The effect of stress concentration on bone remodeling: theoretical predictions

Theoretical predictions of internal bone remodeling around an elliptical hole are studied. The internal remodeling theory due to Cowin and Hegedus is employed. The bone is modeled as an initially homogeneous adaptive elastic plate with an elliptical hole under a superposed steady compressive load. It is shown that there will exist a final inhomogeneous remodeling distribution around the hole that will disappear away from the hole. The remodeling is such that the compressive stress concentration causes the bone structure to evolve to one of greater density and stiffer elastic coefficients. The speed of remodeling around the hole and its variation with respect to distance is investigated and discussed. It is shown that the rate of bone reinforcement in the area of compressive stress concentration is much higher than the rate of bone resorption in the area of existing tensile stress. Special cases of a circular hole and vertical and horizontal slots are studied and discussed.     

A model for resolving the plankton paradox: coexistence in open flows

1. Recent developments in the field of chaotic advection in hydrodynamical/environmental flows encourage us to revisit the population dynamics of competing species in open aquatic systems.
2. We assume that these species are in competition for a common limiting resource in open flows with chaotic advection dynamics. As an illustrative example, we consider a time periodic two-dimensional flow of viscous fluid (water) around a cylindrical obstacle.
3. Individuals accumulate along a fractal set in the wake of the cylinder, which acts as a catalyst for the biological reproduction process. While in homogeneous, well mixed environments only one species could survive this competition, coexistence of competitors is typical in our hydrodynamical system.
4. It is shown that a steady state sets in after sufficiently long times. In this state, the relative density of competitors is determined rather by the fractal nature of the spatial distribution of the advected species, and by their initial conditions, than by their competitive abilities. We argue that two factors, the strong chaotic mixing along a fractal set and the boundary layer around the obstacle, are responsible for the coexistence.     

Inhomogeneous cartilage properties enhance superficial interstitial fluid support and frictional properties, but do not provide a homogeneous state of stress

It has been well established that articular cartilage is compositionally and mechanically inhomogenous through its depth. To what extent this structural inhomogeneity is a prerequisite for appropriate cartilage function and integrity is not well understood. The first hypothesis to be tested in this study was that the depth-dependent inhomogeneity of the cartilage acts to maximize the interstitial fluid load support at the articular surface, to provide efficient frictional and wear properties. The second hypothesis was that the inhomogeneity produces a more homogeneous state of elastic stress in the matrix than would be achieved with uniform properties. We have, for the first time, simultaneously determined depth-dependent tensile and compressive properties of human patellofemoral cartilage from unconfined compression stress relaxation tests. The results show that the tensile modulus increases significantly from 4.1 +/- 1.9 MPa in the deep zone to 8.3 +/- 3.7 MPa at the superficial zone, while the compressive modulus decreases from 0.73 +/- 0.26 MPa to 0.28 +/- 0.16 MPa. The experimental measurements were then implemented with the finite-element method to compute the response of an inhomogeneous and homogeneous cartilage layer to loading. The finite-element models demonstrate that structural inhomogeneity acts to increase the interstitial fluid load support at the articular surface. However, the state of stress, strain, or strain energy density in the solid matrix remained inhomogeneous through the depth of the articular layer, whether or not inhomogeneous material properties were employed. We suggest that increased fluid load support at the articular surface enhances the frictional and wear properties of articular cartilage, but that the tissue is not functionally adapted to produce homogeneous stress, strain, or strain energy density distributions. Interstitial fluid pressurization, but not a homogeneous elastic stress distribution, appears thus to be a prerequisite for the functional and morphological integrity of the cartilage.     

Pulse waves in prestressed arteries

In order to better understand the effect of initial stress in blood flow in arteries, a theoretical analysis of wave propagation in an initially inflated and axially stretched cylindrical thick shell is investigated. For simplicity in the mathematical analysis, the blood is assumed to be an incompressible inviscid fluid while the arterial wall is taken to be an isotropic, homogeneous and incompressible elastic material. Employing the theory of small deformations superimposed on a large initial field the governing differential equations of perturbed solid motions are obtained in cylindrical polar coordinates. Considering the difficulty in obtaining a closed form solution for the field equations, an approximate power series method is utilized. The dispersion relations for the most general case of this approximation and for the thin tube case are thoroughly discussed. The speeds of waves propagating in an unstressed tube are obtained as a special case of our general treatment. It is observed that the speeds of both waves increase with increasing inner pressure and axial stretch.     

