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Degeneration of the lumbar spine, and especially how that degeneration may lead to pain, remains poorly understood. In particular, the mechanics of the facet capsular ligament may contribute to low back pain, but the mechanical changes that occur in this ligament with spinal degeneration are unknown. Additionally, the highly nonlinear, heterogeneous, and anisotropic nature of the facet capsular ligament makes understanding mechanical changes more difficult. Clinically, magnetic resonance imaging (MRI)-based signs of degeneration in the facet joint and the intervertebral disc (IVD) correlate. Therefore, this study examined how the nonlinear, heterogeneous mechanics of the facet capsular ligament change with degeneration of the lumbar spine as characterized using MRI. Cadaveric human spines were imaged via MRI, and the L2-L5 facet joints and IVDs were scored using the Fujiwara and Pfirrmann grading systems. Then, the facet capsular ligament was isolated and biaxially loaded. The nonlinear mechanical properties of the ligament were obtained using a nonlinear generalized anisotropic inverse mechanics analysis (nGAIM). Then a Holzapfel–Gasser–Ogden (HGO) model was fit to the stress–strain data obtained from nGAIM. The facet capsular ligament is stiffer and more anisotropic at larger Pfirrmann grades and higher Fujiwara scores than at lower grades and scores. Analysis of ligament heterogeneity showed all tissues are highly heterogeneous, but no distinct spatial patterns of heterogeneity were found. These results show that degeneration of the lumbar spine including the facet capsular ligament appears to be occurring as a whole joint phenomenon and advance our understanding of lumbar spine degeneration.

The lumbar facet capsular ligament, which surrounds and limits the motion of each facet joint in the lumbar spine, has been recognized as being mechanically significant and has been the subject of multiple mechanical characterization studies in the past. Those studies, however, were performed on isolated tissue samples and thus could not assess the mechanical state of the ligament in vivo, where the constraints of attachment to rigid bone and the force of the joint pressure lead to nonzero strain even when the spine is not loaded. In this work, we quantified these two effects using cadaveric lumbar spines (five spines, 20 total facet joints harvested from L2 to L5). The effect of joint pressure was measured by injecting saline into the joint space and tracking the 3D capsule surface motion via digital image correlation, and the prestrain due to attachment was measured by dissecting a large section of the tissue from the bone and by tracking the motion between the on-bone and free states. We measured joint pressures of roughly 15–40 kPa and local first principal strains of up to 25–50% when 0.3 mL of saline was injected into the joint space; the subsequent increase in pressure and strain were more modest for further increases in injection volume, possibly due to leakage of fluid from the joint. The largest stretches were in the bone-to-bone direction in the portions of the ligament spanning the joint space. When the ligament was released from the vertebrae, it shrank by an average of 4–5%, with local maximum (negative) principal strain values of up to 30%, on average. Based on these measurements and previous tests on isolated lumbar facet capsular ligaments, we conclude that the normal in vivo state of the facet capsular ligament is in tension, and that the collagen in the ligament is likely uncrimped even when the spine is not loaded.

Aortic aneurysms are inherently unpredictable. One can never be sure whether any given aneurysm may rupture or dissect. Clinically, the criteria for surgical intervention are based on size and growth rate, but it remains difficult to identify a high-risk aneurysm, which may require intervention before the cutoff criteria, versus an aneurysm than can be treated safely by more conservative measures. In this work, we created a computational microstructural model of a medial lamellar unit (MLU) incorporating (1) growth and remodeling laws applied directly to discrete, individual fibers, (2) separate but interacting fiber networks for collagen, elastin, and smooth muscle, (3) active and passive smooth-muscle cell mechanics, and (4) failure mechanics for all three fiber types. The MLU model was then used to study different pathologies and microstructural anomalies that may play a role in vascular growth and failure. Our model recapitulated many aspects of arterial remodeling under hypertension with no underlying genetic syndrome including remodeling dynamics, tissue mechanics, and failure. Syndromic effects (smooth muscle cell (SMC) dysfunction or elastin fragmentation) drastically changed the simulated remodeling process, tissue behavior, and tissue strength. Different underlying pathologies were able to produce similarly dilatated vessels with different failure properties, providing a partial explanation for the imperfect nature of aneurysm size as a predictor of outcome.

