The Ring-Pull Assay for Mechanical Properties of Fibrous Soft Tissues - An Analysis of the Uniaxial Approximation and a Correction for Nonlinear Thick-Walled Tissues.

BACKGROUND: The ring-pull test, where a ring of tissue is hooked via two pins and stretched, is a popular biomechanical test, especially for small arteries. Although convenient and reliable, the ring test produces inhomogeneous strain, making determination of material parameters non-trivial. OBJECTIVE: To determine correction factors between ring-pull-estimated and true tissue properties. METHODS: A finite-element model of ring pulling was constructed for a sample with nonlinear, anisotropic mechanical behavior typical of arteries. The pin force and sample cross-section were used to compute an apparent modulus at small and large strain, which were compared to the specified properties. The resulting corrections were validated with experiments on porcine and ovine arteries. The correction was further applied to experiments on mouse aortic rings to determine material and failure properties. RESULTS: Calculating strain based on centerline stretch rather than inner-wall or outer-wall stretch afforded better estimation of tissue properties. Additional correction factors were developed based on ring wall thickness H, centerline ring radius R c , and pin radius a. The corrected estimates for tissue properties were in good agreement with uniaxial stretch experiments. CONCLUSIONS: The computed corrections improved estimation of tissue material properties for both the small-strain (toe) modulus and the large-strain (lockout) modulus. When measuring tensile strength, one should minimize H/a to ensure that peak stress occurs at the sample midplane rather than near the pin. In this scenario, tensile strength can be estimated accurately by using inner-wall stretch at the midplane and the corrected properties.

Crack Propagation Versus Fiber Alignment in Collagen Gels: Experiments and Multiscale Simulation.

It is well known that the organization of the fibers constituting a collagenous tissue can affect its failure behavior. Less clear is how that effect can be described computationally so as to predict the failure of a native or engineered tissue under the complex loading conditions that can occur in vivo. Toward the goal of a general predictive strategy, we applied our multiscale model of collagen gel mechanics to the failure of a double-notched gel under tension, comparing the results for aligned and isotropic samples. In both computational and laboratory experiments, we found that the aligned gels were more likely to fail by connecting the two notches than the isotropic gels. For example, when the initial notches were 30% of the sample width (normalized tip-to-edge distance = 0.7), the normalized tip-to-tip distance at which the transition occurred from between-notch failure to across-sample failure shifted from 0.6 to 1.0. When the model predictions for the type of failure event (between the two notches versus across the sample width) were compared to the experimental results, the two were found to be strongly covariant by Fisher's exact test (p < 0.05) for both the aligned and isotropic gels with no fitting parameters. Although the double-notch system is idealized, and the collagen gel system is simpler than a true tissue, it presents a simple model system for studying failure of anisotropic tissues in a controlled setting. The success of the computational model suggests that the multiscale approach, in which the structural complexity is incorporated via changes in the model networks rather than via changes to a constitutive equation, has the potential to predict tissue failure under a wide range of conditions.

Cross-Linked Fiber Network Embedded in Elastic Matrix.

The mechanical behavior of a three-dimensional cross-linked fiber network embedded in matrix is studied in this work. The network is composed from linear elastic fibers which store energy only in the axial deformation mode, while the matrix is also isotropic and linear elastic. Such systems are encountered in a broad range of applications, from tissue to consumer products. As the matrix modulus increases, the network is constrained to deform more affinely. This leads to internal forces acting between the network and the matrix, which produce strong stress concentration at the network cross-links. This interaction increases the apparent modulus of the network and decreases the apparent modulus of the matrix. A model is developed to predict the effective modulus of the composite and its predictions are compared with numerical data for a variety of networks.

Multiscale mechanical simulations of cell compacted collagen gels.

Engineered tissues are commonly stretched or compressed (i.e., conditioned) during culture to stimulate extracellular matrix (ECM) production and to improve the mechanical properties of the growing construct. The relationships between mechanical stimulation and ECM remodeling, however, are complex, interdependent, and dynamic. Thus, theoretical models are required for understanding the underlying phenomena so that the conditioning process can be optimized to produce functional engineered tissues. Here, we continue our development of multiscale mechanical models by simulating the effect of cell tractions on developing isometric tension and redistributing forces in the surrounding fibers of a collagen gel embedded with explants. The model predicted patterns of fiber reorganization that were similar to those observed experimentally. Furthermore, the inclusion of cell compaction also changed the distribution of fiber strains in the gel compared to the acellular case, particularly in the regions around the cells where the highest strains were found.

