Jeffrey A. Weiss

Finite element implementation of incompressible, transversely isotropic hyperelasticity

This paper describes a three-dimensional constitutive model for biological soft tissues and its finite element implementation for fully incompressible material behavior. The necessary continuum mechanics background is presented, along with derivations of the stress and elasticity tensors for a transversely isotropic, hyperelastic material. A particular form of the strain energy for biological soft tissues is motivated and a finite element implementation of this model based on a three-field variational principle (deformation, pressure and dilation) is discussed. Numerical examples are presented that demonstrate the utility and effectiveness of this approach for incompressible, transversely isotropic materials.

FEBio: finite elements for biomechanics.

In the field of computational biomechanics, investigators have primarily used commercial software that is neither geared toward biological applications nor sufficiently flexible to follow the latest developments in the field. This lack of a tailored software environment has hampered research progress, as well as dissemination of models and results. To address these issues, we developed the FEBio software suite (http://mrl.sci.utah.edu/software/febio), a nonlinear implicit finite element (FE) framework, designed specifically for analysis in computational solid biomechanics. This paper provides an overview of the theoretical basis of FEBio and its main features. FEBio offers modeling scenarios, constitutive models, and boundary conditions, which are relevant to numerous applications in biomechanics. The open-source FEBio software is written in C++, with particular attention to scalar and parallel performance on modern computer architectures. Software verification is a large part of the development and maintenance of FEBio, and to demonstrate the general approach, the description and results of several problems from the FEBio Verification Suite are presented and compared to analytical solutions or results from other established and verified FE codes. An additional simulation is described that illustrates the application of FEBio to a research problem in biomechanics. Together with the pre- and postprocessing software PREVIEW and POSTVIEW, FEBio provides a tailored solution for research and development in computational biomechanics.

Material characterization of human medial collateral ligament.

The objectives of this study were to determine the longitudinal and transverse material properties of the human medial collateral ligament (MCL) and to evaluate the ability of three existing constitutive models to describe the material behavior of MCL. Uniaxial test specimens were punched from ten human cadaveric MCLs and tensile tested along and transverse to the collagen fiber direction. Using load and optical strain analysis information, the tangent modulus, tensile strength and ultimate strain were determined. The material coefficients for each constitutive model were determined using nonlinear regression. All specimens failed within the substance of the tissue. Specimens tested along the collagen fiber direction exhibited the typical nonlinear behavior reported for ligaments. This behavior was absent from the stress-strain curves of the transverse specimens. The average tensile strength, ultimate strain, and tangent modulus for the longitudinal specimens was 38.6 +/- 4.8 MPa, 17.1 +/- 1.5 percent, and 332.2 +/- 58.3 MPa, respectively. The average tensile strength, ultimate strain, and tangent modulus for the transverse specimens was 1.7 +/- 0.5 MPa, 11.7 +/- 0.9 percent, and 11.0 +/- 3.6 MPa, respectively. All three constitutive models described the longitudinal behavior of the ligament equally well. However, the ability of the models to describe the transverse behavior of the ligament varied.

Subject-Specific Finite Element Model of the Pelvis: Development, Validation and Sensitivity Studies

A better understanding of the three-dimensional mechanics of the pelvis, at the patient-specific level, may lead to improved treatment modalities. Although finite element (FE) models of the pelvis have been developed, validation by direct comparison with subject-specific strains has not been performed, and previous models used simplifying assumptions regarding geometry and material properties. The objectives of this study were to develop and validate a realistic FE model of the pelvis using subject-specific estimates of bone geometry, location-dependent cortical thickness and trabecular bone elastic modulus, and to assess the sensitivity of FE strain predictions to assumptions regarding cortical bone thickness as well as bone and cartilage material properties. A FE model of a cadaveric pelvis was created using subject-specific computed tomography image data. Acetabular loading was applied to the same pelvis using a prosthetic femoral stem in a fashion that could be easily duplicated in the computational model. Cortical bone strains were monitored with rosette strain gauges in ten locations on the left hemipelvis. FE strain predictions were compared directly with experimental results for validation. Overall, baseline FE predictions were strongly correlated with experimental results (r2=0.824), with a best-fit line that was not statistically different than the line y=x (experimental strains = FE predicted strains). Changes to cortical bone thickness and elastic modulus had the largest effect on cortical bone strains. The FE model was less sensitive to changes in all other parameters. The methods developed and validated in this study will be useful for creating and analyzing patient-specific FE models to better understand the biomechanics of the pelvis.

