Modelling of global boundary effects on harmonic motion imaging of soft tissues

Biomechanical imaging techniques have been developed for soft tissue characterisation and detection of breast tumours. Harmonic motion imaging (HMI) uses a focused ultrasound technology to generate a harmonic radiation force in a localised region inside a soft tissue. The resulting dynamic response is used to map the local distribution of the mechanical properties of the tissue. In this study, a finite element (FE) model is developed to investigate the effect of global boundary conditions on the dynamic response of a soft tissue during HMI. The direct-solution steady-state dynamic analysis procedure is used to compute the harmonic displacement amplitude in FE simulations. The model is parameterised in terms of boundary conditions and viscoelastic properties, and the corresponding raster-scan displacement amplitudes are captured to examine its response. The effect of the model's global dimensions on the harmonic response is also investigated. It is observed that the dynamic response of soft tissue with high viscosity is independent of the global boundary conditions for regions remote to the boundary; thus, it can be subjected to local analysis to estimate the underlying mechanical properties. However, the dynamic response is sensitive to global boundary conditions for tissue with low viscosity or regions located near to the boundary.     

A simple way to compute protein dynamics without a mechanical model

We found that in proteins the average atomic fluctuation is linearly related to the square of the atomic distance from the center of mass of the protein. Using this simple relation, we can accurately compute the temperature factors of proteins of a wide range of sizes and folds, and the correlation of the fluctuations in proteins. This simple relation provides a direct link between protein dynamics and the static protein's geometrical shape and offers a simple way to compute protein dynamics without either long time trajectory integration or any matrix operations.     

Size dependent elastic modulus and mechanical resilience of dental enamel

Human tooth enamel exhibits a unique microstructure able to sustain repeated mechanical loading during dental function. Although notable advances have been made towards understanding the mechanical characteristics of enamel, challenges remain in the testing and interpretation of its mechanical properties. For example, enamel was often tested under dry conditions, significantly different from its native environment. In addition, constant load, rather than indentation depth, has been used when mapping the mechanical properties of enamel. In this work, tooth specimens are prepared under hydrated conditions and their stiffnesses are measured by depth control across the thickness of enamel. Crystal arrangement is postulated, among other factors, to be responsible for the size dependent indentation modulus of enamel. Supported by a simple structure model, effective crystal orientation angle is calculated and found to facilitate shear sliding in enamel under mechanical contact. In doing so, the stress build-up is eased and structural integrity is maintained.     

A recruitment-based rheological model for mechanical behavior of soft tissues

When lung tissue is subjected to finite deformations, phenomena appear that can only be described using nonlinear models. This paper considers the lung as a material composed of two elements, a continuous phase that acts uninterruptedly and a second phase composed of fiber elements that are recruited progressively into the mechanical process. Each individual fiber participates in the mechanical response of the set only when the deformation is above a certain value. A nine-parameter model was designed adopting standard viscoelastic elements both for the matrix and for each of the fibers. The mechanical behavior of the lung can be reproduced by a fitting process with standard numerical procedures in both dynamic-mechanical measurements and stress relaxation processes. Mechanical stress relaxation tests and dynamic-mechanical measurements have been carried out on subpleural parenchymal strips from rat lung. The model permits the reproduction of lung behavior in both types of measurements. The results show a recruitment ratio that decreases with deformation and the nonparticipation of the parallel matrix fraction in the lung's mechanical response so that a uniaxial transmission of force in the lung occurs via the recruited elements and the matrix series.     

A theoretical model for the elastic properties of very soft tissues.

A theoretical model is developed to predict the elastic properties of very soft tissues such as glands, tumors and brain. Tissues are represented as regular arrays of polyhedral (cubic or tetrakaidecahedral) cells, surrounded by extracellular spaces of uniform width. Cells are assumed to be incompressible, with very low resistance to shear deformation. Tissue shear rigidity is assumed to result mainly from the extracellular matrix, which is treated as a compressible elastic mesh of interconnected fibers. Small-strain elastic properties of tissue are predicted using a finite-element method and an analytical method. The model can be used to estimate the compressibility of a very soft tissue based on its Young's modulus and extracellular volume fraction.     

