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In innervated collagenous tissues, tissue scale loading may contribute to joint pain by transmitting force through collagen fibers to the embedded mechanosensitive axons. However, the highly heterogeneous collagen structures of native tissues make understanding this relationship challenging. Recently, collagen gels with embedded axons were stretched and the resulting axon signals were measured, but these experiments were unable to measure the local axon strain fields. Computational discrete fiber network models can directly determine axon strain fields due to tissue scale loading. Therefore, this study used a discrete fiber network model to identify how heterogeneous collagen networks (networks with multiple collagen fiber densities) change axon strain due to tissue scale loading. In this model, a composite cylinder (axon) was embedded in a Delaunay network (collagen). Homogeneous networks with a single collagen volume fraction and two types of heterogeneous networks with either a sparse center or dense center were created. Measurements of fiber forces show higher magnitude forces in sparse regions of heterogeneous networks and uniform force distributions in homogeneous networks. The average axon strain in the sparse center networks decreases when compared to homogeneous networks with similar collagen volume fractions. In dense center networks, the average axon strain increases compared to homogeneous networks. The top 1% of axon strains are unaffected by network heterogeneity. Based on these results, the interaction of tissue scale loading, collagen network heterogeneity, and axon strains in native musculoskeletal tissues should be considered when investigating the source of joint pain.

The heterogeneous, nonlinear, anisotropic material behavior of biological tissues makes precise definition of an accurate constitutive model difficult. One possible solution to this issue would be to define microstructural elements and perform fully coupled multiscale simulation. However, for complex geometries and loading scenarios, the computational costs of such simulations can be prohibitive. Ideally then, we should seek a method that contains microstructural detail, but leverages the speed of classical continuum-based finite-element (FE) modeling. In this work, we demonstrate the use of the Holzapfel-Gasser-Ogden (HGO) model (Holzapfel et al. in J. Elast. 61:1–48, 2000; Gasser et al. in J. R. Soc. Interface 3(6):15–35, 2006) to fit the behavior of microstructural network models. We show that Delaunay microstructural networks can be fit to the HGO strain energy function by calculating fiber network strain energy and average fiber stretch ratio. We then use the HGO constitutive model in a FE framework to improve the speed of our hybrid model, and demonstrate that this method, combined with a material property update scheme, can match a full multiscale simulation. This method gives us flexibility in defining complex FE simulations that would be impossible, or at least prohibitively time consuming, in multiscale simulation, while still accounting for microstructural heterogeneity.

The human lumbar facet capsule, with the facet capsular ligament (FCL) that forms its primary constituent, is a common source of lower back pain. Prior studies on the FCL were limited to in-plane tissue behavior, but due to the presence of two distinct yet mechanically different regions, a novel out-of-plane study was conducted to further characterize the roles of the collagen and elastin regions. An experimental technique, called stretch-and-bend, was developed to study the tension-compression asymmetry of the FCL due to varying collagen fiber density throughout the thickness of the tissue. Each healthy excised cadaveric FCL sample was tested in four conditions depending on primary collagen fiber alignment and regional loading. Our results indicate that the FCL is stiffest when the collagen fibers (1) are aligned in the direction of loading, (2) are in tension, and (3) are stretched − 16% from its off-the-bone, undeformed state. An optimization routine was used to fit a four-parameter anisotropic, hyperplastic model to the experimental data. The average elastin modulus, E, and the average collagen fiber modulus, ξ, were 13.15 ± 3.59 kPa and 18.68 ± 13.71 MPa (95% CI), respectively.

