Estimation of material elastic moduli in elastography: a local method, and an investigation of Poisson's ratio sensitivity

The local material stiffness of tissues is a well-known indicator of pathology, with locally stiffer tissue related to the possible presence of an abnormal growth in otherwise compliant tissue. Elastography is a non-invasive technique for measuring displacement distributions in loaded tissues within a medical imaging context. From these measured displacement fields, estimated for local strain have been made using well-studied techniques, but the calculation of elastic modulus has been difficult. In this study we show a method for estimating local tissue elastic modulus that gives numerically stable and robust results in test cases, and that is numerically efficient. The method assumes the tissue is isotropic and it requires an independent estimate of tissue Poisson's ratio, but the method reaches a stable result when the estimated Poisson's ratio is in error, and the resulting estimates are not very sensitive to the assumed value.     

The influence of the fixed negative charges on mechanical and electrical behaviors of articular cartilage under unconfined compression

Unconfined compression test has been frequently used to study the mechanical behaviors of articular cartilage, both theoretically and experimentally. It has also been used in explant and gel-cell-complex studies in tissue engineering. In biphasic and poroelastic theories, the effect of charges fixed on the proteoglycan macromolecules in articular cartilage is embodied in the apparent compressive Young's modulus and the apparent Poisson's ratio of the tissue, and the fluid pressure is considered to be the portion above the osmotic pressure. In order to understand how proteoglycan fixed charges might affect the mechanical behaviors of articular cartilage, and in order to predict the osmotic pressure and electric fields inside the tissue in this experimental configuration, it is necessary to use a model that explicitly takes into account the charged nature of the tissue and the flow of ions within its porous interstices. In this paper, we used a finite element model based on the triphasic theory to study how fixed charges in the porous-permeable soft tissue can modulate its mechanical and electrochemical responses under a step displacement in unconfined compression. The results from finite element calculations showed that: 1) A charged tissue always supports a larger load than an uncharged tissue of the same intrinsic elastic moduli. 2) The apparent Young's modulus (the ratio of the equilibrium axial stress to the axial strain) is always greater than the intrinsic Young's modulus of an uncharged tissue. 3) The apparent Poisson's ratio (the negative ratio of the lateral strain to the axial strain) is always larger than the intrinsic Poisson's ratio of an uncharged tissue. 4) Load support derives from three sources: intrinsic matrix stiffness, hydraulic pressure and osmotic pressure. Under the unconfined compression, the Donnan osmotic pressure can constitute between 13%-22% of the total load support at equilibrium. 5) During the stress-relaxation process following the initial instant of loading, the diffusion potential (due to the gradient of the fixed charge density and the associated gradient of ion concentrations) and the streaming potential (due to fluid convection) compete against each other. Within the physiological range of material parameters, the polarity of the electric potential depends on both the mechanical properties and the fixed charge density (FCD) of the tissue. For softer tissues, the diffusion effects dominate the electromechanical response, while for stiffer tissues, the streaming potential dominates this response. 6) Fixed charges do not affect the instantaneous strain field relative to the initial equilibrium state. However, there is a sudden increase in the fluid pressure above the initial equilibrium osmotic pressure. These new findings are relevant and necessary for the understanding of cartilage mechanics, cartilage biosynthesis, electromechanical signal transduction by chondrocytes, and tissue engineering.     

The distribution of connective tissue content in swimming muscle of cod (Gadus morbua)

The amount and distribution of connective tissue in the swimming muscle of cod (Gadus morhua) have been determined. By biochemical analysis, the collagen was found to make up 1,5 % of the total protein content. Measured by light microscopical morphometric analysis, the areal fraction of the connective tissue elements was found to be 3,0 % of the total muscle area. The areal fraction of myocommatal connective tissue mentioned above was found to be 2.3 %. The thickness of the endomysial sheath was calculated by morphometry based on electron microscopy, and was found to be 0.30 and 0.16 μ for red and white fibres, respectively. The areal fraction of the endomysial sheaths was 2.3 % in red muscle and 0.5% in white muscle. The endomysial sheaths make up 25% of the total connective tissue in the swimming muscles. These sheaths influence the binding properties of fish muscle products.     

