Constitutive relation for red cell membrane. Correction.

The intention of this note is to correct a subtle and somewhat esoteric error that the author discovered in his previous publications on membrane elastic behavior. The consitutive relation between membrane force resultants and large, elastic deformations of a membrane surface involves a strain tensor, characterizing the finite deformations. The original strain tensor that appeared in the equations was the Lagrangian strain tensor; however, the proper strain representation (also Lagrangian in nature because it is "measured" relative to the undeformed material state) is transformed by rotations of coordinates in the deformed material state (whereas the Lagrangian strain tensor is transformed by rotations of coordinates in the undeformed state). The principal membrane tensions are unchanged by this correction; the material elastic constants remain the same; and therefore, the material behavior in shear and isotropic tension is the same. However, the tensor, constitutive relation can be properly applied to coordinate systems other than the principal axis system.     

The iceman under pressure (Part I): A description of skull deformations due to 5100 years of glacial action

Due to its long deposition in the glacier, the 'Iceman' (an ice-mummy from the Hauslabjoch) has been deformed, notably its skull. We introduce various comparative methods that describe these deformations, assuming they can be ascribed--to a large extent--to glacial action. While pressure is a scalar, the deformations must be described via a 2-tensor strain field (which can be represented by a matrix function value at every point throughout the skull). In this paper, we present the assumed deformations in numerous graphical forms and, furthermore, the limitations in interpretation--including an estimate of statistical variability--that can be revealed by this analysis. These methods, although describing the results of glacial action and implying a 2-tensor strain field (which will be presented in a subsequent paper), do not permit a straightforward reconstruction of the original, underformed skull. These methods have wider applications to the general problem of deformation.     

Associating the mesoscale fiber organization of the tongue with local strain rate during swallowing

The tongue is an intricately configured muscular organ that undergoes a stereotypical set of deformations during the course of normal human swallowing. In order to demonstrate quantitatively the relationship between 3D aligned lingual fiber organization and mechanics during swallowing, the tissue's myoarchitecture and strain rate were imaged before and during the propulsive phase of a 3.0ml water bolus swallow. Mesoscale fiber organization was imaged with high-resolution diffusion tensor imaging (DTI) and multi-voxel myofiber tracts generated along maximum diffusion vectors. Tissue compression/expansion was obtained via lingual pressure-gated phase-contrast (PC) MRI, a method which determines local strain rate as a function of the phase shift occurring along an applied gradient vector. The co-alignment of myofiber tract direction and the localized principal strain rate vectors was obtained by translating the strain rate tensor into the reference frame with the primary axis parallel to the maximum diffusion vector using Mohr's circle, resulting in the generation of fiber-aligned strain rate (FASR). DTI tractography displayed the complete fiber anatomy of the tongue, consisting of a core region of orthogonally aligned fibers encased within a longitudinal sheath, which merge with the externally connected styloglossus, hyoglossus, and genioglossus fibers. FASR images obtained in the mid-sagittal plane demonstrated that bolus propulsion was associated with prominent compressive strain aligned with the genioglossus muscle combined with expansive strain aligned with the verticalis and geniohyoid muscles. These data demonstrate that lingual deformation during swallowing involves complex interactions involving intrinsic and extrinsic muscles, whose contractility is directed by the alignment of mesoscale fiber tracts.     

A bimodular polyconvex anisotropic strain energy function for articular cartilage

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex "strong interaction terms" for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson's ratios, as well as axial stress-axial strain responses, in tension and compression for finite deformations.     

A three-dimensional digital image correlation technique for strain measurements in microstructures

A three-dimensional digital image correlation technique is presented for strain measurements in open-cell structures such as trabecular bone. The technique uses high-resolution computed tomography images for displacement measurements in the solid structure. In order to determine the local strain-state within single trabeculae, a tetrahedronization method is used to fill the solid structure with tetrahedrae. Displacements are calculated at the nodes of the tetrahedrae. The displacement data is subsequently converted to a deformation tensor in each of the tetrahedral element centers with a least-squares estimation method. Because the trabeculae are represented by a mesh, it is possible to deform this mesh according to the deformation tensor and, at the same time, visualize the calculated local strain in the deformed mesh with a finite element post-processing tool. In this way, the deformation of a single trabecula from an aluminum foam sample was determined and validated with rendered images of the three-dimensional sample. A precision analysis showed that a rigid translation or rotation does not affect the accuracy. Typical values for the standard deviation in the displacement and strain components are 2.0 microm and 0.01, respectively. Presently, the precision limits the technique to strain measurements beyond the yield strain.     