In sickle cell disease, aberrant blood flow due to oxygen-dependent changes in red cell biomechanics is a key driver of pathology. Most studies to date have focused on the potential role of altered red cell deformability and blood rheology in precipitating vaso-occlusive crises. Numerous studies, however, have shown that sickle blood flow is affected even at high oxygen tensions, suggesting a potentially systemic role for altered blood flow in driving pathologies, including endothelial dysfunction, ischemia, and stroke. In this study, we applied a combined experimental-computation approach that leveraged an experimental platform that quantifies sickle blood velocity fields under a range of oxygen tensions and shear rates. We computationally fitted a continuum model to our experimental data to generate physics-based parameters that capture patient-specific rheological alterations. Our results suggest that sickle blood flow is altered systemically, from the arterial to the venous circulation. We also demonstrated the application of this approach as a tool to design patient-specific transfusion regimens. Finally, we demonstrated that patient-specific rheological parameters can be combined with patient-derived vascular models to identify patients who are at higher risk for cerebrovascular complications such as aneurysm and stroke. Overall, this study highlights that sickle blood flow is altered systemically, which can drive numerous pathologies, and this study demonstrates the potential utility of an experimentally parameterized continuum model as a predictive tool for patient-specific care.

The scar that forms after a myocardial infarction is often characterized by a highly disordered architecture but generally exhibits some degree of collagen fiber orientation, with a resulting mechanical anisotropy. When viewed in finer detail, however, the heterogeneity of the sample is clear, with different subregions exhibiting different fiber orientations. In this work, we used a multiscale finite element model to explore the consequences of the heterogeneity in terms of mechanical behavior. To do so, we used previously obtained fiber alignment maps of rat myocardial scar slices (n = 15) to generate scar-specific finite element meshes that were populated with fiber models based on the local alignment state. These models were then compared to isotropic models with the same sample shape and fiber density, and to homogeneous models with the same sample shape, fiber density, and average fiber alignment as the scar-specific models. All simulations involved equibiaxial extension of the sample with free motion in the third dimension. We found that heterogeneity led to a lower degree of mechanical anisotropy and a higher level of local stress concentration than the corresponding homogeneous model, and also that fibers failed in the heterogeneous model at much lower macroscopic strains than in the isotropic and homogeneous models. Taken together, these results suggest that scar heterogeneity may impair myocardial mechanical function both in terms of anisotropy and strength, and that individual variations in scar heterogeneity could be an important consideration for understanding scar remodeling and designing therapeutic interventions for patients after myocardial infarction.

Excessive deformation of nerve fibres (axons) in the spinal facet capsular ligaments (FCLs) can be a cause of pain. The axons are embedded in the fibrous extracellular matrix (ECM) of FCLs, so understanding how local fibre organization and micromechanics modulate their mechanical behaviour is essential. We constructed a computational discrete-fibre model of an axon embedded in a collagen fibre network attached to the axon by distinct fibre–axon connections. This model was used to relate the axonal deformation to the fibre alignment and collagen volume concentration of the surrounding network during transverse, axial and shear deformations. Our results showed that fibre alignment affects axonal deformation only during transverse and axial loading, but higher collagen volume concentration results in larger overall axonal strains for all loading cases. Furthermore, axial loading leads to the largest stretch of axonal microtubules and induces the largest forces on axon's surface in most cases. Comparison between this model and a multiscale continuum model for a representative case showed that although both models predicted similar averaged axonal strains, strain was more heterogeneous in the discrete-fibre model.

The stress in a cell due to extracellular mechanical stimulus is determined by its mechanical properties, and the structural organization of many adherent cells suggests that their properties are anisotropic. This anisotropy may significantly influence the cells' mechanotransductive response to complex loads, and has important implications for development of accurate models of tissue biomechanics. Standard methods for measuring cellular mechanics report linear moduli that cannot capture large-deformation anisotropic properties, which in a continuum mechanics framework are best described by a strain energy density function (SED). In tissues, the SED is most robustly measured using biaxial testing. Here, we describe a cellular microbiaxial stretching (CμBS) method that modifies this tissue-scale approach to measure the anisotropic elastic behavior of individual vascular smooth muscle cells (VSMCs) with nativelike cytoarchitecture. Using CμBS, we reveal that VSMCs are highly anisotropic under large deformations. We then characterize a Holzapfel–Gasser–Ogden type SED for individual VSMCs and find that architecture-dependent properties of the cells can be robustly described using a formulation solely based on the organization of their actin cytoskeleton. These results suggest that cellular anisotropy should be considered when developing biomechanical models, and could play an important role in cellular mechano-adaptation.

