Strain energy descriptions of biological swelling. II: Multiple fluid compartment models

A series of multicompartmental, biphasic elastic tissue models is developed. In its most general form, the models consist of multiple tubular networks, each with an internal spring network. In addition, another spring network occupies the extratubular compartment. Strain energy functions are derived for the models, as well as expressions for the fluid pressures in each compartment arising from volume expansion or swelling. Calculations also show that the distribution of fluid among compartments is a significant determinant of tissue elasticity.     

A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissues

Biological tissues like intervertebral discs and articular cartilage primarily consist of interstitial fluid, collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans, the total ion concentration inside the tissue is higher than in the surrounding synovial fluid (cation concentration is higher and the anion concentration is lower). This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue. In the last decade several mechano-electrochemical models, which include this mechanism, have been developed. As these models are complex and computationally expensive, it is only possible to analyze geometrically relatively small problems. Furthermore, there is still no commercial finite element tool that includes such a mechano-electrochemical theory. Lanir (Biorheology, 24, pp. 173-187, 1987) hypothesized that electrolyte flux in articular cartilage can be neglected in mechanical studies. Lanir's hypothesis implies that the swelling behavior of cartilage is only determined by deformation of the solid and by fluid flow. Hence, the response could be described by adding a deformation-dependent pressure term to the standard biphasic equations. Based on this theory we developed a biphasic swelling model. The goal of the study was to test Lanir's hypothesis for a range of material properties. We compared the deformation behavior predicted by the biphasic swelling model and a full mechano-electrochemical model for confined compression and 1D swelling. It was shown that, depending on the material properties, the biphasic swelling model behaves largely the same as the mechano-electrochemical model, with regard to stresses and strains in the tissue following either mechanical or chemical perturbations. Hence, the biphasic swelling model could be an alternative for the more complex mechano-electrochemical model, in those cases where the ion flux itself is not the subject of the study. We propose thumbrules to estimate the correlation between the two models for specific problems.     

An evaluation of three-dimensional diarthrodial joint contact using penetration data and the finite element method

We have developed an approximate method for simulating the three-dimensional contact of soft biphasic tissues in diarthrodial joints under physiological loading. Input to the method includes: (i) kinematic information describing an in vitro joint articulation, measured while the cartilage is deformed under physiological loads, (ii) geometric properties for the relaxed (undeformed) cartilage layers, obtained for the analyses in this study via stereophotogrammetry, and (iii) material parameters for the biphasic constitutive relations used to represent cartilage. Solid models of the relaxed tissue layers are assembled in physiological positions, resulting in a mathematical overlap of the cartilage layers. The overlap distribution is quantified and converted via the biphasic governing equations into applied traction boundary conditions for both the solid and fluid phases for each of the contacting layers. Linear, biphasic, three-dimensional, finite element analysis is performed using the contact boundary conditions derived for each of the contacting layers. The method is found to produce results consistent with the continuity requirements of biphasic contact. Comparison with results from independent, biphasic contact analyses of axisymmetric problems shows that the method slightly underestimates the contact area, leading to an overestimation of the total traction, but yields a good approximation to elastic stress and solid phase displacement.     

Equivalence between short-time biphasic and incompressible elastic material responses

Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response deltat相似文献   

Theoretical investigation of an artificial joint with micro-pocket-covered component and biphasic cartilage on the opposite articulating surface

This paper presents a theoretical investigation of a geometrically idealized artificial joint with micro-pocket-covered component and biphasic cartilage on the opposite articulating surface. The fluid that exudes from the biphasic cartilage fills and pressurizes the micro-pockets. In this way, a poro-elasto-hydrodynamic regime of lubrication is developed. Assuming that lower friction would result in lower adhesive wear, and neglecting the fatigue as well as the abrasive wear, the proposed bearing system hypothetically could reduce the amount of wear debris. Equations of the linear biphasic theory are applied for the confined and unconfined compression of the cartilage. The fluid pressure and the elastic deformation of the biphasic cartilage are explicitly presented. The effective and equilibrium friction coefficients are obtained for the particular configuration of this bearing system. The micro-pockets geometrical parameters (depth, radius, surface distribution and edge radius) must be established to reduce the local contact stresses, to assure low friction forces and to minimize the biphasic cartilage damage. The influence of the applied pressure, porosity of the micro-pocket-covered component, filling time, cartilage elasticity, permeability and porosity upon the micro-pockets depth is illustrated. Our results are based upon the previously published data for a biphasic cartilage.     

