A goal function approach to remodeling of arteries uncovers mechanisms for growth instability

A novel, goal function-based formulation for the growth dynamics of arteries is introduced and used for investigating the development of growth instability in blood vessels. Such instabilities would lead to abnormal growth of the vessel, reminiscent of an aneurysm. The blood vessel is modeled as a thin-walled cylindrical tube, and the constituents that form the vessel wall are assumed to deform together as a constrained mixture. The growth dynamics of the composite material of the vessel wall are described by an evolution equation, where the effective area of each constituent changes in the direction of steepest descent of a goal function. This goal function is formulated in such way that the constituents grow toward a target potential energy and a target composition. The convergence of the simulated response of the evolution equation toward a target homeostatic state is investigated for a range of isotropic and orthotropic material models. These simulations suggest that elastin-deficient vessels are more prone to growth instability. Increased stiffness of the vessel wall, on the other hand, gives a more stable growth process. Another important finding is that an increased rate of degradation of materials impairs growth stability.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: I--Simultaneous prediction of reaction force and lateral displacement

This study investigated the ability of the linear biphasic poroelastic (BPE) model and the linear biphasic poroviscoelastic (BPVE) model to simultaneously predict the reaction force and lateral displacement exhibited by articular cartilage during stress relaxation in unconfined compression. Both models consider articular cartilage as a binary mixture of a porous incompressible solid phase and an incompressible inviscid fluid phase. The BPE model assumes the solid phase is elastic, while the BPVE model assumes the solid phase is viscoelastic. In addition, the efficacy of two additional models was also examined, i.e., the transversely isotropic BPE (TIBPE) model, which considers transverse isotropy of the solid matrix within the framework of the linear BPE model assumptions, and a linear viscoelastic solid (LVE) model, which assumes that the viscoelastic behavior of articular cartilage is solely governed by the intrinsic viscoelastic nature of the solid matrix, independent of the interstitial fluid flow. It was found that the BPE model was able to accurately account for the lateral displacement, but unable to fit the short-term reaction force data of all specimens tested. The TIBPE model was able to account for either the lateral displacement or the reaction force, but not both simultaneously. The LVE model was able to account for the complete reaction force, but unable to fit the lateral displacement measured experimentally. The BPVE model was able to completely account for both lateral displacement and reaction force for all specimens tested. These results suggest that both the fluid flow-dependent and fluid flow-independent viscoelastic mechanisms are essential for a complete simulation of the viscoelastic phenomena of articular cartilage.     

Strain-rate dependent stiffness of articular cartilage in unconfined compression

The stiffness of articular cartilage is a nonlinear function of the strain amplitude and strain rate as well as the loading history, as a consequence of the flow of interstitial water and the stiffening of the collagen fibril network. This paper presents a full investigation of the interplay between the fluid kinetics and fibril stiffening of unconfined cartilage disks by analyzing over 200 cases with diverse material properties. The lower and upper elastic limits of the stress (under a given strain) are uniquely established by the instantaneous and equilibrium stiffness (obtained numerically for finite deformations and analytically for small deformations). These limits could be used to determine safe loading protocols in order that the stress in each solid constituent remains within its own elastic limit. For a given compressive strain applied at a low rate, the loading is close to the lower limit and is mostly borne directly by the solid constituents (with little contribution from the fluid). In contrast, however in case of faster compression, the extra loading is predominantly transported to the fibrillar matrix via rising fluid pressure with little increase of stress in the nonfibrillar matrix. The fibrillar matrix absorbs the loading increment by self-stiffening: the quicker the loading the faster the fibril stiffening until the upper elastic loading limit is reached. This self-protective mechanism prevents cartilage from damage since the fibrils are strong in tension. The present work demonstrates the ability of the fibril reinfored poroelastic models to describe the strain rate dependent behavior of articular cartilage in unconfined compression using a mechanism of fibril stiffening mainly induced by the fluid flow.     