Background Context The lumbar facet capsular ligament (FCL) is a posterior spinal ligament with a complex structure and kinematic profile. The FCL has a curved geometry, multiple attachment sites, and preferentially aligned collagen fiber bundles on the posterior surface that are innervated with mechanoreceptive nerve endings. Spinal flexion induces three-dimensional (3D) deformations, requiring the FCL to maintain significant tensile and shear loads. Previous works aimed to study 3D facet joint kinematics during flexion, but to our knowledge none have reported localized FCL surface deformations likely created by this complex structure. Purpose The purpose of this study was to elucidate local deformations of both the posterior and anterior surfaces of the lumbar FCL to understand the distribution and magnitude of in-plane and through-plane deformations, including the prevalence of shear. Study Design/Setting The FCL anterior and posterior surface deformations were quantified through creation of a finite element model simulating facet joint flexion using a realistic geometry, physiological kinematics, and fitted constitutive material. Methods Geometry was obtained from the micro-CT data of a healthy L3–L4 facet joint capsule (n=1); kinematics were extracted from sagittal plane fluoroscopic data of healthy volunteers (n=10) performing flexion; and average material properties were determined from planar biaxial extension tests of L4–L5 FCLs (n=6). All analyses were performed with the non-linear finite element solver, FEBio. A grid of equally spaced 3×3 nodes on the posterior surface identified regional differences within the strain fields and was used to create comparisons against previously published experimental data. This study was funded by the National Institutes of Health and the authors have no disclosures. Results Inhomogeneous in-plane and through-plane shear deformations were prominent through the middle body of the FCL on both surfaces. Anterior surface deformations were more pronounced because of the small width of the joint space, whereas posterior surface deformations were more diffuse because the larger area increased deformability. We speculate these areas of large deformation may provide this proprioceptive system with an excellent measure of spinal motion. Conclusions We found that in-plane and through-plane shear deformations are widely present in finite element simulations of a lumbar FCL during flexion. Importantly, we conclude that future studies of the FCL must consider the effects of both shear and tensile deformations.

Tissue-equivalents (TEs), simple model tissues with tunable properties, have been used to explore many features of biological soft tissues. Absent in most formulations however, is the residual stress that arises due to interactions among components with different unloaded levels of stress, which has an important functional role in many biological tissues. To create a pre-stressed model system, co-gels were fabricated from a combination of hyaluronic acid (HA) and reconstituted Type-I collagen (Col). When placed in solutions of varying osmolarity, HA-Col co-gels swell as the HA imbibes water, which in turn stretches (and stresses) the collagen network. In this way, co-gels with residual stress (i.e., collagen fibers in tension and HA in compression) were fabricated. When the three gel types tested here were immersed in hypotonic solutions, pure HA gels swelled the most, followed by HA-Col co-gels; no swelling was observed in pure collagen gels. The greatest swelling rates and swelling ratios occurred in the lowest salt concentration solutions. Tension on the collagen component of HA-Col co-gels was calculated from a stress balance and increased nonlinearly as swelling increased. The swelling experiment results were in good agreement with the stress predicted by a fibril network + non-fibrillar interstitial matrix computational model.

The importance of tissue remodelling is widely accepted, but the mechanism by which the remodelling process occurs remains poorly understood. At the tissue scale, the concept of tensional homeostasis, in which there exists a target stress for a cell and remodelling functions to move the cell stress towards that target, is an important foundation for much theoretical work. We present here a theoretical model of a cell in parallel with a network to study what factors of the remodelling process help the cell move towards mechanical stability. The cell-network system was deformed and kept at constant stress. Remodelling was modelled by simulating strain-dependent degradation of collagen fibres and four different cases of collagen addition. The model did not lead to complete tensional homeostasis in the range of conditions studied, but it showed how different expressions for deposition and removal of collagen in a fibre network can interact to modulate the cell's ability to shield itself from an imposed stress by remodelling the surroundings. This study also showed how delicate the balance between deposition and removal rates is and how sensitive the remodelling process is to small changes in the remodelling rules.