Stress transfer in collagen fibrils reinforcing connective tissues: effects of collagen fibril slenderness and relative stiffness

Unlike engineering fibre composite materials which comprise of fibres that are uniform cylindrical in shape, collagen fibrils reinforcing the proteoglycan-rich (PG) gel in the extra-cellular matrices (ECMs) of connective tissues are taper-ended (paraboloidal in shape). In an earlier paper we have discussed how taper of a fibril leads to an axial stress up-take which differs from that of a uniform cylindrical fibre and implications for fibril fracture. The present paper focuses on the influence of fibre aspect ratio, q (slenderness), and Young's modulus (stiffness), relative to that of the gel phase, E(R), on the magnitude of the axial tensile stresses generated within a fibril and wider implications on failure at tissue level. Fibre composite models were evaluated using finite element (FE) and mathematical analyses. When the applied force is low, there is elastic stress transfer between the PG gel and a fibril. FE modelling shows that the stress in a fibril increases with E(R) and q. At higher applied forces, there is plastic stress transfer. Mathematical modelling predicts that the stress in a fibril increases linearly with q. For small q values, fibrils may be regarded as fillers with little ability to provide tensile reinforcement. Large q values lead to high stress in a fibril. Such high stresses are beneficial provided they do not exceed the fracture stress of collagen. Modulus difference regulates the strain energy release density, u, for interfacial rupture; large E(R) not only leads to high stress in a fibril but also insures against interfacial rupture by raising the value of u.     

A composites theory predicts the dependence of stiffness of cartilage culture tissues on collagen volume fraction.

The tensile stiffness of tissue grown from chondrocyte culture was both measured experimentally and predicted using a composites model theory relating tissue microstructure to macroscopic material stiffness. The tissue was altered by several treatment protocols to provide a wide range of collagen fibril volume fraction (0.015-0.15). The rate of change of tissue modulus with change in collagen volume fraction predicted by the theory was within 14% of the slope of the linear fit through the experimental data, without the use of fitting parameters for the theoretical value of the slope. Use of the model to simulate cytokine mediated tissue digestion suggests that the action of IL-1beta and retinoic acid is mainly removal of proteoglycans and some removal of collagen. The model also indicates that the matrix and collagen remaining in the tissue has the same elastic properties as the untreated tissue, and is not damaged due to the alteration. Young's modulus of the collagen fibrils is predicted to be 120 MPa, a value in the range of previous studies. This value is dependent mainly on the matrix modulus and collagen fibril volume fraction and not on Poisson's ratio of either matrix or fibril. Poisson's ratio of the tissue depends primarily on the Poisson's ratio of the matrix.     

Stichopin-containing nerves and secretory cells specific to connective tissues of the sea cucumber

Stichopin, a 17-amino acid peptide isolated from a sea cucumber, affects the stiffness change of the body-wall catch connective tissues and the contraction of the body-wall muscles. The localization of stichopin in sea cucumbers was studied by indirect immunohistochemistry using antiserum against stichopin. Double staining was performed with both stichopin antiserum and 1E11, the monoclonal antibody specific to echinoderm nerves. A stichopin-like immunoreactivity (stichopin-LI) was exclusively found in the connective tissues of various organs. Many fibres and cells with processes were stained by both the anti-stichopin antibody and 1E11. They were found in the body-wall dermis and the connective tissue layer of the cloacae and were suggested to be connective tissue-specific nerves. Oval cells with stichopin-LI (OCS) without processes were found in the body-wall dermis, the connective tissue sheath of the longitudinal body-wall muscles, the connective tissue layer of the tube feet and tentacles, and the connective tissue in the radial nerves separating the ectoneural part from the hyponeural part. Electron microscopic observations of the OCSs in the radial nerves showed that they were secretory cells. The OCSs were located either near the well-defined neural structures or near the water-filled cavities, such as the epineural sinus and the canals of the tube feet. The location near the water-filled cavities might suggest that stichopin was secreted into these cavities to function as a hormone.     

