Wolff's law of trabecular architecture at remodeling equilibrium

An elastic constitutive relation for cancellous bone tissue is developed. This relationship involves the stress tensor T, the strain tensor E and the fabric tensor H for cancellous bone. The fabric tensor is a symmetric second rank tensor that is a quantitative stereological measure of the microstructural arrangement of trabeculae and pores in the cancellous bone tissue. The constitutive relation obtained is part of an algebraic formulation of Wolff's law of trabecular architecture in remodeling equilibrium. In particular, with the general constitutive relationship between T, H and E, the statement of Wolff's law at remodeling equilibrium is simply the requirement of the commutativity of the matrix multiplication of the stress tensor and the fabric tensor at remodeling equilibrium, T*H* = H*T*. The asterisk on the stress and fabric tensor indicates their values in remodeling equilibrium. It is shown that the constitutive relation also requires that E*H* = H*E*. Thus, the principal axes of the stress, strain and fabric tensors all coincide at remodeling equilibrium.     

A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone

 The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff’s law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone. Received: 25 February 1997/Revised version: 23 September 1997     

On the monodimensional approach to the estimation of the highest reynolds shear stress in a turbulent flow

The measurement of the Reynolds stress tensor, or at least of some of its components, is a necessary step to assess if the turbulence associated with the flow near prosthetic devices can damage blood constituents. Because of the intrinsic three dimensionality of turbulence, in general, a three-component anemometer should be used to measure directly the components of the Reynolds stress tensor. However, this can be practically unfeasible, especially in vivo; therefore, it is interesting to investigate the possibility of characterizing the turbulent flows that may occur in the circulatory system with the monodimensional data that a less complete equipment (e.g., a pulsed ultrasound Doppler) can yield. From the general expression of the Reynolds stress tensor, the highest shear stress can be deduced, as well as the Reynolds normal stress in the main flow direction. The relation between these two quantities, which is an issue already addressed in previous works, can thus be rigorously formulated in terms of some characteristic parameters of the Reynolds stress tensor, the principal normal stresses and the angles that the directions that define them form with the main flow direction. An experimental verification of the ratio of the two above-mentioned quantitites for the flow across bileaflet valves, investigated by means of two-dimensional laser Doppler anemometry, will illustrate the limitations of the monodimensional approach estimating the maximum load on blood constituents.     

Mass and Energy Transport Through Slit Pores: Application to Planar Poiseuille Flow

Abstract

We develop a simple, efficient and general statistical mechanical technique for calculating the pressure tensor and the heat flux vector in atomic fluids. The method is applied to the case of planar Poiseuille flow through a narrow slit pore and the results indicate that our technique is accurate and relatively efficient. A second method to calculate shear stress is derived from the momentum continuity equation. This mesoscopic method again is seen to be accurate with good computational efficiency.

We also find that the commonly used approximation to the Irving-Kirkwood expression for the heat flux and the pressure tensor (where the Irving-Kirkwood O_ij operator is set equal to unity-the so-called IK1 approximation), leads to incorrect results for highly inhomogeneous fluids. In such cases the pressure tensor and heat flux vector display spurious oscillations.

We calculate the spatially dependent viscosity across a narrow pore and find that it exhibits real but weak oscillations, a consequence of oscillations in the number density. Finally we point out that if the heat flux vector is coupled to the gradient of the square of the strain rate tensor such an effect will only affect the shape of the temperature profile. For planar Poiseuille flow, the temperature profile should deviate from the classical quartic form and include an additional quadratic component. The actual magnitude and shape of the heat flux vector remain exactly as they would if such a coupling did not exist.     