General solution for flow and stress phenomena in elastic vessles (Navier-Stokes)

A method is developed for a full nonlinear evaluation of all velocities and stresses represented in the Navier-Stokes equations and in the general stress tensor. The information required is essentially that for solution of linearized forms. The solution is analytical except for the calculation of the axial velocity, which requires computer assistance to step through time and space. The treatment of the problem, although directed towards solutions involving fluid flow in elastic vessels, is also adaptable to solid deformations (strain vs rate of strain) where the general stress tensor applies. A special case for the distorting ellipse is presented as well as a limited, spatially analytic, solution.     

Optical acquisition and polar decomposition of the full-field deformation gradient tensor within a fracture callus

Tracking tissue deformation is often hampered by material inhomogeneity, so local measurements tend to be insufficient thus lending to the necessity of full-field optical measurements. This study presents a novel approach to factoring heterogeneous deformation of soft and hard tissues in a fracture callus by introducing an anisotropic metric derived from the deformation gradient tensor (F). The deformation gradient tensor contains all the information available in a Green–Lagrange strain tensor, plus the rigid-body rotational components. A recent study [Bottlang et al., Journal of Biomechanics 41(3), 2008] produced full-field strains within ovine fracture calluses acquired through the application of electronic speckle pattern interferometery (ESPI). The technique is based on infinitesimal strain approximation (Engineering Strain) whose scheme is not independent of rigid-body rotation. In this work, for rotation extraction, the stretch and rotation tensors were separately determined from F by the polar decomposition theorem. Interfragmentary motions in a fracture gap were characterized by the two distinct mechanical factors (stretch and rotation) at each material point through full-field mapping. In the composite nature of bone and soft tissue, collagen arrangements are hypothesized such that fibers locally aligned with principal directions will stretch and fibers not aligned with the principal direction will rotate and stretch. This approach has revealed the deformation gradient tensor as an appropriate quantification of strain within callus bony and fibrous tissue via optical measurements.     

Anisotropic hydraulic permeability under finite deformation

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.     

Vacuum-assisted closure: microdeformations of wounds and cell proliferation

The mechanism of action of the Vacuum Assisted Closure Therapy (VAC; KCI, San Antonio, Texas), a recent novel innovation in the care of wounds, remains unknown. In vitro studies have revealed that cells allowed to stretch tend to divide and proliferate in the presence of soluble mitogens, whereas retracted cells remain quiescent. The authors hypothesize that application of micromechanical forces to wounds in vivo can promote wound healing through this cell shape-dependent, mechanical control mechanism. The authors created a computer model (finite element) of a wound and simulated VAC application. Finite element modeling is commonly used to engineer complex systems by breaking them down into simple discrete elements. In this model, the authors altered the pressure, pore diameter, and pore volume fraction to study the effects of vacuum-induced material deformations. The authors compared the morphology of deformation of this wound model with histologic sections of wounds treated with the VAC. The finite element model showed that most elements stretched by VAC application experienced deformations of 5 to 20 percent strain, which are similar to in vitro strain levels shown to promote cellular proliferation. Importantly, the deformation predicted by the model also was similar in morphology to the surface undulations observed in histologic cross-sections of the wounds. The authors hypothesize that this tissue deformation stretches individual cells, thereby promoting proliferation in the wound microenvironment. The application of micromechanical forces may be a useful method with which to stimulate wound healing through promotion of cell division, angiogenesis, and local elaboration of growth factors. Finite element modeling of the VAC device is consistent with this mechanism of action.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Analysis of the passive mechanical properties of rat carotid arteries

The passive mechanical properties of rat carotid arteries were studied in vitro. Using a tensile testing machine and a piston pump, intact segments of carotid arteries were subjected to large deformations both in the longitudinal and circumferential directions. Internal pressure, external diameter, length and longitudinal force were measured during the experiment and compared with the in vivo dimensions of the segments prior to excision. The anisotropic mechanical properties of the vessel wall material were analyzed using incremental elastic moduli and incremental Poisson's ratios. The results suggest that there is a characteristic deformation pattern common to all vessels investigated which is highly correlated with the conditions of loading that occur in vivo. That is, under average physiological deformation of the vessel, the longitudinal force is nearly independent of internal pressure. In this range of loading the circumferential incremental elastic modulus is nearly independent of longitudinal strain. However, the longitudinal and radial incremental elastic moduli vary significantly with deformation in this direction. The values of the moduli in all three directions increase with raising internal pressure. The weak coupling between circumferential and longitudinal direction in the wall material of carotid arteries is shown by the small value of the corresponding incremental Poisson's ratios.     