Prediction of the Time Course of Callus Stiffness as a Function of Mechanical Parameters in Experimental Rat Fracture Healing Studies - A Numerical Study

Numerous experimental fracture healing studies are performed on rats, in which different experimental, mechanical parameters are applied, thereby prohibiting direct comparison between each other. Numerical fracture healing simulation models are able to predict courses of fracture healing and offer support for pre-planning animal experiments and for post-hoc comparison between outcomes of different in vivo studies. The aims of this study are to adapt a pre-existing fracture healing simulation algorithm for sheep and humans to the rat, to corroborate it using the data of numerous different rat experiments, and to provide healing predictions for future rat experiments. First, material properties of different tissue types involved were adjusted by comparing experimentally measured callus stiffness to respective simulated values obtained in three finite element (FE) models. This yielded values for Young’s moduli of cortical bone, woven bone, cartilage, and connective tissue of 15,750 MPa, 1,000 MPa, 5 MPa, and 1 MPa, respectively. Next, thresholds in the underlying mechanoregulatory tissue differentiation rules were calibrated by modifying model parameters so that predicted fracture callus stiffness matched experimental data from a study that used rigid and flexible fixators. This resulted in strain thresholds at higher magnitudes than in models for sheep and humans. The resulting numerical model was then used to simulate numerous fracture healing scenarios from literature, showing a considerable mismatch in only 6 of 21 cases. Based on this corroborated model, a fit curve function was derived which predicts the increase of callus stiffness dependent on bodyweight, fixation stiffness, and fracture gap size. By mathematically predicting the time course of the healing process prior to the animal studies, the data presented in this work provides support for planning new fracture healing experiments in rats. Furthermore, it allows one to transfer and compare new in vivo findings to previously performed studies with differing mechanical parameters.     

Elastic deformation of mineralized collagen fibrils: an equivalent inclusion based composite model

Mineralized collagen fibrils are the basic building blocks of bone tissue at the supramolecular level. Several disease states, manipulation of the expression of specific proteins involved in biomineralization, and treatment with different agents alter the extent of mineralization as well as the morphology of mineral crystals which in turn affect the mechanical function of bone tissue. An experimental assessment of mineralized fibers' mechanical properties is challenged by their small size, leaving analytical and computational models as a viable alternative for investigation of the fibril-level mechanical properties. In the current study the variation of the elastic stiffness tensor of mineralized collagen fibrils with changing mineral volume fraction and mineral aspect ratios was predicted via a micromechanical model. The partitioning of applied stresses between mineral and collagen phases is also predicted for normal and shear loading of fibrils. Model predictions resulted in transversely isotropic collagen fibrils in which the modulus along the longer axis of the fibril was the greatest. All the elastic moduli increased with increasing mineral volume fraction whereas Poisson's ratios decreased with the exception of v12 (=v21). The partitioning of applied stresses were such that the stresses acting on mineral crystals were about 1.5, 15, and 3 times greater than collagen stresses when fibrils were loaded transversely, longitudinally, and in shear, respectively. In the overall the predictions were such that: (a) greatest modulus along longer axis; (b) the greatest mineral/collagen stress ratio along the longer axis of collagen fibers (i.e., greatest relief of stresses acting on collagen); and (c) minimal lateral contraction when fibers are loaded along the longer axis. Overall, the pattern of mineralization as put forth in this model predicts a superior mechanical function along the longer axis of collagen fibers, the direction which is more likely to experience greater stresses.     

Viscoelastic properties of bovine orbital connective tissue and fat: constitutive models