On the Application of Widom's Test Particle Method to Homogeneous and Inhomogeneous Fluids

Abstract

Chemical potentials of a homogeneous and an inhomogeneous Lennard-Jones fluid have been determined by molecular dynamics simulations on the vector computer CYBER 205 by applying essentially the fictitious test particle method of Widom. For the homogeneous fluid we find, contrary to the previous result of Guillot and Guissani, that the simulated chemical potential is independent of the particle number. The crucial point, however, is a sufficiently large cut-off radius in the evaluation of the Boltzmann factor. Comparing with our WCA-type perturbation theory, we get agreement in the chemical potentials within 0.1 kT up to the density n[sgrave]³ = 0.80 and a difference of 0.2 kT at n[sgrave]³ = 0.85. For the inhomogeneous case we consider a fluid in a cylindrical pore and integrate Widom's equation over a certain probe volume as suggested earlier by us. Chemical potentials are then calculated independently in five different probe volumes, which are cylindrical shells. The results agree well from the second to the fourth shell. Inaccuracies in the innermost cylinder can be easily explained by bad statistics. In the shell close to the wall the extremely high local density is responsible for the inaccuracies. Extending the probe volume over all cylindrical shells besides the one closest to the wall is thought to yield rather reliable results for the chemical potential. As a by-product of the simulations we also obtained diffusion coefficients, which are given in an appendix.     

Ladder network prediction of transients in linear spatially inhomogeneous cables

A numerical method is described for finding steady state and transient responses in electrically linear, spatially inhomogeneous cables. Spatial inhomogeneities are incorporated by representing the cable by a number of finite length uniform cylindrical segments, each having the radius and electrical characteristics of a small region along the cable. Input waveforms are approximated by truncated Fourier series of sinusoidal components. Output waveforms are produced by multiplying the input Fourier series sinusoids by their respective transfer functions between input and output points on the cable and summing the resultant output point sinusoids. The transfer functions, representing attenuation and phase shift for each input sinusoid, are obtained by numerical analysis of an electrical ladder network derived from the cylindrical segment model of the cable. Results are shown for application of this method to both cylindrical and expanding radius cable geometries.     

Impedance imaging in upper arm fractures

We used the Aberdeen impedance imaging system to drive a constant current of 1 mA on a 10 kHz sine wave into the upper arm encircled by an elastic belt of 16 equi-spaced strip electrodes. The system was used to examine a normal upper arm, an upper arm with a recent humeral fracture and an upper arm with a clinically united fracture. We approximated the human upper arm to a circular cylinder and assumed bilateral symmetry of normal human limbs. We measured transverse limb resistivity ratios and reconstructed static two-dimensional images of the spatial distribution of log(resistivity) by the equipotential back projection technique using a homogeneous muscle equivalent saline reference. Our results indicate that impedance osteography provides unique information about the changing electrical characteristics at the fracture site. This information could prove a useful adjunct to clinical and radiological tests for fracture union.     

The flow of a viscous liquid in a converging tube

The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube. Supported in part by the American Heart Association (No. 62F4EG). This work was done during the tenure of an Established Investigatorship of the American Heart Association.     

Zero-stress states of arteries

The no-load configuration of a living organ is, in general, not the zero-stress state. The difference can be revealed by cutting up an unloaded organ to such an extent that the stress becomes zero in the tissue everywhere. For the aorta, it is shown that the configuration of the zero-stress state differs considerably from being a cylindrical tube. It is, in fact, an open sector with opening angles varying along the arterial tree. This article presents data on the zero-stress state in the arteries of the rat in normal condition.     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Surface waves in plasma waveguides with a smooth transverse inhomogeneity

A theory of cylindrical surface waves in a circular waveguide filled with a smoothly inhomogeneous plasma is presented. For a special radial profile of the plasma density, dispersion relations for the complex frequencies of surface waves are derived analytically. The dispersion relations are solved numerically (in the long-wavelength limit) and numerically. It is shown that there are two types of surface waves. When passing to the case of a sharply bounded plasma, one of the waves becomes an ordinary surface wave, while the other becomes strongly damped.     

