Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Biomechanical properties of porcine cornea; planar biaxial tests versus uniaxial tensile tests

The cornea is a transparent tissue whose mechanical properties are important for its optical and physiological functions. The mechanical properties of cornea depend on the composition and microstructure of its extracellular matrix, which is composed of collagen fibrils with preferential orientations. The present research was done in order to characterize corneal mechanical response using the biaxial mechanical testing method and to compare biaxial measurements with those found from uniaxial tensile tests. For this purpose, thirty square-shaped specimens excised from the center of porcine cornea were mounted into an ElectroForce TestBench device such that their superior/inferior (SI) and nasal/temporal (NT) meridians were aligned with motor axes. Furthermore, ten corneal strips dissected from the NT direction (n = 5) and SI direction (n = 5) were mounted into an RSA-G2 Solid Analyzer testing machine. The biaxial experiments were performed at stretch ratios of 1:1, 1:0.5, 0.5:1, 1:0.01, and 0.01:1 and displacement rates of 2 mm/min (n = 20) and 10 mm/min (n = 10). The uniaxial experiments were done using the displacement rate of 2 mm/min. The planar square-shaped samples tested under equibiaxial loading showed similar mechanical response in NT and SI directions. Furthermore, uniaxial experiments revealed no significant difference in tensile response of corneal strips excised from NT and SI directions. However, equibiaxial testing tensile stresses were significantly larger than those found from uniaxial tensile measurements. The mechanical behavior of cornea in biaxial tests was dependent on the applied stretch ratio. The differences and similarities between uniaxial and biaxial experimental measurements were discussed and it was concluded that the planar biaxial testing method characterized the mechanical response of cornea by mimicking its in vivo loading state more closely than uniaxial experiments.

Stiffening of a fibrous matrix after recovery of contracted inclusions

Disordered fibrous matrices in living tissues are subjected to forces exerted by cells that contract to pull on matrix fibers. To maintain homeostasis or facilitate disease progression, contracted cells often push on matrix fibers as they recover their original sizes. Recent advances have shown that matrix geometry encodes loading history into mechanical memory independently of plasticity mechanisms such as inter-fiber cohesion or fiber yielding. Conceptualizing cells as inclusions undergoing sequential contraction and recovery, prior work documented matrix remodeling surrounding a solitary recovered inclusion. However, because the remodeling induced by the contraction of multiple inclusions differs from that caused by a single contracted inclusion, we investigate how matrix remodeling occurs when multiple contracted inclusions recover simultaneously, a scenario that more accurately reflects real tissues containing many closely spaced cells. Using mechanics-based computational models of fibrous matrices embedded with clusters of inclusions, we studied the mechanical remodeling of the matrix during the simultaneous recovery of inclusions after contraction. The results revealed permanent mechanical remodeling of the matrix within the cluster, with stiffening observed in areas of the matrix enclosed by closely spaced inclusions. This stiffening was driven by microstructural changes in matrix geometry and was corroborated in experiments, where collagen matrices permanently remodeled by the contraction and recovery of closely spaced embedded cells also exhibited stiffening. By enriching the understanding of memory formation in fibrous matrices, this study opens new possibilities for estimating cell forces on matrix substrates and refining metamaterial design strategies.

Unexpected softening of a fibrous matrix by contracting inclusions

Cells respond to the stiffness of their surrounding environment, but quantifying the stiffness of a fibrous matrix at the scale of a cell is complicated, due to the effects of nonlinearity and complex force transmission pathways resulting from randomness in fiber density and connections. While it is known that forces produced by individual contractile cells can stiffen the matrix, it remains unclear how simultaneous contraction of multiple cells in a fibrous matrix alters the stiffness at the scale of a cell. Here, we used computational modeling and experiments to quantify the stiffness of a random fibrous matrix embedded with multiple contracting inclusions, which mimicked the contractile forces of a cell. The results showed that when the matrix was free to contract as a result of the forces produced by the inclusions, the matrix softened rather than stiffened, which was surprising given that the contracting inclusions applied tensile forces to the matrix. Using the computational model, we identified that the underlying cause of the softening was that the majority of the fibers were under a local state of axial compression, causing buckling. We verified that this buckling-induced matrix softening was sufficient for cells to sense and respond by altering their morphology and force generation. Our findings reveal that the localized forces induced by cells do not always stiffen the matrix; rather, softening can occur in instances wherein the matrix can contract in response to the cell-generated forces. This study opens up new possibilities to investigate whether cell-induced softening contributes to maintenance of homeostatic conditions or progression of disease. STATEMENT OF SIGNIFICANCE: Mechanical interactions between cells and the surrounding matrix strongly influence cellular functions. Cell-induced forces can alter matrix properties, and much prior literature in this area focused on the influence of individual contracting cells. Cells in tissues are rarely solitary; rather, they are interspersed with neighboring cells throughout the matrix. As a result, the mechanics are complicated, leaving it unclear how the multiple contracting cells affect matrix stiffness. Here, we show that multiple contracting inclusions within a fibrous matrix can cause softening that in turn affects cell sensing and response. Our findings provide new directions to determine impacts of cell-induced softening on maintenance of tissue or progression of disease.

