Tissue strain amplification at the osteocyte lacuna: a microstructural finite element analysis

A parametric finite element model of an osteocyte lacuna was developed to predict the microstructural response of the lacuna to imposed macroscopic strains. The model is composed of an osteocyte lacuna, a region of perilacunar tissue, canaliculi, and the surrounding bone tissue. A total of 45 different simulations were modeled with varying canalicular diameters, perilacunar tissue material moduli, and perilacunar tissue thicknesses. Maximum strain increased with a decrease in perilacunar tissue modulus and decreased with an increase in perilacunar tissue modulus, regardless of the thickness of the perilacunar region. An increase in the predicted maximum strain was observed with an increase in canalicular diameter from 0.362 to 0.421 microm. In response to the macroscopic application of strain, canalicular diameters increased 0.8% to over 1.0% depending on the perilacunar tissue modulus. Strain magnification factors of over 3 were predicted. However, varying the size of the perilacunar tissue region had no effect on the predicted perilacunar tissue strain. These results indicate that the application of average macroscopic strains similar to strain levels measured in vivo can result in significantly greater perilacunar tissue strains and canaliculi deformations. A decrease in the perilacunar tissue modulus amplifies the perilacunar tissue strain and canaliculi deformation while an increase in the local perilacunar tissue modulus attenuates this effect.     

Micromechanical modelling of cortical bone

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.     

Transfer of macroscale tissue strain to microscale cell regions in the deformed meniscus

Cells within fibrocartilaginous tissues, including chondrocytes and fibroblasts of the meniscus, ligament, and tendon, regulate cell biosynthesis in response to local mechanical stimuli. The processes by which an applied mechanical load is transferred through the extracellular matrix to the environment of a cell are not fully understood. To better understand the role of mechanics in controlling cell phenotype and biosynthetic activity, this study was conducted to measure strain at different length scales in tissue of the fibrocartilaginous meniscus of the knee joint, and to define a quantitative parameter that describes the strain transferred from the far-field tissue to a microenvironment surrounding a cell. Experiments were performed to apply a controlled uniaxial tensile deformation to explants of porcine meniscus containing live cells. Using texture correlation analyses of confocal microscopy images, two-dimensional Lagrangian and principal strains were measured at length scales representative of the tissue (macroscale) and microenvironment in the region of a cell (microscale) to yield a strain transfer ratio as a measure of median microscale to macroscale strain. The data demonstrate that principal strains at the microscale are coupled to and amplified from macroscale principal strains for a majority of cell microenvironments located across diverse microstructural regions, with average strain transfer ratios of 1.6 and 2.9 for the maximum and minimum principal strains, respectively. Lagrangian strain components calculated along the experimental axes of applied deformations exhibited considerable spatial heterogeneity and intersample variability, and suggest the existence of both strain amplification and attenuation. This feature is consistent with an in-plane rotation of the principal strain axes relative to the experimental axes at the microscale that may result from fiber sliding, fiber twisting, and fiber-matrix interactions that are believed to be important for regulating deformation in other fibrocartilaginous tissues. The findings for consistent amplification of macroscale to microscale principal strains suggest a coordinated pattern of strain transfer from applied deformation to the microscale environment of a cell that is largely independent of these microstructural features in the fibrocartilaginous meniscus.     

Multilevel finite element modeling for the prediction of local cellular deformation in bone

The underlying mechanisms by which bone cells respond to mechanical stimuli or how mechanical loads act on osteocytes housed in lacunae in bone are not well understood. In this study, a multilevel finite element (FE) approach is applied to predict local cell deformations in bone tissue. The local structure of the matrix dictates the local mechanical environment of an osteocyte. Cell deformations are predicted from detailed linear FE analysis of the microstructure, consisting of an arrangement of cells embedded in bone matrix material. This work has related the loads applied to a whole femur during the stance phase of the gait cycle to the strain of a single lacuna and of canaliculi. The predicted bone matrix strains around osteocyte lacunae and canaliculi were nonuniform and differed significantly from the macroscopically measured strains. Peak stresses and strains in the walls of the lacuna were up to six times those in the bulk extracellular matrix. Significant strain concentrations were observed at sites where the process meets the cell body.     

