A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.     

Cartilage matrix proteins. A basic 36-kDa protein with a restricted distribution to cartilage and bone

A non-collagenous quantitatively prominent protein was purified from guanidine hydrochloride extracts of bovine tracheal cartilage. Purification was achieved by cesium chloride density gradient centrifugation and chromatography on DEAE-cellulose at pH 7.0 followed by CM-cellulose at pH 5.0. The protein has a marked tendency to form aggregates in denaturing solutions of high ionic strength, e.g. 6 M guanidine hydrochloride. The purified protein contains a single, Mr 36,000 polypeptide chain, with a particularly high content of leucine. It contains about 1% carbohydrate with a remarkable absence of hexosamines and sialic acid, whereas xylose, galactose, mannose, and fucose were identified in the preparation. The protein was identified in extracts of cartilage and bone and could be shown to be primarily extracellular. Tendon may contain trace amounts of the protein, whereas extracts of several other tissues showed no immunoreactivity in enzyme-linked immunosorbent assay.     

A growth mixture theory for cartilage with application to growth-related experiments on cartilage explants

In this paper, we present a growth mixture model for cartilage. The main features of this model are illustrated in a simple equilibrium boundary-value problem that is chosen to illustrate how a mechanical theory of cartilage growth may be applied to growth-related experiments on cartilage explants. The cartilage growth mixture model describes the independent growth of the proteoglycan and collagen constituents due to volumetric mass deposition, which leads to the remodeling of the composition and the mechanical properties of the solid matrix. The model developed here also describes how the material constants of the collagen constituent depend on a scalar parameter that may change over time (e.g., crosslink density); this leads to a remodeling of the structural and mechanical properties of the collagen constituent. The equilibrium boundary-value problem that describes the changes observed in cartilage explants harvested at different stages of a growth or a degenerative process is formulated. This boundary-value problem is solved using existing experimental data for developing bovine cartilage explants harvested at three developmental stages. The solution of the boundary-value problem in conjunction with existing experimental data suggest the types of experimental studies that need to be conducted in the future to determine model parameters and to further refine the model.     

A continuum model of elephant trunks

A continuum model is presented that relates the trunk parameters of loading, geometry, and muscle structure to the necessary conditions of static equilibrium. Linear theory for stress-strain behavior is used to describe an elephant trunk for an incremental displacement as the animal slowly lifts a weight at the trunk tip. With this analysis and experimental values for the trunk parameters, the apparent trunk stiffness Ea is estimated for the living animal. For an Asian elephant with a maximum compression strain of 33 percent, Ea is of the order of 10(6) N/m2. The continuum model is quite general and may be applied to similar nonskeletal appendages and bodies of other animals.     

A continuum model of protrusion of pseudopod in leukocytes.

The morphology of human leukocytes, the biochemistry of actin polymerization, and the theory of continuum mechanics are used to model the pseudopod protrusion process of leukocytes. In the proposed model, the pseudopod is considered as a porous solid of F-actin network, the pores of which are full of aqueous solution. G-actin is considered as a "solute" transported by convection and diffusion in the fluid phase. The pseudopod grows as actin filaments elongate at their barbed ends at the tip of the pseudopod. The driving force of extension is hypothesized as being provided by the actin polymerization. It is assumed that elongation of actin filaments, powered by chemical energy liberated from the polymerization reaction, does mechanical work against opposing pressure on the membrane. This also gives rise to a pressure drop in the fluid phase at the tip of the pseudopod, which is formulated by an equation relating the work done by actin polymerization to the local state of pressure. The pressure gradient along the pseudopod drives the fluid filtration through the porous pseudopod according to Darcy's Law, which in turn brings more actin monomers to the growing tip. The main cell body serves as a reservoir of G-actin. A modified first-order equation is used to describe the kinetics of polymerization. The rate of pseudopod growth is modulated by regulatory proteins. A one-dimensional moving boundary problem based on the proposed mechanism has been constructed and approximate solutions have been obtained. Comparison of the solutions with experimental data shows that the model is compatible with available observations. The model is also applicable to growth of other cellular systems such as elongation of acrosomal process in sperm cells.     

A fractal continuum model of the pulmonary arterial tree.