Ascending thoracic aortic aneurysms (ATAAs) are anatomically complex in terms of architecture and geometry, and both complexities contribute to unpredictability of ATAA dissection and rupture in vivo. The goal of this work was to examine the mechanism of ATAA failure using a combination of detailed mechanical tests on human tissue and a multiscale computational model. We used (1) multiple, geometrically diverse, mechanical tests to characterize tissue properties; (2) a multiscale computational model to translate those results into a broadly usable form; and (3) a model-based computer simulation of the response of an ATAA to the stresses generated by the blood pressure. Mechanical tests were performed in uniaxial extension, biaxial extension, shear lap, and peel geometries. ATAA tissue was strongest in circumferential extension and weakest in shear, presumably because of the collagen and elastin in the arterial lamellae. A multiscale, fiber-based model using different fiber properties for collagen, elastin, and interlamellar connections was specified to match all of the experimental data with one parameter set. Finally, this model was used to simulate ATAA inflation using a realistic geometry. The predicted tissue failure occurred in regions of high stress, as expected; initial failure events involved almost entirely interlamellar connections, consistent with arterial dissection—the elastic lamellae remain intact, but the connections between them fail. The failure of the interlamellar connections, paired with the weakness of the tissue under shear loading, is suggestive that shear stress within the tissue may contribute to ATAA dissection.

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The role of computational modeling for biomechanics research and related clinical care will be increasingly prominent. The biomechanics community has been developing computational models routinely for exploration of the mechanics and mechanobiology of diverse biological structures. As a result, a large array of models, data, and discipline-specific simulation software has emerged to support endeavors in computational biomechanics. Sharing computational models and related data and simulation software has first become a utilitarian interest, and now, it is a necessity. Exchange of models, in support of knowledge exchange provided by scholarly publishing, has important implications. Specifically, model sharing can facilitate assessment of reproducibility in computational biomechanics and can provide an opportunity for repurposing and reuse, and a venue for medical training. The community's desire to investigate biological and biomechanical phenomena crossing multiple systems, scales, and physical domains, also motivates sharing of modeling resources as blending of models developed by domain experts will be a required step for comprehensive simulation studies as well as the enhancement of their rigor and reproducibility. The goal of this paper is to understand current perspectives in the biomechanics community for the sharing of computational models and related resources. Opinions on opportunities, challenges, and pathways to model sharing, particularly as part of the scholarly publishing workflow, were sought. A group of journal editors and a handful of investigators active in computational biomechanics were approached to collect short opinion pieces as a part of a larger effort of the IEEE EMBS Computational Biology and the Physiome Technical Committee to address model reproducibility through publications. A synthesis of these opinion pieces indicates that the community recognizes the necessity and usefulness of model sharing. There is a strong will to facilitate model sharing, and there are corresponding initiatives by the scientific journals. Outside the publishing enterprise, infrastructure to facilitate model sharing in biomechanics exists, and simulation software developers are interested in accommodating the community's needs for sharing of modeling resources. Encouragement for the use of standardized markups, concerns related to quality assurance, acknowledgement of increased burden, and importance of stewardship of resources are noted. In the short-term, it is advisable that the community builds upon recent strategies and experiments with new pathways for continued demonstration of model sharing, its promotion, and its utility. Nonetheless, the need for a long-term strategy to unify approaches in sharing computational models and related resources is acknowledged. Development of a sustainable platform supported by a culture of open model sharing will likely evolve through continued and inclusive discussions bringing all stakeholders at the table, e.g., by possibly establishing a consortium.

The von Mises (VM) stress is a common stress measure for finite element models of tissue mechanics. The VM failure criterion, however, is inherently isotropic, and therefore may yield incorrect results for anisotropic tissues, and the relevance of the VM stress to anisotropic materials is not clear. We explored the application of a well-studied anisotropic failure criterion, the Tsai–Hill (TH) theory, to the mechanically anisotropic porcine aorta. Uniaxial dogbones were cut at different angles and stretched to failure. The tissue was anisotropic, with the circumferential failure stress nearly twice the axial (2.67 ± 0.67 MPa compared to 1.46 ± 0.59 MPa). The VM failure criterion did not capture the anisotropic tissue response, but the TH criterion fit the data well (R2 = 0.986). Shear lap samples were also tested to study the efficacy of each criterion in predicting tissue failure. Two-dimensional failure propagation simulations showed that the VM failure criterion did not capture the failure type, location, or propagation direction nearly as well as the TH criterion. Over the range of loading conditions and tissue geometries studied, we found that problematic results that arise when applying the VM failure criterion to an anisotropic tissue. In contrast, the TH failure criterion, though simplistic and clearly unable to capture all aspects of tissue failure, performed much better. Ultimately, isotropic failure criteria are not appropriate for anisotropic tissues, and the use of the VM stress as a metric of mechanical state should be reconsidered when dealing with anisotropic tissues.