Reported mechanical properties of orbital connective tissue and fat have been too sparse to model strain–stress relationships underlying biomechanical interactions in strabismus. We performed rheological tests to develop a multi-mode upper convected Maxwell (UCM) model of these tissues under shear loading. From 20 fresh bovine orbits, 30 samples of connective tissue were taken from rectus pulley regions and 30 samples of fatty tissues from the posterior orbit. Additional samples were defatted to determine connective tissue weight proportion, which was verified histologically. Mechanical testing in shear employed a triborheometer to perform: strain sweeps at 0.5–2.0 Hz; shear stress relaxation with 1% strain; viscometry at 0.01−0.5 s⁻¹ strain rate; and shear oscillation at 1% strain. Average connective tissue weight proportion was 98% for predominantly connective tissue and 76% for fatty tissue. Connective tissue specimens reached a long-term relaxation modulus of 668 Pa after 1,500 s, while corresponding values for fatty tissue specimens were 290 Pa and 1,100 s. Shear stress magnitude for connective tissue exceeded that of fatty tissue by five-fold. Based on these data, we developed a multi-mode UCM model with variable viscosities and time constants, and a damped hyperelastic response that accurately described measured properties of both connective and fatty tissues. Model parameters differed significantly between the two tissues. Viscoelastic properties of predominantly connective orbital tissues under shear loading differ markedly from properties of orbital fat, but both are accurately reflected using UCM models. These viscoelastic models will facilitate realistic global modeling of EOM behavior in binocular alignment and strabismus.     

Elastic fibres in health and disease

Elastic fibres are a major class of extracellular matrix fibres that are abundant in dynamic connective tissues such as arteries, lungs, skin and ligaments. Their structural role is to endow tissues with elastic recoil and resilience. They also act as an important adhesion template for cells, and they regulate growth factor availability. Mutations in major structural components of elastic fibres, especially elastin, fibrillins and fibulin-5, cause severe, often life-threatening, heritable connective tissue diseases such as Marfan syndrome, supravalvular aortic stenosis and cutis laxa. Elastic-fibre function is also frequently compromised in damaged or aged elastic tissues. The ability to regenerate or engineer elastic fibres and tissues remains a significant challenge, requiring improved understanding of the molecular and cellular basis of elastic-fibre biology and pathology, and ability to regulate the spatiotemporal expression and assembly of its molecular components.     

Magnitudes of local stress and strain along bony surfaces predict the course and type of fracture healing

A new quantitative tissue differentiation theory which relates the local tissue formation in a fracture gap to the local stress and strain is presented. Our hypothesis proposes that the amounts of strain and hydrostatic pressure along existing calcified surfaces in the fracture callus determine the differentiation of the callus tissue. The study compares the local strains and stresses in the callus as calculated from a finite element model with histological findings from an animal fracture model. The hypothesis predicts intramembranous bone formation for strains smaller approximately +/- 5% and hydrostatic pressures smaller than +/- 0.15 MPa. Endochondral ossification is associated with compressive pressures larger than about -0.15 MPa and strains smaller than +/- 15%. All other conditions seemed to lead to connective tissue or fibrous cartilage. The hypothesis enables a better understanding of the complex tissue differentiation seen in histological images and the mechanical conditions for healing delayed healing or nonunions.     

Dynamic mechanical properties of body-wall dermis in various mechanical states and their implications for the behavior of sea cucumbers

The dermis of the sea cucumber body wall is a typical catch connective tissue that rapidly changes its mechanical properties in response to various stimuli. Dynamic mechanical properties were measured in stiff, standard, and soft states of the sea cucumber Actinopyga mauritiana. Sinusoidal deformations were applied, either at a constant frequency of 0.1 Hz with varying maximum strain of 2%-20% or at a fixed maximum strain of 1.8% with varying frequency of 0.0005-50 Hz. The dermis showed viscoelasticity with both strain and strain-rate dependence. The dermis in the standard state showed a J-shaped stress-strain curve with a stiffness of 1 MPa and a dissipation ratio of 60%; the curve of the stiff dermis was linear with high stiffness (3 MPa) and a low dissipation ratio (30%). Soft dermis showed a J-shaped curve with low stiffness (0.3 MPa) and a high dissipation ratio (80%). The strain-induced softening was observed in the soft state. Stiff samples had a higher storage modulus and a lower tangent delta than soft ones, implying a larger contribution of the elastic component in the stiff state. A simple molecular model was proposed that accounted for the mechanical behavior of the dermis. The model suggested that stiffening stimulation increased inter-molecular bonds, whereas softening stimulation affected intra-molecular bonds. The adaptive significance of each mechanical state in the behavior of sea cucumbers is discussed.     