A model for the nonlinear elastic response of large arteries

The nonlinear elastic response of large arteries subjected to finite deformations due to action of biaxial principal stresses, is described by simple constitutive equations. Generalized measures of strain and stress are introduced to account for material nonlinearity. This also ensures the existence of a strain energy density function. The orthotropic elastic response is described via quasi-linear relations between strains and stresses. One nonlinear parameter which defines the measures of strain and stress, and three elastic moduli are assumed to be constants. The lateral strain parameters (equivalent to Poisson's ratios in infinitesimal deformations) are deformation dependent. This dependence is defined by empirical relations developed via the incompressibility condition, and by the introduction of a fifth material parameter. The resulting constitutive model compares well with biaxial experimental data of canine carotid arteries.     

Mechanical properties of the red cell membrane skeleton: analysis of axisymmetric deformations

The mechanical properties of erythrocyte membrane composed of a membrane bilayer and membrane skeleton are considered. Two membrane models are described: the model of free boundaries (MFB) and the model of immobilized boundaries (MIB). In MFB, the skeleton is assumed to be attached to the bilayer at a finite number of points, whereas MIB allows the interaction of each spectrin filament with the bilayer along its whole length. For MFB an estimate was made of the mechanical strain generated in the membrane by sucking erythrocytes into a micropipette. The existence of the deformation threshold is demonstrated, below which no mechanical strain, except that of bending, appears in the membrane. Thus only deformations exceeding this threshold result in strain. The relationship between the applied tension and the height of erythrocyte "tongue" sucked into a micropipette was determined. The MIB characteristics correspond to the model of Evans: strains in the membrane are generated at any deformation, however small, i.e. the threshold is equal to zero. A basic feature of this model is quite a different distribution of the skeleton deformations in the membrane. A comparison of the theoretical models and experimental data demonstrated the possibility of either MFB or MIB occurring, depending on the characteristic measurement time.     

Isotropy and anisotropy of the arterial wall

The passive biomechanical response of intact cylindrical rat carotid arteries is studied in vitro and compared with the mechanical response of rubber tubes. Using true stress and natural strain in the definition of the incremental modulus of elasticity, the tissue wall properties are analyzed over wide ranges of simultaneous circumferential and longitudinal deformations. The type of loading chosen is 'physiological' i.e. symmetric: the cylindrical segments are subjected to internal pressure and axial prestretch without torsion or shear. Several aspects pertaining to the choice of parameters characterizing the material are discussed and the analysis pertaining to the deformational behavior of a hypothetical compliant tube with Hookean wall material is presented. The experimental results show that while rubber response can be adequately represented as linearly elastic and isotropic, the overall response of vascular tissue is highly non-linear and anisotropic. However, for states of deformation that occur in vivo, the elasticity of arteries is quite similar to that of rubber tubes and as such the arterial wall may be viewed as incrementally isotropic for the range of deformations that occur in vivo.     

Glycosaminoglycan network geometry may contribute to anisotropic hydraulic permeability in cartilage under compression.

Resistance to fluid flow within cartilage extracellular matrix is provided primarily by a dense network of rod-like glycosaminoglycans (GAGs). If the geometrical organization of this network is random, the hydraulic permeability tensor of cartilage is expected to be isotropic. However, experimental data have suggested that hydraulic permeability may become anisotropic when the matrix is mechanically compressed, contributing to cartilage biomechanical functions such as lubrication. We hypothesized that this may be due to preferred GAG rod orientations and directionally-dependent reduction of inter-GAG spacings which reflect molecular responses to tissue deformations. To examine this hypothesis, we developed a model for effects of compression which allows the GAG rod network to deform consistently with tissue-scale deformations but while still respecting limitations imposed by molecular structure. This network deformation model was combined with a perturbation analysis of a classical analytical model for hydraulic permeability based on molecular structure. Finite element analyses were undertaken to ensure that this approach exhibited results similar to those emerging from more exact calculations. Model predictions for effects of uniaxial confined compression on the hydraulic permeability tensor were consistent with previous experimental results. Permeability decreased more rapidly in the direction perpendicular to compression than in the parallel direction, for matrix solid volume fractions associated with fluid transport in articular cartilage. GAG network deformations may therefore introduce anisotropy to the permeability (and other GAG-associated matrix properties) as physiological compression is applied, and play an important role in cartilage lubrication and other biomechanical functions.     

