Adaptive Finite Element Analysis of the Anisotropic Biphasic Theory of Tissue-Equivalent Mechanics

The nonlinear partial differential equations of the anisotropic biphasic theory of tissue-equivalent mechanics are solved with axial symmetry by an adaptive finite element system. The adaptive procedure operates within a method-of-lines framework using finite elements in space and backward difference software in time. Spatial meshes are automatically refined, coarsened, and relocated in response to error indications and material deformation. Problems with arbitrarily complex two-dimensional regions may be addressed. With meshes graded in high-error regions, the adaptive solutions have fewer degrees of freedom than solutions with comparable accuracy obtained on fixed quasi-uniform meshes. The adaptive software is used to address problems involving an isometric cell traction assay, where a cylindrical tissue equivalent is adhered at its end to fixed circular platens; a prototypical bioartificial artery; and a novel configuration that is intended as an initial step in a study to determine bioartificial arteries having optimal collagen and cell concentrations.     

Adaptive Finite Element Analysis of the Anisotropic Biphasic Theory of Tissue-Equivalent Mechanics

Abstract

The nonlinear partial differential equations of the anisotropic biphasic theory of tissue-equivalent mechanics are solved with axial symmetry by an adaptive finite element system. The adaptive procedure operates within a method-of-lines framework using finite elements in space and backward difference software in time. Spatial meshes are automatically refined, coarsened, and relocated in response to error indications and material deformation. Problems with arbitrarily complex two-dimensional regions may be addressed. With meshes graded in high-error regions, the adaptive solutions have fewer degrees of freedom than solutions with comparable accuracy obtained on fixed quasi-uniform meshes. The adaptive software is used to address problems involving an isometric cell traction assay, where a cylindrical tissue equivalent is adhered at its end to fixed circular platens; a prototypical bioartificial artery; and a novel configuration that is intended as an initial step in a study to determine bioartificial arteries having optimal collagen and cell concentrations.     

Bifurcation analysis of a nonlinear field model of regeneration

The analysis of bifurcating solutions in the Totafurno and Trainor [23] model of supernumerary limb production in salamanders is re-examined using the symmetry analysis developed by Totafurno [22]. In particular, we show analytically that the appearance of field solutions possessing 2 and 4 singularities (the 2- and 4-centered solutions, respectively) also correspond to true bifurcations with reduced symmetries, just as had been previously found for a solution to the field equations not possessing such singularities (the twist solution). While the results have significance primarily for the biological problem, this work serves as an instructive example of the application of symmetry groups to the bifurcation analysis of nonlinear field equations arising from a variational principle. The relationship between the solutions of the nonlinear equations and the corresponding linear equations is discussed.Supported by the National Sciences and Engineering Research Council and the Medical Research Council of CanadaTo whom correspondence should be sent     

Differential polarization imaging. II. Symmetry properties and calculations

Various differential polarization images or Mueller images of model objects are generated using the equations derived in the previous paper (paper I of this series). These calculated images include models of the higher-order organization of metaphase chromosomes, and show the applicability of the differential polarization imaging method to the elucidation of complex molecular organizations. Then, the symmetry behavior of the Mueller matrix elements upon infinitesimal rotations of the optical components about the optical axis of the imaging system is presented. It is shown that the rotational properties of the Mueller images can be used to eliminate the linear polarization contributions to the M14 and M44 images, which appear when these images are generated with imperfect circular polarizations. The relationships between the 16 bright-field Mueller images for four different media, i.e., linearly and circularly isotropic, circularly anisotropic, linearly anisotropic, and linearly and circularly anisotropic, are also derived. For the first three cases simple relationships between the Mueller images are found and phenomenological equations in terms of the optical coefficients are derived. In the last case there are no specific relationships between the Mueller images and instead we briefly present Schellman and Jensen's method for treating this type of medium. The criterion of spatial resolution between adjacent domains of different optical anisotropy is then derived. It is found that in transitions between domains of opposite anisotropy the classical Rayleigh limit must be replaced by a magnitude criterion which depends on the limits of the sensitivity of the detection. Finally, the feasibility of optical sectioning in differential polarization imaging is demonstrated.     

Electromagnetic van Kampen waves

The theory of van Kampen waves in plasma with an arbitrary anisotropic distribution function is developed. The obtained solutions are explicitly expressed in terms of the permittivity tensor. There are three types of perturbations, one of which is characterized by the frequency dependence on the wave vector, while for the other two, the dispersion relation is lacking. Solutions to the conjugate equations allowing one to solve the initial value problem are analyzed.     

Mappings of the plane that simulate prey-predator systems

A prey-predator system that is spatially dependent is built around the familiar logistic equation. The result is a set of coupled integro-differential equations. If at least one of the populations reproduces periodically in a time that is short to other characteristic times, these equations can be transformed to a set of coupled maps, each map representing a spatial point. Various models are developed that include predator ranging as well predator migration via diffusion. In two dimensions the coupling among the spatial points will have the symmetry of the regular polyhedra if the infinite plane is viewed in polar coordinates, or toroidal symmetry if periodic boundary conditions are imposed. Numerical examples are given. Depending on the value of the parameters and on the initial conditions the system can oscillate in a variety of different spatial modes, giving rise to unusual bifurcation portraits, or it can behave chaotically. The solutions show that a patchy distribution of population is more stable against the influences of environmental noise than is a smoothly distributed population. The numerical results in two dimensions are in constrast with the results of a previous study in one dimension.     