The ascending thoracic aorta is poorly understood mechanically, especially its risk of dissection. To make better predictions of dissection risk, more information about the multidimensional failure behavior of the tissue is needed, and this information must be incorporated into an appropriate theoretical/computational model. Toward the creation of such a model, uniaxial, equibiaxial, peel, and shear lap tests were performed on healthy porcine ascending aorta samples. Uniaxial and equibiaxial tests showed anisotropy with greater stiffness and strength in the circumferential direction. Shear lap tests showed catastrophic failure at shear stresses (150–200 kPa) much lower than uniaxial tests (750–2500 kPa), consistent with the low peel tension (∼60 mN/mm). A novel multiscale computational model, including both prefailure and failure mechanics of the aorta, was developed. The microstructural part of the model included contributions from a collagen-reinforced elastin sheet and interlamellar connections representing fibrillin and smooth muscle. Components were represented as nonlinear fibers that failed at a critical stretch. Multiscale simulations of the different experiments were performed, and the model, appropriately specified, agreed well with all experimental data, representing a uniquely complete structure-based description of aorta mechanics. In addition, our experiments and model demonstrate the very low strength of the aorta in radial shear, suggesting an important possible mechanism for aortic dissection.

The spinal facet capsular ligament (FCL) is primarily comprised of heterogeneous arrangements of collagen fibers. This complex fibrous structure and its evolution under loading play a critical role in determining the mechanical behavior of the FCL. A lack of analytical tools to characterize the spatial anisotropy and heterogeneity of the FCL's microstructure has limited the current understanding of its structure–function relationships. Here, the collagen organization was characterized using spatial correlation analysis of the FCL's optically obtained fiber orientation field. FCLs from the cervical and lumbar spinal regions were characterized in terms of their structure, as was the reorganization of collagen in stretched cervical FCLs. Higher degrees of intra- and intersample heterogeneity were found in cervical FCLs than in lumbar specimens. In the cervical FCLs, heterogeneity was manifested in the form of curvy patterns formed by collections of collagen fibers or fiber bundles. Tensile stretch, a common injury mechanism for the cervical FCL, significantly increased the spatial correlation length in the stretch direction, indicating an elongation of the observed structural features. Finally, an affine estimation for the change of correlation length under loading was performed which gave predictions very similar to the actual values. These findings provide structural insights for multiscale mechanical analyses of the FCLs from various spinal regions and also suggest methods for quantitative characterization of complex tissue patterns.

The Pacinian corpuscle (PC) is a cutaneous mechanoreceptor that senses low-amplitude, high-frequency vibrations. The PC contains a nerve fiber surrounded by alternating layers of solid lamellae and interlamellar fluid, and this structure is hypothesized to contribute to the PC's role as a band-pass filter for vibrations. In this study, we sought to evaluate the relationship between the PC's material and geometric parameters and its response to vibration. We used a spherical finite element mechanical model based on shell theory and lubrication theory to model the PC's outer core. Specifically, we analyzed the effect of the following structural properties on the PC's frequency sensitivity: lamellar modulus (E), lamellar thickness (h), fluid viscosity (μ), PC outer radius (Ro), and number of lamellae (N). The frequency of peak strain amplification (henceforth “peak frequency”) and frequency range over which strain amplification occurred (henceforth “bandwidth”) increased with lamellar modulus or lamellar thickness and decreased with an increase in fluid viscosity or radius. All five structural parameters were combined into expressions for the relationship between the parameters and peak frequency, ωpeak=1.605×10−6N3.475(Eh/μRo), or bandwidth, B=1.747×10−6N3.951(Eh/μRo). Although further work is needed to understand how mechanical variability contributes to functional variability in PCs and how factors such as PC eccentricity also affect PC behavior, this study provides two simple expressions that can be used to predict the impact of structural or material changes with aging or disease on the frequency response of the PC.

