On the effects of residual stress in microindentation tests of soft tissue structures

Microindentation methods are commonly used to determine material properties of soft tissues at the cell or even sub-cellular level. In determining properties from force-displacement (FD) data, it is often assumed that the tissue is initially a stress-free, homogeneous, linear elastic half-space. Residual stress, however, can strongly influence such results. In this paper, we present a new microindentation method for determining both elastic properties and residual stress in soft tissues that, to a first approximation, can be regarded as a pre-stressed layer embedded in or adhered to an underlying relatively soft, elastic foundation. The effects of residual stress are shown using two linear elastic models that approximate specific biological structures. The first model is an axially loaded beam on a relatively soft, elastic foundation (i.e., stress-fiber embedded in cytoplasm), while the second is a radially loaded plate on a foundation (e.g., cell membrane or epithelium). To illustrate our method, we use a nonlinear finite element (FE) model and experimental FD and surface contour data to find elastic properties and residual stress in the early embryonic chick heart, which, in the region near the indenter tip, is approximated as an isotropic circular plate under tension on a foundation. It is shown that the deformation of the surface in a microindentation test can be used along with FD data to estimate material properties, as well as residual stress, in soft tissue structures that can be regarded as a plate under tension on an elastic foundation. This method may not be as useful, however, for structures that behave as a beam on a foundation.     

Tensile behavior of cortical bone: dependence of organic matrix material properties on bone mineral content

A porous composite model is developed to analyze the tensile mechanical properties of cortical bone. The effects of microporosity (volksman's canals, osteocyte lacunae) on the mechanical properties of bone tissue are taken into account. A simple shear lag theory, wherein tensile loads are transferred between overlapped mineral platelets by shearing of the organic matrix, is used to model the reinforcement provided by mineral platelets. It is assumed that the organic matrix is elastic in tension and elastic-perfectly plastic in shear until it fails. When organic matrix shear stresses at the ends of mineral platelets reach their yield values, the stress-strain curve of bone tissue starts to deviate from linear behavior. This is referred as the microscopic yield point. At the point where the stress-strain behavior of bone shows a sharp curvature, the organic phase reaches its shear yield stress value over the entire platelet. This is referred as the macroscopic yield point. It is assumed that after macroscopic yield, mineral platelets cannot contribute to the load bearing capacity of bone and that the mechanical behavior of cortical bone tissue is determined by the organic phase only. Bone fails when the principal stress of the organic matrix is reached. By assuming that mechanical properties of the organic matrix are dependent on bone mineral content below the macroscopic yield point, the model is used to predict the entire tensile mechanical behavior of cortical bone for different mineral contents. It is found that decreased shear yield stresses and organic matrix elastic moduli are required to explain the mechanical behavior of bones with lowered mineral contents. Under these conditions, the predicted values (elastic modulus, 0.002 yield stress and strain, and ultimate stress and strain) are within 15% of experimental data.     

Determination of mechanical properties of soft tissue scaffolds by atomic force microscopy nanoindentation

While the determination of mechanical properties of a hard scaffold is relatively straightforward, the mechanical testing of a soft tissue scaffold poses significant challenges due in part to its fragility. Here, we report a new approach for characterizing the stiffness and elastic modulus of a soft scaffold through atomic force microscopy (AFM) nanoindentation. Using collagen-chitosan hydrogel scaffolds as model soft tissue scaffolds, we demonstrated the feasibility of using AFM nanoindentation to determine a force curve of a soft tissue scaffold. A mathematical model was developed to ascertain the stiffness and elastic modulus of a scaffold from its force curve obtained under different conditions. The elastic modulus of a collagen-chitosan (80%/20%, v/v) scaffold is found to be 3.69 kPa. The scaffold becomes stiffer if it contains more chitosan. The elastic modulus of a scaffold composed of 70% collagen and 30% chitosan is about 11.6 kPa. Furthermore, the stiffness of the scaffold is found to be altered significantly by extracellular matrix deposited from cells that are grown inside the scaffold. The elastic modulus of collagen-chitosan scaffolds increased from 10.5 kPa on day 3 to 63.4 kPa on day 10 when human foreskin fibroblast cells grew inside the scaffolds. Data acquired from these measurements will offer new insights into understanding cell fate regulation induced by physiochemical cues of tissue scaffolds.     

A rheological network model for the continuum anisotropic and viscoelastic behavior of soft tissue

The mechanical behavior of soft tissue demonstrates a number of complex features including nonlinearity, anisotropy, viscoelasticity, and growth. Characteristic features of the time-dependent and anisotropic behavior are related to the properties of various components of the tissue such as fibrous collagen and elastin networks, large proteins and sugars attached to these networks, and interstitial fluid. Attempts to model the elastic behavior of these tissues based on assumptions about the behavior of the underlying constituents have been reasonably successful, but the essential addition of viscoelasticity to these models has been met with varying success. Here, a new rheological network model is proposed using, as its basis, an orthotropic hyperelastic constitutive model for fibrous tissue and a viscoelastic reptation model for soft materials. The resulting model has been incorporated into numerical and computational models, and is shown to capture the mechanical behavior of soft tissue in various modes of deformation including uniaxial and biaxial tension and simple shear.     