The extant morphometric data from the intrapulmonary arteries of dog, human, and cat lungs produce graphs of the log of the vessel number, (N) or length (l) in each level vs. the log of the mean diameter (D) in each level that are sufficiently linear to suggest that a scale-independent self-similar or fractal structure may underlie the observed relationships. These data can be correlated by the following formulas: Nj = a1Dj-beta 1, and lj = a2Dj beta 2, where j denotes the level (order or generation) number measured from the largest vessel at the entrance to the arterial tree to the smallest vessel at the entrance to the capillary bed. With the hemodynamic resistance (R) represented by Rj = 128 microliterj/(Nj pi Dj4) and the vascular volume (Q) by Qj = Nj pi Dj2lj/4, the continuous cumulative distribution of vascular resistance (Rcum) vs. cumulative vascular volume (Qcum) (where Rcum and Qcum represent the total resistance or volume, respectively, upstream from the jth level) can be calculated from [formula: see text] where r = Dj/Dj+1 is a constant independent of j. Analogous equations are developed for the inertance and compliance distributions, providing simple formulas to represent the hemodynamic consequences of the pulmonary arterial tree structure.     

A continuum model for root growth

This paper presents a study of a simple one-dimensional continuum model for growth of the plant root. A fundamental constitutive equation is derived. The model is studied by means of various special cases of increasing complexity. Asymptotic expansions are used to derive approximate solutions to the equation of the model under the fundamental assumption that cell wall thickness is small in comparison with the diameter of the cell. The basic results of the study may be summarized as follows. The observed growth pattern of the root cannot be modelled by a mechanical system whose properties are independent of position on the root. The observed pattern can be modelled by a simple mechanical system in which, for example, cell wall yield stress first decreases and then increases. Two fundamental observations are made based on the modelling study. The first is that any mechanical model must take into account the convective displacement from the tip of points along the root. The second is that in describing growth, data on cell wall mechanical properties are meaningless without corresponding data on cell water potential, and vice versa.     

A continuum model for coupled cells

A continuum model of diffusion-coupled cells that more accurately reflects the presence of low-permeability gap junctions between cells is analyzed. It is shown by a multi-scale analysis that to lowest order the slow evolution of the mean concentration is described by the usual ordinary differential equations for a discrete model. Furthermore, stable non-uniform steady solutions are shown to exist in the continuum model of a one component system, whereas this is impossible for the standard reaction-diffusion model of this system. It is also shown how to average the equations in this continuum model to obtain a system of reaction-diffusion equations with constant coefficients.     

A continuum theory and an experiment for the ion-induced swelling behavior of articular cartilage

Swelling of normal bovine articular cartilage equilibrated in NaCl solutions was dimensionally measured in thin strips of tissue. The ion-induced strains show that free swelling of articular cartilage is anisotropic and inhomogeneous. For the molar concentrations used, contraction increased linearly with concentration, defining a "coefficient of chemical contraction" (alpha c). Isometrically constrained specimens registered a rise in tensile force followed by stress relaxation. An extension of the biphasic theory incorporating this ion-induced strain is proposed. This theory can describe the equilibrium anisotropic swelling behavior of cartilage and explain the transient force history observed in the isometric experiment.     

A hyperelastic biphasic fibre-reinforced model of articular cartilage considering distributed collagen fibre orientations: continuum basis,computational aspects and applications

Cartilage is a multi-phase material composed of fluid and electrolytes (68–85% by wet weight), proteoglycans (5–10% by wet weight), chondrocytes, collagen fibres and other glycoproteins. The solid phase constitutes an isotropic proteoglycan gel and a fibre network of predominantly type II collagen, which provides tensile strength and mechanical stiffness. The same two components control diffusion of the fluid phase, e.g. as visualised by diffusion tensor MRI: (i) the proteoglycan gel (giving a baseline isotropic diffusivity) and (ii) the highly anisotropic collagenous fibre network. We propose a new constitutive model and finite element implementation that focus on the essential load-bearing morphology: an incompressible, poroelastic solid matrix reinforced by an inhomogeneous, dispersed fibre fabric, which is saturated with an incompressible fluid residing in strain-dependent pores of the collagen–proteoglycan solid matrix. The inhomogeneous, dispersed fibre fabric of the solid further influences the fluid permeability, as well as an intrafibrillar portion that cannot be ‘squeezed out’ from the tissue. Using representative numerical examples on the mechanical response of cartilage, we reproduce several features that have been demonstrated experimentally in the cartilage mechanics literature.     

