Peculiarities of the Structure and Mechanical Properties of Biological Tissues*

Five basic principles that determine the structure and mechanical behavior of almost any biological tissue have been formulated. They are the principle of hierarchy, principle of helicality, principle of feedback, principle of universality, and principle of an optimum. It is shown that there are also two types of factors – biological and mechanical – that must be taken into account in determining the mechanical properties of biological tissue. As an example of complexity of the mechanical behavior of a biological tissue the data on compact bone tissue are presented.     

A mathematical model of cortical bone remodeling at cellular level under mechanical stimulus

A bone cell population dynamics model for cortical bone remodeling under mechanical stimulus is developed in this paper. The external experiments extracted from the literature which have not been used in the creation of the model are used to test the validity of the model. Not only can the model compare reasonably well with these experimental results such as the increase percentage of final values of bone mineral content (BMC) and bone fracture energy (BFE) among different loading schemes (which proves the validity of the model), but also predict the realtime development pattern of BMC and BFE, as well as the dynamics of osteoblasts (OBA), osteoclasts (OCA), nitric oxide (NO) and prostaglandin E₂ (PGE₂) for each loading scheme, which can hardly be monitored through experiment. In conclusion, the model is the first of its kind that is able to provide an insight into the quantitative mechanism of bone remodeling at cellular level by which bone cells are activated by mechanical stimulus in order to start resorption/formation of bone mass. More importantly, this model has laid a solid foundation based on which future work such as systemic control theory analysis of bone remodeling under mechanical stimulus can be investigated. The to-be identified control mechanism will help to develop effective drugs and combined nonpharmacological therapies to combat bone loss pathologies. Also this deeper understanding of how mechanical forces quantitatively interact with skeletal tissue is essential for the generation of bone tissue for tissue replacement purposes in tissue engineering.     

On stability of stationary motion of a nonconservative nonholonomic rheonomic system

One considers, in this paper, the motion of a mechanical system in a nonstationary field of potential and positional forces, subject to the action of rheonomic holonomic and nonholonomic linear homogeneous constraints. Assuming that differential equations of motion of the system considered satisfy the conditions for the existence of Painlevé's integral of energy, formulated in [Painlevé, P., 1897. Leçons sur l'intégration des équations de la Mécanique, Paris] and [Appell, P., 1911. Traité de mécanique rationnelle, T. II, Dynamique des systémes – Mécanique analitique, Gauthier-Villars, Paris] and generalized in [Čović, V., Vesković, M., 2004. On stability of motion of a rheonomic system in the field of potential and positional forces, BAMM-1720/2004, No-2233, 93–100] and [Čović, V., Vesković, M., 2005. Hagedorn's theorem in some special cases of rheonomic systems. Mechanics Research Communications 32 (3), 265–280], the original mechanical system is substituted by an equivalent one whose Lagrangian function, nontransformed with respect to nonholonomic constraints, does not depend on time explicitly. Using the properties of the equivalent system, which, in contrast to the original one, moves in a stationary field of potential forces and in a nonstationary field of gyroscopic forces, the definition of cyclic coordinates is generalized, as well as sufficient conditions for the existence of (cyclic) first integrals, corresponding to coordinates mentioned and linear in velocities are established. Further, the conditions for the existence of steady motion of the system considered are found. In the case of existence of such a motion of the system, the Theorem of Routh's type on stability of that motion, based on the minimum of reduced potential for which it is shown that, in contrast to known cases (see, for example, [Gantmacher, F., 1975. Lectures in Analytical Mechanics. Mir Publisher, Moscow; Neimark, J., Fufaev, N., 1972. Dynamics of Nonholonomic Systems. Amer. Math. Soc., Providence, RI; Pars, L., 1962. An Introduction to Calculus of Variations. Heinemann, London; Karapetyan, A., Rumyantsev, V., 1983. Stability of conservative and dissipative systems. In: Itogi Nauki I Tekhniki: Obschaya Mekh., vol. 6, VINITI, Moscow, pp. 3–128 (in Russian)]), it includes the influence of the positional forces field, is formulated. Thus, the Routh's Theorem on stability of steady motion of a conservative mechanical system is extended to the case of a nonconservative system.     