A Mathematical Model of Spatial Self-Organization in a Mechanically Active Cellular Medium

A general continual model of a medium composed of mechanically active cells is proposed. The medium is considered to be formed by three phases: cells, extracellular fluid, and an additional phase that is responsible for active interaction forces between cells and, for instance, may correspond to a system of protrusions that provide the development of active contractile forces. The deformation of the medium, which is identified with the deformation of the cell phase, consists of two components: elastic deformation of individual cells and cell rearrangements. The elastic deformation is associated with stresses in the cell phase. The spherical component of the stress tensor describes the nonlinear resistance of the cellular medium, which leads to the impossibility of its excessive compression. The constitutive equation for pressure in the cell phase is taken in the form of a nonlinear dependence on the volume cell density. The rearrangement of cells is considered as a flow controlled by stresses in the cell phase, active stresses, and fluid pressure. The tensor of active stresses is assumed to be spherical and nonlocally dependent on the cell density. Assuming that the process of biological tissue deformation is slow, we obtained a reduced model that neglects the elastic deformation of cells, compared to the inelastic deformation. A linear stability analysis of a spatially uniform steady-state solution was performed. The hydrostatic pressure of fluid is present among the parameters that are responsible for the loss of stability of the steady-state solution: an increase in it has a destabilizing effect owing to the action of the component of the interphase interaction force that is determined by the fluid pressure. The model we obtained can be used to describe the process of cavity formation in an initially homogeneous cell spheroid. The role of local and nonlocal mechanisms of active stress generation in the formation of cavity is investigated.     

Geometrical aspects of surface morphogenesis

This paper is concerned with the morphogenesis of structures which form thin deformable sheets. A general formalism is presented for the deformation of a sheet in the presence of an isotropic local body stress. This formalism leads to a set of equations, based on the theory of shells, in which corrections are made in the geometry due to large deformations. Under certain conditions the equations may be solved to give the surface metric tensor as a function of the local tension. A numerical example based on a simple "threshold" model is also presented.     

Crack-Tip Strain Field Mapping and the Toughness of Metallic Glasses

We have used high-energy x-ray scattering to map the strain fields around crack tips in fracture specimens of a bulk metallic glass under load at room temperature and below. From the measured strain fields we can calculate the components of the stress tensor as a function of position and determine the size and shape of the plastic process zone around the crack tip. Specimens tested at room temperature develop substantial plastic zones and achieve high stress intensities () prior to fracture. Specimens tested at cryogenic temperatures fail at reduced but still substantial stress intensities () and show only limited evidence of crack-tip plasticity. We propose that the difference in behavior is associated with changes in the flow stress and elastic constants, which influence the number density of shear bands in the plastic zone and thus the strain required to initiate fracture on an individual band. A secondary effect is a change in the triaxial state of stress around the crack tip due to the temperature dependence of Poisson''s ratio. It is likely that this ability to map elastic strains on the microscale will be useful in other contexts, although interpreting shifts in the position of the scattering peaks in amorphous materials in terms of elastic strains must be done with caution.     

A hybrid approach to theoretical analysis of bovine pancreatic trypsin inhibitor.

A static analysis of bovine pancreatic trypsin inhibitor (BPTI) is presented based on a new discrete/continuum approach to modeling the dynamics of biomolecules. This hybrid method utilizes knowledge of the intramolecular potential and molecular configuration to generate a field of elastic modulus tensors. These tensors, which relate the local stress and strain for each atom in the biomolecule, can be used to judge the local rigidity as well as indicate regions of high stress. Comparing the tensor fields for an unrelaxed and a relaxed configuration, the microscopic structure of BPTI is found to be anisotropic and to have regions of stress even when it is relaxed in the potential field. However, when these fields are averaged over the whole protein or over individual residues the structure becomes more isotropic and the stressed regions vanish. Using these averaged tensors, we calculated bulk properties such as Young's modulus and the Lamé constants and they agreed with previously reported values.     