Osteocyte lacunae tissue strain in cortical bone

Current theories suggest that bone modeling and remodeling are controlled at the cellular level through signals mediated by osteocytes. However, the specific signals to which bone cells respond are still unknown. Two primary theories are: (1) osteocytes are stimulated via the mechanical deformation of the perilacunar bone matrix and (2) osteocytes are stimulated via fluid flow generated shear stresses acting on osteocyte cell processes within canaliculi. Recently, much focus has been placed on fluid flow theories since in vitro experiments have shown that bone cells are more responsive to analytically estimated levels of fluid shear stress than to direct mechanical stretching using macroscopic strain levels measured on bone in vivo. However, due to the complex microstructural organization of bone, local perilacunar bone tissue strains potentially acting on osteocytes cannot be reliably estimated from macroscopic bone strain measurements. Thus, the objective of this study was to quantify local perilacunar bone matrix strains due to macroscopically applied bone strains similar in magnitude to those that occur in vivo. Using a digital image correlation strain measurement technique, experimentally measured bone matrix strains around osteocyte lacunae resulting from macroscopic strains of approximately 2000 microstrain are significantly greater than macroscopic strain on average and can reach peak levels of over 30,000 microstrain locally. Average strain concentration factors ranged from 1.1 to 3.8, which is consistent with analytical and numerical estimates. This information should lead to a better understanding of how bone cells are affected by whole bone functional loading.     

Inferring spatial variations of microstructural properties from macroscopic mechanical response

Disease alters tissue microstructure, which in turn affects the macroscopic mechanical properties of tissue. In elasticity imaging, the macroscopic response is measured and is used to infer the spatial distribution of the elastic constitutive parameters. When an empirical constitutive model is used, these parameters cannot be linked to the microstructure. However, when the constitutive model is derived from a microstructural representation of the material, it allows for the possibility of inferring the local averages of the spatial distribution of the microstructural parameters. This idea forms the basis of this study. In particular, we first derive a constitutive model by homogenizing the mechanical response of a network of elastic, tortuous fibers. Thereafter, we use this model in an inverse problem to determine the spatial distribution of the microstructural parameters. We solve the inverse problem as a constrained minimization problem and develop efficient methods for solving it. We apply these methods to displacement fields obtained by deforming gelatin–agar co-gels and determine the spatial distribution of agar concentration and fiber tortuosity, thereby demonstrating that it is possible to image local averages of microstructural parameters from macroscopic measurements of deformation.     

A theoretical model to study the effects of cellular stiffening on the damage evolution in deep tissue injury

Pressure induced deep tissue injury (DTI) is a severe form of pressure ulcers that is hard to detect in early stages and difficult to prevent and treat. High prevalence figures are partly due to a lack of understanding of pathological pathways involved in DTI. The aim of this study was to investigate, whether changes in material properties of damaged tissue can play a role in DTI aetiology. A numerical model was developed based on muscle microstructure and tissue engineering experiments. A time dependent damage law was proposed and stiffening of dead cells incorporated. The results obtained in the microstructural investigations were used to include the stiffening information in a pre-existing macroscopic model based on animal experiments, which correlated strains to tissue damage measured in the tibialis anterior muscle in rat limbs. With the modelling approach employed in this paper, the damaged area in the rat limb models increased up to 1.65-fold and the rate of damage progression was up to 2.1 times higher in microstructural simulations when stiffening was included.     

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.     

Key parameters controlling stiffness variability within trees: a multiscale experimental–numerical approach

Microstructural properties of wood vary considerably within a tree. Knowledge of these properties and a better understanding of their relationship to the macroscopic mechanical performance of wood are crucial to optimize the yield and economic value of forest stocks. This holds particularly for the end-use requirements in engineering applications. In this study the microstructure–stiffness relationships of Scots pine are examined with a focus on the effects of the microstructural variability on the elastic properties of wood at different length scales. For this purpose, we have augmented microstructural data acquired using SilviScan-3? (namely wood density, cell dimensions, earlywood and latewood proportion, microfibril angle) with local measurements of these quantities and of the chemical composition derived from wide-angle X-ray scattering, light microscopy, and thermogravimetric analysis, respectively. The stiffness properties were determined by means of ultrasonic tests at the clear wood scale and by means of nanoindentation at the cell wall scale. In addition, micro-mechanical modeling was applied to assess the causal relations between structural and mechanical properties and to complement the experimental investigations. Typical variability profiles of microstructural and mechanical properties are shown from pith to bark, across a single growth ring and from earlywood to latewood. The clear increase of the longitudinal stiffness as well as the rather constant transverse stiffness from pith to bark could be explained by the variation in microfibril angle and wood density over the entire radial distance. The dependence of local cell wall stiffness on the local microfibril angle was also demonstrated. However, the local properties did not necessarily follow the trends observed at the macroscopic scale and exhibited only a weak relationship with the macroscopic mechanical properties. While the relationship between silvicultural practice and wood microstructure remains to be modeled using statistical techniques, the influence of microstructural properties on the macroscopic mechanical behavior of wood can now be described by a physical model. The knowledge gained by these investigations and the availability of a new micromechanical model, which allows transferring these findings to non-tested material, will be valuable for wood quality assessment and optimization in timber engineering.     