A constitutive formulation of vascular tissue mechanics including viscoelasticity and softening behaviour

Nearly all soft tissues, among which the vascular tissue is included, present a certain degree of viscoelastic response. This behaviour may be attributed in part to fluid transport within the solid matrix, and to the friction between its fluid and solid constituents. After being preconditioned, the tissue displays highly repetitive behaviour, so that it can be considered pseudo-elastic, that is, elastic but behaving differently in loading and unloading. Because of this reason, very few constitutive laws accounting for the viscoelastic behaviour of the tissue have been developed. Nevertheless, the consideration of this inelastic effect is of crucial importance in surgeries—like vascular angioplasty—where the mentioned preconditioning cannot be considered since non-physiological deformation is applied on the vessel which, in addition, can cause damage to the tissue. A new constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these two inelastic phenomena is presented here and used to fit experimental stress–stretch curves from simple tension loading–unloading tests and relaxation test on porcine and ovine vascular samples.     

Trabeculated embryonic myocardium shows rapid stress relaxation and non-quasi-linear viscoelastic behavior

Passive viscoelastic behavior is important in embryonic cardiovascular function, influencing the rate and magnitude of contraction and relaxation. We hypothesized that if viscoelastic behavior is influenced by interstitial fluid flow, then the stage-21 (312d) and stage-24 (4d) chick myocardium with large intertrabecular spaces will exhibit much different viscoelastic behavior than stage-16 (212d) and stage-18 (3d) compact myocardium and a non-quasi-linear response. Excised left ventricular sections were tested with ramp-and-hold stress relaxation tests at axial stretch ratios of 1.05:1.1:1.2:1.3. The measured stress relaxation was much more rapid than previously observed in the compact, non-trabeculated myocardium. The reduced relaxation curves depended significantly on the stretch level. A continuous-spectrum quasi-linear relaxation function described their shape well but the model-fit parameters also depended on the stretch level. Sinusoidal stretching of ventricular sections at rates from 0.2 to 25Hz showed that the steepening of stress-strain curves with increasing strain rate was half as much as predicted by a quasi-linear model. Hysteresis ranged from 25-35%, varied little with loading rate from 0.2 to 8Hz, and was twice that predicted from a quasi-linear model. Doubling the viscosity of the perfusate in stress-relaxation tests produced increased stiffness and decreased relaxation rate. These results demonstrate that the passive viscoelastic behavior of the trabeculated embryonic myocardium is markedly different from that of younger, compact myocardium and is not quasi-linear.     

Hydro-mechanical coupling in the periodontal ligament: A porohyperelastic finite element model

Harmonic tension–compression tests at 0.1, 0.5 and 1 Hz on hydrated bovine periodontal ligament (PDL) were numerically simulated. The process was modeled by finite elements (FE) within the framework of poromechanics, with the objective of isolating the contributions of the solid- and fluid phases. The solid matrix was modeled as a porous hyperelastic material (hyperfoam) through which the incompressible fluid filling the pores flowed in accordance with the Darcy’s law. The hydro-mechanical coupling between the porous solid matrix and the fluid phase circulating through it provided an apparent time-dependent response to the PDL, whose rate of deformation depended on the permeability of the porous solid with respect to the interstitial fluid. Since the PDL was subjected to significant deformations, finite strains were taken into account and an exponential dependence of PDL permeability on void ratio – and therefore on the deformation state – was assumed. PDL constitutive parameters were identified by fitting the simulated response to the experimental data for the tests at 1 Hz. The values thus obtained were then used to simulate the tests at 0.1 and 0.5 Hz. The results of the present simulation demonstrate that a porohyperelastic model with variable permeability is able to describe the two main aspects of the PDL’s response: (1) the dependency on strain-rate—the saturated material can develop volumetric strains by only exchanging fluid and (2) the asymmetry between tension and compression, which is due to the effect of both the permeability and the elastic properties on deformation.     