The role of fabric in the large strain compressive behavior of human trabecular bone

Osteoporosis-related vertebral body fractures involve large compressive strains of trabecular bone. The small strain mechanical properties of the trabecular bone such as the elastic modulus or ultimate strength can be estimated using the volume fraction and a second order fabric tensor, but it remains unclear if similar estimations may be extended to large strain properties. Accordingly, the aim of this work is to identify the role of volume fraction and especially fabric in the large strain compressive behavior of human trabecular bone from various anatomical locations. Trabecular bone biopsies were extracted from human T12 vertebrae (n=31), distal radii (n=43), femoral head (n=44), and calcanei (n=30), scanned using microcomputed tomography to quantify bone volume fraction (BV/TV) and the fabric tensor (M), and tested either in unconfined or confined compression up to very large strains (～70%). The mechanical parameters of the resulting stress-strain curves were analyzed using regression models to examine the respective influence of BV/TV and fabric eigenvalues. The compressive stress-strain curves demonstrated linear elasticity, yielding with hardening up to an ultimate stress, softening toward a minimum stress, and a steady rehardening followed by a rapid densification. For the pooled experiments, the average minimum stress was 1.89 ± 1.77 MPa, while the corresponding mean strain was 7.15 ± 1.84%. The minimum stress showed a weaker dependence with fabric as the elastic modulus or ultimate strength. For the confined experiments, the stress at a logarithmic strain of 1.2 was 8.08 ± 7.91 MPa, and the dissipated energy density was 5.67 ± 4.42 MPa. The latter variable was strongly related to the volume fraction (R(2)=0.83) but the correlation improved only marginally with the inclusion of fabric (R(2)=0.84). The influence of fabric on the mechanical properties of human trabecular bone decreases with increasing strain, while the role of volume fraction remains important. In particular, the ratio of the minimum versus the maximum stress, i.e., the relative amount of softening, decreases strongly with fabric, while the dissipated energy density is dominated by the volume fraction. The collected results will prove to be useful for modeling the softening and densification of the trabecular bone using the finite element method.     

A three-dimensional digital image correlation technique for strain measurements in microstructures

A three-dimensional digital image correlation technique is presented for strain measurements in open-cell structures such as trabecular bone. The technique uses high-resolution computed tomography images for displacement measurements in the solid structure. In order to determine the local strain-state within single trabeculae, a tetrahedronization method is used to fill the solid structure with tetrahedrae. Displacements are calculated at the nodes of the tetrahedrae. The displacement data is subsequently converted to a deformation tensor in each of the tetrahedral element centers with a least-squares estimation method. Because the trabeculae are represented by a mesh, it is possible to deform this mesh according to the deformation tensor and, at the same time, visualize the calculated local strain in the deformed mesh with a finite element post-processing tool. In this way, the deformation of a single trabecula from an aluminum foam sample was determined and validated with rendered images of the three-dimensional sample. A precision analysis showed that a rigid translation or rotation does not affect the accuracy. Typical values for the standard deviation in the displacement and strain components are 2.0 microm and 0.01, respectively. Presently, the precision limits the technique to strain measurements beyond the yield strain.     

Anisotropic hydraulic permeability under finite deformation

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.     

Theory of the Sphering of Red Blood Cells

A rigorous mathematical solution of the sphering of a red blood cell is obtained under the assumptions that the red cells is a fluid-filled shell and that it can swell into a perfect sphere in an appropriate hypotonic medium. The solution is valid for finite strain of the cell membrane provided that the membrane is isotropic, elastic and incompressible. The most general nonlinear elastic stress-strain law for the membrane in a state of generalized plane stress is used. A necessary condition for a red cell to be able to sphere is that its extensional stiffness follow a specific distribution over the membrane. This distribution is strongly influenced by the surface tension in the cell membrane. A unique relation exists between the extensional stiffness, pressure differential, surface tension, and the ratio of the radius of the sphere to that of the undeformed red cell. The functional dependence of this stiffness distribution on various physical parameters is presented. A critique of some current literature on red cell mechanics is presented.     

Structural integration at the shoot apical meristem: models, measurements, and experiments

The shoot apical meristem (SAM) produces stem and initiates leaves. Its structure is maintained despite a continuous flow of cells basipetally from the distal portion of the meristem. The apoplasm and symplasm are the obvious means of cell integration, and their role in chemical cell-to-cell signaling is known. However, the cell wall apoplasm is most likely also involved in a mechanical integration mode, in which mechanical stress and strains (elastic and plastic strain, i.e., growth) are putative signaling factors. Shoot apex cells grow symplastically and their growth is in general anisotropic. Therefore tensor of growth rates that depends on the displacements caused by growth is the most suitable physical entity to describe growth. The tensor approach introduces the concept of principal directions of growth, i.e., the directions in which growth rates attain extremal values. Because of the symplastic mode of growth, the cell wall pattern within the shoot apical meristem informs us about the sequence and planes of cell divisions and about the deformation of existing walls. In consequence, within the meristem, periclines and anticlines can be recognized, both representing the principal directions of growth.     