Background: Collagen gels are important as platforms for in vitro study of cell behavior and as prototypical bioartificial tissues, but their mechanical behavior, particularly on the microscopic scale, is still poorly understood. Method of Approach: Collagen gels were studied in step (10% strain in 0.05 s) and ramp (0.1%/s strain rate for 100 s) confined compression. Real-time birefringence mapping gave the local collagen concentration and orientation along with piston stress. Variations in the retardation allowed material-point tracking and qualitative determination of the strain distribution. Results: Ramp tests showed classical poroelastic behavior: compression near the piston and relaxation to a uniform state. Step tests, however, showed an irreversibly collapsed region near the piston. Conclusions: Our results suggest that interstitial flow and fibril bending at crosslinks are the dominant mechanical processes during compression, and that fibril bending is reversible before collapse.

A method to impose and measure a one dimensional strain field via confined compression of a tissue-equivalent and measure the resulting cell and collagen fibril alignment was developed. Strain was determined locally by the displacement of polystyrene beads dispersed and entrapped within the network of collagen fibrils along with the cells, and it was correlated to the spatial variation of collagen network birefringence and concentration. Alignment of fibroblasts and smooth muscle cells was determined based on the long axis of elongated cells. Cell and collagen network alignment were observed normal to the direction of compression after a step strain and increased monotonically up to 50% strain. These results were independent of time after straining over 24 hr despite continued cell motility after responding instantly to the step strain with a change in alignment by deforming/convecting with the strained network. Since the time course of cell alignment followed that of strain and not stress which, due to the viscoelastic fluid-like nature of the network relaxes completely within the observation period, these results imply cell alignment in a compacting tissue-equivalent is due to fibril alignment associated with anisotropic network strain. Estimation of a contact guidance sensitivity parameter indicates that both cell types align to a greater extent than the surrounding fibrils.

Soft-tissue characterization using planar biaxial testing and nominal stress–strain curves usually relies on certain conditions1: The sample should have a shape, (e.g., square or cruciform) that tends to produce homogeneous strain fields in the central region. If the sample is fibrous, the fiber orientation should be known and aligned with the axes of the test system. Strain should be measured far from any rigid boundary, such as a grip or attachment to bone. These criteria are often met (e.g., Refs. [3–9]), but for some tissue types, meeting one or more criteria is impossible. The tissue may, for example, be too small to allow isolation of a sample that is large enough for biaxial testing and is shaped to create a homogeneous strain region in the center. Aligning the material axes does not allow a biaxial test to generate planar shear strain [10]. Furthermore, if the material axes are improperly aligned, the variation in tissue response will increase, which could hinder elucidation of complex behaviors such as coupling between the two directions [10]. Other groups [e.g., Ref. 10] have used biaxial testing to good effect for off-axis fibrous samples. This is only effective, however, when the fiber orientation is known. When fiber orientation is unknown and/or the objective is to determine the fiber orientation, as in the current study, force-stretch data alone are insufficient. It also may be undesirable or even impossible to remove the tissue from bone, which restricts one's ability to align the tissue fiber direction with the testing apparatus and leads to inhomogeneity of the strain field. An example of these challenges is found in the AF of the intervertebral disk, particularly if one seeks to do single lamella experiments. Although some single lamella experiments have been performed in uniaxial [11–13] and biaxial modes [14–16], testing with intact bone [12–14,16] is attractive both because of direct relevance to in vivo loading and because of minimized tissue damage. Figure ​Figure11 shows a lamella of AF attached to axial vertebral bone. This geometry is not conducive to a homogeneous strain field, and the principal fiber direction does not align with the axes of the testing apparatus. Fig. 1 Lamella of the AF. The sample is attached to bone, labeled, and is anisotropic with fibers aligned 30 deg from the horizontal testing axis, along the dotted line. The dissected tissue is too small to be removed from the bone and cut to align the ... Sample geometries that produce inhomogeneous strain fields pose a considerable challenge to the investigator, but the advent of image-correlation-based methods for tracking motion over the entire tissue during testing [17–24] present new opportunities. The approach of simulating the experiment and then iterating over the model parameters to determine a best fit has been used in elastography [25,26] as well as indentation [27] and can be applied to tissue testing as well. In this study, we demonstrate the use of combined displacement field data, grip force-stretch data, and FE modeling to determine tissue properties from a biaxial test.