Recruitment of tendon crimp with applied tensile strain

The tensile stress-strain behavior of ligaments and tendons begins with a toe region that is believed to result from the straightening of crimped collagen fibrils. The in situ mechanical function is mostly confined to this toe region and changes in crimp morphology are believed to be associated with pathological conditions. A relatively new imaging technique, optical coherence tomography (OCT), provides a comparatively inexpensive method for nondestructive investigation of tissue ultrastructure with resolution on the order of 15 microm and the potential for use in a clinical setting. The objectives of this work were to assess the utility of OCT for visualizing crimp period, and to use OCT to determine how crimp period changed as a function of applied tensile strain in rat tail tendon fascicles. Fascicles from rat tail tendons were subjected to 0.5 percent strain increments up to 5 percent and imaged at each increment using OCT. A comparison between OCT images and optical microscopy images taken between crossed polarizing lenses showed a visual correspondence between features indicative of crimp pattern. Crimp pattern always disappeared completely before 3 percent axial strain was reached. Average crimp period increased as strain increased, but both elongation and shortening occurred within single crimp periods during the application of increasing strain to the fascicle.

Subject-specific finite element analysis of the human medial collateral ligament during valgus knee loading

The objectives of this study were (1) to develop subject‐specific experimental and finite element (FE) techniques to study the three‐dimensional stress–strain behavior of ligaments, with application to the human medial collateral ligament (MCL), and (2) to determine the importance of subject‐specific material properties and initial (in situ) strain distribution for prediction of the strain distribution in the MCL under valgus loading. Eight male knees were subjected to varus–valgus loading at flexion angles of 0°, 30°, and 60°. Three‐dimensional joint kinematics and MCL strains were recorded during kinematic testing. Following testing, the MCL of each knee was removed to allow measurement of the in situ strain distribution and to perform material testing. A FE model of the femur–MCL–tibia complex was constructed for each knee to simulate valgus loading at each flexion angle, using subject‐specific bone and ligament geometry, material properties, and joint kinematics. A transversely isotropic hyperelastic material model was used to represent the MCL. The MCL in situ strain distribution at full extension was used to apply in situ strain to each MCL FE model. FE predicted MCL strains during valgus loading were compared to experimental measurements using regression analysis. The subject‐specific FE predictions of strain correlated reasonably well with experimentally measured MCL strains (R2 = 0.83, 0.72, and 0.66 at 0°, 30°, and 60°, respectively). Despite large inter‐subject variation in MCL material properties, MCL strain distributions predicted by individual FE models that used average MCL material properties were strongly correlated with subject‐specific FE strain predictions (R2 = 0.99 at all flexion angles). However, predictions by FE models that used average in situ strain distributions yielded relatively poor correlations with subject‐specific FE predictions (R2 = 0.44, 0.35, and 0.33 at flexion angles of 0°, 30°, and 60°, respectively). The strain distribution within the MCL was nonuniform and changed with flexion angle. The highest MCL strains occurred at full extension in the posterior region of the MCL proximal to the joint line during valgus loading, suggesting this region may be most vulnerable to injury under these loading conditions. This work demonstrates that subject‐specific FE models can predict the complex, nonuniform strain fields that occur in ligaments due to external loading of the joint.

Validation of finite element predictions of cartilage contact pressure in the human hip joint.