Random fiber networks are assemblies of elastic elements connected in random configurations. They are used as models for a broad range of fibrous materials including biopolymer gels and synthetic nonwovens. Although the mechanics of networks made from the same type of fibers has been studied extensively, the behavior of composite systems of fibers with different properties has received less attention. In this work we numerically and theoretically study random networks of beams and springs of different mechanical properties. We observe that the overall network stiffness decreases on average as the variability of fiber stiffness increases, at constant mean fiber stiffness. Numerical results and analytical arguments show that for small variabilities in fiber stiffness the amount of network softening scales linearly with the variance of the fiber stiffness distribution. This result holds for any beam structure and is expected to apply to a broad range of materials including cellular solids.

A variety of macromolecules accumulate in the glomerular mesangium in many different diseases, but the physics of the transport of these molecules within the mesangial matrix has not been extensively studied. We present a computational model of convection and diffusion within the porous mesangial matrix and apply this model to the specific instance of immunoglobulin A (IgA) transport in IgA nephropathy. We examine the influence of physiological factors including glomerular basement membrane (GBM) thickness and mesangial matrix density on the total accumulation of IgA. Our results suggest that IgA accumulation can be understood by relating convection and diffusion, thus demonstrating the importance of intrinsic glomerular factors.

ATAA are dilatations of the arterial wall that result primarily from a weakening in the middle smooth muscle layer, or media, of the artery [1–3]. Aneurysm dissection and rupture pose serious risk of mortality and are the primary concerns for patients with an ATAA [1–4]. Cystic medial degeneration (which can be caused by disease, injury, or hereditary condition [2,3]), weakens the aortic wall, allowing for the formation of aneurysms and subsequent dissections. Aneurysm growth and dissection is further driven by increased hemodynamic loading within the media [5]. Ascending aortic dissection, the most prominent failure mode, occurs when the hemodynamic stresses within the media exceed the capacity of its collagen and elastin constituents allowing blood to pass into the medial lamella and peel it apart (delaminate) [6]. Ultimately, this tear can propagate axially along the length of the aorta, either proximally or distally, and in the absence of clinical intervention, may lead to further complications such as ischemia, aortic regurgitation, cardiac tamponade, aortic rupture, and death [1]. Ascending aortic aneurysms have an incidence rate of 10.4 per 100,000 person-years [7], with significant risk of dissection or rupture [7]. Clinically, ATAA management balances the risk of aneurysm rupture with the risk of surgery-related complications and death [2,3]. Risk of rupture is correlated with aneurysm diameter [2,7]; aneurysms greater than 6 cm in diameter exhibit a significantly higher risk of rupture [2]. Thus, for patients without valvular disorders, current guidelines indicate surgical intervention for ATAA diameters greater than 5.5 cm and monitoring for those less than 5.5 cm [2]. In spite of clinical guidelines and what is known about the pathogenesis of aneurysms, dissection and rupture occur in a significant number of patients whose aneurysms are less than 5.5 cm in diameter [8], suggesting that a better understanding of the dissection process is needed. Experimental studies have investigated the mechanical failure of aortic media [6,9,10], but only a few theoretical models have attempted to emulate this response [11–13]. The most notable work to date, by Gasser and Holzapfel [12], employs constitutive FE modeling with two independent continuous and cohesive zones to model the fiber network and ground matrix, respectively. While the Gasser–Holzapfel model is capable of describing the mechanical behavior of the aorta, it fails to capture complex fiber-fiber and fiber-matrix interactions or address failure at the microscopic scale. A model employing microscale mechanics and structure to determine the macroscale response would have potential to incorporate greater anatomical detail and could be applied to a broader range of conditions. This work represents an initial attempt to apply such multiscale concepts to the mechanics of the ATA. The objectives of this study were (1) to characterize the prefailure and failure response of healthy porcine ascending aorta tissue under uniaxial and biaxial loading and (2) to describe that response with a multiscale, FE damage model. The model has been shown to be effective for prefailure and failure behavior of collagen gels [14] and has been applied to prefailure behavior of decellularized arterial wall [15].