Development and instability of steady convective flows in a square cavity heated from below and a field of vertically directed vibrational forces

The present paper is devoted to numerical investigation of the spatial structure and stability of secondary vibrational convective flows resulting from instability of the equilibrium of a fluid heated from below. Vibrations parallel to the vector of the gravitational force (vertical vibrations) are considered. As in earlier work [7–9], a region of finite size is used — a square cavity heated from below. It is shown that enhancement of the vibrational disturbance of the natural convective flow may either stabilize or destabilize flows with different spatial structures; it may also stabilize certain solutions of the system of convection equations that are unstable in the absence of vibrational forces. In addition, increase of the vibrational Rayleigh number can lead to a change of the mechanisms responsible for equilibrium instability and oscillatory instability of the secondary steady flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–18, March–April, 1991.I thank G. Z. Gershuni for assistance and extremely fruitful discussions of the results of the paper.     

Mechanical model of lung parenchyma as a two-phase porous medium

The anatomy and geometry of the lung at the micro- and macroscopic level have been described briefly. A notion of lung parenchyma — a macroscopically continuous medium whose mechanical properties result from those of microstructural components — has been adapted. Simplifying assumptions propounded in the constitutive model have been discussed. Two phases have been distinguished in the medium: the solid phase — a highly deformable, nonlinearly elastic skeleton in the form of a thin-walled tissue structure on the micro-scale — and the fluid phase — perfect gas (air) filterating through the structure. General constitutive relations for both phases and their mechanical interactions have ben formulated. Further, the fundamental set of differential equations of the quasi-static coupled problem has been developed. Large deformations, material nonlinearities, and dependence of permeability on skeleton deformation have been included. Matrix formulation of the problem has been presented from the point of view of the finite element method. An implicit iterative time integration scheme has been proposed. The algorithm has been illustrated with results of simple numerical tests.     

Thermocapillary instability of the equilibrium of a flat layer containing internal heat sources

In the absence of body forces, a factor which has a strong influence on the equilibrium stability of a nonuniformly heated liquid is the dependence of the coefficient of surface tension on the temperature and the thermocapillary effect generated by it. If the equilibrium temperature gradient is sufficiently great, then the presence of the thermocapillary forces on the free surface can lead to the occurrence of convective motion. The monotonie instability of the equilibrium of a flat layer was investigated in [1–3]. Analysis of nonmonotonic disturbances [4] showed that in the case of an undeformable free surface there is no oscillatory instability. In [5] it was found that oscillatory instability is possible if there is a nonlinear dependence of the coefficient of surface tension on the temperature. The present paper is devoted to numerical investigation of the equilibrium stability of a flat layer with respect to arbitrary disturbances. It is shown that for a deformable free boundary there appears an additional neutral curve, which corresponds to monotonie capillary instability. In addition, when the capillary convection mechanism is taken into account, there appears an oscillatory instability, which becomes the most dangerous in the region of small Prandtl and wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–31, March–April, 1991.I thank V. K. Andreev for a helpful discussion of the work.     