An analysis of the unconfined compression of articular cartilage

Analytical solutions have been obtained for the internal deformation and fluid-flow fields and the externally observable creep, stress relaxation, and constant strain-rate behaviors which occur during the unconfined compression of a cylindrical specimen of a fluid-filled, porous, elastic solid, such as articular cartilage, between smooth, impermeable plates. Instantaneously, the "biphasic" continuum deforms without change in volume and behaves like an incompressible elastic solid of the same shear modulus. Radial fluid flow then allows the internal fluid pressure to equilibrate with the external environment. The equilibrium response is controlled by the Young's modulus and Poisson's ratio of the solid matrix.     

Fluid transport and mechanical properties of articular cartilage: a review

This review is aimed at unifying our understanding of cartilage viscoelastic properties in compression, in particular the role of compression-dependent permeability in controlling interstitial fluid flow and its contribution to the observed viscoelastic effects. During the previous decade, it was shown that compression causes the permeability of cartilage to drop in a functional manner described by k = ko exp (epsilon M) where ko and M were defined as intrinsic permeability parameters and epsilon is the dilatation of the solid matrix (epsilon = tr delta u). Since permeability is inversely related to the diffusive drag coefficient of relative fluid motion with respect to the porous solid matrix, the measured load-deformation response of the tissue must therefore also depend on the non-linearly permeable nature of the tissue. We have summarized in this review our understanding of this non-linear phenomenon. This understanding of these flow-dependent viscoelastic effects are put into the historical perspective of a comprehensive literature review of earlier attempts to model the compressive viscoelastic properties of articular cartilage.     

Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage

The dominant mechanism giving rise to the viscoelastic response of articular cartilage during compression is the nonlinear diffusive interaction of the fluid and solid phases of the tissue as they flow relative to one another. The present study is concerned with the role of this interaction under uniaxial stress relaxation in compression. The model is a biphasic mixture of fluid and solid which incorporates the strain-dependent permeability found earlier from permeation experiments. When a ramp-displacement is imposed on the articular surface, simple, but accurate, asymptotic approximations are derived for the deformation and stress fields in the tissue for slow and moderately fast rates of compression. They are shown to agree very well with experiment and they provide a simple means for determining the material parameters. Moreover, they lead to important insights into the role of the flow-dependent viscoelastic nature of articular cartilage and other hydrated biological tissues.     

Tendon exhibits complex poroelastic behavior at the nanoscale as revealed by high-frequency AFM-based rheology

Tendons transmit load from muscle to bone by utilizing their unique static and viscoelastic tensile properties. These properties are highly dependent on the composition and structure of the tissue matrix, including the collagen I hierarchy, proteoglycans, and water. While the role of matrix constituents in the tensile response has been studied, their role in compression, particularly in matrix pressurization via regulation of fluid flow, is not well understood. Injured or diseased tendons and tendon regions that naturally experience compression are known to have alterations in glycosaminoglycan content, which could modulate fluid flow and ultimately mechanical function. While recent theoretical studies have predicted tendon mechanics using poroelastic theory, no experimental data have directly demonstrated such behavior. In this study, we use high-bandwidth AFM-based rheology to determine the dynamic response of tendons to compressive loading at the nanoscale and to determine the presence of poroelastic behavior. Tendons are found to have significant characteristic dynamic relaxation behavior occurring at both low and high frequencies. Classic poroelastic behavior is observed, although we hypothesize that the full dynamic response is caused by a combination of flow-dependent poroelasticity as well as flow-independent viscoelasticity. Tendons also demonstrate regional dependence in their dynamic response, particularly near the junction of tendon and bone, suggesting that the structural and compositional heterogeneity in tendon may be responsible for regional poroelastic behavior. Overall, these experiments provide the foundation for understanding fluid-flow-dependent poroelastic mechanics of tendon, and the methodology is valuable for assessing changes in tendon matrix compressive behavior at the nanoscale.     

