Development of a three-dimensional unit cell to model the micromechanical response of a collagen-based extracellular matrix

Effect of random fiber networks on bubble growth in gelatin hydrogels

In hydrodynamics, the event of dynamic bubble growth in a pure liquid under tensile pressure is known as cavitation. The same event can also be observed in soft materials (e.g., elastomers and hydrogels). However, for soft materials, bubble/cavity growth is either defined as cavitation if the bubble growth is elastic and reversible or as fracture if the cavity growth is by material failure and irreversible. In any way, bubble growth can cause damage to soft materials (e.g., tissue) by inducing high strain and strain-rate deformation. Additionally, a high-strength pressure wave is generated upon the collapse of the bubble. Therefore, it is crucial to identify the critical condition of spontaneous bubble growth in soft materials. Experimental and theoretical observations have agreed that the onset of bubble growth in soft materials requires higher tensile pressure than pure water. The extra tensile pressure is required since the cavitating bubble needs to overcome the elastic and surface energy in soft materials. In this manuscript, we developed two models to study and quantify the extra tensile pressure for different gelatin concentrations. Both the models are then compared with the existing cavitation onset criteria of rubber-like materials. Validation is done with the experimental results of threshold tensile pressure for different gelatin concentrations. Both models can moderately predict the extra tensile pressure within the intermediate range of gelatin concentrations (3-7% [w/v]). For low concentration (∼1%), the network's non-affinity plays a significant role and must be incorporated. On the other hand, for higher concentrations (∼10%), the entropic deformation dominates, and the strain energy formulation is not adequate.

Shear-induced force transmission in a multicomponent, multicell model of the endothelium

Haemodynamic forces applied at the apical surface of vascular endothelial cells (ECs) provide the mechanical signals at intracellular organelles and through the inter-connected cellular network. The objective of this study is to quantify the intracellular and intercellular stresses in a confluent vascular EC monolayer. A novel three-dimensional, multiscale and multicomponent model of focally adhered ECs is developed to account for the role of potential mechanosensors (glycocalyx layer, actin cortical layer, nucleus, cytoskeleton, focal adhesions (FAs) and adherens junctions (ADJs)) in mechanotransmission and EC deformation. The overriding issue addressed is the stress amplification in these regions, which may play a role in subcellular localization of mechanotransmission. The model predicts that the stresses are amplified 250-600-fold over apical values at ADJs and 175-200-fold at FAs for ECs exposed to a mean shear stress of 10 dyne cm(-2). Estimates of forces per molecule in the cell attachment points to the external cellular matrix and cell-cell adhesion points are of the order of 8 pN at FAs and as high as 3 pN at ADJs, suggesting that direct force-induced mechanotransmission by single molecules is possible in both. The maximum deformation of an EC in the monolayer is calculated as 400 nm in response to a mean shear stress of 1 Pa applied over the EC surface which is in accord with measurements. The model also predicts that the magnitude of the cell-cell junction inclination angle is independent of the cytoskeleton and glycocalyx. The inclination angle of the cell-cell junction is calculated to be 6.6° in an EC monolayer, which is somewhat below the measured value (9.9°) reported previously for ECs subjected to 1.6 Pa shear stress for 30 min. The present model is able, for the first time, to cross the boundaries between different length scales in order to provide a global view of potential locations of mechanotransmission.

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 Poisson's 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.

Oligomers modulate interfibril branching and mass transport properties of collagen matrices

Mass transport within collagen-based matrices is critical to tissue development, repair, and pathogenesis, as well as the design of next-generation tissue engineering strategies. This work shows how collagen precursors, specified by intermolecular cross-link composition, provide independent control of collagen matrix mechanical and transport properties. Collagen matrices were prepared from tissue-extracted monomers or oligomers. Viscoelastic behavior was measured in oscillatory shear and unconfined compression. Matrix permeability and diffusivity were measured using gravity-driven permeametry and integrated optical imaging, respectively. Both collagen types showed an increase in stiffness and permeability hindrance with increasing collagen concentration (fibril density); however, different physical property–concentration relationships were noted. Diffusivity was not affected by concentration for either collagen type over the range tested. In general, oligomer matrices exhibited a substantial increase in stiffness and only a modest decrease in transport properties when compared with monomer matrices prepared at the same concentration. The observed differences in viscoelastic and transport properties were largely attributed to increased levels of interfibril branching within oligomer matrices. The ability to relate physical properties to relevant microstructure parameters, including fibril density and interfibril branching, is expected to advance the understanding of cell–matrix signaling, as well as facilitate model-based prediction and design of matrix-based therapeutic strategies.