Simulated remodeling of loaded collagen networks via strain-dependent enzymatic degradation and constant-rate fiber growth.

Recent work has demonstrated that enzymatic degradation of collagen fibers exhibits strain-dependent kinetics. Conceptualizing how the strain dependence affects remodeling of collagenous tissues is vital to our understanding of collagen management in native and bioengineered tissues. As a first step towards this goal, the current study puts forward a multiscale model for enzymatic degradation and remodeling of collagen networks for two sample geometries we routinely use in experiments as model tissues. The multiscale model, driven by microstructural data from an enzymatic decay experiment, includes an exponential strain-dependent kinetic relation for degradation and constant growth. For a dogbone sample under uniaxial load, the model predicted that the distribution of fiber diameters would spread over the course of degradation because of variation in individual fiber load. In a cross-shaped sample, the central region, which experiences smaller, more isotropic loads, showed more decay and less spread in fiber diameter compared to the arms. There was also a slight shift in average orientation in different regions of the cruciform.

Initial fiber alignment pattern alters extracellular matrix synthesis in fibroblast-populated fibrin gel cruciforms and correlates with predicted tension.

Human dermal fibroblasts entrapped in fibrin gels cast in cross-shaped (cruciform) geometries with 1:1 and 1:0.5 ratios of arm widths were studied to assess whether tension and alignment of the cells and fibrils affected ECM deposition. The cruciforms of contrasting geometry (symmetric vs. asymmetric), which developed different fiber alignment patterns, were harvested at 2, 5, and 10 weeks of culture. Cruciforms were subjected to planar biaxial testing, polarimetric imaging, DNA and biochemical analyses, histological staining, and SEM imaging. As the cruciforms compacted and developed fiber alignment, fibrin was degraded, and elastin and collagen were produced in a geometry-dependent manner. Using a continuum mechanical model that accounts for direction-dependent stress due to cell traction forces and cell contact guidance with aligned fibers that occurs in the cruciforms, the mechanical stress environment was concluded to influence collagen deposition, with deposition being the greatest in the narrow arms of the asymmetric cruciform where stress was predicted to be the largest.

Posterior chamber volume does not change significantly during dilation.

BACKGROUND/AIMS: To determine whether the volume of the posterior chamber changes during pupillary dilation. METHODS: Eyes with anatomically narrow angles underwent ultrasound biomicroscopy of the posterior chamber and pupillary margin under dark- and light-room conditions to assess changes in posterior chamber anatomy and volume. All examinations were stored as real-time video. A frame-by-frame analysis was performed using a macro written for the ImageJ image-processing software. RESULTS: Thirteen eyes were assessed. The mean patient age was 63.0 (SD 10.0) years, and the mean refractive error was 1.1 (1.9) dioptres. The horizontal mean pupillary diameter was 2.3 (0.6) mm and 3.5 (0.5) mm under light- and dark-room conditions, respectively (p<10(-7), paired t test). The mean posterior chamber volumes were unchanged under light and dark conditions (3.76 (1.09) vs 3.63 (0.78) mm(3), p = 0.22, paired t test). Volumes were greater under light conditions in eight eyes and under dark conditions in five eyes. CONCLUSIONS: The volume of the posterior chamber does not change significantly during dilation.

Comparison of 2D fiber network orientation measurement methods.