Effects of ischemia on epicardial deformation in the passive rabbit heart

BACKGROUND: Surgically induced ischemia in the arrested heart can result in changes in the mechanical properties of the myocardium. Regions of ischemia may be characterized based on the amount of epicardial deformation for a given load. Computer aided speckle interferometry (CASI), which tracks the movement of clusters of particles, is developed as a technique for measuring epicardial deformation, thereby determining the perfusion status of the passive heart. MATERIALS AND METHODS: Silicone carbide particles and retroreflective beads were dispersed randomly onto the epicardial surface of 11 isolated rabbit hearts to form speckle images. The hearts were arrested with hyperkalemic Krebs-Henseleit buffered solution. Each heart was then exposed to a series of intracavitary pressures, and at each pressure speckle images were acquired with a charge-coupled device (CCD) camera. Nine hearts were exposed to global ischemia, and two hearts were exposed to regional ischemia by occluding the second diagonal branch of the left anterior descending artery (LAD). The hearts were again loaded and the speckle images were acquired. CASI was used to determine the distribution of deformation field. RESULTS: CASI was able to determine displacements with a spatial resolution of about 50 microns. Global ischemia resulted in a significant increase in the maximum principle strain and the first invariant of the 2-D strain tensor. In the regionally ischemic heart, a large difference in deformation between the ischemic and perfused regions was clearly observed. CONCLUSION: Based on epicardial deformation, CASI is able to distinguish between perfused and ischemic myocardium, with a spatial resolution of 50 microns.     

Geometrical aspects of surface morphogenesis

This paper is concerned with the morphogenesis of structures which form thin deformable sheets. A general formalism is presented for the deformation of a sheet in the presence of an isotropic local body stress. This formalism leads to a set of equations, based on the theory of shells, in which corrections are made in the geometry due to large deformations. Under certain conditions the equations may be solved to give the surface metric tensor as a function of the local tension. A numerical example based on a simple "threshold" model is also presented.     

Stress-relaxation and microscopic dynamics of rabbit periodontal ligament

The aim of the present study was to examine the structural basis for the stress-relaxation behaviour of the periodontal ligament (PDL). Seventeen 4-month-old rabbits were used. A tooth-PDL-bone segment was cut in a rectangular prism from the incisor of a dissected mandible. The specimen was mounted in a testing machine built on a video stereomicroscope. Following preconditioning, each specimen was stretched to a deformation of 35 microm and then the deformation was kept constant for 300 s to obtain a stress-relaxation curve. Thereafter, stress-relaxation tests were repeated sequentially at deformations of 55, 75, and 95 microm. Polarised-light video-stereomicroscopic images of the specimens were simultaneously recorded and analysed with the stress-relaxation curves. The image analysis revealed that during stress-relaxation, the brightness of the birefringent fibres tended to initially increase rapidly and then do so gradually. There were negative correlations between the brightness and relaxation modulus at the four deformations. The decreases of normalised relaxation modulus for 300 s were less at greater deformation levels. The stress-relaxation process was well described by a function with three exponential decay terms and a constant. These findings suggest that during stress-relaxation of the PDL, the alignment of the collagen molecules and fibrils within the stretched fibres may occur, which could be driven by the strain energy imparted to the specimen on initial stretching.     

A fibre reorientation model for orthotropic multiplicative growth

The main goal of this contribution consists in the development of a remodelling framework for orthotropic continua whereby the underlying symmetry group is incorporated via two fibre families. Special emphasis is placed on the modelling of biological tissues at finite deformations. Besides the incorporation of a referential mass source, anisotropic growth is addressed by means of a multiplicative decomposition of the overall deformation gradient into an elastic and a growth distortion. Projected quantities of a configurational growth stress tensor are advocated as driving forces for time-dependent saturation–type evolution of the principal values of the growth distortion. Moreover, the reorientation of both fibre families, which directly affects the strain energy as well as the growth distortion itself, is guided by analyzing critical energy points. In particular, a time-dependent formulation is developed which aligns the fibre directions according to the principal stretch directions. Finally, the proposed framework is embedded into a finite element context so that representative numerical examples, examining growth and resorption in volume and density together with fibre reorientation, close this study.