Toward single cell traction microscopy within 3D collagen matrices

Mechanical interaction between the cell and its extracellular matrix (ECM) regulates cellular behaviors, including proliferation, differentiation, adhesion, and migration. Cells require the three-dimensional (3D) architectural support of the ECM to perform physiologically realistic functions. However, current understanding of cell-ECM and cell-cell mechanical interactions is largely derived from 2D cell traction force microscopy, in which cells are cultured on a flat substrate. 3D cell traction microscopy is emerging for mapping traction fields of single animal cells embedded in either synthetic or natively derived fibrous gels. We discuss here the development of 3D cell traction microscopy, its current limitations, and perspectives on the future of this technology. Emphasis is placed on strategies for applying 3D cell traction microscopy to individual tumor cell migration within collagen gels.

Avalanche process of the fiber-bundle model with stick-slip dynamics and a variable Young modulus

In order to more accurately describe the fracture process of extensive biological fibers, a fiber-bundle model with stick-slip dynamics and a variable Young modulus is constructed. In this model, the Young modulus of a fiber is assumed to increase or decrease by multiplying with a changing ratio after local sliding events. So, the maximum number of stick-slip events of a single fiber and the changing ratio of the Young modulus are the two key parameters of the model. By means of analytical theory and numerical simulation, the constitutive law, the critical stress, the average size of the largest avalanche, and the avalanche size distribution are shown against the two parameters of the model. From a macroscopic viewpoint, the constitutive curves show different morphologies varying from a local plastic state to a unimodal parabola, while from a microscopic viewpoint, the avalanche size distributions can be well fitted into a power law relationship, which is in accord with the classical fiber-bundle model.

Prediction of equibiaxial loading stress in collagen-based extracellular matrix using a three-dimensional unit cell model

Mechanical signals are important factors in determining cell fate. Therefore, insights as to how mechanical signals are transferred between the cell and its surrounding three-dimensional collagen fibril network will provide a basis for designing the optimum extracellular matrix (ECM) microenvironment for tissue regeneration. Previously we described a cellular solid model to predict fibril microstructure-mechanical relationships of reconstituted collagen matrices due to unidirectional loads (Acta Biomater 2010;6:1471-86). The model consisted of representative volume elements made up of an interconnected network of flexible struts. The present study extends this work by adapting the model to account for microstructural anisotropy of the collagen fibrils and a biaxial loading environment. The model was calibrated based on uniaxial tensile data and used to predict the equibiaxial tensile stress-stretch relationship. Modifications to the model significantly improved its predictive capacity for equibiaxial loading data. With a comparable fibril length (model 5.9-8μm, measured 7.5μm) and appropriate fibril anisotropy the anisotropic model provides a better representation of the collagen fibril microstructure. Such models are important tools for tissue engineering because they facilitate prediction of microstructure-mechanical relationships for collagen matrices over a wide range of microstructures and provide a framework for predicting cell-ECM interactions.

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 new flow co-culture system for studying mechanobiology effects of pulse flow waves

Artery stiffening is known as an important pathological change that precedes small vessel dysfunction, but underlying cellular mechanisms are still elusive. This paper reports the development of a flow co-culture system that imposes a range of arterial-like pulse flow waves, with similar mean flow rate but varied pulsatility controlled by upstream stiffness, onto a 3-D endothelial-smooth muscle cell co-culture. Computational fluid dynamics results identified a uniform flow area critical for cell mechanobiology studies. For validation, experimentally measured flow profiles were compared to computationally simulated flow profiles, which revealed percentage difference in the maximum flow to be <10, <5, or <1% for a high, medium, or low pulse flow wave, respectively. This comparison indicated that the computational model accurately demonstrated experimental conditions. The results from endothelial expression of proinflammatory genes and from determination of proliferating smooth muscle cell percentage both showed that cell activities did not vary within the identified uniform flow region, but were upregulated by high pulse flow compared to steady flow. The flow system developed and characterized here provides an important tool to enhance the understanding of vascular cell remodeling under flow environments regulated by stiffening.

Multiscale strain analysis of tissue equivalents using a custom-designed biaxial testing device

Mechanical signals transferred between a cell and its extracellular matrix play an important role in regulating fundamental cell behavior. To further define the complex mechanical interactions between cells and matrix from a multiscale perspective, a biaxial testing device was designed and built. Finite element analysis was used to optimize the cruciform specimen geometry so that stresses within the central region were concentrated and homogenous while minimizing shear and grip effects. This system was used to apply an equibiaxial loading and unloading regimen to fibroblast-seeded tissue equivalents. Digital image correlation and spot tracking were used to calculate three-dimensional strains and associated strain transfer ratios at macro (construct), meso, matrix (collagen fibril), cell (mitochondria), and nuclear levels. At meso and matrix levels, strains in the 1- and 2-direction were statistically similar throughout the loading-unloading cycle. Interestingly, a significant amplification of cellular and nuclear strains was observed in the direction perpendicular to the cell axis. Findings indicate that strain transfer is dependent upon local anisotropies generated by the cell-matrix force balance. Such multiscale approaches to tissue mechanics will assist in advancement of modern biomechanical theories as well as development and optimization of preconditioning regimens for functional engineered tissue constructs.