Shawn P. Reese

Validation of computational models in biomechanics

Abstract The topics of verification and validation have increasingly been discussed in the field of computational biomechanics, and many recent articles have applied these concepts in an attempt to build credibility for models of complex biological systems. Verification and validation are evolving techniques that, if used improperly, can lead to false conclusions about a system under study. In basic science, these erroneous conclusions may lead to failure of a subsequent hypothesis, but they can have more profound effects if the model is designed to predict patient outcomes. While several authors have reviewed verification and validation as they pertain to traditional solid and fluid mechanics, it is the intent of this paper to present them in the context of computational biomechanics. Specifically, the task of model validation will be discussed, with a focus on current techniques. It is hoped that this review will encourage investigators to engage and adopt the verification and validation process in an effort to increase peer acceptance of computational biomechanics models.

Micromechanical models of helical superstructures in ligament and tendon fibers predict large Poisson's ratios

Experimental measurements of the Poissons ratio in tendon and ligament tissue greatly exceed the isotropic limit of 0.5. This is indicative of volume loss during tensile loading. The microstructural origin of the large Poissons ratios is unknown. It was hypothesized that a helical organization of fibrils within a fiber would result in a large Poissons ratio in ligaments and tendons, and that this helical organization would be compatible with the crimped nature of these tissues, thus modeling their classic nonlinear stress-strain behavior. Micromechanical finite element models were constructed to represent crimped fibers with a super-helical organization, composed of fibrils embedded within a matrix material. A homogenization procedure was performed to determine both the effective Poissons ratio and the Poisson function. The results showed that helical fibril organization within a crimped fiber was capable of simultaneously predicting large Poissons ratios and the nonlinear stress-strain behavior seen experimentally. Parametric studies revealed that the predicted Poissons ratio was strongly dependent on the helical pitch, crimp angle and the material coefficients. The results indicated that, for physiologically relevant parameters, the models were capable of predicting the large Poissons ratios seen experimentally. It was concluded that helical organization within a crimped fiber can produce both the characteristic nonlinear stress-strain behavior and large Poissons ratios, while fiber crimp alone could only account for the nonlinear stress-strain behavior.

Effects of decorin proteoglycan on fibrillogenesis, ultrastructure, and mechanics of type I collagen gels.

The proteoglycan decorin is known to affect both the fibrillogenesis and the resulting ultrastructure of in vitro polymerized collagen gels. However, little is known about its effects on mechanical properties. In this study, 3D collagen gels were polymerized into tensile test specimens in the presence of decorin proteoglycan, decorin core protein, or dermatan sulfate (DS). Collagen fibrillogenesis, ultrastructure, and mechanical properties were then quantified using a turbidity assay, 2 forms of microscopy (SEM and confocal), and tensile testing. The presence of decorin proteoglycan or core protein decreased the rate and ultimate turbidity during fibrillogenesis and decreased the number of fibril aggregates (fibers) compared to control gels. The addition of decorin and core protein increased the linear modulus by a factor of 2 compared to controls, while the addition of DS reduced the linear modulus by a factor of 3. Adding decorin after fibrillogenesis had no effect, suggesting that decorin must be present during fibrillogenesis to increase the mechanical properties of the resulting gels. These results show that the inclusion of decorin proteoglycan during fibrillogenesis of type I collagen increases the modulus and tensile strength of resulting collagen gels. The increase in mechanical properties when polymerization occurs in the presence of the decorin proteoglycan is due to a reduction in the aggregation of fibrils into larger order structures such as fibers and fiber bundles.

Continuum description of the Poisson׳s ratio of ligament and tendon under finite deformation

Ligaments and tendons undergo volume loss when stretched along the primary fiber axis, which is evident by the large, strain-dependent Poissons ratios measured during quasi-static tensile tests. Continuum constitutive models that have been used to describe ligament material behavior generally assume incompressibility, which does not reflect the volumetric material behavior seen experimentally. We developed a strain energy equation that describes large, strain dependent Poissons ratios and nonlinear, transversely isotropic behavior using a novel method to numerically enforce the desired volumetric behavior. The Cauchy stress and spatial elasticity tensors for this strain energy equation were derived and implemented in the FEBio finite element software (www.febio.org). As part of this objective, we derived the Cauchy stress and spatial elasticity tensors for a compressible transversely isotropic material, which to our knowledge have not appeared previously in the literature. Elastic simulations demonstrated that the model predicted the nonlinear, upwardly concave uniaxial stress-strain behavior while also predicting a strain-dependent Poissons ratio. Biphasic simulations of stress relaxation predicted a large outward fluid flux and substantial relaxation of the peak stress. Thus, the results of this study demonstrate that the viscoelastic behavior of ligaments and tendons can be predicted by modeling fluid movement when combined with a large Poissons ratio. Further, the constitutive framework provides the means for accurate simulations of ligament volumetric material behavior without the need to resort to micromechanical or homogenization methods, thus facilitating its use in large scale, whole joint models.

Micromechanical model of a surrogate for collagenous soft tissues: development, validation and analysis of mesoscale size effects

Aligned, collagenous tissues such as tendons and ligaments are composed primarily of water and type I collagen, organized hierarchically into nanoscale fibrils, microscale fibers and mesoscale fascicles. Force transfer across scales is complex and poorly understood. Since innervation, the vasculature, damage mechanisms and mechanotransduction occur at the microscale and mesoscale, understanding multiscale interactions is of high importance. This study used a physical model in combination with a computational model to isolate and examine the mechanisms of force transfer between scales. A collagen-based surrogate served as the physical model. The surrogate consisted of extruded collagen fibers embedded within a collagen gel matrix. A micromechanical finite element model of the surrogate was validated using tensile test data that were recorded using a custom tensile testing device mounted on a confocal microscope. Results demonstrated that the experimentally measured macroscale strain was not representative of the microscale strain, which was highly inhomogeneous. The micromechanical model, in combination with a macroscopic continuum model, revealed that the microscale inhomogeneity resulted from size effects in the presence of a constrained boundary. A sensitivity study indicated that significant scale effects would be present over a range of physiologically relevant inter-fiber spacing values and matrix material properties. The results indicate that the traditional continuum assumption is not valid for describing the macroscale behavior of the surrogate and that boundary-induced size effects are present.

