Victor K. Lai

Mechanics of a fiber network within a non-fibrillar matrix: Model and comparison with collagen-agarose co-gels

While collagen is recognized as the predominant mechanical component of soft connective tissues, the role of the non-fibrillar matrix (NFM) is less well understood. Even model systems, such as the collagen-agarose co-gel, can exhibit complex behavior, making it difficult to identify relative contributions of specific tissue constituents. In the present study, we developed a two-component microscale model of collagen-agarose tissue analogs and used it to elucidate the interaction between collagen and NFM in uniaxial tension. Collagen fibers were represented with Voronoi networks, and the NFM was modeled as a neo-Hookean solid. Model predictions of total normal stress and Poisson’s ratio matched experimental observations well (including high Poisson’s values of ~3), and the addition of NFM led to composition-dependent decreases in volume change and increases in fiber stretch. Because the NFM was more resistant to volume change than the fiber network, extension of the composite led to pressurization of the NFM. Within a specific range of parameter values (low shear modulus and moderate Poisson’s ratio), the magnitude of the reaction force decreased relative to this pressurization component resulting in a negative (compressive) NFM stress in the loading direction, even though the composite tissue was in tension.

A Coupled Fiber-Matrix Model Demonstrates Highly Inhomogeneous Microstructural Interactions in Soft Tissues Under Tensile Load

A soft tissues macroscopic behavior is largely determined by its microstructural components (often a collagen fiber network surrounded by a nonfibrillar matrix (NFM)). In the present study, a coupled fiber-matrix model was developed to fully quantify the internal stress field within such a tissue and to explore interactions between the collagen fiber network and nonfibrillar matrix (NFM). Voronoi tessellations (representing collagen networks) were embedded in a continuous three-dimensional NFM. Fibers were represented as one-dimensional nonlinear springs and the NFM, meshed via tetrahedra, was modeled as a compressible neo-Hookean solid. Multidimensional finite element modeling was employed in order to couple the two tissue components and uniaxial tension was applied to the composite representative volume element (RVE). In terms of the overall RVE response (average stress, fiber orientation, and Poissons ratio), the coupled fiber-matrix model yielded results consistent with those obtained using a previously developed parallel model based upon superposition. The detailed stress field in the composite RVE demonstrated the high degree of inhomogeneity in NFM mechanics, which cannot be addressed by a parallel model. Distributions of maximum/minimum principal stresses in the NFM showed a transition from fiber-dominated to matrix-dominated behavior as the matrix shear modulus increased. The matrix-dominated behavior also included a shift in the fiber kinematics toward the affine limit. We conclude that if only gross averaged parameters are of interest, parallel-type models are suitable. If, however, one is concerned with phenomena, such as individual cell-fiber interactions or tissue failure that could be altered by local variations in the stress field, then the detailed model is necessary in spite of its higher computational cost.

Microstructural and mechanical differences between digested collagen-fibrin co-gels and pure collagen and fibrin gels

Collagen and fibrin are important extracellular matrix (ECM) components in the body, providing structural integrity to various tissues. These biopolymers are also common scaffolds used in tissue engineering. This study investigated how co-gelation of collagen and fibrin affected the properties of each individual protein network. Collagen-fibrin co-gels were cast and subsequently digested using either plasmin or collagenase; the microstructure and mechanical behavior of the resulting networks were then compared with the respective pure collagen or fibrin gels of the same protein concentration. The morphologies of the collagen networks were further analyzed via three-dimensional network reconstruction from confocal image z-stacks. Both collagen and fibrin exhibited a decrease in mean fiber diameter when formed in co-gels compared with the pure gels. This microstructural change was accompanied by an increased failure strain and decreased tangent modulus for both collagen and fibrin following selective digestion of the co-gels. In addition, analysis of the reconstructed collagen networks indicated the presence of very long fibers and the clustering of fibrils, resulting in very high connectivities for collagen networks formed in co-gels.