Elasticity imaging of polymeric media

Viscoelastic properties of soft tissues and hydropolymers depend on the strength of molecular bonding forces connecting the polymer matrix and surrounding fluids. The basis for diagnostic imaging is that disease processes alter molecular-scale bonding in ways that vary the measurable stiffness and viscosity of the tissues. This paper reviews linear viscoelastic theory as applied to gelatin hydrogels for the purpose of formulating approaches to molecular-scale interpretation of elasticity imaging in soft biological tissues. Comparing measurements acquired under different geometries, we investigate the limitations of viscoelastic parameters acquired under various imaging conditions. Quasi-static (step-and-hold and low-frequency harmonic) stimuli applied to gels during creep and stress relaxation experiments in confined and unconfined geometries reveal continuous, bimodal distributions of respondance times. Within the linear range of responses, gelatin will behave more like a solid or fluid depending on the stimulus magnitude. Gelatin can be described statistically from a few parameters of low-order rheological models that form the basis of viscoelastic imaging. Unbiased estimates of imaging parameters are obtained only if creep data are acquired for greater than twice the highest retardance time constant and any steady-state viscous response has been eliminated. Elastic strain and retardance time images are found to provide the best combination of contrast and signal strength in gelatin. Retardance times indicate average behavior of fast (1-10 s) fluid flows and slow (50-400 s) matrix restructuring in response to the mechanical stimulus. Insofar as gelatin mimics other polymers, such as soft biological tissues, elasticity imaging can provide unique insights into complex structural and biochemical features of connectives tissues affected by disease.     

The apparent viscoelastic behavior of articular cartilage--the contributions from the intrinsic matrix viscoelasticity and interstitial fluid flows

Articular cartilage was modeled rheologically as a biphasic poroviscoelastic material. A specific integral-type linear viscoelastic model was used to describe the constitutive relation of the collagen-proteoglycan matrix in shear. For bulk deformation, the matrix was assumed either to be linearly elastic, or viscoelastic with an identical reduced relaxation spectrum as in shear. The interstitial fluid was considered to be incompressible and inviscid. The creep and the rate-controlled stress-relaxation experiments on articular cartilage under confined compression were analyzed using this model. Using the material data available in the literature, it was concluded that both the interstitial fluid flow and the intrinsic matrix viscoelasticity contribute significantly to the apparent viscoelastic behavior of this tissue under confined compression.     

Relative contribution of articular cartilage’s constitutive components to load support depending on strain rate

Cartilage is considered a biphasic material in which the solid is composed of proteoglycans and collagen. In biphasic tissue, the hydraulic pressure is believed to bear most of the load under higher strain rates and its dissipation due to fluid flow determines creep and relaxation behavior. In equilibrium, hydraulic pressure is zero and load bearing is transferred to the solid matrix. The viscoelasticity of the collagen network also contributes to its time-dependent behavior, and the osmotic pressure to load bearing in equilibrium. The aim of the present study was to determine the relative contributions of hydraulic pressure, viscoelastic collagen stress, solid matrix stiffness and osmotic pressure to load carriage in cartilage under transient and equilibrium conditions. Unconfined compression experiments were simulated using a fibril-reinforced poroviscoelastic model of articular cartilage, including water, fibrillar viscoelastic collagen and non-fibrillar charged glycosaminoglycans. The relative contributions of hydraulic and osmotic pressures and stresses in the fibrillar and non-fibrillar network were evaluated in the superficial, middle and deep zone of cartilage under five different strain rates and after relaxation. Initially upon loading, the hydraulic pressure carried most of the load in all three zones. The osmotic swelling pressure carried most of the equilibrium load. In the surface zone, where the fibers were loaded in tension, the collagen network carried 20 % of the load for all strain rates. The importance of these fibers was illustrated by artificially modifying the fiber architecture, which reduced the overall stiffness of cartilage in all conditions. In conclusion, although hydraulic pressure dominates the transient behavior during cartilage loading, due to its viscoelastic nature the superficial zone collagen fibers carry a substantial part of the load under transient conditions. This becomes increasingly important with higher strain rates. The interesting and striking new insight from this study suggests that under equilibrium conditions, the swelling pressure generated by the combination of proteoglycans and collagen reinforcement accounts cartilage stiffness for more than 90 % of the loads carried by articular cartilage. This finding is different from the common thought that load is transferred from fluid to solid and is carried by the aggregate modulus of the solid. Rather, it is transformed from hydraulic to osmotic swelling pressure. These results show the importance of considering both (viscoelastic) collagen fibers as well as swelling pressure in studies of the (transient) mechanical behavior of cartilage.     

