-line with published values [13] and our relative mechanical anisotropy metric (Sim. R = 0; Exp. R = 0) indicates isotropy, as expected (Fig. 1). Consistent with increased thickness, thick PDMS gels exhibited stiffness increases from left to right for simulated (not shown) and experimental samples. Relative anisotropy results were muddled, suggesting a limitation of GAIM’s ability to detect mechanical isotropy in variably thick samples. GAIM produced first Kelvin moduli of 14.7 and 12.4 N/mm for simulated and experimental TissueMend samples, respectively, agreeing with uniaxial metrics of TissueMend’s stiffness [14], [15]. Simulated TissueMend produced a high anisotropy metric (R = 0.63) aligning with the difference between E1 and E2 input into ANSYS, whereas the experimental sample exhibited more moderate anisotropy (R = 0.35). As expected, both samples exhibited a preference for their long-axis, which was aligned with the x-axis of our testing system (φ = 0°).
Discussion:To refine our quantifications of soft tissue mechanics, we introduced a generally orthotropic constraint to our GAIM method. This constraint requires positive-definite stiffness tensors and strain energies, improving the physiological significance of our solutions. Validation revealed excellent agreement between our results and expected stiffnesses of PDMS samples. GAIM slightly underestimated the stiffness of the TissueMend sample, likely due to our use of nonlinear kinetics. In contrast, mechanical anisotropy was detected well in samples of relatively constant thickness and poorly in the PDMS sample with variable thickness. To address this issue, we may use a smoothness constraint limiting anisotropy differences between neighboring partitions.
Conclusions: Here, we updated our inverse method, then validated it using simulated and experimental datasets from samples of varying mechanical complexity. We also highlighted the utility of our biaxial testing protocols and inverse method for quantifying mechanical anisotropy, producing one of the first biaxial characterizations of TissueMend’s anisotropic mechanical behavior. In the future, we plan to implement this method in a piecewise manner accounting for nonlinear kinetics, then use this method to study spatially heterogenous soft tissues.
Acknowledgements (Optional): : This work was funded by a grant from the National Science Foundation Division of Civil, Mechanical and Manufacturing Innovation (ID, 2030173) to CMW. The authors would also like to thank Mark Nemcek, Michael Chiariello, Shreya Sreedhar, Elizabeth Gunderson, Connor Link, Riley Pieper, and Jiujiu Pan for their assistance on various projects that have made this work possible.
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