Collagen mechanics are crucial to the function and dysfunction of many tissues, including blood vessels and articular cartilage, and bioartificial tissues. Previous attempts to develop computer simulations of collagenous tissue based on macroscopic property descriptions have often been limited in application by the simplicity of the model; simulations based on microscopic descriptions, in contrast, have numerical limitations imposed by the size of the mathematical problem. We present a method that combines the tractability of the macroscopic approach with the flexibility of the microstructural approach. The macroscopic domain is divided into finite elements (as in standard FEM). Each element contains a microscopic scale network. Instead of a stress constitutive equation; the macroscopic problem is distributed over the microscopic scale network and solved in each element to satisfy the weak formulation of Cauchy’s stress continuity equation over the macroscopic domain. The combined method scales by order 1.1 as the overall number of degrees of freedom is increased, allowing it to handle larger problems than a direct microstructural approach. Model predictions agree qualitatively with tensile tests on isotropic and aligned reconstituted type I collagen gels.

A tissue's microstructure determines its failure properties at larger length scales, however, the specific relationship between microstructure and macroscopic failure in native and engineered soft tissues (such as capsular ligaments, aortic aneurysms, or vascular grafts) has proven elusive. In this study, variations in the microscale fiber alignment in collagen gel tissue analogs were modeled in order to understand their effects on macroscale damage and failure outcomes. The study employed a multiscale finite-element (FE) model for damage and failure in collagen-based materials. The model relied on microstructural representative volume elements (RVEs) that consisted of stochastically-generated networks of discrete type-I collagen fibers. Fiber alignment was varied within RVEs and between layers of RVEs in a macroscopic FE model of a notched dogbone geometry. The macroscale stretch and the microscale response of fibers for each of the differently aligned cases were compared as the dogbone was uniaxially extended to failure. Networks with greater fiber alignment parallel to the direction of extension failed at smaller strains (with a 6–22% reduction in the Green strain at failure), however, at greater grip forces (a 28–60% increase) than networks with fibers aligned perpendicular to the extension. Alternating layers of crisscrossed network alignments (aligned ±45 deg to the direction of extension) failed at smaller strains but at greater grip forces than those created using one fiber alignment type. In summary, variations in microscale structure via fiber alignment produced different macroscale failure trends. To conclude, these findings may be significant in the realm of tissue engineering and in soft tissue biomechanics.

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.

In Part 1 of this work, we formulated and analyzed a mathematical model for our fibroblast-populated collagen microsphere (FPCM) assay of cell traction forces (Moon and Tranquillo, 1993). In this assay, the FPCM diameter decreases with time as the cells compact the gel by exerting traction on collagen fibrils. In Part I we demonstrated that the diameter reduction profiles for varied initial cell concentration and varied initial FPCM diameter are qualitatively consistent with the model predictions. We show here in Part 2 how predictions of a model similar to that of Part 1, along with the determination of the growth parameters of the cells and the viscoelastic parameters of the gel, allow us to estimate the magnitude of a cell traction parameter, the desired objective index of cell traction forces. The model is based on a monophasic continuum-mechanical theory of cell-extracellular matrix (ECM) mechanical interactions, with a species conservation equation for cells (1), a mass conservation equation for ECM (2), and a mechanical force balance for the cell/ECM composite (3). Using a constant-stress rheometer and a fluids spectrometer in creep and oscillatory shear modes, respectively, we establish and characterize the linear viscoelastic regime for the reconstituted type I collagen gel used in our FPCM traction assay and in other assays of cell-collagen mechanical interactions. Creep tests are performed on collagen gel specimens in a state resembling that in our FPCM traction assay (initially uncompacted, and therefore nearly isotropic and at a relatively low collagen concentration of 2.1 mg/ml), yielding measurements of the zero shear viscosity, μ 0 (7.4 x 10 6 Poise), and the steady-state creep compliance, J c 0 . The shear modulus, G (155 dynes/cm 2 ), is then determined from the inverse of J e 0 in the linear viscoelastic regime. Oscillatory shear tests are performed in strain sweep mode, indicating linear viscoelastic behavior up to shear strains of approximately 10 percent. We discuss the estimation of Poisson's ratio, v, which along with G and μ 0 specifies the assumed isotropic, linear viscoelastic stress tensor for the cell/collagen gel composite which appears in (3). The proliferation rate of fibroblasts in free floating collagen gel (appearing in (1)) is characterized by direct cell counting, yielding an estimate of the first-order growth rate constant, k (5.3 X 10 -6 s -1 ). These independently measured and estimated parameter values allow us to estimate that the cell traction parameter, τ 0 , defined in the active stress tensor which also appears in (3), is in the range of 0.00007-0.0002 dyne cm 4 /mg collagen cell. This value is in agreement with a reported measure of traction obtained directly via isometric force measurement across a slab of fibroblast-containing collagen gel.