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.

To investigate the contribution of three anatomical and physiologic factors (dilator thickness, dynamic pupillary block, and iris compressibility) to changes in iris configuration and anterior chamber angle during pupil dilation.A MATHEMATICAL MODEL OF THE ANTERIOR SEGMENT BASED ON THE AVERAGE VALUES OF OCULAR DIMENSIONS WAS DEVELOPED TO SIMULATE PUPIL DILATION. TO CHANGE THE PUPIL DIAMETER FROM 3.0 TO 5.4 MM IN 10 SECONDS, ACTIVE DILATOR CONTRACTION WAS APPLIED BY IMPOSING STRESS IN THE DILATOR REGION. THREE SETS OF PARAMETERS WERE VARIED IN THE SIMULATIONS: (1) a thin (4 μm, 1% of full thickness) versus a thick dilator (covering the full thickness iris) to quantify the effects of dilator anatomy, (2) in the presence (+PB) versus absence of pupillary block (-PB) to quantify the effect of dynamic motion of aqueous humor from the posterior to the anterior chamber, and (3) a compressible versus an incompressible iris to quantify the effects of iris volume change. Changes in the apparent iris-lens contact and angle open distance (AOD500) were calculated for each case.The thin case predicted a significant increase (average 700%) in iris curvature compared with the thick case (average 70%), showing that the anatomy of dilator plays an important role in iris deformation during dilation. In the presence of pupillary block (+PB), AOD500 decreased 25% and 36% for the compressible and incompressible iris, respectively.Iris bowing during dilation was driven primarily by posterior location of the dilator muscle and by dynamic pupillary block, but the effect of pupillary block was not as large as that of the dilator anatomy according to the quantified values of AOD500. Incompressibility of the iris, in contrast, had a relatively small effect on iris curvature but a large effect on AOD500; thus, we conclude that all three effects are important.

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

While several studies have evaluated how the degree of collagen alignment affects the response of soft tissues to tensile loading, the role of fibrillar organization in indentation is less understood. Collagen-based tissue-equivalents (TEs) provide a convenient model system to explore structure-function relationships since their microstructural properties can be easily controlled during fabrication. The purpose of this study was to evaluate the role of initial collagen alignment on the mechanical and structural behavior of soft tissues subjected to indentation using TEs as a model system. Cell-compacted TEs with either isotropic or highly anisotropic fiber alignment were subjected to four-step incremental stress-relaxation indentation tests. The mechanical properties, collagen reorganization and 2D strain patterns were quantified at each indentation step and compared between groups. While no differences were seen in the peak force response, significant differences were seen in relaxation behavior, fiber kinematics and tissue strain. Specifically, highly aligned samples exhibited a slower relaxation rate, smaller changes in collagen fiber orientation, larger changes in strength of alignment, and larger strain magnitudes compared to isotropic samples. Results demonstrate the significant role that microstructural organization plays in mediating the response of soft tissues to a non-tensile (i.e., indentation) mechanical stimulus.

