An extended biphasic model for charged hydrated tissues with application to the intervertebral disc

Finite element models for hydrated soft biological tissue are numerous but often exhibit certain essential deficiencies concerning the reproduction of relevant mechanical and electro-chemical responses. As a matter of fact, singlephasic models can never predict the interstitial fluid flow or related effects like osmosis. Quite a few models have more than one constituent, but are often restricted to the small-strain domain, are not capable of capturing the intrinsic viscoelasticity of the solid skeleton, or do not account for a collagen fibre reinforcement. It is the goal of this contribution to overcome these drawbacks and to present a thermodynamically consistent model, which is formulated in a very general way in order to reproduce the behaviour of almost any charged hydrated tissue. Herein, the Theory of Porous Media (TPM) is applied in combination with polyconvex Ogden-type material laws describing the anisotropic and intrinsically viscoelastic behaviour of the solid matrix on the basis of a generalised Maxwell model. Moreover, other features like the deformation-dependent permeability, the possibility to include inhomogeneities like varying fibre alignment and behaviour, or osmotic effects based on the simplifying assumption of Lanir are also included. Finally, the human intervertebral disc is chosen as a representative for complex soft biological tissue behaviour. In this regard, two numerical examples will be presented with focus on the viscoelastic and osmotic capacity of the model.     

The correspondence between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage

Mixture models have been successfully used to describe the response of articular cartilage to various loading conditions. Mow et al. (J. Biomech. Eng. 102 (1980) 73) formulated a biphasic mixture model of articular cartilage where the collagen-proteoglycan matrix is modeled as an intrinsically incompressible porous-permeable solid matrix, and the interstitial fluid is modeled as an incompressible fluid. Lai et al. (J. Biomech. Eng. 113 (1991) 245) proposed a triphasic model of articular cartilage as an extension of their biphasic theory, where negatively charged proteoglycans are modeled to be fixed to the solid matrix, and monovalent ions in the interstitial fluid are modeled as additional fluid phases. Since both models co-exist in the cartilage literature, it is useful to show how the measured properties of articular cartilage (the confined and unconfined compressive and tensile moduli, the compressive and tensile Poisson's ratios, and the shear modulus) relate to both theories. In this study, closed-form expressions are presented that relate biphasic and triphasic material properties in tension, compression and shear. These expressions are then compared to experimental findings in the literature to provide greater insight into the measured properties of articular cartilage as a function of bathing solutions salt concentrations and proteoglycan fixed-charge density.     

Collagen fiber orientation and geometry effects on the mechanical properties of secondary osteons.

The effects of collagen fiber orientation and osteon geometry on the mechanical properties of secondary osteons under axial compression/tension and combined loadings (compression, bending and torsion) were investigated using a composite-beam finite-element model. Three cross-sectional shapes of secondary osteons were studied to show the effect of geometry. The results of stiffness are presented using the tension and compression properties for each lamella. The model shows that the mechanical properties of osteons are enhanced in bending and torsion when collagen fibers are oriented within 30 degrees of the loading axis. Osteons with alternating lamellar orientation are not well adapted to resist torsional moments, but alternate collagen fiber orientation has virtually no effect on the bending stiffness of osteons. Fiber orientation affects the mechanical properties less significantly when osteons are non-circular. Collagen fiber orientation and osteon geometry interact to determine the mechanical behavior of the osteon, and may act in a compensatory manner in the adaptive process.     

Collagen fiber orientation at the tendon to bone insertion and its influence on stress concentrations

The tendon to bone insertion serves the mechanical role of transferring loads from a relatively compliant tendon to a relatively rigid bone. The details of the mechanism of load transfer are of great importance, since current surgical procedures for tendon reattachment have high failure rates. We hypothesized that the microscopic structure of the insertion is optimized to minimize stress concentrations associated with this load transfer. To explore this, collagen fiber orientation distributions were measured in the supraspinatus tendons of rats. The angular deviation of fibers was fairly uniform across the insertion, and the mean angles of the local distributions deviated mildly from the tendon axis. To explore how these observed property distributions could influence load transfer, these distributions were used to derive material properties for an idealized two-dimensional mechanical model of an insertion. Comparison between stress concentrations in this idealized model and those in three comparison models suggests that the microstructure serves to (1) simultaneously reduce stress concentrations and material mass, and (2) shield the insertion's outward splay from the highest stresses.     