A biphasic, anisotropic model of the aortic wall

A biphasic, anisotropic elastic model of the aortict wall is developed and compared to literature values of experimental measurements of vessel wall radii, thickness, and hvdraulic conductivity as a function of intraluminal pressure. The model gives good predictions using a constant wall modulus for pressures less than 60 mmHg, but requires a strain-dependent modulus for pressures greater than this. In both bovine and rabbit aorta, the tangential modulus is found to be approximately 20 times greater than the radial modulus. These moduli lead to predictions that, when perfused in a cylindrical geometry, the aortic volume and its specific hydraulic coonductivity are relatively independent of perfusion pressure, in agreement with experimental measurements. M, the parameter that relates specific hydraulic conductivy, to tissue dilation, is found to be a positive quantity correcting a previous error in the literature.     

Seasonal variation of leaf tissue elasticity in four forest tree species

Measures of leaf tissue elasticity were determined by analyzing the turgor pressure-water content relation developed from pressure-volume experiments, in four naturally occurring forest tree species. The relation between the bulk tissue elastic modulus and tissue turgor pressure varied by species and varied with state of leaf maturation through the growing season. Although the values of the elastic modulus reported here agree in magnitude with values reported in general for higher plant tissue, the functional dependency of the apparent elasticity on tissue water status was more complex in this study than can be accounted for by current models.     

An approach for the stress analysis of transversely isotropic biphasic cartilage under impact load.

Stress analysis of contact models for isotropic articular cartilage under impacting loads shows high shear stresses at the interface with the subchondral bone and normal compressive stresses near the surface of the cartilage. These stress distributions are not consistent, with lesions observed on the cartilage surface of rabbit patellae from blunt impact, for example, to the patello-femoral joint. The purpose of the present study was to analyze, using the elastic capabilities of a finite element code, the stress distribution in more morphologically realistic transversely isotropic biphasic contact models of cartilage. The elastic properties of an incompressible material, equivalent to those of the transversely isotropic biphasic material at time zero, were derived algebraically using stress-strain relations. Results of the stress analysis showed the highest shear stresses on the surface of the solid skeleton of the cartilage and tensile stresses in the zone of contact. These results can help explain the mechanisms responsible for surface injuries observed during blunt insult experiments.     

A microstructural model of elastostatic properties of articular cartilage in confined compression

A microstructural model of cartilage was developed to investigate the relative contribution of tissue matrix components to its elastostatic properties. Cartilage was depicted as a tensed collagen lattice pressurized by the Donnan osmotic swelling pressure of proteoglycans. As a first step in modeling the collagen lattice, two-dimensional networks of tensed, elastic, interconnected cables were studied as conceptual models. The models were subjected to the boundary conditions of confined compression and stress-strain curves and elastic moduli were obtained as a function of a two-dimensional equivalent of swelling pressure. Model predictions were compared to equilibrium confined compression moduli of calf cartilage obtained at different bath concentrations ranging from 0.01 to 0.50 M NaCl. It was found that a triangular cable network provided the most consistent correspondence to the experimental data. The model showed that the cartilage collagen network remained tensed under large confined compression strains and could therefore support shear stress. The model also predicted that the elastic moduli increased with increasing swelling pressure in a manner qualitatively similar to experimental observations. Although the model did not preclude potential contributions of other tissue components and mechanisms, the consistency of model predictions with experimental observations suggests that the cartilage collagen network, prestressed by proteoglycan swelling pressure, plays an important role in supporting compression.     

Modelling cartilage mechanobiology

The growth, maintenance and ossification of cartilage are fundamental to skeletal development and are regulated throughout life by the mechanical cues that are imposed by physical activities. Finite element computer analyses have been used to study the role of local tissue mechanics on endochondral ossification patterns, skeletal morphology and articular cartilage thickness distributions. Using single-phase continuum material representations of cartilage, the results have indicated that local intermittent hydrostatic pressure promotes cartilage maintenance. Cyclic tensile strains (or shear), however, promote cartilage growth and ossification. Because single-phase material models cannot capture fluid exudation in articular cartilage, poroelastic (or biphasic) solid/fluid models are often implemented to study joint mechanics. In the middle and deep layers of articular cartilage where poroelastic analyses predict little fluid exudation, the cartilage phenotype is maintained by cyclic fluid pressure (consistent with the single-phase theory). In superficial articular layers the chondrocytes are exposed to tangential tensile strain in addition to the high fluid pressure. Furthermore, there is fluid exudation and matrix consolidation, leading to cell 'flattening'. As a result, the superficial layer assumes an altered, more fibrous phenotype. These computer model predictions of cartilage mechanobiology are consistent with results of in vitro cell and tissue and molecular biology experiments.     