Excessive tissue-level forces communicated to the microstructure and extracellular matrix of soft tissues can lead to damage and failure through poorly understood physical processes that are multiscale in nature. In this work, we propose a multiscale mechanical model for the failure of collagenous soft tissues that incorporates spatial heterogeneity in the microstructure and links the failure of discrete collagen fibers to the material response of the tissue. The model, which is based on experimental failure data derived from different collagen gel geometries, was able to predict the mechanical response and failure of type I collagen gels, and it demonstrated that a fiber-based rule (at the micrometer scale) for discrete failure can strongly shape the macroscale failure response of the gel (at the millimeter scale). The model may be a useful tool in predicting the macroscale failure conditions for soft tissues and engineered tissue analogs. In addition, the multiscale model provides a framework for the study of failure in complex fiber-based mechanical systems in general.

A model is developed and analyzed for type IV collagen turnover in the kidney glomerular basement membrane (GBM), which is the primary structural element in the glomerular capillary wall. The model incorporates strain dependence in both deposition and removal of the GBM, leading to an equilibrium tissue strain at which deposition and removal are balanced. The GBM thickening decreases tissue strain per unit of transcapillary pressure drop according to the law of Laplace, but increases the transcapillary pressure drop required to maintain glomerular filtration. The model results are in agreement with the observed GBM alterations in Alport syndrome and thin basement membrane disease, and the model-predicted linear relation between the inverse capillary radius and inverse capillary thickness at equilibrium is consistent with published data on different mammals. In addition, the model predicts a minimum achievable strain in the GBM based on the geometry, properties, and mechanical environment; that is, an infinitely thick GBM would still experience a finite strain. Although the model assumptions would be invalid for an extremely thick GBM, the minimum achievable strain could be significant in diseases, such as Alport syndrome, characterized by focal GBM thickening. Finally, an examination of reasonable values for the model parameters suggests that the oncotic pressure drop—the osmotic pressure difference between the plasma and the filtrate due to large molecules—plays an important role in setting the GBM strain and, thus, leakage of protein into the urine may be protective against some GBM damage.

Fibrin and collagen, biopolymers occurring naturally in the body, are biomaterials commonly-used as scaffolds for tissue engineering. How collagen and fibrin interact to confer macroscopic mechanical properties in collagen-fibrin composite systems remains poorly understood. In this study, we formulated collagen-fibrin co-gels at different collagen-to-fibrin ratios to observe changes in the overall mechanical behavior and microstructure. A modeling framework of a two-network system was developed by modifying our micro-scale model, considering two forms of interaction between the networks: (a) two interpenetrating but noninteracting networks (“parallel”), and (b) a single network consisting of randomly alternating collagen and fibrin fibrils (“series”). Mechanical testing of our gels show that collagen-fibrin co-gels exhibit intermediate properties (UTS, strain at failure, tangent modulus) compared to those of pure collagen and fibrin. The comparison with model predictions show that the parallel and series model cases provide upper and lower bounds, respectively, for the experimental data, suggesting that a combination of such interactions exists between the collagen and fibrin in co-gels. A transition from the series model to the parallel model occurs with increasing collagen content, with the series model best describing predominantly fibrin co-gels, and the parallel model best describing predominantly collagen co-gels.