A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage.

The indentation problem of a thin layer of hydrated soft tissue such as cartilage or meniscus by a circular plane-ended indenter is investigated. The tissue is represented by a biphasic continuum model consisting of a solid phase (collagen and proteoglycan) and a fluid phase (interstitial water). A finite element formulation of the linear biphasic continuum equations is used to solve an axisymmetric approximation of the indentation problem. We consider stress-relaxation problems for which analytic solution is intractable; where the indenter is impermeable (solid) and/or when the interface between the indenter and tissue is perfectly adhesive. Thicknesses corresponding to a thin and thick specimen are considered to examine the effects of tissue thickness. The different flow, pressure, stress and strain fields which are predicted within the tissue, over time periods typically used in the mechanical testing of soft tissues, will be presented. Results are compared with the case of a porous free-draining indenter with a perfectly lubricated tissue-indenter interface, for which an analytic solution is available, to show the effects of friction at the tissue-indenter interface, and the effects of an impermeable indenter. While these effects are present for both thin and thick tissues, they are shown to be more significant for the thin tissue. We also examine the effects of the stiffness of the subchondral bone on the response of the soft tissue and demonstrate that the subchondral bone substrate can be modeled as a rigid, impermeable boundary. The effects of a curved tissue-subchondral bone interface, and the early time response are also studied. For physiologically reasonable levels of curvature, we will show that the curved tissue-subchrondal bone interface has negligible influence on the tissue response away from the interface. In addition, the short-time stress-relaxation responses of the tissue (e.g., at times less than 1s) demonstrate the essential role of the fluid phase in supporting the load applied to the tissue, and by extrapolation to shorter times characteristics of normal joint motion, suggest the essential role of a biphasic model in representing soft tissue behavior in joint response.     

A biphasic model for micro-indentation of a hydrogel-based contact lens

The stiffness and hydraulic permeability of soft contact lenses may influence its clinical performance, e.g., on-eye movement, fitting, and wettability, and may be related to the occurrence of complications; e.g., lesions. It is therefore important to determine these properties in the design of comfortable contact lenses. Micro-indentation provides a nondestructive means of measuring mechanical properties of soft, hydrated contact lenses. However, certain geometrical and material considerations must be taken into account when analyzing output force-displacement (F-D) data. Rather than solely having a solid response, mechanical behavior of hydrogel contact lenses can be described as the coupled interaction between fluid transport through pores and solid matrix deformation. In addition, indentation of thin membranes ( approximately 100 microm) requires special consideration of boundary conditions at lens surfaces and at the indenter contact region. In this study, a biphasic finite element model was developed to simulate the micro-indentation of a hydrogel contact lens. The model accounts for a curved, thin hydrogel membrane supported on an impermeable mold. A time-varying boundary condition was implemented to model the contact interface between the impermeable spherical indenter and the lens. Parametric studies varying the indentation velocities and hydraulic permeability show F-D curves have a sensitive region outside of which the force response reaches asymptotic limits governed by either the solid matrix (slow indentation velocity, large permeability) or the fluid transport (high indentation velocity, low permeability). Using these results, biphasic properties (Young's modulus and hydraulic permeability) were estimated by fitting model results to F-D curves obtained at multiple indentation velocities (1.2 and 20 microm/s). Fitting to micro-indentation tests of Etafilcon A resulted in an estimated permeability range of 1.0 x 10(-15) to 5.0 x 10(-15) m(4)N s and Young's modulus range of 130 to 170 kPa.     