The mechanical properties of tissues, tissue analogs, and biomaterials are dependent on their underlying microstructure. As such, many mechanical models incorporate some aspect of microstructure, but a robust protocol for characterizing fiber architecture remains a challenge. A number of image-based methods, including mean intercept length (MIL), line fraction deviation (LFD), and Fourier transform methods (FTM), have been applied to microstructural images to describe material heterogeneity and orientation, but a performance comparison, particularly for fiber networks, has not been conducted. In this study, we constructed 40 two-dimensional test images composed of simulated fiber networks varying in fiber number, orientation, and anisotropy index. We assessed the accuracy of each method in measuring principal direction (theta) and anisotropy index (alpha). FTM proved to be the superior method because it was more reliable in measurement accuracy (Deltatheta = 2.95 degrees +/- 6.72 degrees , Deltaalpha = 0.03 +/- 0.02), faster in execution time, and flexible in its application. MIL (Deltatheta = 6.23 degrees +/- 10.68 degrees , Deltaalpha = 0.08 +/- 0.06) was not significantly less accurate than FTM but was much slower. LFD (Deltatheta = 9.97 degrees +/- 11.82 degrees , Deltaalpha = 0.24 +/- 0.13) consistently underperformed. FTM results agreed qualitatively with fibrin gel SEM micrographs, suggesting that FTM can be used to obtain image-based statistical measurements of microstructure.

Confined compression of a tissue-equivalent: collagen fibril and cell alignment in response to anisotropic strain.

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.

Coupled macroscopic and microscopic scale modeling of fibrillar tissues and tissue equivalents.

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.

Modeling passive mechanical interaction between aqueous humor and iris.

Certain forms of glaucoma are associated with displacement of the iris from its normal contour. We present here a mathematical model of the coupled aqueous humor-iris system that accountsfor the contribution of aqueous humor flow and passive iris deformability to the iris contour. The aqueous humor is modeled as a Newtonian fluid, and the iris is modeled as a linear elastic solid. The resulting coupled equation set is solved by the finite element method with mesh motion in response to iris displacement accomplished by tracking a pseudo-solid overlying the aqueous humor. The model is used to predict the iris contour in healthy and diseased eyes. The results compare favorably with clinical observations, supporting the hypothesis that passive iris deformation can produce the iris contours observed using ultrasound biomicroscopy.

Permeability and diffusion in vitreous humor: implications for drug delivery.

PURPOSE: Previous experimental work suggests that convection may be important in determining the biodistribution of drugs implanted or injected in the vitreous humor. To develop accurate biodistribution models, the relative importance of diffusion and convection in intravitreal transport must be assessed. This requires knowledge of both the diffusivity of candidate drugs and the hydraulic conductivity of the vitreous humor. METHODS: Hydraulic conductivity of cadaveric bovine vitreous humor was measured by confined compression tests at constant loads of 0.15 and 0.2 N and analyzed numerically using a two-phase model. Diffusion coefficient of acid orange 8, a model compound, in the same medium was measured in a custom-built diffusion cell. RESULTS: Acid orange 8 diffusivity within vitreous humor is about half that in free solution. When viscous effects are properly accounted for, the hydraulic conductivity of bovine vitreous humor is 8.4+/-4.5 x 10(-7) cm2/Pa x s. CONCLUSIONS: We predict that convection does not contribute significantly to transport in the mouse eye, particularly for low-molecular-weight compounds. For delivery to larger animals, such as humans we conclude that convection accounts for roughly 30% of the total intravitreal drug transport. This effect should be magnified for higher-molecular-weight compounds, which diffuse more slowly, and in glaucoma, which involves higher intraocular pressure and thus potentially faster convective flow. Thus, caution should be exercised in the extrapolation of small-animal-model biodistribution data to human scale.

Mechanical characterization of the bovine iris.

Quantification of the mechanical properties of the iris is necessary to assess the clinical significance of passive iris deformation, which has been suggested as a mechanism for certain forms of glaucoma. Extension tests were performed on isolated bovine irises to determine the passive mechanical behavior of the iris and the contribution of each of its two constituent muscles, the sphincter iridis and the dilator pupillae, to the overall properties. Because of the shape of the iris and our desire to use intact tissue, a "loop" experiment was performed in which the iris was stretched by hooking the sample and pulling. A simple mathematical model was used to account for the geometry of the experiment and the progressive recruitment of tissue. Radial extension experiments on samples dissected from the iris were also performed. The iris was found to be anisotropic, elastic, and incompressible. The average azimuthal Young's modulus of the sphincter was found to be 340 kPa; the average azimuthal Young's modulus of the dilator was found to be 890 kPa, which was significantly higher (p < 0.01). The radial Young's modulus of the dilator was found to be 9.6 kPa, much lower than the azimuthal value.

