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The mechanical behaviors of biological soft tissues are challenging to describe abstractly, with each individual tissue potentially characterized by its own unique nonlinear, anisotropic, and viscoelastic properties. These complexities are exacerbated by patient to patient variability, both mechanically and anatomically, and by inherent constitutive heterogeneity. Despite these challenges, computational models of whole knee biomechanics can be instrumental in describing the onset and progression of injury and disease. In this work, a three-dimensional whole knee computational model was developed using patient-specific anatomy, containing tissues with constitutive relationships built from relevant experimental investigations. In an effort to address the common assumption of linear elastic descriptions of articular cartilage in whole knee models, this work investigates the implications, with respect to macroscopic kinematics and local deformation, of incorporating physiologically motivated and mechanically accurate constitutive heterogeneity in articular cartilage, highlighting the sensitivities of each corresponding level of constitutive complexity. We show how the inclusion of representative cartilage material models affects deformation distributions within the joint, as well as relative joint motion. In particular, the assumption of linear elasticity in articular cartilage results in an overprediction of joint motion and significantly affects predicted local cartilage strains, while full-field, mechanically heterogeneous cartilage descriptions have a less drastic effect at both the tissue and joint levels. Nonetheless, joints containing complete descriptions of articular cartilage heterogeneity may be an integral component in building comprehensive computational tools to advance our understanding of injury and disease mechanisms.

The inverse Langevin function is an integral component to network models of rubber elasticity with networks assembled using non-Gaussian descriptions of chain statistics. The non-invertibility of the inverse Langevin often requires the implementation of approximations. A variety of approximant forms have been proposed, including series, rational, and trigonometric divided domain functions. In this work, we develop an error-minimizing framework for determining inverse Langevin approximants. This method can be generalized to approximants of arbitrary form, and the approximants produced through the proposed framework represent the error-minimized forms of the particular base function. We applied the error-minimizing approach to Pade approximants, reducing the average and maximum relative errors admitted by the forms of the approximants. The error-minimization technique was extended to improve standard Pade approximants by way of understanding the error admitted by the specific approximant and using error-correcting functions to minimize the residual relative error. Tailored approximants can also be constructed by appreciating the evaluation domain of the application implementing the inverse Langevin function. Using a non-Gaussian, eight-chain network model of rubber elasticity, we show how specifying locations of zero error and reducing the minimization domain can shrink the associated error of the approximant and eliminate numerical discontinuities in stress calculations at small deformations.

Based on a reader comment, it has become clear that, in the originally published version of the article, Fig. 1 was published with incorrect anatomical labels.