Determination of the Poisson's ratio of the cell: recovery properties of chondrocytes after release from complete micropipette aspiration

Chondrocytes in articular cartilage are regularly subjected to compression and recovery due to dynamic loading of the joint. Previous studies have investigated the elastic and viscoelastic properties of chondrocytes using micropipette aspiration techniques, but in order to calculate cell properties, these studies have generally assumed that cells are incompressible with a Poisson's ratio of 0.5. The goal of this study was to measure the Poisson's ratio and recovery properties of the chondrocyte by combining theoretical modeling with experimental measures of complete cellular aspiration and release from a micropipette. Chondrocytes isolated from non-osteoarthritic and osteoarthritic cartilage were fully aspirated into a micropipette and allowed to reach mechanical equilibrium. Cells were then extruded from the micropipette and cell volume and morphology were measured throughout the experiment. This experimental procedure was simulated with finite element analysis, modeling the chondrocyte as either a compressible two-mode viscoelastic solid, or as a biphasic viscoelastic material. By fitting the experimental data to the theoretically predicted cell response, the Poisson's ratio and the viscoelastic recovery properties of the cell were determined. The Poisson's ratio of chondrocytes was found to be 0.38 for non-osteoarthritic cartilage and 0.36 for osteoarthritic chondrocytes (no significant difference). Osteoarthritic chondrocytes showed an increased recovery time following full aspiration. In contrast to previous assumptions, these findings suggest that chondrocytes are compressible, consistent with previous studies showing cell volume changes with compression of the extracellular matrix.     

Experimental measurement of the stiffness of the cupula.

An experimental procedure is described which consists of cutting the canal duct, inserting a micropipette and administering known volumetric displacements to the cupula. The cupula is made visible by dying the endolymph. Known displacements are administered to the cupula, and the time constant of the return to its equilibrium position is measured. With this information, the stiffness of the cupula is calculated. The experiment was successfully carried out on five White King pigeons. The mean stiffness found in somewhat less than other results reported in the literature, and reasons for this discrepancy are noted.     

Dynamics of the head-neck system in response to small perturbations: Analysis and modeling in the frequency domain

The response of the head-neck system to forces of small amplitude (up to 15 N) is described. Sinusoidal (0.6–20 Hz) and impulsive (duration: 100 msec) forces are applied in the sagittal plane to the head of the subject who is instructed to resist the disturbancy. In the case of sinusoidal forces of frequency less than about 2 Hz the active effort to resist the disturbancy results in a largely distorted sinusoidal displacement. Above this frequency the response becomes almost linear. The variations with frequency of the amplitude and the phase of the linear response relative to the applied force (transfer function) are used to characterize the dynamics of the system. The transfer functions evaluated from the impulse response are very similar in shape to those obtained with sinusoidal forces. In both cases the results suggest that the system behaves as a quasi-linear second order system with two degrees of freedom. The most prominent nonlinearities, beyond those present in the low frequency range, are related to the properties of the neck muscles. In particular, the transfer functions clearly show that the rigidity of the system increases as a function of the continuous value of the applied force. On the basis of previous work, both the passive properties of the muscles and those pertaining to the neuronal control system are pooled together in the form of viscoelastic parameters. A simple model of the system is introduced and applied to the experimental results. Its main features are 1) the presence of two centers of rotation. 2) the dependency of the viscoelastic parameters (stiffness and viscosity) on the frequency. It is suggested that both these features are necessary and sufficient to account for the observed behaviour above 2 Hz.     

Modelling the Brain for Rotational Loading: Shear Effects and Other Complications

This study considers modelling the brain due to rotation of the skull where, at lower frequencies, the shear property of the material is important. Investigations reported here cover the effect of elastic and viscoelastic (lossy) cerebral material, the effect of the Falx protruding into the brain, the gap around the Falx and the brain filled with non viscous fluid in addition to different models of the Falx with bending or membrane stiffness. Analytical benchmark formulations are also described for the simple 2D plane strain in a cylinder produced by a half-sine rotation on the outer periphery which allows numerical (Finite Element) models to be validated. The results show the importance of the material properties, duration of loading and amplitude of loading as well as the influence of the partition. The results are shown for predicted maximum Principal strains in the models, as this may well be indicative of whether damage of the brain tissue occurs.