Comparison of material models for anterior cruciate ligament in tension: from poroelastic to a novel fibril-reinforced nonlinear composite model

Abstract

Computational models of the knee joint are useful for evaluating stresses and strains within the joint tissues. However, the outcome of those models is sensitive to the material model and material properties chosen for ligaments, the collagen reinforced tissues connecting bone to bone. The purpose of this study was to investigate different compositionally motivated material models and further to develop a model that can accurately reproduce experimentally measured stress-relaxation data of bovine anterior cruciate ligament (ACL).

Tensile testing samples were extracted from ACLs of bovine knee joints (N = 10) and subjected to a three-step stress-relaxation test at the toe region. Data from the experiments was averaged and one average finite element model was generated to replicate the experiment. Poroelastic and different fibril-reinforced poro(visco)elastic material models were applied, and their material parameters were optimized to reproduce the experimental force-time response.

Material models with only fluid flow mediated relaxation were not able to capture the stress-relaxation behavior (R² = 0.806, 0.803 and 0.938). The inclusion of the viscoelasticity of the fibrillar network improved the model prediction (R² = 0.978 and 0.976), but the complex stress-relaxation behavior was best captured by a poroelastic model with a nonlinear two-relaxation-time strain-recruited viscoelastic fibrillar network (R² = 0.997).

The results suggest that in order to replicate the multi-step stress-relaxation behavior of ACL in tension, the fibrillar network formulation should include the complex nonlinear viscoelastic phenomena.

Introduction

Computational models of the whole knee joint provide a useful tool for evaluating mechanical parameters of the knee, such as stresses and strains in articular cartilage under different loading scenarios and disease states. However, the outcome and reliability of such models are sensitive to the material model and material properties chosen for tissues, such as ligaments (Beidokhti et al., 2017, Dhaher et al., 2010, Orozco et al., 2018b). Compositionally motivated material models provide a feasible option for accurate and realistic modeling of ligaments at the tissue-level, as opposed to phenomenological models without physical meaning for the model parameters. Moreover, compositionally motivated material models provide a possibility for studying changes in the constituent-specific mechanical material properties, e.g., due to tissue degeneration or regeneration, and may be helpful in the design of tissue engineered ACL grafts.

Ligaments are collagenous connective tissues that attach bone to bone. In the musculoskeletal system, their primary function is to transmit forces in tension, restrict and guide movements of joints, stabilize joints and act as mechanical dampers. Well adapted to the physiological function, they exhibit complex nonlinear and viscoelastic material behavior. Ligaments are mainly composed of water (65 to 80% of total weight), highly organized type I collagen fibers (70 to 90% of dry weight), elastin (few percent), proteoglycans (few percent) and fibroblast cells (Eleswarapu et al., 2011, Woo et al., 2005, Woo et al., 1999).

Previous composition-based tissue-level continuum models of ligaments and tendons have included the poroelastic and fiber-reinforced nature of the tissue (Ahmadzadeh et al., 2015, Khayyeri et al., 2016, Khayyeri et al., 2015, Notermans et al., 2019, Oftadeh et al., 2018). The reinforcing collagen fibrils themselves exhibit viscoelastic behavior (Shen et al., 2011, Svensson et al., 2012, Svensson et al., 2010, Yang et al., 2012), which was incorporated in the fibrillar network of an Achilles tendon model (Khayyeri et al., 2016, Khayyeri et al., 2015, Notermans et al., 2019). While the model performed well in their studies, it inherently fails to capture the two-relaxation-time behavior observed experimentally within the collagen fibrillar network (Gupta et al., 2010, Screen et al., 2013, Shen et al., 2011, Yang et al., 2012). The tissue or fascicle-level viscoelasticity has been observed to be dependent on strain, i.e. higher relaxation at higher strains (Elliott et al., 2003, Gupta et al., 2010, Lynch et al., 2003, Sarver et al., 2003, Screen et al., 2013). The strain-dependent nature of relaxation may originate from the fibrillar network (Raz and Lanir, 2009, Screen et al., 2013), which was not taken into account in previous fiber-reinforced models that distinguish fluid, matrix and fibers.

The purpose of this study was to investigate the ability of different tissue-level compositionally motivated material models to capture anterior cruciate ligament (ACL) stress-relaxation behavior in tension and to further develop a model to account for the complex viscoelastic behavior. We hypothesize that the previously used poroelastic or fibril-reinforced poro(visco)elastic models cannot accurately represent the complex viscoelastic behavior of ACL, as they fail to capture the two-relaxation-time behavior or strain-dependent nature of relaxation. We further hypothesize that a model with a more complex fibrillar network formulation is required to capture this behavior.