Bioinspired 3D microprinted cell scaffolds: Integration of graph theory to recapitulate complex network wiring in lymph nodes

Physical networks are ubiquitous in nature, but many of them possess a complex organizational structure that is difficult to recapitulate in artificial systems. This is especially the case in biomedical and tissue engineering, where the microstructural details of 3D cell scaffolds are important. Studies of biological networks-such as fibroblastic reticular cell (FRC) networks-have revealed the crucial role of network topology in a range of biological functions. However, cell scaffolds are rarely analyzed, or designed, using graph theory. To understand how networks affect adhered cells, 3D culture platforms capturing the complex topological properties of biologically relevant networks would be needed. In this work, we took inspiration from the small-world organization (high clustering and low path length) of FRC networks to design cell scaffolds. An algorithmic toolset was created to generate the networks and process them to improve their 3D printability. We employed tools from graph theory to show that the networks were small-world (omega factor, ω = -0.10 ± 0.02; small-world propensity, SWP = 0.74 ± 0.01). 3D microprinting was employed to physicalize networks as scaffolds, which supported the survival of FRCs. This work, therefore, represents a bioinspired, graph theory-driven approach to control the networks of microscale cell niches.

Stiffening mechanisms in stochastic athermal fiber networks

Stochastic athermal networks composed of fibers that deform axially and in bending strain stiffen much faster than thermal networks of axial elements, such as elastomers. Here we investigate the physical origin of stiffening in athermal network materials. To this end, we use models of stochastic networks subjected to uniaxial deformation and identify the emergence of two subnetworks, the stress path subnetwork (SPSN) and the bending support subnetwork (BSSN), which carry most of the axial and bending energies, respectively. The BSSN controls lateral contraction and modulates the organization of the SPSN during deformation. The SPSN is preferentially oriented in the loading direction, while the BSSN's preferential orientation is orthogonal to the SPSN. In nonaffine networks stiffening is exponential, while in close-to-affine networks it is quadratic. The difference is due to a much more modest lateral contraction in the approximately affine case and to a stiffer BSSN. Exponential stiffening emerges from the interplay of the axial and bending deformation modes at the scale of individual or small groups of fibers undergoing large deformations and being subjected to the constraint of rigid cross-links, and it is not necessarily a result of complex interactions involving many connected fibers. An apparent third regime of quadratic stiffening may be evidenced in nonaffinely deforming networks provided the nominal stress is observed. This occurs at large stretches, when the BSSN contribution of stiffening vanishes. However, this regime is not present if the Cauchy stress is used, in which case stiffening is exponential throughout the entire deformation. These results shed light on the physical nature of stiffening in a broad class of materials including connective tissue, the extracellular matrix, nonwovens, felt, and other athermal network materials.

Materials science and mechanosensitivity of living matter

Living systems are composed of molecules that are synthesized by cells that use energy sources within their surroundings to create fascinating materials that have mechanical properties optimized for their biological function. Their functionality is a ubiquitous aspect of our lives. We use wood to construct furniture, bacterial colonies to modify the texture of dairy products and other foods, intestines as violin strings, bladders in bagpipes, and so on. The mechanical properties of these biological materials differ from those of other simpler synthetic elastomers, glasses, and crystals. Reproducing their mechanical properties synthetically or from first principles is still often unattainable. The challenge is that biomaterials often exist far from equilibrium, either in a kinetically arrested state or in an energy consuming active state that is not yet possible to reproduce de novo. Also, the design principles that form biological materials often result in nonlinear responses of stress to strain, or force to displacement, and theoretical models to explain these nonlinear effects are in relatively early stages of development compared to the predictive models for rubberlike elastomers or metals. In this Review, we summarize some of the most common and striking mechanical features of biological materials and make comparisons among animal, plant, fungal, and bacterial systems. We also summarize some of the mechanisms by which living systems develop forces that shape biological matter and examine newly discovered mechanisms by which cells sense and respond to the forces they generate themselves, which are resisted by their environment, or that are exerted upon them by their environment. Within this framework, we discuss examples of how physical methods are being applied to cell biology and bioengineering.