The stability of a cylindrical elastic membrane of biological tissue and the effect of internal fluid flow

Summary Some stability problems for finitely deformed elastic membranes are considered. The first part treats the stability of a stretched elastic membrane which is orthotropic in its reference state. The stability criteria for a uniformly deformed orthotropic plane sheet under conditions of dead loading are obtained. The potential energy for a uniformly extended and inflated circular cylindrical membrane (the preferred directions being axial and circumferential) with axially symmetric deformations and under constant pressure is found to have the same form as that for an isotropic material. Stability criteria for a long tube are obtained and some numerical results are given through stability diagrams using strain energy functions appropriate for biological tissues. The effect on stability of the flow of an incompressible fluid through the tube is considered next. To model the internal fluid motion, small perturbations superimposed on a state of steady flow with velocity profiles of a slug flow and a Poiseuille flow are assumed, and a one-dimensional analysis is used as a third approach. Equations for the fundamental frequencies of vibration are derived. The fluid motion is found to have a stabilizing or destabilizing influence on the membrane depending on the approach used. The results are compared to Rayleigh’s results for a circular liquid jet and they are found to agree for the slug flow and the Poiseuille flow models.

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übersicht Es werden einige Stabilit?tsprobleme für Membranen mit endlichen Deformationen betrachtet. Im ersten Teil wird die Stabilit?t einer gedehnten Membran betrachtet, die im Ausgangszustand orthotrop ist. Es werden die Stabilit?tsbedingungen für ein gleichf?rmig deformiertes ebenes Blatt erhalten. Die potentielle Energie für eine gleichf?rmig gedehnte und aufgeblasene kreiszylindrische Membran mit achsialsymmetrischer Deformation unter konstantem Druck hat dann dieselbe Form wie bei isotropem Material. Es werden Stabilit?tskriterien für lange R?hren abgeleitet und einige numerische Ergebnisse für biologische Gewebe mitgeteilt. Als n?chstes wird der Einflu? auf die Stabilit?t der Str?mung eines inkompressiblen Fluids durch die R?hre untersucht. Dabei werden kleine St?rungen der station?ren Str?mung überlagert, wobei als Geschwindigkeitsprofil Schleichstr?mung und Poiseuille-Str?mung angenommen werden. Die eindimensionale Analyse wird bis zur dritten Ordnung durchgeführt. Es werden Gleichungen für die Frequenzen der Grundschwingungen abgeleitet. Dabei zeigt sich, da? die Fluidstr?mung je nach der Art der N?herungen einen stabilisierenden oder auch destabilisierenden Einflu? haben kann. Die Ergebnisse werden mit denen von Rayleigh für einen Fluidstrahl mit kreisf?rmigem Querschnitt verglichen. Dabei wird übereinstimmung sowohl für Schleich- wie auch für Poiseuille-Str?mung festgestellt.

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Dedicated to Hans Ziegler     

Measurement of rheological properties of soft biological tissue with a novel torsional resonator device

A new device for measuring the rheological properties of soft biological tissues is presented. The mechanical response is characterized for harmonic shear deformations at high frequencies (up to 10 kHz) and small strains (up to 0.2%). Experiments are performed using a cylindrical rod excited to torsional resonance. One extremity of the rod is in contact with the soft tissue and adherence is ensured by vacuum clamping. The damping characteristics and the resonance frequency of the vibrating system are inferred from the control variables of a phase stabilization loop. Due to the contact with the soft tissue, and depending on the rheological properties of the tissue, changes occur in the Q-factor and in the resonance frequency of the system. The shear modulus of the soft tissue is determined from the experimental results with an analytical model. The reliability of the proposed technique is evaluated through repeatability tests and comparative measurements with synthetic materials. The results of measurements on bovine organs demonstrate the suitability of the experimental procedure for the characterization of biological tissues and provide some insight in their rheological properties at frequencies in the range 1–10 kHz.     

Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network

A new class of micromechanically motivated chain network models for soft biological tissues is presented. On the microlevel, it is based on the statistics of long chain molecules. A wormlike chain model is applied to capture the behavior of the collagen microfibrils. On the macrolevel, the network of collagen chains is represented by a transversely isotropic eight chain unit cell introducing one characteristic material axis. Biomechanically induced remodeling is captured by allowing for a continuous reorientation of the predominant unit cell axis driven by a biomechanical stimulus. To this end, we adopt the gradual alignment of the unit cell axis with the direction of maximum principal strain. The evolution of the unit cell axis’ orientation is governed by a first-order rate equation. For the temporal discretization of the remodeling rate equation, we suggest an exponential update scheme of Euler-Rodrigues type. For the spatial discretization, a finite element strategy is applied which introduces the current individual cell orientation as an internal variable on the integration point level. Selected model problems are analyzed to illustrate the basic features of the new model. Finally, the presented approach is applied to the biomechanically relevant boundary value problem of an in vitro engineered functional tendon construct.     