Mechanical anisotropy of inflated elastic tissue from the pig aorta

Uniaxial and biaxial mechanical properties of purified elastic tissue from the proximal thoracic aorta were studied to understand physiological load distributions within the arterial wall. Stress–strain behaviour was non-linear in uniaxial and inflation tests. Elastic tissue was 40% stiffer in the circumferential direction compared to axial in uniaxial tests and～100% stiffer in vessels at an axial stretch ratio of 1.2 or 1.3 and inflated to physiological pressure. Poisson’s ratio v_θz averaged 0.2 and v_zθ increased with circumferential stretch from ～0.2 to ～0.4. Axial stretch had little impact on circumferential behaviour. In intact (unpurified) vessels at constant length, axial forces decreased with pressure at low axial stretches but remained constant at higher stretches. Such a constant axial force is characteristic of incrementally isotropic arteries at their in vivo dimensions. In purified elastic tissue, force decreased with pressure at all axial strains, showing no trend towards isotropy. Analysis of the force–length–pressure data indicated a vessel with v_θz≈0.2 would stretch axially 2–4% with the cardiac pulse yet maintain constant axial force. We compared the ability of 4 mathematical models to predict the pressure-circumferential stretch behaviour of tethered, purified elastic tissue. Models that assumed isotropy could not predict the stretch at zero pressure. The neo-Hookean model overestimated the non-linearity of the response and two non-linear models underestimated it. A model incorporating contributions from orthogonal fibres captured the non-linearity but not the zero-pressure response. Models incorporating anisotropy and non-linearity should better predict the mechanical behaviour of elastic tissue of the proximal thoracic aorta.     

Finite element modeling for predicting the contact pressure between a foam mattress and the human body in a supine position

In this paper, we generated finite element (FE) models to predict the contact pressure between a foam mattress and the human body in a supine position. Twenty-year-old males were used for three-dimensional scanning to produce the FE human models, which was composed of skin and muscle tissue. A linear elastic isotropic material model was used for the skin, and the Mooney–Rivlin model was used for the muscle tissue because it can effectively represent the nonlinear behavior of muscle. The contact pressure between the human model and the mattress was predicted by numerical simulation. The human models were validated by comparing the body pressure distribution obtained from the same human subject when he was lying on two different mattress types. The experimental results showed that the slope of the lower part of the mattress caused a decrease in the contact pressure at the heels, and the effect of bone structure was most pronounced in the scapula. After inserting a simple structure to function as the scapula, the contact pressure predicted by the FE human models was consistent with the experimental body pressure distribution for all body parts. These results suggest that the models proposed in this paper will be useful to researchers and designers of products related to the prevention of pressure ulcers.     

A cellular model of lung elasticity

The mechanics of the lung parenchyma is studied using models comprised of line members interconnected to form 3-D cellular structures. The mechanical properties are represented as elastic constants of a continuum. These are determined by perturbing each individual cell from a reference state by an increment in stress which is superimposed upon the uniform stretching forces initially present in the members due to the transpulmonary pressure. A force balance on the distorted structure, together with a force-deformation law for the members, leads to a calculation of the strain increments of the members. Predictions based on the analysis of the 3-D isotropic dodecahedron are in good agreement with experimental values for the Young's, shear, and bulk moduli reported in the literature. The model provides an explanation for the dependence of the elastic moduli on transpulmonary pressure, the geometrical details of the structure, and the stress-strain law of the tissue.     

Confined compression experiments on bovine nucleus pulposus and annulus fibrosus: sensitivity of the experiment in the determination of compressive modulus and hydraulic permeability

The biphasic material properties for nucleus pulposus tissue in confined compression have not been reported previously, and are required for a better understanding of intervertebral disc function and to provide material properties for use in finite-element models. The aims of this study were to determine linear and non-linear material properties for nucleus pulposus and annulus fibrosus tissues in confined compression, to define the influence of swelling conditions on these properties, and to determine the changes in the compressive modulus and hydraulic permeability induced by the repetition of the stress-relaxation experiment after a return to swelling pressure equilibrium. Specimens from caudal bovine nucleus and annulus were tested in confined compression stress-relaxation experiments and analyzed to quantify the compressive modulus and hydraulic permeability using linear and non-linear biphasic models. Our results suggested the use of confined swelling pre-test condition and non-linear biphasic model, which provided the material parameters with lowest relative variance and water content most representative of physiological conditions. Smaller compressive modulus and higher hydraulic permeability were obtained for the nucleus (H(A0)=0.31+/-0.04 MPa, k(0)=0.67+/-0.09 x 10(-15)m(4)/Ns) than for the annulus (H(A0)=0.74+/-0.13 MPa, k(0)=0.23+/-0.19 x 10(-15)m(4)/Ns), with relatively weak non-linearities. Strains up to 20% resulted in material properties that were significantly altered upon retesting. These altered material properties are an effort to quantify non-recoverable damage that occurs in disc tissue and suggest that in vivo exposure of disc tissues to low strain-rate and high-deformation loading conditions which outpace biological repair may result in altered mechanical behaviors.     