A theoretically-based experimental approach for identifying vascular constitutive relations

We employ a structurally-motivated phenomenological formulation to identify biomechanical experiments which can be used to determine a vascular constitutive relation directly from data. Large deformations, nonlinear material behavior, load-dependent anisotropy, material heterogeneity and incompressibility are accounted for in the analysis. For purposes of illustration, we outline a procedure for studying elastic arteries wherein the behavior of the media and adventitia is considered separately. This general approach for identifying vascular constitutive relations can be applied to any vessel or airway, however, and should provide certain advantages over previous microstructural or purely phenomenological formulations.     

Action Potential Duration Heterogeneity of Cardiac Tissue Can Be Evaluated from Cell Properties Using Gaussian Green's Function Approach

Action potential duration (APD) heterogeneity of cardiac tissue is one of the most important factors underlying initiation of deadly cardiac arrhythmias. In many cases such heterogeneity can be measured at tissue level only, while it originates from differences between the individual cardiac cells. The extent of heterogeneity at tissue and single cell level can differ substantially and in many cases it is important to know the relation between them. Here we study effects from cell coupling on APD heterogeneity in cardiac tissue in numerical simulations using the ionic TP06 model for human cardiac tissue. We show that the effect of cell coupling on APD heterogeneity can be described mathematically using a Gaussian Green''s function approach. This relates the problem of electrotonic interactions to a wide range of classical problems in physics, chemistry and biology, for which robust methods exist. We show that, both for determining effects of tissue heterogeneity from cell heterogeneity (forward problem) as well as for determining cell properties from tissue level measurements (inverse problem), this approach is promising. We illustrate the solution of the forward and inverse problem on several examples of 1D and 2D systems.     

Strain transfer through the aortic valve

The complex structural organization of the aortic valve (AV) extracellular matrix (ECM) enables large and highly nonlinear tissue level deformations. The collagen and elastin (elastic) fibers within the ECM form an interconnected fibrous network (FN) and are known to be the main load-bearing elements of the AV matrix. The role of the FN in enabling deformation has been investigated and documented. However, there is little data on the correlation between tissue level and FN-level strains. Investigating this correlation will help establish the mode of strain transfer (affine or nonaffine) through the AV tissue as a key feature in microstructural modeling and will also help characterize the local FN deformation across the AV sample in response to applied tissue level strains. In this study, the correlation between applied strains at tissue level, macrostrains across the tissue surface, and local FN strains were investigated. Results showed that the FN strain distribution across AV samples was inhomogeneous and nonuniform, as well as anisotropic. There was no direct transfer of the deformation applied at tissue level to the fibrous network. Loading modes induced in the FN are different than those applied at the tissue as a result of different local strains in the valve layers. This nonuniformity of local strains induced internal shearing within the FN of the AV, possibly exposing the aortic valve interstitial cells (AVICs) to shear strains and stresses.     

Dynamic imaging of cell, extracellular matrix, and tissue movements during avian vertebral axis patterning

Vertebrate axis patterning depends on cell and extracellular matrix (ECM) repositioning and proper cell-ECM interactions. However, there are few in vivo data addressing how large-scale tissue deformations are coordinated with the motion of local cell ensembles or the displacement of ECM constituents. Combining the methods of dynamic imaging and experimental biology allows both cell and ECM fate-mapping to be correlated with ongoing tissue deformations. These fate-mapping studies suggest that the axial ECM components "move" both as a composite meshwork and as autonomous particles, depending on the length scale being examined. Cells are also part of this composite, and subject to passive displacements resulting from tissue deformations. However, in contrast to the ECM, cells are self-propelled. The net result of cell and ECM displacements, along with proper ECM-cell adhesion, is the assembly of new tissue architecture. Data herein show that disruption of normal cell-ECM interactions during axis formation results in developmental abnormalities and a disorganization of the ECM. Our goal in characterizing the global displacement patterns of axial cells and ECM is to provide critical information regarding existing strain fields in the segmental plate and paraxial mesoderm. Deducing the mechanical influences on cell behavior is critical, if we are to understand vertebral axis patterning. Supplementary material for this article is available online at http://www.mrw.interscience.wiley.com/suppmat/1542-975X/suppmat/72/v72.266.html.     