Engineered alignment in media equivalents: magnetic prealignment and mandrel compaction.

We predicted and measured the evolution of smooth muscle cell (SMC) orientation in media-equivalents (MEs) for four fabrication conditions (F-, M-, F+, M+) under Free or Mandrel compaction (F/M) with and without magnetic prealignment of the collagen fibrils in the circumferential direction (+/-). Mandrel compaction refers to SMC-induced compaction of the ME that is constrained by having a nonadhesive mandrel placed in the ME lumen. Predictions were made using our anisotropic biphasic theory (ABT) for tissue-equivalent mechanics. Successful prediction of trends of the SMC orientation data for all four fabrication cases was obtained: maintenance of the initial isotropic state for F-, loss of initial circumferential alignment for F+, development of circumferential alignment for M-, and enhancement of initial circumferential alignment for M+. These results suggest two mechanisms by which the presence of the mandrel leads to much greater mechanical stiffness in the circumferential direction reported for mandrel compacted MEs relative to free compacted MEs: (1) by inducing an increasing circumferential alignment of the SMC and collagen, and (2) by inducing a large stress on the SMC, resulting in secretion and accumulation of stiffening components.

A finite element solution for the anisotropic biphasic theory of tissue-equivalent mechanics: the effect of contact guidance on isometric cell traction measurement.

We present a method for solving the governing equations from our anisotropic biphasic theory of tissue-equivalent mechanics (Barocas and Tranquillo, 1997) for axisymmetric problems. A mixed finite element method is used for discretization of the spatial derivatives, and the DASPK subroutine (Brown et al., 1994) is used to solve the resulting differential-algebraic equation system. The preconditioned GMRES algorithm, using a preconditioner based on an extension of Dembo's (1994) adaptation of the Uzawa algorithm for viscous flows, provides an efficient and scaleable solution method, with the finite element method discretization being first-order accurate in space. In the cylindrical isometric cell traction assay, the chosen test problem, a cylindrical tissue equivalent is adherent at either end to fixed circular platens. As the cells exert traction on the collagen fibrils, the force required to maintain constant sample length, or load, is measured. However, radial compaction occurs during the course of the assay, so that the cell and network concentrations increase and collagen fibrils become aligned along the axis of the cylinder, leading to cell alignment along the axis. Our simulations predict that cell contact guidance leads to an increase in the load measured in the assay, but this effect is diminished by the tendency of contact guidance to inhibit radial compaction of the sample, which in turn reduces concentrations and hence the measured load.

An anisotropic biphasic theory of tissue-equivalent mechanics: the interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance.

We present a general mathematical theory for the mechanical interplay in tissue-equivalents (cell-populated collagen gels): Cell traction leads to compaction of the fibrillar collagen network, which for certain conditions such as a mechanical constraint or inhomogeneous cell distribution, can result in inhomogeneous compaction and consequently fibril alignment, leading to cell contact guidance, which affects the subsequent compaction. The theory accounts for the intrinsically biphasic nature of collagen gel, which is comprised of collagen network and interstitial solution. The theory also accounts for fibril alignment due to inhomogeneous network deformation, that is, anisotropic strain, and for cell alignment in response to fibril alignment. Cell alignment results in anisotropic migration and traction, as modeled by a cell orientation tensor that is a function of a fiber orientation tensor, which is defined by the network deformation tensor. Models for a variety of tissue-equivalents are shown to predict qualitatively the alignment that arises due to inhomogeneous compaction driven by cell traction.

The fibroblast-populated collagen microsphere assay of cell traction force--Part 2: Measurement of the cell traction parameter.

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 1 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 1 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, mu 0 7.4 x 10(6) Poise), and the steady-state creep compliance, J0e. The shear modulus, G (155 dynes/cm2), is then determined from the inverse of J0e 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 mu 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, tau 0, defined in the active stress tensor which also appears in (3), is in the range of 0.00007-0.0002 dyne.cm4/mg collagen.cell.(ABSTRACT TRUNCATED AT 400 WORDS)