A Multiscale Approach to Modeling the Passive Mechanical Contribution of Cells in Tissues

In addition to their obvious biological roles in tissue function, cells often play a significant mechanical role through a combination of passive and active behaviors. This study focused on the passive mechanical contribution of cells in tissues by improving our multiscale model via the addition of cells, which were treated as dilute spherical inclusions. The first set of simulations considered a rigid cell, with the surrounding ECM modeled as (1) linear elastic, (2) Neo-Hookean, and (3) a fiber network. Comparison with the classical composite theory for rigid inclusions showed close agreement at low cell volume fraction. The fiber network case exhibited nonlinear stress-strain behavior and Poissons ratios larger than the elastic limit of 0.5, characteristics similar to those of biological tissues. The second set of simulations used a fiber network for both the cell (simulating cytoskeletal filaments) and matrix, and investigated the effect of varying relative stiffness between the cell and matrix, as well as the effect of a cytoplasmic pressure to enforce incompressibility of the cell. Results showed that the ECM network exerted negligible compression on the cell, even when the stiffness of fibers in the network was increased relative to the cell. Introduction of a cytoplasmic pressure significantly increased the stresses in the cell filament network, and altered how the cell changed its shape under tension. Findings from this study have implications on understanding how cells interact with their surrounding ECM, as well as in the context of mechanosensation.

Multiscale Mechanical Model of the Pacinian Corpuscle Shows Depth and Anisotropy Contribute to the Receptor’s Characteristic Response to Indentation

Cutaneous mechanoreceptors transduce different tactile stimuli into neural signals that produce distinct sensations of touch. The Pacinian corpuscle (PC), a cutaneous mechanoreceptor located deep within the dermis of the skin, detects high frequency vibrations that occur within its large receptive field. The PC is comprised of lamellae that surround the nerve fiber at its core. We hypothesized that a layered, anisotropic structure, embedded deep within the skin, would produce the nonlinear strain transmission and low spatial sensitivity characteristic of the PC. A multiscale finite-element model was used to model the equilibrium response of the PC to indentation. The first simulation considered an isolated PC with fiber networks aligned with the PC’s surface. The PC was subjected to a 10 μm indentation by a 250 μm diameter indenter. The multiscale model captured the nonlinear strain transmission through the PC, predicting decreased compressive strain with proximity to the receptor’s core, as seen experimentally by others. The second set of simulations considered a single PC embedded epidermally (shallow) or dermally (deep) to model the PC’s location within the skin. The embedded models were subjected to 10 μm indentations at a series of locations on the surface of the skin. Strain along the long axis of the PC was calculated after indentation to simulate stretch along the nerve fiber at the center of the PC. Receptive fields for the epidermis and dermis models were constructed by mapping the long-axis strain after indentation at each point on the surface of the skin mesh. The dermis model resulted in a larger receptive field, as the calculated strain showed less indenter location dependence than in the epidermis model.

Mechanics of a two-fiber model with one nested fiber network, as applied to the collagen-fibrin system

The mechanical behavior of collagen-fibrin (col-fib) co-gels is both scientifically interesting and clinically relevant. Collagen-fibrin networks are a staple of tissue engineering research, but the mechanical consequences of changes in co-gel composition have remained difficult to predict or even explain. We previously observed fundamental differences in failure behavior between collagen-rich and fibrin-rich co-gels, suggesting an essential change in how the two components interact as the co-gels composition changes. In this work, we explored the hypothesis that the co-gel behavior is due to a lack of percolation by the dilute component. We generated a series of computational models based on interpenetrating fiber networks. In these models, the major network component percolated the model space but the minor component did not, instead occupying a small island embedded within the larger network. Each component was assigned properties based on a fit of single-component gel data. Island size was varied to match the relative concentrations of the two components. The model predicted that networks rich in collagen, the stiffer component, would roughly match pure-collagen gel behavior with little additional stress due to the fibrin, as seen experimentally. For fibrin-rich gels, however, the model predicted a smooth increase in the overall network strength with added collagen, as seen experimentally but not consistent with an additive parallel model. We thus conclude that incomplete percolation by the low-concentration component of a co-gel is a major determinant of its macroscopic properties, especially if the low-concentration component is the stiffer component. STATEMENT OF SIGNIFICANCE Models for the behavior of fibrous networks have useful applications in many different fields, including polymer science, textiles, and tissue engineering. In addition to being important structural components in soft tissues and blood clots, these protein networks can serve as scaffolds for bioartificial tissues. Thus, their mechanical behavior, especially in co-gels, is both interesting from a materials science standpoint and significant with regard to tissue engineering.