根压的本质

在植物弱蒸腾情况下,木质部的水分处于正压状态,这种压力叫根压。关于根压有不少相互矛盾的假说,大致可分为两大类：单相模型和双相模型。单相模型不强调水流：疗向,基本上只将根中的共质体作为一层连续的半透膜看待。单相模型包括分泌理论、电渗理论、渗透理论和渗透持续梯度理论。而根压的双相模型认为根中有具有不同反弹系数的双膜系统,即高反弹系数的外膜和低反弹系数的内膜。本文对这些根压理论进行了总结和分析,并结合一些新的研究成果,特别是对根中薄壁细胞有节律性地收缩和松弛及对水通道蛋白的研究,对根压的本质和产生机理进行了深入的讨论。     

Mechanics of interstitial-lymphatic fluid transport: theoretical foundation and experimental validation

Interstitial fluid movement is intrinsically linked to lymphatic drainage. However, their relationship is poorly understood, and associated pathologies are mostly untreatable. In this work we test the hypothesis that bulk tissue fluid movement can be evaluated in situ and described by a linear biphasic theory which integrates the regulatory function of the lymphatics with the mechanical stresses of the tissue. To accomplish this, we develop a novel experimental and theoretical model using the skin of the mouse tail. We then use the model to demonstrate how interstitial–lymphatic fluid movement depends on a balance between the elasticity, hydraulic conductivity, and lymphatic conductance as well as to demonstrate how chronic swelling (edema) alters the equipoise between tissue fluid balance parameters. Specifically, tissue fluid equilibrium is perturbed with a continuous interstitial infusion of saline into the tip of the tail. The resulting gradients in tissue stress are measured in terms of interstitial fluid pressure using a servo-null system. These measurements are then fit to the theory to provide in vivo estimates of the tissue hydraulic conductivity, elastic modulus, and overall resistance to lymphatic drainage. Additional experiments are performed on edematous tails to show that although chronic swelling causes an increase in the hydraulic conductivity, its greatly increased distensibility (due to matrix remodeling) dampens the driving forces for fluid movement and leads to fluid stagnation. This model is useful for examining potential treatments for edema and lymphatic disorders as well as substances which may alter tissue fluid balance and/or lymphatic drainage.     

Mechanical compression-induced pressure sores in rat hindlimb: muscle stiffness, histology, and computational models.

Pressure sores affecting muscles are severe injuries associated with ischemia, impaired metabolic activity, excessive tissue deformation, and insufficient lymph drainage caused by prolonged and intensive mechanical loads. We hypothesize that mechanical properties of muscle tissue change as a result of exposure to prolonged and intensive loads. Such changes may affect the distribution of stresses in soft tissues under bony prominences and potentially expose additional uninjured regions of muscle tissue to intensified stresses. In this study, we characterized changes in tangent elastic moduli and strain energy densities of rat gracilis muscles exposed to pressure in vivo (11.5, 35, or 70 kPa for 2, 4, or 6 h) and incorporated the abnormal properties that were measured in finite element models of the head, shoulders, pelvis, and heels of a recumbent patient. Using in vitro uniaxial tension testing, we found that tangent elastic moduli of muscles exposed to 35 and 70 kPa were 1.6-fold those of controls (P < 0.05, for strains /=5%). Histological (phosphotungstic acid hematoxylin) evaluation showed that this stiffening accompanied extensive necrotic damage. Incorporating these effects into the finite element models, we were able to show that the increased muscle stiffness in widening regions results in elevated tissue stresses that exacerbate the potential for tissue necrosis. Interfacial pressures could not predict deep muscle (e.g., longissimus or gluteus) stresses and injuring conditions. We conclude that information on internal muscle stresses is required to establish new criteria for pressure sore prevention.     

The hydraulic conductivity of Matrigel

In this study, we measured the specific hydraulic conductivity (K) of Matrigel at 1% and 2% concentrations as a function of perfusion pressure (0 to 100 mmHg) and compared the results to predictions from two models: a fiber matrix model that predicted K of the gel based upon its composition, and a biphasic model that predicted changes in K caused by pressure induced compaction of the gels. The extent of gel compaction as a function of perfusion pressure was also assessed, allowing us to estimate the stiffness of the gels. As expected, 2% Matrigel had a lower K and a higher stiffness than did 1% Matrigel. Measured values of K of both 1% and 2% Matrigel samples showed good agreement with the predictions of the fiber matrix model. Pressure-induced changes in K were better described by the biphasic model than a model in which uniform compression of the gel was assumed. We conclude that K of multi-component gels, such as Matrigel can be well characterized by fiber matrix models, and that pressure-induced changes in K of such gels can be well characterized by biphasic models.