Human leg impact: energy dissipation of wobbling masses

In terrestrial locomotion, the soft-tissue masses of the body undergo damped oscillations following leg impacts with the ground. Appropriate biomechanical models, therefore, describe gross soft-tissue dynamics by “wobbling masses”. We calculated mechanical energy balances of shank and thigh wobbling masses of the stance leg for the first 90 ms after touch-down in human heel-toe running. Thereto, we re-visited a data set on wobbling mass kinematics which had formerly been gained non-invasively by acquiring the motion of grids of lines painted on the skin of the corresponding muscle masses with high-speed cameras. We found frequencies ranging from 3 Hz to 55 Hz and maximum wobbling mass excursions relative to the bone ranging from 3 mm to 4 cm for the centres of mass and from 2.2° to 11.4° for the rotations. The rotational energy balance is practically neutral (±1 J). Usually, there is clearly more energy that is dissipated by wobbling mass movement in horizontal (thigh: <50 J) than in vertical direction (thigh: <15 J). There is less energy dissipated in the shank (horizontal: <10 J, vertical: <5 J). We argue that the energetic costs of separating significant wobbling masses from the skeleton may be over-compensated by avoiding metabolic costs of active impact reduction and by decreasing loads on passive skeletal structures, in particular when distal leg masses are functional, as in humans. Within reasonable biological limits, impacts are known to be even necessary for structural strengthening of bones. Beyond that, impacts might also be useful for stabilising locomotion, both by increasing basins of attraction and by providing simple mechanical signals for control.     

An Experimental–Numerical Methodology for a Rapid Prototyped Application Combined with Finite Element Models in Vertebral Trabecular Bone

In the present study, a novel evaluation method involving rapid prototyped (RP) technology and finite element (FE) analysis was used to study the elastic mechanical characteristics of human vertebral trabecular bone. Three-dimensional (3D) geometries of the RP and FE models were obtained from the central area of vertebral bones of female cadavers, age 70 and 85. RP and FE models were generated from the same high-resolution micro-computed tomography (μCT) scan data. We utilized RP technology along with FE analysis based on μCT for high-resolution vertebral trabecular bone specimens. RP models were used to fabricate complex 3D objects of vertebral trabecular bone that were created in a fused deposition modeling machine. RP models of vertebral trabecular bone are advantageous, particularly considering the repetition, risks, and ethical issues involved in using real bone from cadaveric specimens. A cubic specimen with a side length of 6.5 mm or a cylindrical specimen with a 7 mm diameter and 5 mm length proved better than a universal cubic specimen with a side length of 4 mm for the evaluation of elastic mechanical characteristics of vertebral trabecular bones through experimental and simulated compression tests. The results from the experimental compression tests of RP models closely matched those predicted by the FE models, and thus provided substantive corroboration of all three approaches (experimental tests using RP models and simulated tests using FE models with ABS and trabecular bone material properties). The RP technique combined with FE analysis has potential for widespread biomechanical use, such as the fabrication of dummy human skeleton systems for the investigation of elastic mechanical characteristics of various bones.     

Analytic investigation of a nonlinear problem of the effect of buoyancy on the evolution of an annular vortex

The evolution of a heavy axisymmetric vortex whose density differs from that of the surrounding irrotational ideal fluid is investigated analytically. If the vortex had no buoyancy (i.e., if the densities were identical), it would preserve its shape and velocity. An approximate analytic solution of the problem is obtained. This solution describes two types of evolution of the vortex shape corresponding to different values of the initial velocity and small buoyancy. The spectrum of a nonlinear wave developing on the vortex boundary is estimated.Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 56–66, January–February, 1996.     

人的密质骨的力学性能

本文把骨组织看作是一种材料和骨骼系统的粘弹性结构元件,评述了它们的力学性能.首先描述了骨的结构,包括宏观结构、微结构和超微结构.随后是骨的力学性能.第三部分介绍了哈弗氏骨的力学模型.笔者指出,单个哈弗氏骨可考虑为正交铺设的胶原纤维增强的粘弹性厚壁筒.最后讨论了哈弗氏骨的应力-应变关系.显然,简单的弹性关系是不可用于实际的.      

Nonlinear Electrohydrodynamic Stability of a Poiseuille Two–Layer Flow

The long–wave stability of the Poiseuille two–layer flow of homogeneous viscous dielectrics between plate electrodes under a constant potential difference is studied in an electrohydrodynamic approximation. A linear asymptotic stability analysis shows that surface polarization forces are a destabilizing factor, in addition to viscous stratification. The method of many scales is used to obtain the Kuramoto—Sivashinsky equation governing the weakly nonlinear evolution of the interface between the dielectrics. Within the framework of the approaches used, it is shown that nonlinear interactions limit perturbation growth and the interface does not fail even for a rather large potential difference.     