The effect of neutral and charged micelles on the acid-base dissociation of the local anesthetic tetracaine

The influence of surfactant micelles on the acid-base dissociation of the charged tertiary amino group of the local anesthetic, tetracaine, has been investigated. From measurements of tetracaine fluorescence as a function of bulk pH, apparent $\text{p}\text{K}$ values of 6.88, 7.58 and 9.92 were found in the presence of cationic, neutral and anionic micelles, respectively, in 10 mM NaCl. These values are considerably displaced with respect to the $\text{p}\text{K}$ in aqueous solution which is 8.26. Such large shifts can be attributed to the effect of the surface polarity and electrical potential on the dissociation behavior of the anesthetic bound to micelles. It can be expected that the acid-base dissociation of a local anesthetic adsorbed to nerve fibers will also be affected by the properties of the membrane surface. Thus, it is suggested that the influence of the interfacial region on the $\text{p}\text{K}$ of surface-bound molecules should not be disregarded when estimating the proportion of charged and uncharged forms of local anesthetics interacting with axonal membranes.     

Effect of fiber orientation and strain rate on the nonlinear uniaxial tensile material properties of tendon

Tendons are exposed to complex loading scenarios that can only be quantified by mathematical models, requiring a full knowledge of tendon mechanical properties. This study measured the anisotropic, nonlinear, elastic material properties of tendon. Previous studies have primarily used constant strain-rate tensile tests to determine elastic modulus in the fiber direction. Data for Poisson's ratio aligned with the fiber direction and all material properties transverse to the fiber direction are sparse. Additionally, it is not known whether quasi-static constant strain-rate tests represent equilibrium elastic tissue behavior. Incremental stress-relaxation and constant strain-rate tensile tests were performed on sheep flexor tendon samples aligned with the tendon fiber direction or transverse to the fiber direction to determine the anisotropic properties of toe-region modulus (E0), linear-region modulus (E), and Poisson's ratio (v). Among the modulus values calculated, only fiber-aligned linear-region modulus (E1) was found to be strain-rate dependent. The E1 calculated from the constant strain-rate tests were significantly greater than the value calculated from incremental stress-relaxation testing. Fiber-aligned toe-region modulus (E(1)0 = 10.5 +/- 4.7 MPa) and linear-region modulus (E1 = 34.0 +/- 15.5 MPa) were consistently 2 orders of magnitude greater than transverse moduli (E(2)0 = 0.055 +/- 0.044 MPa, E2 = 0.157 +/- 0.154 MPa). Poisson's ratio values were not found to be rate-dependent in either the fiber-aligned (v12 = 2.98 +/- 2.59, n = 24) or transverse (v21 = 0.488 +/- 0.653, n = 22) directions, and average Poisson's ratio values in the fiber-aligned direction were six times greater than in the transverse direction. The lack of strain-rate dependence of transverse properties demonstrates that slow constant strain-rate tests represent elastic properties in the transverse direction. However, the strain-rate dependence demonstrated by the fiber-aligned linear-region modulus suggests that incremental stress-relaxation tests are necessary to determine the equilibrium elastic properties of tendon, and may be more appropriate for determining the properties to be used in elastic mathematical models.     

Role of electrostatics in the binding of charged metallophthalocyanines to neutral and charged phospholipid membranes