Dynamics of biological soft tissue and rubber: internally pressurized spherical membranes surrounded by a fluid

The behavior of a family of dynamical systems representing the elastodynamic response of an internally pressurized, non-linearly elastic spherical membrane lying in an incompressible external fluid is governed primarily by the strain energy function for the membrane, the specific forcing function due to the internal pressure, and the viscosity of the external fluid. It is shown that such systems with an inviscid external fluid and having a constant internal pressure are integrable but not Hamiltonian. Under periodic internal loading, and for a small spherical radius and constitutive relations typical of many biological soft tissues, a periodic orbit in phase space exists near a static equilibrium. A viscous external fluid causes the periodic orbit to be an attractor. The dynamical system is robust under small loading perturbations common in normal biological systems. Rubber models, on the other hand, may admit structural catastrophes. For small initial sphere radii, a jump from one periodic orbit to another is possible for rubber models but not for the classical soft tissue models. It is dangerous, therefore, to model soft biological tissue as a rubber either mathematically or physically in experiments because the predicted instabilities may not exist in tissue.     

Photoviscoelastic stress analysis of a plate with a central hole

An epoxy resin containing excessive plasticizer was developed and characterized. The material, which deforms viscously at room temperature, has optical properties that depend on stress and strain. A tensile specimen was prepared from the epoxy resin so that the mechanical and optical properties of the epoxy resin could be characterized. The elastic and plastic behavior was determined at 37°C using tensile stresses between 4 and 26 MPa. The birefringence was also recorded as a function of time and stress. From these results, a photoviscoelastic constitutive equation was constructed to describe the dependence of the birefringence on stress and strain. The constitutive equation was then applied to study the deformation of a tensile specimen containing a central circular hole. By using the isochromatic fringes in combination with the isoclinic, the time-dependent variation of the stress field in the specimen was solved.     

Feasibility study on strengthening heating effect of high power short pulse laser on biological tissue by micro/nano metal particles

A new method for enhancing the heating effect of high power short pulse laser on biological tissue by micro/nano metal particles was proposed. Theoretical analysis of the influences of the micro/nano particle kind, the concentration and the microcosmic distribution of micro/nano particles on the temperature response was carried out with a multi-layer hyperbolic heat conduction model with volumetric heat generation. The results indicate that embedding micro/nano particles could improve the surface temperature increase of biological tissue with short duration and reduce the deeper material temperature under the same heating condition, which would help strengthen the heating effects of high power short pulse laser on biological tissue. This study may open a new technical approach for improving laser applications.     

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.     

Elastoplastic analysis of a bent circular plate with a central hole

A circular aluminum plate with a small concentric hole (1/10 the plate thickness) and supported on its outer edge by a ring was subjected to a concentrated load at its center, applied through a rigid ball of radius equal to the plate thickness. Strains were determined using grids, moiré, and electrical strain gages on the top and bottom surfaces of the plate for loads up to and including the one associated with the appearance of the first crack in the plate. The investigation is related to the development of specimens to be used to determine fracture characteristics of materials used in lightweight construction.     

Stresses in a deep beam with a central concentrated load

The stress distribution in deep beams has received the attention of mathematicians an research workers since the early days of Wilson, Stokes and Filon.^{1, 2} A general theoretical solution is not possible and particular solutions of the biharmonic equatiion for the Airy stress function have been worked out to satisfy certain loadings. In the case of a deep beam under a symmetrical pressure on the top edge, solutions by (1) Durant and Garwood,³ (2) Chow, Conway and Winter⁴ have been put forward. Systematic photoelastic experimental work appears to be scarce and the object of this article is to present for comparison with existing theories the results of three photoelastic tests on simply supported deep beams carrying a central point load. The experimental stresses and those obtained from the solutions referred to above agreed reasonably well. The normal stresses σ _x calculated from the Wilson-Stokes theory were in error for the case under consideration (the theory is not valid for (H/L)>0.4), and the simple theory of bending is adequate for the maximum tensile stress provided the span exceeds 1.5 times the depth. Results are summarized in figures showing the magnitude and direction of the principal stresses; magnitudes are in a dimensionless from, hence it is possible to adapt them fior any beam of similar depth to span ratio under the same type of loading.     