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

Blood vessel buckling within soft surrounding tissue generates tortuosity

The stability of blood vessel under lumen pressure load is essential to the maintenance of normal arterial function. Previous mechanical models showed that blood vessels may buckle into a half sine wave but arteries and veins in vivo often demonstrate tortuous paths with multiple waves. The objective of this study was to analyze the buckling of blood vessels under lumen pressure with surrounding tissue support. Blood vessels were modeled as elastic cylindrical vessels within an elastic substrate. Buckling equations were established to determine the critical pressure and the wavelength. These equations and simulation results demonstrated that blood vessels do take higher order mode shapes when buckling inside an elastic substrate while they take the basal mode shape without the substrate. The wave number increases i.e. blood vessels take a higher mode shape, as the stiffness of the substrate increases. These results suggest that mechanical buckling is a possible mechanism for the development of tortuous blood vessels. The current model provides a powerful tool for further studying the tortuosity of arteries and veins.     

A model of stress-induced geometrical remodeling of vessel segments adjacent to stents and artery/graft anastomoses

The mismatch between the elastic properties and initial geometry of a host artery and an implanted stent or graft cause significant stress concentration at the zones close to junctions. This may contribute to the often observed intimal hyperplasia, resulting in late lumen loss and eventual restenosis. This study proposes a mathematical model for stress-induced thickening of the arterial wall at the zones close to an implanted stent or graft. The host artery was considered initially as a cylindrical shell with constant thickness that was clamped to the stent or graft, which was assumed to be non-deformable in the circumferential direction. It was assumed that the abnormal circumferential and axial stresses due to the bending of the arterial wall cause wall thickening that tends to restore the stress state close to that existing far from the junction. The linear equations of a cylindrical shell with variable thickness were coupled to an evolution equation for the wall thickness. These equations were solved numerically and a parametric study was performed using finite difference method and explicit time step. The results show that the remodeling process is self-limiting and leads to local thickening that gradually decreases with distance from the edge of the stent/graft. Model predictions were tested against morphological findings existing in the literature. Recommendations on stent designs that reduce stress concentrations are discussed.     

On the effects of residual stress in microindentation tests of soft tissue structures

Microindentation methods are commonly used to determine material properties of soft tissues at the cell or even sub-cellular level. In determining properties from force-displacement (FD) data, it is often assumed that the tissue is initially a stress-free, homogeneous, linear elastic half-space. Residual stress, however, can strongly influence such results. In this paper, we present a new microindentation method for determining both elastic properties and residual stress in soft tissues that, to a first approximation, can be regarded as a pre-stressed layer embedded in or adhered to an underlying relatively soft, elastic foundation. The effects of residual stress are shown using two linear elastic models that approximate specific biological structures. The first model is an axially loaded beam on a relatively soft, elastic foundation (i.e., stress-fiber embedded in cytoplasm), while the second is a radially loaded plate on a foundation (e.g., cell membrane or epithelium). To illustrate our method, we use a nonlinear finite element (FE) model and experimental FD and surface contour data to find elastic properties and residual stress in the early embryonic chick heart, which, in the region near the indenter tip, is approximated as an isotropic circular plate under tension on a foundation. It is shown that the deformation of the surface in a microindentation test can be used along with FD data to estimate material properties, as well as residual stress, in soft tissue structures that can be regarded as a plate under tension on an elastic foundation. This method may not be as useful, however, for structures that behave as a beam on a foundation.     

Influence of fibril taper on the function of collagen to reinforce extracellular matrix

Collagen fibrils provide tensile reinforcement for extracellular matrix. In at least some tissues, the fibrils have a paraboloidal taper at their ends. The purpose of this paper is to determine the implications of this taper for the function of collagen fibrils. When a tissue is subjected to low mechanical forces, stress will be transferred to the fibrils elastically. This process was modelled using finite element analysis because there is no analytical theory for elastic stress transfer to a non-cylindrical fibril. When the tissue is subjected to higher mechanical forces, stress will be transferred plastically. This process was modelled analytically. For both elastic and plastic stress transfer, a paraboloidal taper leads to a more uniform distribution of axial tensile stress along the fibril than would be generated if it were cylindrical. The tapered fibril requires half the volume of collagen than a cylindrical fibril of the same length and the stress is shared more evenly along its length. It is also less likely to fracture than a cylindrical fibril of the same length in a tissue subjected to the same mechanical force.     