Binding of the cationic tetra(tributylammoniomethyl)-substituted hydroxoaluminum phthalocyanine (AlPcN₄) to bilayer lipid membranes was studied by fluorescence correlation spectroscopy (FCS) and intramembrane field compensation (IFC) methods. With neutral phosphatidylcholine membranes, AlPcN₄ appeared to bind more effectively than the negatively charged tetrasulfonated aluminum phthalocyanine (AlPcS₄), which was attributed to the enhancement of the coordination interaction of aluminum with the phosphate moiety of phosphatidylcholine by the electric field created by positively charged groups of AlPcN₄. The inhibitory effect of fluoride ions on the membrane binding of both AlPcN₄ and AlPcS₄ supported the essential role of aluminum-phosphate coordination in the interaction of these phthalocyanines with phospholipids. The presence of negative or positive charges on the surface of lipid membranes modulated the binding of AlPcN₄ and AlPcS₄ in accord with the character (attraction or repulsion) of the electrostatic interaction, thus showing the significant contribution of the latter to the phthalocyanine adsorption on lipid bilayers. The data on the photodynamic activity of AlPcN₄ and AlPcS₄ as measured by sensitized photoinactivation of gramicidin channels in bilayer lipid membranes correlated well with the binding data obtained by FCS and IFC techniques. The reduced photodynamic activity of AlPcN₄ with neutral membranes violating this correlation was attributed to the concentration quenching of singlet excited states as proved by the data on the AlPcN₄ fluorescence quenching.     

Role of electrostatics in the binding of charged metallophthalocyanines to neutral and charged phospholipid membranes

Binding of the cationic tetra(tributylammoniomethyl)-substituted hydroxoaluminum phthalocyanine (AlPcN(4)) to bilayer lipid membranes was studied by fluorescence correlation spectroscopy (FCS) and intramembrane field compensation (IFC) methods. With neutral phosphatidylcholine membranes, AlPcN(4) appeared to bind more effectively than the negatively charged tetrasulfonated aluminum phthalocyanine (AlPcS(4)), which was attributed to the enhancement of the coordination interaction of aluminum with the phosphate moiety of phosphatidylcholine by the electric field created by positively charged groups of AlPcN(4). The inhibitory effect of fluoride ions on the membrane binding of both AlPcN(4) and AlPcS(4) supported the essential role of aluminum-phosphate coordination in the interaction of these phthalocyanines with phospholipids. The presence of negative or positive charges on the surface of lipid membranes modulated the binding of AlPcN(4) and AlPcS(4) in accord with the character (attraction or repulsion) of the electrostatic interaction, thus showing the significant contribution of the latter to the phthalocyanine adsorption on lipid bilayers. The data on the photodynamic activity of AlPcN(4) and AlPcS(4) as measured by sensitized photoinactivation of gramicidin channels in bilayer lipid membranes correlated well with the binding data obtained by FCS and IFC techniques. The reduced photodynamic activity of AlPcN(4) with neutral membranes violating this correlation was attributed to the concentration quenching of singlet excited states as proved by the data on the AlPcN(4) fluorescence quenching.     

The influence of fiber dispersion on the mechanical response of aortic tissues in health and disease: a computational study

Changes in the structural components of aortic tissues have been shown to play a significant role in the pathogenesis of aortic degeneration. Therefore, reliable stress analyses require a suitable and meaningful constitutive model that captures micro-structural changes. As recent data show, in-plane and out-of-plane collagen fiber dispersions vary significantly between healthy and aneurysmatic aortic walls. The aim of this study is to computationally investigate the influence of fiber dispersion on the mechanical response of aortic tissues in health and disease. In particular, the influence of three different fiber dispersions is studied: (i) non-rotationally symmetric dispersion, the most realistic assumption for aortic tissues; (ii) transversely isotropic dispersion, a special case; (iii) perfectly aligned fibers (no dispersion in either plane), another special case. Explicit expressions for the stress and elasticity tensors as needed for the implementation in a finite element code are provided. Three representative numerical examples are studied: planar biaxial extension, inflation of residually stressed and pre-stretched aortic segments and inflation of an idealized abdominal aortic aneurysm (AAA) geometry. For the AAA geometry the case of isotropic dispersion is additionally analyzed. Documented structural and mechanical parameters are taken from human aortas (healthy media/adventitia and AAA). The influence of fiber dispersions upon magnitudes and distributions of stresses and deformations are presented and analyzed. Stresses vary significantly, especially in the AAA case, where material stiffening is significantly influenced by fiber dispersion. The results highlight the need to incorporate the structural differences into finite element simulations to obtain more accurate stress predictions. Additionally, results show the capability of one constitutive model to represent different scenarios of aortic micro-structures allowing future studies of collagen reorientation during disease progression.     