Micromechanics of fibroblast contraction of a collagen-GAG matrix

The contractile force developed by fibroblasts has been studied by measuring the macroscopic contraction of porous collagen-GAG matrices over time. We have identified the microscopic deformations developed by individual fibroblasts which lead to the observed macroscopic matrix contraction. Observation of live cells attached to the matrix revealed that matrix deformation occurred as a result of cell elongation. The time dependence of the increase in average fibroblast aspect ratio over time corresponded with macroscopic matrix contraction, further linking cell elongation and matrix contraction. The time dependence of average fibroblast aspect ratio and macroscopic matrix contraction was found to be the result of the stochastic nature of cell elongation initiation and of the time required for cells to reach a final morphology (2-4 h). The proposed micromechanics associated with observed buckling or bending of individual struts of the matrix by cells may, in part, explain the observation of a force plateau during macroscopic contraction. These findings indicate that the macroscopic matrix contraction measured immediately following cell attachment is related to the extracellular force necessary to support cell elongation, and that macroscopic time dependence is not directly related to microscopic deformation events.     

In situ chondrocyte viscoelasticity

It has been proposed, based on theoretical considerations, that the strain rate-dependent viscoelastic response of cartilage reduces local tissue and cell deformations during cyclic compressions. However, experimental studies have not addressed the in situ viscoelastic response of chondrocytes under static and dynamic loading conditions. In particular, results obtained from experimental studies using isolated chondrocytes embedded in gel constructs cannot be used to predict the intrinsic viscoelastic responses of chondrocytes in situ or in vivo. Therefore, the purpose of this study was to investigate the viscoelastic response of chondrocytes in their native environment under static and cyclic mechanical compression using a novel in situ experimental approach. Cartilage matrix and chondrocyte recovery in situ following mechanical compressions was highly viscoelastic. The observed in situ behavior was consistent with a previous study on in vivo chondrocyte mechanics which showed that it took 5-7min for chondrocytes to recover shape and volume following virtually instantaneous cell deformations during muscular loading of the knee in live mice. We conclude from these results that the viscoelastic properties of cartilage minimize chondrocyte deformations during cyclic dynamic loading as occurs, for example, in the lower limb joints during locomotion, thereby allowing the cells to reach mechanical and metabolic homeostasis even under highly dynamic loading conditions.     

Microscale methods to assemble mammalian cells into tissue-like structures

Different cell types make up tissues and organs hierarchically and communicate within a complex, three-dimensional (3D) environment. The in vitro recapitulation of tissue-like structures is meaningful, not only for fundamental cell biology research, but also for tissue engineering (TE). Currently, TE research adopts either the top-down or bottom-up approach. The top-down approach involves defining the macroscopic tissue features using biomaterial scaffolds and seeding cells into these scaffolds. Conversely, the bottom-up approach aims at crafting small tissue building blocks with precision-engineered structural and functional microscale features, using physical and/or chemical approaches. The bottom-up strategy takes advantage of the repeating structural and functional units that facilitate cell-cell interactions and cultures multiple cells together as a functional unit of tissue. In this review, we focus on currently available microscale methods that can control mammalian cells to assemble into 3D tissue-like structures.     

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.     

Estimation of disruption of animal cells by turbulent capillary flow

Disruption of animal cells in turbulent capillary flows has been predicted from a model of cell-hydrodynamic interactions using cell mechanical properties determined by micromanipulation. Eddies of sizes similar to or smaller than the cells are presumed to interact with those cells, causing local surface deformations. The proposed mechanism of cell damage is that such deformations result in an increase in membrane tension and surface energy and that a cell disrupts when its bursting membrane tension and bursting surface energy are exceeded. The surface energy of the cells is estimated from the kinetic energy of appropriately sized eddies. To test the model, cells were disrupted in turbulent flows in capillaries at mean energy dissipation rates up to 2 x 10(4) m(2)/s(3). In all cases the model underestimated the cell disruption by about 15%. Such good agreement implies that the approach of the model to the complicated phenomena of cell turbulence interactions is reasonable. (c) 1993 John Wiley & Sons, Inc.     

Microstructure-stiffness relationships of ten European and tropical hardwood species

Hardwood species exhibit a huge anatomical variability. This makes them perfect study objects for exploring relations between structural features at different length scales and corresponding stiffness properties of wood. We carry out microscopic analysis, nanoindentation tests, as well as macroscale ultrasonic and quasi-static tension tests and build a complete set of microstructural and corresponding micromechanical data of ten different (European and tropical) hardwood species. In addition, we apply micromechanical modeling to further elucidate the individual influences of particular structural features, which might appear only in a superimposed manner in experiments. The test results confirm the dominant influences of the microfibril angle on the stiffness at cell wall level and of density at the macroscopic scale. Vessels and ray cells affect the macroscopic stiffness of the wood tissue not only through their content, but also through their arrangement and shape: A ring-porous structure results in comparably higher longitudinal but lower radial stiffness than a diffuse-porous one. As for ray cells, large and particularly compactly shaped bundles might reduce the stiffness in tangential direction because of the fiber deviations they cause. Moreover, vessel and ray content might affect the relation between nanoindentation modulus and density-corrected macroscopic longitudinal stiffness.