A constitutive model for the mechanical behaviour of soft connective tissues

A model of the mechanical behaviour of soft connective tissue has been developed by considering the role of the collagen and glycosaminoglycan (GAG) components within the tissue in order to examine the mechanism by which a variation in the GAG components may exert a control over the mechanical properties of the tissue. It is proposed that the strain energy stored within the collagen fibrils of the loaded tissue can be transferred into a potential field created by the charged GAG components and their electrostatic interaction with the collagen fibrils. A fundamental mechanical unit is described to simulate this energy transfer and a combination of such units is used to represent the tissue. The computer implementation of the proposed tissue model shows it to reproduce many features which have been recognised in the rate dependent mechanical behaviour of soft tissues. These include the characteristic non-linearity of the force-deformation behaviour and the approximate invariance of the stress relaxation behaviour with deformation. The model is also consistent with earlier constitutive representations of tissue behaviour.     

A continuous method to compute model parameters for soft biological materials

Developing appropriate mathematical models for biological soft tissues such as ligaments, tendons, and menisci is challenging. Stress-strain behavior of these tissues is known to be continuous and characterized by an exponential toe region followed by a linear elastic region. The conventional curve-fitting technique applies a linear curve to the elastic region followed by a separate exponential curve to the toe region. However, this technique does not enforce continuity at the transition between the two regions leading to inaccuracies in the material model. In this work, a Continuous Method is developed to fit both the exponential and linear regions simultaneously, which ensures continuity between regions. Using both methods, three cases were evaluated: idealized data generated mathematically, noisy idealized data produced by adding random noise to the idealized data, and measured data obtained experimentally. In all three cases, the Continuous Method performed superiorly to the conventional technique, producing smaller errors between the model and data and also eliminating discontinuities at the transition between regions. Improved material models may lead to better predictions of nonlinear biological tissues' behavior resulting in improved the accuracy for a large array of models and computational analyses used to predict clinical outcomes.     

Alignment maps of tissues: II. Fast harmonic analysis for imaging

A methodology for generating polarized light retardation and alignment direction images is presented. A rotated quarter-wave plate changes the linear polarized light to a polarized probe with various degrees of ellipticity by which samples are imaged with the use of a circular analyzer. A harmonic representation of image intensity allows simple analysis, requiring only simple image operations and realizing four orders-of-magnitude computational savings for strongly aligned tissues, where linear birefringence is the dominant optical property. The method is demonstrated for a porcine heart valve leaflet.     

A simple model for estimating the active reactions of embryonic tissues to a deforming mechanical force

Active reactions of embryonic tissues to mechanical forces play an important role in morphogenesis. To study these reactions, experimental models that enable to evaluate the applied forces and the deformations of the tissues are required. A model based upon the active intrusion of a living early gastrula Xenopus embryo into a tube half the embryo in diameter is described. The intrusion is initially triggered by a suction force of several dozen Pa but then continues in the absence of external driving force, stopping immediately after the entire embryo has penetrated into the tube. The process can be stopped by cytoskeletal drugs or by the damage of the part of the embryo still non-aspirated and is associated with the transversal contraction and meridional elongation of the non-aspirated part of the embryo surface and quasi-periodic longitudinal contractions/extensions of the cells within the part already aspirated. We suggest that this reaction is an active response to the embryo deformation and discuss its morphogenetic role. The problem of estimating the elastic modules of embryonic tissues is also discussed.     

Incremental elastic modulus for ventricles in diastole

Utilizing the formulation of so-called 'small deformations superimposed on a large initial deformation' the incremental pressure modulus of a ventricle in diastole is studied and the explicit expression of it is obtained as a function of intraventricular pressure. In the analysis the ventricular wall material is assumed to be homogeneous, incompressible, isotropic and the stress-strain relation is exponential. The numerical results for a dog left ventricle indicate that above a critical value of inner pressure the incremental pressure modulus increases with increasing intra-ventricular pressure. Furthermore, the relationship between the stiffness and pressure is seen to be curvilinear (particularly for low pressure level), but for large values of inner pressure the behavior of the curve may be approximated by a set of straight lines.     