Experimental and numerical study of a gas cyclone with a central filter

This paper studies a novel gas cyclone with a cylindrical filter face installed in the center from the vortex finder to the bottom hopper. The experimental results show that this composite cyclone has a higher collection efficiency and a lower pressure drop than the original cyclone. The mechanisms for the improvement are analyzed by both physical experiments and numerical simulations. By measuring dust samples collected at different places it is revealed that the center filter can prevent fine particles from entering the inner vortex and escaping, which accounts for the increase of the collection efficiency. In addition, the flow field of the composite cyclone is simulated by computational fluid dynamics and compared with that of the original cyclone. The analysis shows that with the filter layer installed, the swirling flow disappears in the vortex finder, which decreases the kinetic energy dissipation and hence lowers the pressure drop.     

Experimental and numerical study of a gas cyclone with a central filter

This paper studies a novel gas cyclone with a cylindrical filter face installed in the center from the vortex finder to the bottom hopper.The experimental resul...     

Nonlinear effect of biological feedback on brain attentional state

Nonlinear Dynamics - A nonlinear effect of biological feedback on visual perception is studied when a brain–computer interface is applied. The implemented algorithm for estimation of visual...     

Diffusion of magnetic nanoparticles in a multi-site injection process within a biological tissue during magnetic fluid hyperthermia using lattice Boltzmann method

In this study the lattice Boltzmann model (LBM) has been used to simulate diffusion of magnetic nanoparticles (MNPs) injected at multiple sites inside a biological tissue during magnetic fluid hyperthermia (MFH). To validate the numerical results, diffusion in infinite one and two dimensional domains have been compared with the analytical solutions. Agreement were excellent. Also diffusion of a water based nanofluid containing magnetite MNPs (ferrofluid) for mono and multi-site injection in the tissue has been studied. Moreover, the effects of ferrofluid injection volume as well as infusion flow rate of ferrofluid on the distribution of MNPs have been investigated.     

A high precision element with a central circular hole

An eight-node rectangular membrane element with a central circular hole has been developed using complex potentials to define stress and strain functions. This new element type can be combined with the conventional elements, e.g., isoparametric elements, with no modifications required. Two numerical examples were employed to illustrate the superiority and versatility of the proposed element type. The results obtained using the said element type are closer to the theoretical results as compared with those obtained by similar types of elements.     

Bifurcations in a birhythmic biological system with time-delayed noise

We study the effects of recycled noise on the dynamics of a birhythmic biological system. This noise is generated by the superposition of a primary Gaussian white noise source with a second component (its replicas delayed of time τ). We find that under the influence of this kind of noise, the dynamics of the birhythmic biological system can be well characterized through the concept of stochastic bifurcation, consisting in a qualitative change of the stationary probability distribution. Analytical results are obtained following the quasiharmonic assumption through the Langevin and Fokker–Planck equations. Comparing the analytical and numerical results, we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. We also find that the increase of noise intensity leads to coherence resonance.     

Generalisations of the strain-energy function of linear elasticity to model biological soft tissue

Strain measures consistent with the linear, infinitesimal form of the strain-energy function are obtained within the context of isotropic, homogeneous, compressible, and non-linear elasticity. It will be shown that there are two distinct families of such measures. One family has already been much studied in the literature, the most important member being the strains whose principal values are a function only of the corresponding principal stretches. The second family of strains appears new. The motivation for studying such strains is the intuitive expectation that, for at least moderate deformations, a good fit with experimental data from material characterisation tests will be obtained with the corresponding strain-energy functions. In particular, there is the expectation that such models could prove useful for the modelling of biological soft tissue, whose physiological response is characterised by moderate strains. It will be shown that this is indeed the case for simple tension tests on porcine brain tissue.     

Detonation combustion of hydrogen in a convergent-divergent nozzle with a central coaxial cylinder

The feasibility of steady detonation combustion of a hydrogen-air mixture entering at a supersonic velocity in an axisymmetric convergent-divergent nozzle with a central coaxial cylinder is considered. The problem of the nozzle starting and the initiation of detonation combustion is numerically solved with account for the interaction of the outflowing gas with the external supersonic flow. The modeling is based on the gasdynamic Euler equations for an axisymmetric flow. The calculations are carried out using the Godunov scheme on a fine fixed grid which allows one to study in detail the interaction of an oblique shock wave formed in the diffuser with the nozzle axis. It is shown that a central coaxial cylinder ensures the starting with the formation of supersonic flow throughout the entire nozzle and stable detonation combustion of a stoichiometric hydrogen-air mixture in the divergent section of the nozzle.