The influence of substrate contact on gravity orientation

Summary Tilting of the animal with respect to gravity results in compensatory eyestalk movements and in leg counterforce reactions which vary with the number and sequential position of legs touching the body-fixed substrate board (Figs. 2, 3). The gravity response is reduced with increasing number of legs touching the substrate. The results fit an interpretation that the weight of the substrate input interacting with gravity signals results from superposition of the weighted effects of the single legs involved (Figs. 2, 4).Acknowledgements are due to the Max-Planck-Institut (D. M. Neil, N. Scapini) and to the Carnegie Trust (D. M. Neil) for financial support. We want to thank renate Alton for her co-operative help.     

Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues

Structural constitutive models integrate information on tissue composition and structure, avoiding ambiguities in material characterization. However, critical structural information (such as fiber orientation) must be modeled using assumed statistical distributions, with the distribution parameters estimated from fits to the mechanical test data. Thus, full realization of structural approaches continues to be limited without direct quantitative structural information for direct implementation or to validate model predictions. In the present study, fiber orientation information obtained using small angle light scattering (SALS) was directly incorporated into a structural constitutive model based on work by Lanir (J. Biomech., v. 16, pp. 1-12, 1983). Demonstration of the model was performed using existing biaxial mechanical and fiber orientation data for native bovine pericardium (Sacks and Chuong, ABME, v.26, pp. 892-902, 1998). The structural constitutive model accurately predicted the complete measured biaxial mechanical response. An important aspect of this approach is that only a single equibiaxial test to determine the effective fiber stress-strain response and the SALS-derived fiber orientation distribution were required to determine the complete planar biaxial mechanical response. Changes in collagen fiber crimp under equibiaxial strain suggest that, at the meso-scale, fiber deformations follow the global tissue strains. This result supports the assumption of affine strain to estimate the fiber strains. However, future evaluations will have to be performed for tissue subjected to a wider range of strain to more fully validate the current approach.     

The effect of calcium on the properties of charged phospholipid bilayers

We have performed molecular dynamics simulations to investigate the structure and dynamics of charged bilayers as well as the distribution of counterions at the bilayer interface. For this, we have considered the negatively charged di-myristoyl-phosphatidyl-glycerol (DMPG) and di-myristoyl-phosphatidyl-serine (DMPS) bilayers as well as a protonated di-myristoyl-phosphatidyl-serine (DMPSH) bilayer. We were particularly interested in calcium ions due to their important role in biological systems. Simulations performed in the presence of calcium ions (DMPG, DMPS) or sodium ions (DMPS) were run for 45-60 ns. Simulation results for DMPG are compared with fluorescence measurements. The average areas per molecule were 47.4 ± 0.5 Å² (DMPG with calcium), 47.3 ± 0.5 Å² (DMPS with calcium), 51.3 ± 1.0 Å² (DMPS with sodium) and 45.3 ± 0.5 Å² (DMPSH). The structure of the negatively charged lipids is significantly affected by the counterions, where calcium ions have a more pronounced effect than sodium ions. Calcium ions were found to be tightly bound to the anionic groups of the lipid molecules and as such appear to constitute an integral part of the membrane interface on nanoseconds time scales. In contrast to sodium ions, calcium ions are localised in a narrow (∼ 10 Å) band around the phosphate group. The interaction of calcium with the lipid molecules enhances the molecular packing of the PG and PS lipids. This observation is in good agreement with emission spectra of the membrane partitioning probe Laurdan in DMPG multilamellar vesicles that indicate an increase in the ordering of the DMPG bilayer due to the presence of calcium. Our results indicate that calcium ions, which often function as a second messengers in living cells have a pronounced effect on membrane structures, which may have implications during signal transduction events.     