A mathematical model for the elastic and fluid mechanical behavior of the human eye

A simplified mathematical model for the eye is proposed. This idealized model simulates the aquaous flow and intraocular pressure behavior of the human eye. Starting from elementary concepts in elasticity and fluid mechanics, one can derive differential equations governing the behavior of the mathematical model. When integrated, these equations yield algebraic relationships which are closely related to some of the widely used empirical formulae in ophthalmology, e.g., Friedenwald's (1948) formula for the scleral rigidity coefficient and Grant's (1950) equation for the facility of aqueous outflow. The eye's intraocular pressure variations are simply related to its aqueous and blood volume changes if one assumes that ocular tissue has essentially linear elastic properties. St. Helen's and McEwen's (1961) experiments indicate that a linear approximation is reasonably accurate if the standard Hookian stress strain law is modified to take into account the anelastic or time-dependent elastic behavior of the corneo-scleral membrane. The last part of the paper discusses transient phenomena in an externally disturbed eye, e.g., when a tonometer is applied. One important result is a theoretical equation to describe the mean curve that one records in a tonographic tracing. This paper represents the first attempt to formulate a mathematical model relating the over-all elastic and fluid mechanical behavior of the eye. It is hoped that the model will stimulate interst and prove useful to the medical profession in motivating future experiments and in suggesting improvements for existing empirical formulae.     

A mathematical model for the elastic and fluid mechanical behavior of the human eye

A simplified mathematical model for the eye is proposed. This idealized model simulates the aquaous flow and intraocular pressure behavior of the human eye. Starting from elementary concepts in elasticity and fluid mechanics, one can derive differential equations governing the behavior of the mathematical model. When integrated, these equations yield algebraic relationships which are closely related to some of the widely used empirical formulae in ophthalmology, e.g., Friedenwald's (1948) formula for the scleral rigidity coefficient and Grant's (1950) equation for the facility of aqueous outflow. The eye's intraocular pressure variations are simply related to its aqueous and blood volume changes if one assumes that ocular tissue has essentially linear elastic properties. St. Helen's and McEwen's (1961) experiments indicate that a linear approximation is reasonably accurate if the standard Hookian stress strain law is modified to take into account the anelastic or time-dependent elastic behavior of the corneo-scleral membrane. The last part of the paper discusses transient phenomena in an externally disturbed eye, e.g., when a tonometer is applied. One important result is a theoretical equation to describe the mean curve that one records in a tonographic tracing. This paper represents the first attempt to formulate a mathematical model relating the over-all elastic and fluid mechanical behavior of the eye. It is hoped that the model will stimulate interst and prove useful to the medical profession in motivating future experiments and in suggesting improvements for existing empirical formulae.     

A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement

In this paper the mechanical response of soft collagenous tissues with regular fiber arrangement (RSCTs) is described by means of a nanoscale model and a two-step micro–macro homogenization technique. The non-linear collagen constitutive behavior is modeled at the nanoscale by a novel approach accounting for entropic mechanisms as well as stretching effects occurring in collagen molecules. Crimped fibers are reduced to equivalent straight ones at the microscale and the constitutive response of RSCTs at the macroscale is formulated by homogenizing a fiber reinforced material. This approach has been applied to different RSCTs (tendon, periodontal ligament and aortic media), resulting effective and accurate as proved by the excellent agreement with available experimental data. The model is based on few parameters, directly related to histological and morphological evidences and whose sensitivity has been widely investigated. Applications to simulation of some physiopathological mechanisms are also proposed, providing confirmation of clinical evidences and quantitative indications helpful for clinical practice.     

Elastic modulus of hard tissues

This work aims at evaluating the elastic modulus of hard biological tissues by considering their staggered platelet micro-structure. An analytical expression for the effective modulus along the stagger direction is formulated using three non-dimensional structural variables. Structures with a single staggered hierarchy (e.g. collagen fibril) are first studied and predictions are compared with the experimental results and finite element simulations from the literature. A more complicated configuration, such as an array of fibrils, is analyzed next. Finally, a mechanical model is proposed for tooth dentin, in which variations in the multi-scale structural hierarchy are shown to significantly affect the macroscopic mechanical properties.     

Machine learning method for extracting elastic modulus of cells

Biomechanics and Modeling in Mechanobiology - The Hertz contact mechanics model is commonly used to extract the elastic modulus of the cell, but the basic assumptions of the model are often not met...