Occurrence of "elastic" fibres in the invertebrates

The connective tissues of a wide range of invertebrates have been investigated using a potassium permanganate/spirit blue staining technique. The method demonstrates fibres which are distinct from collagen, vertebrate elastic fibres, and reticular fibres. The fibres are variously associated with subepidermal collagen, muscles, basement membrances, the walls of blood vessels and the sheaths around nerve cords; the method also gives a positive reaction with the arborescent mesogloeal fibres of medusoid coelenterates. Their electron microscope appearance is distinctive and they exhibit a physical elasticity. Chemical tests indicate that following pretreatment with acidified potassium permanganate they show many, although not all, of the properties of vertebrate "elastin". It is concluded that they should be regarded as a special category of "elastic" fibre.     

Thin bio-artificial tissues in plane stress: the relationship between cell and tissue strain, and an improved constitutive model

Constitutive models are needed to relate the active and passive mechanical properties of cells to the overall mechanical response of bio-artificial tissues. The Zahalak model attempts to explicitly describe this link for a class of bio-artificial tissues. A fundamental assumption made by Zahalak is that cells stretch in perfect registry with a tissue. We show this assumption to be valid only for special cases, and we correct the Zahalak model accordingly. We focus on short-term and very long-term behavior, and therefore consider tissue constituents that are linear in their loading response (although not necessarily linear in unloading). In such cases, the average strain in a cell is related to the macroscopic tissue strain by a scalar we call the "strain factor". We incorporate a model predicting the strain factor into the Zahalak model, and then reinterpret experiments reported by Zahalak and co-workers to determine the in situ stiffness of cells in a tissue construct. We find that, without the modification in this article, the Zahalak model can underpredict cell stiffness by an order of magnitude.     

A note on adhesion of bacteria to chicken muscle connective tissue

Laboratory cultures of bacteria were tested for the ability to attach to collagen fibres of intact chicken muscle connective tissue. All salmonellas, fimbriate strains of Escherichia coli and a strain of Campylobacter coli were able to attach to tissue only when suspended in distilled water. Prior immersion of tissue in sterile water for 20 min or extended immersion in these bacterial suspensions was a prerequisite for adhesion. Attachment could be prevented by the addition of physiological levels of sodium chloride to the attachment medium. Furthermore, attached bacteria could be removed by rinsing tissues in saline media.     

Morphology of pancreatic tissue-containing accessory spleen in an APA hamster

Accessory spleen (AS) was observed in 19 out of 71 (27%) golden hamsters of APA strain autopsied in the Institute of Physical and Chemical Research. When these AS were examined histologically, clusters of pancreatic tissues, such as islets and acini, were detected in the parenchyma of one AS located in the tail of pancreas of a 15-week-old male animal. This heterotopic pancreatic tissue was considered to penetrate deep into the AS from the pancreas through the adhesion site without any connective tissue border during the ontogenesis.     

Topographical variation of the elastic properties of articular cartilage in the canine knee

Equilibrium response of articular cartilage to indentation loading is controlled by the thickness (h) and elastic properties (shear modulus, mu, and Poisson's ratio, nu) of the tissue. In this study, we characterized topographical variation of Poisson's ratio of the articular cartilage in the canine knee joint (N=6). Poisson's ratio was measured using a microscopic technique. In this technique, the shape change of the cartilage disk was visualized while the cartilage was immersed in physiological solution and compressed in unconfined geometry. After a constant 5% axial strain, the lateral strain was measured during stress relaxation. At equilibrium, the lateral-to-axial strain ratio indicates the Poisson's ratio of the tissue. Indentation (equilibrium) data from our prior study (Arokoski et al., 1994. International Journal of Sports Medicine 15, 254-260) was re-analyzed using the Poisson's ratio results at the test site to derive values for shear and aggregate moduli. The lowest Poisson's ratio (0.070+/-0.016) located at the patellar surface of femur (FPI) and the highest (0.236+/-0.026) at the medial tibial plateau (TMI). The stiffest cartilage was found at the patellar groove of femur (micro=0.964+/-0.189MPa, H(a)=2.084+/-0. 409MPa) and the softest at the tibial plateaus (micro=0.385+/-0. 062MPa, H(a)=1.113+/-0.141MPa). Comparison of the mechanical results and the biochemical composition of the tissue (Jurvelin et al., 1988. Engineering in Medicine 17, 157-162) at the matched sites of the canine knee joint indicated a negative correlation between the Poisson's ratio and collagen-to-PG content ratio. This is in harmony with our previous findings which suggested that, in unconfined compression, the degree of lateral expansion in different tissue zones is related to collagen-to-PG ratio of the zone.     