The effect of calcium on the properties of charged phospholipid bilayers

We have performed molecular dynamics simulations to investigate the structure and dynamics of charged bilayers as well as the distribution of counterions at the bilayer interface. For this, we have considered the negatively charged di-myristoyl-phosphatidyl-glycerol (DMPG) and di-myristoyl-phosphatidyl-serine (DMPS) bilayers as well as a protonated di-myristoyl-phosphatidyl-serine (DMPSH) bilayer. We were particularly interested in calcium ions due to their important role in biological systems. Simulations performed in the presence of calcium ions (DMPG, DMPS) or sodium ions (DMPS) were run for 45-60 ns. Simulation results for DMPG are compared with fluorescence measurements. The average areas per molecule were 47.4+/-0.5 A2 (DMPG with calcium), 47.3+/-0.5 A2 (DMPS with calcium), 51.3+/-1.0 A2 (DMPS with sodium) and 45.3+/-0.5 A2 (DMPSH). The structure of the negatively charged lipids is significantly affected by the counterions, where calcium ions have a more pronounced effect than sodium ions. Calcium ions were found to be tightly bound to the anionic groups of the lipid molecules and as such appear to constitute an integral part of the membrane interface on nanoseconds time scales. In contrast to sodium ions, calcium ions are localised in a narrow (approximately 10 A) band around the phosphate group. The interaction of calcium with the lipid molecules enhances the molecular packing of the PG and PS lipids. This observation is in good agreement with emission spectra of the membrane partitioning probe Laurdan in DMPG multilamellar vesicles that indicate an increase in the ordering of the DMPG bilayer due to the presence of calcium. Our results indicate that calcium ions, which often function as a second messengers in living cells have a pronounced effect on membrane structures, which may have implications during signal transduction events.     

The influence of fetoabdominal tissues on fetal ECGs and MCGs

Both fetal electrocardiography and fetal magnetocardiography are influenced by the volume conduction within the abdomen of the pregnant woman. In this paper, various models are used to simulate this influence. Such models are helpful to determine where to attach electrodes at the maternal abdomen in case fetal ECGs are measured and where to position the magnetocardiograph in case fetal MCGs are measured. Another goal is to assess the influence of individual differences, such as the amount of amniotic fluid. Seven models based on MR-images have been created, four for the third trimester of gestation, with the fetus in left occiput position, and three for the second trimester. The models consist of four compartments; the fetus, the vernix caseosa, the amniotic fluid, and the remainder of the maternal abdomen. It turns out that individual differences have a large impact on the fetal MCG and that the best measurement positions are expected over the centre of the abdomen near the fetal heart. The fetal ECG is dependent on the vernix caseosa and when this layer is present, the fetal ECG is best measured by two electrodes, one over the fetal mouth and the other over the bottom of the fetus.     

The influence of niche and neutral processes on a neotropical anuran metacommunity

One of the most important questions in ecology is the relative importance of local conditions (niche processes) and dispersal ability (neutral processes) in driving metacommunity structure. Although many studies have been conducted in recent years, there is still much debate. We evaluated the processes (niche and neutral) responsible for variation in anuran composition in 28 lentic water bodies in southeastern Brazil. Because anurans depend heavily on environmental conditions, we hypothesized that environmental variables (niche processes) are the most important drivers of community composition. Additionally, as anurans have limited dispersal abilities, and the study region presents harsh conditions (high forest fragmentation, low rainfall and long dry season), we expected a lower, but significant, spatial signature in metacommunity structure, due to neutral dynamics. We used a partial redundancy analysis with variation partitioning to evaluate the relative influence of environmental and spatial variables as drivers of metacommunity structure. Additionally, we used a recently developed spatial autocorrelation analysis to test if neutral dynamics can be attributed to the pure spatial component. This analysis is based on predictions that species abundances are independent but similarly spatially structured, with correlograms similar in shape. Therefore, under neutral dynamics there is no expectation of a correlation between the pairwise distance of spatial correlograms and the pairwise correlation of species abundances predicted by the pure spatial component. We found that the environmental component explained 21.5%, the spatial component 10.2%, and the shared component 6.4% of the metacommunity structure. We found no correlation between correlograms and correlation of abundances predicted by the pure spatial component (Mantel test = ?0.109, P = 0.961). In our study, niche‐based processes are the dominant process that explained community composition. However, neutral processes are important because spatial variation can be attributed to pure neutral dynamics rather than to missing spatially structured environmental factors.