Multiscale Modeling of Ligaments and Tendons

Ligaments and tendons are composed primarily of water and fibrillar type I collagen, which is hierarchically organized into complex structures that span multiple physical scales. Forces at the macroscopic joint level are transmitted via interactions at the mesoscale, microscale and nanoscale. Tissue repair and growth is mediated by fibroblasts and tenocytes, which are subjected to a unique microscale mechanical environment. The burgeoning field of multiscale modeling holds promise in filling the gaps in our understanding of structure–function relationships and mechanotransduction in these tissues, and these questions are difficult or impossible to address using experimental techniques alone. This article reviews the state of the art in multiscale modeling of ligaments and tendons, while providing sufficient background on the structure and function of these tissues to allow a reader who is new to the area to proceed without substantial outside reading. The multiscale structure of ligaments and tendons is described in detail. The available data on material characterization at different physical scales is reviewed as well. The final section of the chapter summarizes the efforts at developing and validating multiscale models that are relevant to ligament and tendon mechanics, and identifies future directions for research. Multiscale modeling of tendon and ligament holds considerable promise in advancing our understanding regarding the complex mechanisms of multiscale force transfer within these tissues.

Extracellular Matrix Stiffness Modulates Microvascular Morphology During Early Sprouting Angiogenesis In Vitro

Sprouting angiogenesis is associated with changes in matrix stiffness[1]. Neovessel growth and morphology are in turn affected by the changes in matrix orientation or forces acting on the matrix[2]. Matrix rigidity influences the formation of cord like structures[3, 4] and could play a role in development of tissue specific vascular morphology or inhibit cellular functions in diseases. The effect of matrix stiffness on neovessel growth from preformed vasculature has not been examined. Matrix stiffness could be increased both by an increase in matrix density[5] as well as increased crosslink formation, as in hyperglycemia[6]. It is thus essential to first identify the effect of increase in local stiffness alone, in the absence of artificially induced crosslinks, which may interfere with matrix orientation. Our aim is to characterize changes in early angiogenesis associated with ECM of different densities and relate these to changes in matrix orientation.Copyright

The effect of fiber microstructure on the observed poisson's ratio of tendon and ligament tissue

During tensile testing along the predominant collagen fiber direction, ligament and tendon tissue exhibit large Poisson’s ratios ranging from 1.3 in capsular ligament to 2.98 in flexor tendon [1][2]. Although the microstructure of these tissues (especially fiber crimp) has been characterized, the relationship between microstructure and Poisson’s ratio is relatively unexplored. There has been debate regarding the exact nature of the characteristic crimp within tendon fibers, however the two views most present in the literature are that of planar crimp and helical crimp. The aim of this study was to perform a finite element analysis on prototypical models of fibril bundles for both forms of crimp under tensile loading conditions. It was hypothesized that planar crimp alone would be insufficient for generating large Poisson’s ratios, and that some other microstructure (such as a helix) would be required.Copyright

Fabrication of dense anisotropic collagen scaffolds using biaxial compression

We developed a new method to manufacture dense, aligned, and porous collagen scaffolds using biaxial plastic compression of type I collagen gels. Using a novel compression apparatus that constricts like an iris diaphragm, low density collagen gels were compressed to yield a permanently densified, highly aligned collagen material. Micro-porosity scaffolds were created using hydrophilic elastomer porogens that can be selectively removed following biaxial compression, with porosity modulated by using different porogen concentrations. The resulting scaffolds exhibit collagen densities that are similar to native connective tissues (∼10% collagen by weight), pronounced collagen alignment across multiple length scales, and an interconnected network of pores, making them highly relevant for use in tissue culture, the study of physiologically relevant cell-matrix interactions, and tissue engineering applications. The scaffolds exhibited highly anisotropic material behavior, with the modulus of the scaffolds in the fiber direction over 100 times greater than the modulus in the transverse direction. Adipose-derived mesenchymal stem cells were seeded onto the biaxially compressed scaffolds with minimal cell death over seven days of culture, along with cell proliferation and migration into the pore spaces. This fabrication method provides new capabilities to manufacture structurally and mechanically relevant cytocompatible scaffolds that will enable more physiologically relevant cell culture studies. Further improvement of manufacturing techniques has the potential to produce engineered scaffolds for direct replacement of dense connective tissues such as meniscus and annulus fibrosus. STATEMENT OF SIGNIFICANCE In vitro studies of cell-matrix interactions and the engineering of replacement materials for collagenous connective tissues require biocompatible scaffolds that replicate the high collagen density (15-25%/wt), aligned fibrillar organization, and anisotropic mechanical properties of native tissues. However, methods for creating scaffolds with these characteristics are currently lacking. We developed a new apparatus and method to create high density, aligned, and porous collagen scaffolds using a biaxial compression with porogens technique. These scaffolds have a highly directional structure and mechanical properties, with the tensile strength and modulus up to 100 times greater in the direction of alignment. We also demonstrated that the scaffolds are a suitable material for cell culture, promoting cell adhesion, viability, and an aligned cell morphology comparable to the cell morphology observed in native aligned tissues.

Tendons and Ligaments: Current State and Future Directions

Tendons and ligament are soft connective tissues that transmit load and support movement and joint articulation within the musculoskeletal system.