Swelling of collagen-hyaluronic acid tissue-equivalents: An experimental model to evaluate residual stress in soft tissues

Collagen gel tissue-equivalents (TEs), which are simple model tissues with tunable properties, have been used to explore many properties of soft tissues, such as how structural and compositional properties affect mechanical function [1–4]. One aspect not captured in previous TE formulations is residual stress due to interactions among components, which has an important functional role in many tissues (e.g., blood vessels [5], ligaments [6], annulus fibrosus [7]). Since the in vivo stress state of native tissues is not easily replicated in TE fabrication, a different method for “pre-stressing” collagen networks of TEs was necessary. To this end, co-gel TEs were fabricated by adding hyaluronic acid (HA) to reconstituted Type-I collagen (Col) gels. When placed in solutions of varying osmolarity, HA-Col TEs swell as the HA binds water, which in turn will stretch (and stress) the collagen network. In this way, TEs with residual stress (i.e., pre-stressed collagen fibers) can be fabricated and evaluated in order to elucidate relationships between residual stress and functional properties. Therefore, the goals of the present study were to fabricate HA-Col TEs, make initial measurements of their swelling properties, and quantify the mechanical response and changes in microstructural organization under applied tensile load.Copyright

Elucidation of Microstructural Interactions Between Collagen and Non-Fibrillar Matrix in Soft Tissue Using a Coupled Fiber-Matrix Model

The mechanical properties of soft connective tissues are governed by their collagen fiber network and surrounding non-fibrillar matrix (e.g., proteoglycans, cells, elastin, etc.). In order to understand how healthy tissues function, and how properties change in injury and disease, it is necessary to quantify the mechanical response of both the collagen network and the non-fibrillar matrix (NFM), as well as the nature of the interaction between these tissue constituents. Using collagen-agarose co-gels as a simple experimental tissue analog system, we have demonstrated how NFM contributes to the mechanical and organizational properties of soft tissues in indentation and tension [1–2]. Furthermore, we used a network-based microscale model to examine how specific NFM properties alter the response of fiber-matrix composites under load [3]. This model fit our experimental data well and provided insight into the role of NFM in tensile mechanics. Since it was constructed according to the conventional approach of superposition of the two constituents (collagen network and NFM), however, the model could not specifically examine local interactions between collagen fibers and the surrounding NFM, which could be critical in assessing tissue damage or cell-matrix interactions. Therefore, we developed and evaluated a fiber-matrix modeling scheme to characterize the microstructural interactions between tissue constituents, as well as to quantify the role of individual tissue components in the behavior of soft tissues under tensile load. For validation, the new model (‘coupled’) was compared to our previous model (‘parallel’) and to experimental co-gel data.Copyright

Collagen Network Architecture Varies Between Pure Collagen and Collagen-Fibrin Co-Gels

Naturally-occurring extracellular matrix (ECM) proteins, e.g. collagen I and fibrin, play an important role in tissues, conferring structural integrity and providing a biochemical environment for eliciting important cellular responses (e.g. migration). Tissue engineers use a variety of matrix polymers as initial scaffolds for seeding cells, sometimes in combination with one another (e.g. collagen-fibrin [1]). For example, our group fabricates arterial tissue equivalents (TEs) by seeding cells in a fibrin gel, which is gradually degraded over time and replaced by cell-produced collagen [2]. While the structure and mechanics of individual ECM proteins have been studied extensively, how multiple fibrillar networks interact to confer overall mechanical behavior remains poorly understood. Narrowing this gap in knowledge of scaffolds comprising multiple fibril networks is crucial in allowing for more rational design in tissue engineering, as cells react differently according to their mechanical environments. For collagen-fibrin networks in particular, early efforts in elucidating interactions between these two fibril networks in co-gels have proven inconclusive due to inconsistent findings from various groups. Recent modeling efforts by our group have shown that simple “series” and “parallel” type interactions provide bounds for the mechanical behavior of collagen-fibrin co-gels [3]. In addition, experiments on pure collagen and fibrin vs. their respective networks from collagen-fibrin co-gels after digestion showed slight differences in mechanical behavior [4]. These previous studies have focused on the composition-function relationship between collagen and fibrin. The objective of the current work is to explore how collagen network architecture changes in the presence of the fibrin network in collagen-fibrin co-gels, thereby providing an added dimension to our understanding of collagen-fibrin systems by elucidating structure-composition-function relationships between collagen and fibrin.© 2012 ASME

A microscale collagen-fibrin interacting network model with comparison to experimental results

Many native and bioengineered soft tissues are composed of two or more types of biopolymer networks that mechanically define and support the material [1]. Modeling the response of multi-network soft tissues to mechanical loading can be difficult due to the heterogeneous nature of these materials and the large strains (>1) involved. As tissues deform, the different biopolymer networks interact with one another and determine the overall stress-strain outcome for the tissue. Capturing this interaction could help improve the accuracy of a computer model to simulate the microscale behavior of soft tissues under load. We have developed a two-network model to reflect interactions between collagen and fibrin biopolymer networks loaded in uniaxial extension. The model can help improve our understanding of native and engineered tissue mechanics.© 2012 ASME