A stress-strain relation for a rat abdominal aorta

Assuming the arterial wall is homogeneous, incompressible, isotropic and elastic, a stress-strain relation has been presented for a rat's abdominal aorta. As an illustrating example, the problem of simultaneous inflation and the axial stretch of a cylindrical artery under physiological loading has been solved and then the material coefficients are determined by comparing theoretical results with the existing experiments. The result indicates that the maximum deviation between the theory and experiment for various pressure levels is 3.7% which seems to be a good approximation of theory to the experiments. The variation of circumferential stress and the incremental pressure modulus with inner pressure are also depicted in the work.     

Tissue dynamics of steady state growth in Hydra littoralis. II. Patterns of tissue movement

In the column of hydra, tissues continually move away from a region located just underthe whorl of tentacles. Above this subtentacular region, tissues proceed into the hypostome and tentacles; below it tissues pass into the buds or continue down the stalk. These movements represent a steady state pattern of tissue renewal in which column growth is balanced by tissue loss at the body extremities. But the existence of a subtentacular zone in which tissue appears stationary does not necessarily indicate that growth is restrictedto this region, as is commonly stated. The body column of hydra can be viewed as an expanding cylinder whose elongation is balanced by tissue loss at the two ends. In such a body there must be one region from whichtissue appears to emanate, regardless of how growth is distributed along the cylinder. Only the rates at which tissues move will be characteristic of the underlying growthpattern. In Hydra littoralis, the measured rates of tissue movement down the gastric column are consistent with the distributions of mitotic figures, which indicate that growth is spread out along the column.     

Analysis of the Scattering of Surface Plasmon Polariton Waves from a Perfectly Conducting Circular Cylinder

Surface plasmon polariton (SPP) waves are the most extensively studied waves among various types of surface waves because they are easy to excite and have been used in many optical applications particularly for plasmonic communication, sensing, and harvesting solar energy. In all these applications, especially on-chip plasmonic communication, scattering can be an important issue to deal with. Therefore, this paper aimed to theoretically inspect the scattering pattern of SPP waves from a perfect electric conductor (PEC) cylindrical scatterer. The cylindrical wave approach is used to solve the scattering problem by a cylindrical object placed below a metallic layer. The scattering is investigated thoroughly by changing the size of the scatterer and its distance from the interface along which the SPP wave is excited. As the size of the scatterer increases, the scattering increases significantly. On the other hand, when the distance of the scatterer from the interface is increased, the scattered field becomes small. Two-dimensional field maps are produced for the incident angle at which SPP is excited. Numerical results are also presented for other incident angles to make a comparison. Furthermore, the forward and backward far-fields are significantly enhanced if the SPP wave is scattered in comparison with when the SPP wave is not present.

     

Analysis of deformation and contraction in cylindrical crystals due to dispirations or dislocations.

Cylindrical crystal structures are common in biology. The shape changes and movements of cylindrical crystals are basic to the understanding of the contractile mechanisms in biological systems such as tobacco mosaic viruses and the tail sheath of T-even bacteriophages. It has been suggested that the concept of defects in crystal physics can be applied to study these contractile mechanisms. The defect believed to be responsible for the shape changes of cylindrical crystals is known as a dispiration. Dispirations are characterized by the shear displacement on the slip plane through a screw symmetry operation. The elastic field of a dispiration can be decomposed into its translational (dislocation) and rotational (disclination) components. The magnitude of the translational and rotational displacements in a cylindrical crystal has been related to the crystal structural parameters. The passage of a dispiration along a helical plane in a cylindrical crystal can induce one of two types of shape changes. In one type, only the disclination component of the dispiration contributes to contraction, whereas in the other type, both the disclination and dislocation components are responsible for the shape change. Estimates of the magnitude of contraction are made in terms of the dimensional and structural parameters of the cylindrical crystal. The reversal of the direction of helical slip results in extension instead of contraction of the cylindrical crystal. The local elastic deformation of a dispiration dipole situated on the helical plane of a cylindrical crystal is examined. It has been shown that, for the first type of deformation mentioned above, closed form solutions of the stress field can be obtained by superposing the stress fields of two dispiration dipoles with slip planes parallel and normal to the cylinder axis, respectively. The approximations of shallow shell theory are adopted in the analysis. Future problems of biological interests are identified.