Deviatoric and hydrostatic mode interaction in hard and soft tissue

It has been established that many hard and soft tissues have anisotropic material symmetry. It is noted here that the deviatoric and hydrostatic modes interact with each other in a general anisotropic elastic material. In the special case of isotropic, linear elastic, materials these modes are non-interactive. As a consequence of the interaction of these modes encountered in anisotropic materials, the decomposition into hydrostatic and deviatoric modes, and deviatoric mode concepts such as the von Mises effective stress are not appropriate for anisotropic materials in general. The implications of this observation for the presentation of computationally generated stress contours for hard and soft tissues are discussed. It is also pointed out that the mode coupling and mode interaction raise the question of whether anisotropic living tissues respond directly to stress or to some other physical quantity such as strain or strain energy, in view of the recent hypothesis concerning the proliferation and ossification of cartilage.     

On the theory of the nematic phase and its possible relation to the mitotic spindle structure

Onsager's method of studying the nematic phase is developed for general molecular interactions. It is shown that the symmetry of the molecule helps determine the type of transition that occurs in passing from the isotropic phase to the anisotropic phase. The possible relation between the nematic phase and spindle structure is briefly discussed.     

On the theory of Cherenkov emission in a plasma

The Cherenkov emission of transverse-longitudinal waves in an anisotropic plasma is considered by applying a Hamiltonian method and by drawing an analogy between the equations for the Cherenkov emission of purely transverse and purely longitudinal waves in isotropic media and the equations for the emission of transverse-longitudinal electromagnetic waves in a highly anisotropic medium (a magnetized plasma). A formula for the emitted power is derived, as well as an expression for the directional pattern of the emitted waves in an anisotropic plasma.     

Mathematical modeling of the excitation process in myocardial tissue: influence of fiber rotation on wavefront propagation and potential field

In our macroscopic model the heart tissue is represented as a bidomain coupling the intra- and extracellular media. Owing to the fiber structure of the myocardium, these media are anisotropic, and their conductivity tensors have a principal axis parallel to the local fiber direction. A reaction-diffusion system is derived that governs the distribution and evolution of the extracellular and transmembrane potentials during the depolarization phase of the heart beat. To investigate frontlike solutions, the system is rescaled and transformed into a system dependent on a small parameter. Subsequently a perturbation analysis is carried out that yields zero- and first-order approximations called eikonal equations. The effects of the transmural fiber rotation on wavefront propagation and the corresponding potential field, elicited by point stimulations, are investigated by means of numerical simulations.     

Internal structural anisotropy of spherical viruses studied with magnetic birefringence

Six so called spherical viruses (four plant and two animal) are shown to exhibit magnetically induced birefringence in solution. They must therefore be magnetically and optically anisotropic. This is attributed to static structural anisotropy of the interiors as neither natural shape nor field-induced deformations are likely causes. Thus at least part of these virus cores have a symmetry differing from that of their capsids. An estimate of the average orientation of the RNA bases is given for the plant viruses: turnip yellow mosaic, bromegrass mosaic, tomato bushy stunt and turnip crinkle. The packing geometry of the nucleic acid/protein cores of adenovirus and probably influenza virus are anisotropic but to an extent that cannot be quantified.     

Comments on the use of the order parameters obtained from 2H-NMR to describe the anisotropic motions of the methylene groups of the fatty acyl chains in lipid bilayer membranes

The informational content of a deuteron quadrupole splitting obtained from a methylene group undergoing anisotropic motion is inversely proportional to the degree of symmetry underlying this motion. To accurately assess the power and the limitation of ²H-NMR, the motional symmetry of the methylene groups was strictly examined. From the two orthogonal and geometrical planes of symmetry of a methylene group one was found to remain a plane of symmetry for the motion. The other plane was shown, by a quantitative evaluation, to keep its symmetry property only in first approximation. The direction defined by the intersection of the two orthogonal planes was not found to be a motional axis of axial symmetry (even as a poor assumption). However, to a good approximation the motion of this direction with respect to the bilayer normal can be specified by the deuteron quadrupole splitting arising from the particular geometry of a given methylene group.     

A theory of the symmetries of filamentous bacteriophages.