As an alternative to purely mechanical methods, optical tracking of passive osmotic swelling was used to assess mechanical properties of the porcine lens capsule. A simple model was developed accounting for the permeability of the lens fiber cells and capsule to water, the concentration of fixed charges in the fiber cells, and the capsule’s resistance to the swelling of fiber cells. Fitting the model solution to experimental data provided an estimate of the elastic modulus of the lens capsule under the assumption of linear isotropic elasticity. The calculated elastic modulus at a fixed charge density of 20 mol m−3 was 2.0±0.5 MPa (mean±95% confidence interval; n=15) for 0.1% saline solution, 0.64±0.3 MPa(n=10) for 0.2% saline solution, and 0.28±0.5 MPa(n=6) for 0.5% saline solution. These values are comparable to previously reported moduli of elasticity for the porcine lens capsule at small strains (

Elastography, which is the imaging of soft tissues on the basis of elastic modulus (or, more generally, stiffness) has become increasingly popular in the last decades and holds promise for application in many medical areas. Most of the attention has focused on inhomogeneous materials that are locally isotropic, the intent being to detect a (stiff) tumor within a (compliant) tissue. Many tissues of mechanical interest, however, are anisotropic, so a method capable of determining material anisotropy would be attractive. We present here an approach to determine the mechanical anisotropy of inhomogeneous, anisotropic tissues, by directly solving the finite element representation of the Cauchy stress balance in the tissue. The method divides the sample domain into subdomains assumed to have uniform properties and solves for the material constants in each subdomain. Two-dimensional simulated experiments on linear anisotropic inhomogeneous systems demonstrate the ability of the method, and simulated experiments on a nonlinear model demonstrate the ability of the method to capture anisotropy qualitatively even though only a linear model is used in the inverse problem. As with any inverse problem, ill-posedness is a serious concern, and multiple tests may need to be done on the same sample to determine the properties with confidence.

The fibroblast-populated collagen lattice is an attractive model tissue for in vitro studies of cell behavior and as the basis for bioartificial tissues. In spite of its simplicity—containing only collagen and cells—the system is surprisingly difficult to describe mechanically because of the ability of the cells to remodel the matrix, including compaction at short times and synthesis and/or degradation (and cell proliferation) at longer times. The objectives of this work were to measure the equilibrium modulus of fibroblast-populated gels with different collagen and cell concentrations, and to use that characterization as the basis for a theoretical model that could be used to predict gel mechanics based on conditions. Although many observations were as expected (e.g., the gel compacts more when there are more cells in it, and the gel is stiffer when there is more collagen in it), an unexpected result arose: the final modulus of the gel was not dependent solely on the final composition. Even if it compacted more than a gel that was originally at a high collagen concentration, a gel that started at a low collagen concentration remained less stiff than the higher-concentration gel. In light of these results and experimental studies by others, we propose a model in which the gel compaction is not homogeneous but consists instead of extreme densification near the cells in an otherwise unchanged matrix. By treating the dense regions as spherical inclusions, we used classical composite material theory to develop an expression for the modulus of a compacted gel based on the initial collagen density and the final inclusion (i.e., cell) density. The new model fit the data for moderately compacted gels well but broke down, as expected, for larger volume fractions at which the underlying model assumptions did not apply.

Though it is widely accepted that fiber alignment has a great influence on the mechanical anisotropy of tissues, a systematic study of the influence of fiber alignment on the macroscopic mechanical behavior by native tissues is precluded due to their predefined microstructure and heterogeneity. Such a study is possible using collagen-based bioartificial tissues that allow for alignment to be prescribed during their fabrication. To generate a systemic variation of strength of fiber alignment, we made cruciform tissue constructs in Teflon molds that had arms of different aspect ratios. We implemented our anisotropic biphasic theory of tissue-equivalent mechanics to simulate the compaction by finite element analysis. Prior to tensile testing, the construct geometry was standardized by cutting test samples with a 1:1 cruciform punch after releasing constructs from the molds. Planar biaxial testing was performed on these samples, after stretching them to their in-mold dimensions to recover in-mold alignment, to observe the macroscopic mechanical response with simultaneous fiber alignment imaging using a polarimetry system. We found that the strength of fiber alignment of the samples prior to release from the molds linearly increased with anisotropy of the mold. In testing after release, modulus ratio (modulus in fiber direction/modulus in normal direction) was greater as the initial strength of fiber alignment increased, that is, as the aspect ratio increased. We also found that the fiber alignment strength and modulus ratio increased in a hyperbolic fashion with stretching for a sample of given aspect ratio.