The flow of a dielectric liquid

The nature of the forces acting on a dielectric liquid in an electric field in the presence of a temperature gradient is investigated. It is shown that the electrical forces produced by nonuniformity in the electrical conductivity of the medium can give rise to a hydrodynamic flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 136–137, November–December, 1973.     

Development of hydrodynamic instability in film condensation on a cylindrical tube in weightlessness

In the process of film condensation on an heat exchanger tube in weightlessness the evolution of small perturbations of the free surface is determined not only by capillary forces but also by the nature of the heat and mass exchange in the liquid-vapor system. In the present paper it is shown that simultaneously taking all these factors into account, even within the framework of a very simple model, leads to results which differ qualitatively from the known formulas for the case of thermal equilibrium. Under the assumption that the velocity profile is quasi-steady, both analytic formulas and corrections to these formulas associated with the unsteady term in the equation of motion are obtained.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–110, November–December, 1995.     

Reptation Motion of Animals in a Fluid

The plane problem of reptation motion of a biological object in a viscous fluid is solved analytically in a long-wave approximation. The motion if laminar. Computational expressions and asymptotic estimates are obtained for the axial and shear forces, expended energy, and motion trajectory. Results of a numerical analysis of the solution are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 106–115, September–October, 2005.     

The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.     

The Newton drag law for bodies formed by intersecting surfaces

The Newtonian shock layer theory for smooth surfaces has been developed in [1–7]. However, if a body consists of two intersecting surfaces, on each of which there is a regular shock layer, these two layers will interact at the line of intersection. Considering that the shock layer is hypersonic, we can again apply the Newton scheme to the impact of the particles with the second surface. As a result, concentrated forces arise at the line of intersection. An indication of the possibility of the appearance of such forces is given in [2]. In the following we present a method of calculating the concentrated forces and indicate the real flow of which they are the analogy.     

Acceleration of a gas compressed under conditions of acuteangled geometry

In obtaining high velocities (10–100 km/sec) by methods employing the mechanical motion of compressible media it is customary to use the flow scheme obtained with shaped charges [1–4]. This paper presents the results of a study of an apparatus that can be used to obtain gas jets in the same velocity range but at higher densities by means of a different type of flow.     

Instability and flow of a weakly conducting fluid in the presence of oxidation-reduction reactions at electrodes and recombination

Experiments show that a weakly conducting fluid in a plane-parallel system of electrodes is set into motion if the field intensity is sufficiently great [1–5]. The loss of stability is due to the formation of charges near the electrodes and the influence of the Coulomb forces on these charges. The formation of the space charges is usually attributed to oxidation-reduction electrode reactions and bulk recombination of the ions formed at the electrodes [1–4]. In the present paper, the stability of a weakly conducting fluid in a plane-parallel system of electrodes with symmetric distribution of the space charge is studied. The methods of the theory of solution bifurcation are used to construct the stationary flow which arises after the loss of stability and to investigate the stability of this flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 20–26, July–August, 1981.     

A non-equilibrium multiscale simulation of shock wave propagation

A novel non-equilibrium multiscale dynamics (NEMSD) is proposed to simulate non-equilibrium thermal–mechanical processes. The model couples coarse-grain thermodynamics with a fine scale molecular dynamics. A Distributed Nośe-Hoover Thermostat Network is used, which regulates the temperature in each coarse scale Voronoi cell according to the finite element (FE) nodal temperature. The atoms in each element-cell, namely Voronoi cell-ensemble, are assumed to be in a local equilibrium state within one coarse scale time step. The change of FE nodal temperature provides a source of random forces, which drive the system out of equilibrium. The proposed NEMSD can successfully simulate shock wave propagation in a cubic lattice.