A mathematical model is presented which explains the symmetries observed for the protein coats of filamentous bacterial viruses. Three viruses (Ff, IKe, and If1) all have five-start helices with rotation angles of 36 degrees and axial translations of 16 A (Type I symmetry), and three other viruses (Pf1, Xf, and Pf3) all have one-start helices with rotation angles of approximately equal to 67 degrees and translations of approximately 3 A (Type II symmetry). The coat protein subunits in each group diverge from each other in amino acid sequence, and Type II viruses differ dramatically in DNA structure. Regardless of the differences, both Type I and Type II symmetry can be understood as direct, natural consequences of the close-packing of alpha-helical protein subunits. In our treatment, an alpha-helical subunit is modeled as consisting of two interconnected, flexible tubular segments that follow helical paths around the DNA, one in an inner layer and the other in an outer layer. The mathematical model is a set of algebraic equations describing the disposition of the flexible segments. Solutions are described by newly introduced symmetry indices and other parameters. An exhaustive survey over the range of indices has produced a library of all structures that are geometrically feasible within our modeling scheme. Solutions which correspond in their rotation angles to Type I and Type II viruses occur over large ranges of the parameter space. A few solutions with other symmetries are also allowed, and viruses with these symmetries may exist in nature. One solution to the set of equations, obtained without any recourse to the x-ray data, yields a calculated x-ray diffraction pattern for Pf1 which compares reasonably with experimental patterns. The close-packing geometry we have used helps explain the near constant linear mass density of known filamentous phages. Helicoid, rigid cylinder, and maximum entropy structure models proposed by others for Pf1 are reconciled with the flexible tube models and with one another.     

Single-mode magnetic structures in a plasma with anisotropic pressure

The equations of vortex electron anisotropic hydrodynamics are used to show that, in a plasma with anisotropic pressure, the Weibel instability of short-wavelength perturbations gives rise to a large-amplitude quasi-harmonic magnetic field varying periodically as a function of time. The computed field parameters agree well with the proposed analytic estimates.     

Photoelectric response of polarization sensitive bacteriorhodopsin films

Polarization sensitivity is introduced into oriented bacteriorhodopsin (BR) films through a photochemical bleaching process, which chemically modifies the structure of the purple membrane by breaking the intrinsic symmetry of the membrane-bound BR trimers. The resulting photovoltage generated in an indium-tin oxide (ITO)/BR/ITO detector is found to be anisotropic with respect to cross-polarized probe beams. A model, based on the polarization dependent photoselection of the BR molecules qualitatively explains the photochemical bleaching process and the observed anisotropic response. The effect reported here can be used to construct a polarization sensitive BR-based bio-photoreceiver.     

Symmetry, orientation and rotational mobility in the a3 heme of cytochrome c oxidase in the inner membrane of mitochondria.

The photoinduced linear dichroism of absorption changes resulting from photolysis of the complex between heme a3 of the cytochrome oxidase and CO is studied. The experiments started from isotropic solutions or suspensions of the enzyme both in its isolated form and in mitochondria. The anisotropy responsible for the linear dichroism was induced by excitation with a flash of linearly polarized light. The dichroic ratios observed with various systems; polymerized enzyme in solution, enzyme in mitochondria and in submitochondrial particles (at 20 degrees C as well as at liquid N2-temperature) all approached a value of 4/3 which characterizes a chromophore which is circularly degenerate. Therefrom we conclude that the interaction of heme a3 with its microenvironment within the protein does not break its four-fold symmetry. The experiments with mitochondria and submitochondrial particles suspended in aqueous buffer revealed similarly high dichoric ratios without any dichroic relaxation other than a rather slow one which could be attributed to the rotation of the whole organelle in the suspending medium. Therefrom we conclude that the cytochrome oxidase either is totally immobilized in the membrane, or that it carries out only limited rotational diffusion around a single axis coinciding with the symmetry axis of heme a3. In the light of independent evidence for a transmembrane arrangement of the oxidase and for the general fluidity of the inner mitochondrial membrane we consider anisotropic mobility of the cytochrome oxidase around an axis normal to the plane of the membrane as the most likely interpretation. Then our experimental results imply that the plane of heme a3 is coplanar to the membrane.     

Quantum-mechanical theory of CD for infinite helical polymers

Quantum-mechanical equations are derived that are particularly well suited to actual computations of the CD for helical polymers. They make use of cyclic boundary conditions and helical symmetry, so that only two matrices with a size equal to the number of transitions considered need be diagonalized. The final equations are expressed directly in terms of monomer properties and helical parameters to invite the same input as earlier calculations, and are given as a rotational strength times a shape function for ease of comparison with the earlier work. The shape of the helix term is expressed as a derivative with respect to ω and depends on the distance between monomers along the helix axis. Other terms involving two electric transition dipoles depend on the distance from the helix axis to the transition center. These equations are directly comparable to the classical equations derived for cyclic boundary conditions and helical symmetry. We present an outline of the derivation and enough intermediate steps to clarify how the equations arise.     

Solutions to axon equations

The solutions to a general class of axon partial differential equations proposed by FitzHugh which includes the Hodgkin-Huxley equations are studied. It is shown that solutions to the partial differential equations are exactly the solutions to a related set of integral equations. An iterative procedure for constructing the solutions based on standard methods for ordinary differential equations is given and each set of initial values is shown to lead to a unique solution. Continuous dependence of the solutions on the initial values is established and solutions with initial values in a restricted (physiological) range are shown to remain in that range for all time. The iterative procedure is not suggested as the basis for numerical integration.