The vitreous humor liquefies with age and readily sloshes during eye motion. The objective was to develop a computational model to determine the effect of sloshing on intravitreal drug transport for transscleral and intra-vitreal drug sources at various locationsA finite element model based on a telescopic implicit envelope tracking scheme was developed to model drug dispersion. Flow velocities due to saccadic oscillations were solved for and were used to simulate drug dispersion.Saccades induced a three-dimensional flow field that indicates intense drug dispersion in the vitreous. Model results showed that the time scale for transport decreased for the sloshing vitreous when compared to static vitreous. Macular concentrations for the sloshing vitreous were found be much higher than that for the static vitreous. For low viscosities the position of the intravitreal source did not have a big impact on drug distribution.Model results show that care should be taken when extrapolating animal data, which are mostly done on intact vitreous, to old patients whose vitreous might be a liquid. The decrease in drug transport time scales and changes in localized concentrations should be considered when deciding on treatment modalities and dosing strategies.

The purpose of this study was to quantify how the elastic modulus of the ex vivo iris changes following stimulation by pilocarpine (PILO), phenylephrine (PE), and tropicamide (TROP). Irides (n = 20) were dissected from porcine eyes within 4 h post-mortem and tested uniaxially. Either the entire iris or sector thereof was used. The samples were stretched up to 40% Green strain. The radial modulus was calculated from the linear portion of the stress-strain curve, and the azimuthal modulus was fitted to a model treating the iris as a collection of circular elastic bands. One of the three drugs (n = 6 or 7) of interest was added (80 microg/ml) to the bath surrounding the tissue, and the test was repeated. Changes in pupil diameter of free-floating samples and isometric force of mounted samples confirmed that the tissue was responsive to the drugs. The untreated iris modulus for cut sections in radial extension was 4.0 +/- 0.9 kPa (mean +/- s.d., n = 20), and treated iris modulus was 7.7 +/- 2.0 kPa (PILO, n = 7), 6.9 +/- 2.2 kPa (PE, n = 6), and 8.4 +/- 1.7 kPa (TROP, n = 7). Intact irides (n = 10) gave similar trends but values approximately 25% higher, presumably due to support from the nominally unloaded tissue. The azimuthal modulus of the untreated iris was 2.97 +/- 1.3 kPa (n = 5), and that of the treated iris (PILO) was 5.34 +/- 2.1 kPa. Although PILO, PE, and TROP work by different mechanisms, all three had similar results - an increase of modulus by a factor of two. These results suggest that in most normal situations the iris remains compliant at all pupil diameters.

Eyes were imaged using anterior segment ultrasound biomicroscopy to compare alterations in iris contour following the onset of accommodation in eyes with narrow angles, pigment dispersion syndrome, and controls. A radial perpendicular image in the horizontal temporal meridian was obtained for one eye while the subject focused on a distant target (~6m, unaccommodated state) with the fellow eye. The subject then focused steadily on a near target (~0.33m, accommodated state) for three minutes. Images were acquired at 0, 1, 2, and 3 minutes. Iris curvature was determined by measuring the maximum distance between the posterior iris surface and a line from the iris root to the first point of contact between the iris and lens. In control subjects (n = 22), iris curvature decreased immediately after the onset of accommodation, but not significantly (p = 0.49), from 246 ± 37 µm (mean ± SEM) to 205 ± 82 µm; curvature increased after three minutes of accommodation to 298 ± 57 µm (p = 0.10 vs. onset of accommodation). Eyes with pigment dispersion syndrome (n = 15) exhibited curvatures of 60 ± 79 µm when unaccommodated, −3 ± 83 µm immediately after accommodation (p = 0.12), and 146 ± 94 µm (p = 0.01) three minutes later. Eyes with narrow angles (n = 16) exhibited curvatures of 449 ± 45 µm when unaccommodated, 414 ± 46 µm immediately after accommodation (p = 0.37), and 523 ± 40 µm (p < 0.01) three minutes later. The results confirm the time-dependent nature of iris contour response, with significant differences observed between the initial observation after accommodation and the observation three minutes later. The largest drop in curvature immediately after accommodation and the most rapid increase in curvature during subsequent observation were seen in the PDS subjects. We suspect that the more rapid increase in curvature in the PDS subjects is due to the elevated anterior-chamber pressure caused by the "reverse pupillary block" effect.