Three-dimensional finite element modeling of ligaments: Technical aspects

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Abstract

The objective of this paper is to describe strategies for addressing technical aspects of the computational modeling of ligaments with the finite element (FE) method. Strategies for FE modeling of ligament mechanics are described, differentiating between whole-joint models and models of individual ligaments. Common approaches to obtain three-dimensional ligament geometry are reviewed, with an emphasis on techniques that rely on volumetric medical image data. Considerations for the three-dimensional constitutive modeling of ligaments are reviewed in the context of ligament composition and structure. A novel approach to apply in situ strain to FE models of ligaments is described, and test problems are presented that demonstrate the efficacy of the approach. Approaches for the verification and validation of ligament FE models are outlined. The paper concludes with a discussion of future research directions.

Introduction

The skeletal ligaments are short bands of tough fibrous connective tissue that bind bones together across joints. Their mechanical function is to guide normal joint motion and restrict abnormal joint movement. These functions are assisted by the congruent geometry of the articulating joint surfaces and musculotendinous forces. Ligaments can be subjected to extreme stress while performing their role in restricting abnormal joint motions and can be damaged or completely disrupted when overloaded. Excessive stretching or disruption can result in gross joint instability, resulting in altered joint kinematics, altered load distribution, and increased vulnerability to injury of other ligaments and musculoskeletal tissues. Eventually, degenerative joint disease may result from alterations in load bearing and joint kinematics.

Because ligamentous instability can greatly restrict the activity level of an individual and may result in degenerative disease, basic and applied research efforts have examined ligament injury mechanisms, techniques for ligament repair and reconstruction, and rehabilitation methods for use during the healing period. These studies have helped to elucidate details of the natural history of ligament injury and healing from biomechanical, histological, and biochemical viewpoints. However, fundamental mechanical questions regarding the role of individual ligaments, the mechanisms of ligament injury, and the efficacy of reparative/reconstructive procedures persist. This is partially due to inherent limitations of experimental studies such as their high cost, low sensitivity, and the difficulties associated with accurate measurement of basic kinematic and mechanical quantities, both in vivo and in vitro. The use of computational methods for the study of joint mechanics can elucidate ligament function and yield information that is difficult or impossible to obtain experimentally [2], [3], [4], [5], [6]. In particular, the finite element (FE) method offers the ability to predict spatial and temporal variations in stress, strain, and contact area/forces. The FE method also provides a standardized framework for parameter studies, such as evaluation of multiple clinical treatments. Further, subject-specific FE modeling of ligament stress–strain behavior can potentially accommodate the large intersubject variability in joint kinematics and resting ligament tensions, which can limit the sensitivity of experimental and clinical investigations [7].

The vast majority of studies that have employed computational methods to examine ligament mechanics have used a one-dimensional representation of ligament geometry [3], [8], [9], [10]. This entails using either single- or multiple-line elements [10] while allowing load transfer to bones at single or multiple points [11]. A one-dimensional representation requires only a few parameters to control load-elongation behavior, and overall in situ tension can be specified with a single scalar value. This approach has proved useful for predicting joint kinematics under the application of external loads (e.g., [12]), but it possesses several significant shortcomings: (1) nonuniform, 3D stresses and strains cannot be predicted, and (2) multiple sets of parameters and initial tensions routinely produce nearly identical predictions of joint kinematics. Ligaments are subjected to highly nonuniform deformations in vivo that result from a combination of tension, shear, bending, and compression [13], [14], and the regional contribution of a ligament to joint stability changes with joint orientation [15], [16], [17], [18], [19], [20]. A three-dimensional FE modeling approach is required to capture these characteristics.

Three-dimensional FE modeling of ligament stress–strain behavior is complicated by highly anisotropic, nonlinear material behavior, large deformations and complex geometry and boundary conditions. The objectives of this paper are to describe strategies for addressing these important technical aspects of the computational modeling of ligaments with the FE method. In particular, this paper describes strategies for FE modeling of ligament mechanics, methods for obtaining ligament geometry for computational models, considerations for the constitutive modeling of ligaments, the representation of in situ strains in FE models of ligaments, and the verification and validation of ligament FE models. Focus is placed on techniques that can be used when representing ligaments with three-dimensional continuum or shell elements.

Section snippets

Strategies for representing ligaments in joint models

Two strategies have been used for the three-dimensional FE analysis of ligament mechanics. In the first approach, a model of the entire joint is constructed, including all supporting soft tissue structures [21], [22]. The influence of arbitrary external loads and/or displacements on joint kinematics and ligament mechanics can then be studied. This approach can predict joint kinematics, ligament stresses, strains, insertion site forces and load transfer to the bones via contact. However, the

Geometry acquisition

The acquisition of accurate geometry for the ligament(s) and possibly the bones is a fundamental requirement for the construction of three-dimensional FE models of ligaments. Laser scanning and medical imaging are the primary techniques that have been used for this purpose. Laser scanning can be very accurate, but cannot differentiate between the ligament of interest and surrounding bone and soft tissue structures. Further, it can only digitize geometry that is visible directly from the laser

General considerations

Constitutive equations are used to describe the mechanical behavior of ideal materials through specification of the dependence of stress on variables, such as the deformation gradient, rate of deformation, temperature, and pressure. The accurate description and prediction of the three-dimensional mechanical behavior of ligaments by constitutive equations remains one of the challenges for computational modeling. The development and application of these constitutive models relies on an

In situ strain

When a ligament is separated from one or both of its insertions to bone, it will retract. The strain distribution that corresponds to that tension will be referred to herein as the in situ strain [19], [132]. This terminology is used to differentiate it from residual strain/stress, which results from internal forces that are self-equilibrated without any externally applied boundary conditions. In the ligaments of diarthrodial joints, typical in situ strains are approximately 3–10% [17], [19],

Verification and validation

The phrase “verification and validation” has become popular in the recent literature on computational mechanics (see e.g [139], [140]). In the context of the present paper, verification refers to the process of determining whether or not an FE model of a ligament can be used to represent the underlying principles of continuum mechanics with sufficient accuracy. Verification has two parts: (1) testing the ability of constitutive models, element technology, contact algorithms, etc. in an FE

Discussion and future directions

The objective of this paper was to describe techniques that can facilitate the construction, analysis and validation of FE models of ligaments. The authors hope that this information will assist other investigators in their research and provide guidelines for the development and critical assessment of ligament FE models. The methodologies described in this work can be readily adapted to the study of many different ligamentous structures and joints. This should provide a solid foundation for

Acknowledgments

Financial support from NIH grants #AR47369 and #AR050218 is gratefully acknowledged. The authors thank their collaborators at the University of Pittsburgh, Dr. Richard Debski and Ms. Susan Moore, for providing the CT data of the inferior glenohumeral ligament, Drs. Dennis Parker and S.-E. Kim of the Department of Radiology at the University of Utah for implementing the MR acquisition sequences that were used for short T2 imaging of ligaments, and Dr. David Weinstein of the Scientific Computing

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