Solute concentration effect on osmotic reflection coefficient

A theory for the effect of concentration on osmotic reflection coefficient, correct to first order, was developed at the molecular level by considering the effect of solute-solute interactions on solute concentration and the fluid stress tensor within a solvent-filled pore. The solvent was modeled as a continuous fluid and potential energies between solute molecules and the pore wall were assumed to be pairwise additive. Although the theory is more general, calculations are presented only for excluded volume effects (hard-sphere for solute, hard-wall for pore). The relationship between the first-order concentration effect and the infinite dilution value of reflection coefficient appears to be geometry independent. The theory is discussed in light of experimental studies of osmotic flow that have recently appeared in the literature.     

Hydrogen peroxide resistance of Acetobacter pasteurianus NBRC3283 and its relationship to acetic acid fermentation

The bacterium Acetobacter pasteurianus can ferment acetic acid, a process that proceeds at the risk of oxidative stress. To understand the stress response, we investigated catalase and OxyR in A. pasteurianus NBRC3283. This strain expresses only a KatE homolog as catalase, which is monofunctional and growth dependent. Disruption of the oxyR gene increased KatE activity, but both the katE and oxyR mutant strains showed greater sensitivity to hydrogen peroxide as compared to the parental strain. These mutant strains showed growth similar to the parental strain in the ethanol oxidizing phase, but their growth was delayed when cultured in the presence of acetic acid and of glycerol and during the acetic acid peroxidation phase. The results suggest that A. pasteurianus cells show different oxidative stress responses between the metabolism via the membrane oxidizing pathway and that via the general aerobic pathway during acetic acid fermentation.     

Constitutive relation for red cell membrane. Correction.

The intention of this note is to correct a subtle and somewhat esoteric error that the author discovered in his previous publications on membrane elastic behavior. The consitutive relation between membrane force resultants and large, elastic deformations of a membrane surface involves a strain tensor, characterizing the finite deformations. The original strain tensor that appeared in the equations was the Lagrangian strain tensor; however, the proper strain representation (also Lagrangian in nature because it is "measured" relative to the undeformed material state) is transformed by rotations of coordinates in the deformed material state (whereas the Lagrangian strain tensor is transformed by rotations of coordinates in the undeformed state). The principal membrane tensions are unchanged by this correction; the material elastic constants remain the same; and therefore, the material behavior in shear and isotropic tension is the same. However, the tensor, constitutive relation can be properly applied to coordinate systems other than the principal axis system.     

Numerical simulations of plasma equilibrium in a one-dimensional current sheet with a nonzero normal magnetic field component

The force balance in a thin collisionless current sheet in the Earth’s magnetotail with a given constant magnetic field component B _z across the sheet is numerically studied for the first time in a self-consistent formulation of the problem. The current sheet is produced by oppositely directed plasma flows propagating from the periphery of the sheet toward the neutral plane. A substantially improved version of a macroparticle numerical model is used that makes it possible to simulate on the order of 10⁷ macroparticles even with a personal computer and to calculate equilibrium configurations with a sufficiently low discrete noise level in the first-and second-order moments of the distribution function, which determine the stress tensor elements. Quasisteady configurations were calculated numerically for several sets of plasma parameters in some parts of the magnetotail. The force balance in the sheet was checked by calculating the longitudinal and transverse pressures as well as the elements of the full stress tensor. The stress tensor in the current sheet is found to be nondiagonal and to differ appreciably from the gyrotropic stress tensor in the Chew-Goldberger-Low model, although the Chew-Goldberger-Low theory and numerical calculations yield close results for large distances from the region of reversed magnetic field.