Methods to predict contact stresses in the hip can provide an improved understanding of load distribution in the normal and pathologic joint. The objectives of this study were to develop and validate a three-dimensional finite element (FE) model for predicting cartilage contact stresses in the human hip using subject-specific geometry from computed tomography image data, and to assess the sensitivity of model predictions to boundary conditions, cartilage geometry, and cartilage material properties. Loads based on in vivo data were applied to a cadaveric hip joint to simulate walking, descending stairs, and stair-climbing. Contact pressures and areas were measured using pressure sensitive film. CT image data were segmented and discretized into FE meshes of bone and cartilage. FE boundary and loading conditions mimicked the experimental testing. Fair to good qualitative correspondence was obtained between FE predictions and experimental measurements for simulated walking and descending stairs, while excellent agreement was obtained for stair-climbing. Experimental peak pressures, average pressures, and contact areas were 10.0 MPa (limit of film detection), 4.4-5.0 MPa, and 321.9-425.1 mm(2), respectively, while FE-predicted peak pressures, average pressures, and contact areas were 10.8-12.7 MPa, 5.1-6.2 MPa, and 304.2-366.1 mm(2), respectively. Misalignment errors, determined as the difference in root mean squared error before and after alignment of FE results, were less than 10%. Magnitude errors, determined as the residual error following alignment, were approximately 30% but decreased to 10-15% when the regions of highest pressure were compared. Alterations to the cartilage shear modulus, bulk modulus, or thickness resulted in +/-25% change in peak pressures, while changes in average pressures and contact areas were minor (+/-10%). When the pelvis and proximal femur were represented as rigid, there were large changes, but the effect depended on the particular loading scenario. Overall, the subject-specific FE predictions compared favorably with pressure film measurements and were in good agreement with published experimental data. The validated modeling framework provides a foundation for development of patient-specific FE models to investigate the mechanics of normal and pathological hips.

Ligament material behavior is nonlinear, viscoelastic and rate-independent under shear loading

The material behavior of ligament is determined by its constituents, their organization and their interaction with each other. To elucidate the origins of the multiaxial material behavior of ligaments, we investigated ligament response to shear loading under both quasi-static and rate-dependent loading conditions. Stress relaxation tests demonstrated that the tissue was highly viscoelastic in shear, with peak loads dropping over 40% during 30 min of stress relaxation. The stress relaxation response was unaffected by three decades of change in shear strain rate (1.3, 13 and 130%/s). A novel parameter estimation technique was developed to determine material coefficients that best described the experimental response of each test specimen to shear. The experimentally measured clamp displacements and reaction forces from the simple shear tests were used with a nonlinear optimization strategy based around function evaluations from a finite element program. A transversely isotropic material with an exponential matrix strain energy provided an excellent fit to experimental load-displacement curves. The shear modulus of human MCL showed a significant increase with increasing shear strain (p<0.001), reaching a maximum of 1.72+/-0.4871 MPa. The results obtained from this study suggest that viscoelasticity in shear does not likely result from fluid flow. Gradual loading of transversely oriented microstructural features such as intermolecular collagen crosslinks or collagen-proteoglycan crosslinking may be responsible for the stiffening response under shear loading.

Finite element implementation of anisotropic quasi-linear viscoelasticity using a discrete spectrum approximation.

The objective of this work was to develop a theoretical and computational framework to apply the finite element method to anisotropic, viscoelastic soft tissues. The quasilinear viscoelastic (QLV) theory provided the basis for the development. To allow efficient and easy computational implementation, a discrete spectrum approximation was developed for the QLV relaxation function. This approximation provided a graphic means to fit experimental data with an exponential series. A transversely isotropic hyperelastic material model developed for ligaments and tendons was used for the elastic response. The viscoelastic material model was implemented in a general-purpose, nonlinear finite element program. Test problems were analyzed to assess the performance of the discrete spectrum approximation and the accuracy of the finite element implementation. Results indicated that the formulation can reproduce the anisotropy and time-dependent material behavior observed in soft tissues. Application of the formulation to the analysis of the human femur-medial collateral ligament-tibia complex demonstrated the ability of the formulation to analyze large three-dimensional problems in the mechanics of biological joints.

Verification, validation and sensitivity studies in computational biomechanics

Computational techniques and software for the analysis of problems in mechanics have naturally moved from their origins in the traditional engineering disciplines to the study of cell, tissue and organ biomechanics. Increasingly complex models have been developed to describe and predict the mechanical behavior of such biological systems. While the availability of advanced computational tools has led to exciting research advances in the field, the utility of these models is often the subject of criticism due to inadequate model verification and validation (V&V). The objective of this review is to present the concepts of verification, validation and sensitivity studies with regard to the construction, analysis and interpretation of models in computational biomechanics. Specific examples from the field are discussed. It is hoped that this review will serve as a guide to the use of V&V principles in the field of computational biomechanics, thereby improving the peer acceptance of studies that use computational modeling techniques.