The use of small angle light scattering in assessing strain induced collagen degradation in arterial tissue ex vivo

Collagen is the predominant load bearing component in many soft tissues including arterial tissue and is therefore critical in determining the mechanical integrity of such tissues. Degradation of collagen fibres is hypothesized to be a strain dependent process whereby the rate of degradation is affected by the magnitude of strain applied to the collagen fibres. The aim of this study is to investigate the ability of small angle light scattering (SALS) imaging to identify strain dependent degradation of collagen fibres in arterial tissue ex vivo, and determine whether a strain induced protection mechanism exists in arterial tissue as observed in pure collagen and other collagenous tissues. SALS was used in combination with histological and second harmonic generation (SHG) analysis to determine the collagen fibre architecture in arterial tissue subjected to strain directed degradation. SALS alignment analysis identified statistically significant differences in fibre alignment depending on the strain magnitude applied to the tissue. These results were also observed using histology and SHG. Our findings suggest a strain protection mechanism may exist for arterial collagen at intermediate strain magnitudes between 0% and 25%. These findings may have implications for the onset and progression of arterial disease where changes in the mechanical environment of arterial tissue may lead to changes in the collagen degradation rate.     

A mechano-regulation model for tissue differentiation during fracture healing: analysis of gap size and loading

Bone has a capability to repair itself when it is fractured. Repair involves the generation of intermediate tissues, such as fibrous connective tissue, cartilage and woven bone, before final bone healing can occur. The intermediate tissues serve to stabilise the mechanical environment and provide a scaffold for differentiation of new tissues. The repair process is fundamentally affected by mechanical loading and by the geometric configuration of the fracture fragments. Biomechanical analyses of fracture healing have previously computed the stress distribution within the callus and identified the components of the stress tensor favouring or inhibiting differentiation of particular tissue phenotypes. In this paper, a biphasic poroelastic finite element model of a fracture callus is used to simulate the time-course of tissue differentiation during fracture healing. The simulation begins with granulation tissue (post-inflammation phase) and finishes with bone resorption. The biomechanical regulatory model assumes that tissue differentiation is controlled by a combination of shear strain and fluid flow acting within the tissue. High shear strain and fluid flows are assumed to deform the precursor cells stimulating formation of fibrous connective tissue, lower levels stimulate formation of cartilage, and lower again allows ossification. This mechano-regulatory scheme was tested by simulating healing in fractures with different gap sizes and loading magnitudes. The appearance and disappearance of the various tissues found in a callus was similar to histological observation. The effect of gap size and loading magnitude on the rate of reduction of the interfragmentary strain was sufficiently close to confirm the hypothesis that tissue differentiation phenomena could be governed by the proposed mechano-regulation model.     

A fuzzy logic model of fracture healing

A quantitative biomechanical model describes the tissue transformation during healing of a transverse osteotomy of a sheep metatarsal. The model predicts bridging of the bone ends through cartilage, followed by the growth of a callus cuff, and finally, the resorption of callus after ossification of the interfragmentary gap. We suggest bone density or the modulus of elasticity do not sufficiently characterize healing tissue for predictive purposes. In addition to the stimulus reflected by strain energy density we introduce a new osteogenic factor based upon stress gradients and which predicts areas of a high osteogenic capacity. Our model distinguishes three basic types of tissue, namely bone, cartilage and fibrous tissue. A fuzzy controller is proposed to model the tissue reaction. A set of fuzzy rules derived from medical knowledge has been implemented to describe tissue transformation such as intramembraneous or chondral ossification, atrophy or destruction. Fuzzy logic is able to model tissue transformation processes within the numerical simulation of remodeling processes. This approach improves the simulation tools and affords the potential to optimize planning of animal experiments and conduct parametric studies.