Mechanics of collagen gels, like that of many tissues, is governed by events occurring on a length scale much smaller than the functional scale of the material. To deal with the challenge of incorporating deterministic micromechanics into a continuous macroscopic model, we have developed an averaging-theory-based modeling framework for collagen gels. The averaging volume, which is constructed around each integration point in a macroscopic finite-element model, is assumed to experience boundary deformations homogeneous with the macroscopic deformation field, and a micromechanical problem is solved to determine the average stress at the integration point. A two-dimensional version was implemented with the microstructure modeled as a network of nonlinear springs, and 500 segments were found to be sufficient to achieve statistical homogeneity. The method was then used to simulate the experiments of Tower et al. (Ann. Biomed. Eng., 30, pp. 1221–1233) who performed uniaxial extension of prealigned collagen gels. The simulation captured many qualitative features of the experiments, including a toe region and the realignment of the fibril network during extension. Finally, the method was applied to an idealized wound model based on the characterization measurements of Bowes et al. (Wound Repair Regen., 7, pp. 179–186). The model consisted of a strongly aligned “wound” region surrounded by a less strongly aligned “healthy” region. The alignment of the fibrils in the wound region led to reduced axial strains, and the alignment of the fibrils in the healthy region, combined with the greater effective stiffness of the wound region, caused rotation of the wound region during uniaxial stretch. Although the microscopic model in this study was relatively crude, the multiscale framework is general and could be employed in conjunction with any microstructural model.

The microstructure of tissues and tissue equivalents (TEs) plays a critical role in determining the mechanical properties thereof. One of the key challenges in constitutive modeling of TEs is incorporating the kinematics at both the macroscopic and the microscopic scale. Models of fibrous microstructure commonly assume fibrils to move homogeneously, that is affine with the macroscopic deformation. While intuitive for situations of fibril-matrix load transfer, the relevance of the affine assumption is less clear when primary load transfer is from fibril to fibril. The microstructure of TEs is a hydrated network of collagen fibrils, making its microstructural kinematics an open question. Numerical simulation of uniaxial extensile behavior in planar TE networks was performed with fibril kinematics dictated by the network model and by the affine model. The average fibril orientation evolved similarly with strain for both models. The individual fibril kinematics, however, were markedly different. There was no correlation between fibril strain and orientation in the network model, and fibril strains were contained by extensive reorientation. As a result, the macroscopic stress given by the network model was roughly threefold lower than the affine model. Also, the network model showed a toe region, where fibril reorientation precluded the development of significant fibril strain. We conclude that network fibril kinematics are not governed by affine principles, an important consideration in the understanding of tissue and TE mechanics, especially when load bearing is primarily by an interconnected fibril network.

The Ahmed glaucoma valve (AGV) is a popular glaucoma drainage device, allowing maintenance of normal intraocular pressure in patients with reduced trabecular outflow facility. The uniquely attractive feature of the AGV, in contrast to other available drainage devices, is its variable resistance in response to changes in flow rate. As a result of this variable resistance, the AGV maintains a pressure drop between 7 and 12 mm Hg for a wide range of aqueous humor flow rates. In this paper, we demonstrate that the nonlinear behavior of the AGV is a direct result of the flexibility of the valve material. Due to the thin geometry of the system, the leaflets of the AGV were modeled using the von Karman plate theory coupled to a Reynolds lubrication theory model of the aqueous humor flow through the valve. The resulting two-dimensional coupled steady-state partial differential equation system was solved by the finite element method. The Poisson's ratio of the valve was set to 0.45, and the modulus was regressed to experimental data, giving a best-fit value 4.2 MPa. Simulation results compared favorably with previous experimental studies and our own pressure-drop/flow-rate data. For an in vitro flow of 1.6 microL/min, we calculated a pressure drop of 5.8 mm Hg and measured a pressure drop of 5.2 +/- 0.4 mm Hg. As flow rate was increased, pressure drop rose in a strongly sublinear fashion, with a flow rate of 20 microL/min giving a predicted pressure drop of only 10.9 mm Hg and a measured pressure drop of 10.5 +/- 1.1 mm Hg. The AGV model was then applied to simulate in vivo conditions. For an aqueous humor flow rate of 1.5-3.0 microL/min, the calculated pressure drops were 5.3 and 6.3 mm Hg.