A continuum model for surface contraction waves

The periodic travelling waves which appear on some animal eggs after fertilization are considered here. These are thought to be caused by a calcium initiated calcium release on the surface, causing calcium waves. A continuum model is developed where the cell is treated as a small viscous droplet with a surface contamination. When a periodic source of surfactant acts at one pole and propagates down the cell surface to the opposite pole, the drop responds by forming constriction rings which move from pole to pole.     

The osmotic properties of sulphoethyl-Sephadex. A model for cartilage

1. Observations are reported on the variation of the swelling of sulphoethyl-Sephadex C-50 and C-25, and of the partition of NaCl between solution and gel, with the concentration of NaCl. 2. The results were interpreted in terms of Manning's (1969) treatment of the interactions between polyionic polymers and simple electrolytes and of Flory's (1953) treatment of the swelling of gels. 3. The results indicate net inner osmotic pressures as high as 3.6x10(5)Pa (140 mosmolar=3.6x10(5)N/m(2)) in 0.15m-NaCl. 4. It is suggested that cartilage may have net inner osmotic pressure of the order of 10(5)Pa.     

A new model for the continuum concept

A reformulation of the continuum concept is presented after considering the implications of the community/continuum controversy and current niche theory. Community is a spatial concept dependent on landscape pattern while the continuum is an environmental concept referring to an abstract space. When applying niche theory to plants, the mechanisms of competition are ill-defined and the assumption of bell-shaped response curves for species unrealistic.Eight testable propositions on the pattern of response of vegetation to environmental gradients are presented 1. Environmental gradients are of two types. a) resource gradients or b) direct physiological gradients. 2. The fundamental niche response of species to resource gradients is a series of similar nested response curves. 3. The fundamental niche response of species to direct gradients is a series of separate, independent, overlapping response curves. 4. Species fundamental response curves are such that they have a relative performance advantage in some part of the environmental space. 5. The shape of the realized niche is variable even bimodal but predictable from the fundamental response given the other species present. Propositions 6–8 describe the response shapes of emergent community properties to environmental gradient; species richness is bimodal, dominance trimodal and standing crop unimodal. Detailed comparisons of these propositions are made with the alternative theories of Ellenberg, Gauch and Whittaker, Grime, and Tilman. These theories are incomplete lacking several generally accepted properties of plants and vegetation.     

A model for dispersion experiments

Summary A new model is proposed for the dispersion of animals and other organisms and its use is discussed for the analysis of the data from experiments on dispersion. The model is a generalisation of the random walk model, but because of its flexibility it should be much more widely applicable than the random walk model.The new model has been found to fit the results of many dispersion experiments and examples are given of its use with data for millipedes and Drosophila.     

A dynamic continuum model for molecular regulation of the humoral immune response.

Based on current concepts of the nature of specific immunocompetent cell surface receptors, hematopoietic stem cell differentiation and membrane dynamics, a simple model is proposed by which the event of immunogen binding by cell surface receptors specifically stimulates a constitutive process. The concept that enhancement of process flows in two directions with time allows for correlation of recent experimental findings into a molecular theory for antibody induction. The Model helps explain antibody specificity, heterogeneity and affinity changes and rationalizes immune memory.     

A cartilage growth mixture model for infinitesimal strains: solutions of boundary-value problems related to in vitro growth experiments

A cartilage growth mixture (CGM) model is linearized for infinitesimal elastic and growth strains. Parametric studies for equilibrium and nonequilibrium boundary-value problems representing the in vitro growth of cylindrical cartilage constructs are solved. The results show that the CGM model is capable of describing the main biomechanical features of cartilage growth. The solutions to the equilibrium problems reveal that tissue composition, constituent pre-stresses, and geometry depend on collagen remodeling activity, growth symmetry, and differential growth. Also, nonhomogeneous growth leads to nonhomogeneous tissue composition and constituent pre-stresses. The solution to the nonequilibrium problem reveals that the tissue is nearly in equilibrium at all time points. The results suggest that the CGM model may be used in the design of tissue engineered cartilage constructs for the repair of cartilage defects; for example, to predict how dynamic mechanical loading affects the development of nonuniform properties during in vitro growth. Furthermore, the results lay the foundation for future analyses with nonlinear models that are needed to develop realistic models of cartilage growth.