In vivo characterization of mechanical tissue properties of internal organs using endoscopic microscopy and inverse finite element analysis

Knowledge about mechanical tissue properties is required for functional modelling and simulating of tissue and organ responses to external mechanical stress. To get the right properties especially for functional modelling of organs, tissue properties have to be determined in vivo. There are only few described methods for characterization of internal organ's tissue mechanics that can be applied in vivo. We introduce and evaluate a method to determine mechanical tissue properties, especially those of lung tissue, endoscopically. Inverse finite element analysis (utilizing a Neo-Hookean model for hyperelastic materials) and image processing algorithms are used to determine the shear modulus of a soft tissue. The resulting values for shear moduli were normally distributed. The shear modulus of the artificial tissue sample was determined with a relative error of 0.47% compared to the value obtained by uniaxial tensile test.     

Optical measurements of vocal fold tensile properties: implications for phonatory mechanics

In voice research, in vitro tensile stretch experiments of vocal fold tissues are commonly employed to determine the tissue biomechanical properties. In the standard stretch-release protocol, tissue deformation is computed from displacements applied to sutures inserted through the thyroid and arytenoid cartilages, with the cartilages assumed to be rigid. Here, a non-contact optical method was employed to determine the actual tissue deformation of vocal fold lamina propria specimens from three excised human larynges in uniaxial tensile tests. Specimen deformation was found to consist not only of deformation of the tissue itself, but also deformation of the cartilages, as well as suture alignment and tightening. Stress-stretch curves of a representative load cycle were characterized by an incompressible Ogden model. The initial longitudinal elastic modulus was found to be considerably higher if determined based on optical displacement measurements than typical values reported in the literature. The present findings could change the understanding of the mechanics underlying vocal fold vibration. Given the high longitudinal elastic modulus the lamina propria appeared to demonstrate a substantial level of anisotropy. Consequently, transverse shear could play a significant role in vocal fold vibration, and fundamental frequencies of phonation should be predicted by beam theories accounting for such effects.     

Modelling of non-linear elastic tissues for surgical simulation

Realistic modelling of the interaction between surgical instruments and human organs has been recognised as a key requirement in the development of high-fidelity surgical simulators. Primarily due to computational considerations, most of the past real-time surgical simulation research has assumed linear elastic behaviour for modelling tissues, even though human soft tissues generally possess non-linear properties. For a non-linear model, the well-known Poynting effect developed during shearing of the tissue results in normal forces not seen in a linear elastic model. Using constitutive equations of non-linear tissue models together with experiments, we show that the Poynting effect results in differences in force magnitude larger than the absolute human perception threshold for force discrimination in some tissues (e.g. myocardial tissues) but not in others (e.g. brain tissue simulants).     

A method to estimate the elastic properties of the extracellular matrix of articular cartilage

In this work we propose a method to estimate the elastic properties of the extracellular matrix of articular cartilage, once the elastic properties of the chondrocytes and the whole tissue are known. The influence of the elastic properties of the tissue and the cell concentration on the estimated elastic properties of the matrix are investigated.     

The shear mechanical properties of diabetic and non-diabetic plantar soft tissue

Changes in the plantar soft tissue shear properties may contribute to ulceration in diabetic patients, however, little is known about these shear parameters. This study examines the elastic and viscoelastic shear behavior of both diabetic and non-diabetic plantar tissue. Previously compression tested plantar tissue specimens (n=54) at six relevant plantar locations (hallux, first, third, and fifth metatarsal heads, lateral midfoot, and calcaneus) from four cadaveric diabetic feet and five non-diabetic feet were utilized. Per in vivo data (i.e., combined deformation patterns of compression followed by shear), an initial static compressive strain (36-38%) was applied to the tissue followed by target shear strains of 50% and 85% of initial thickness. Triangle waves were used to quantify elastic parameters at both strain levels and a stress relaxation test (0.25 s ramp and 300 s hold) was used to quantify the viscoelastic parameters at the upper strain level. Several differences were found between test groups including a 52-62% increase in peak shear stress, a 63% increase in toe shear modulus, a 47% increase in final shear modulus, and a 67% increase in middle slope magnitude (sharper drop in relaxation) in the diabetic tissue. Beyond a 54% greater peak compressive stress in the third metatarsal compared to the lateral midfoot, there were no differences in shear properties between plantar locations. Notably, this study demonstrates that plantar soft tissue with diabetes is stiffer than healthy tissue, thereby compromising its ability to dissipate shear stresses borne by the foot that may increase ulceration risk.     

Sensitivity of damage predictions to tissue level yield properties and apparent loading conditions

High-resolution finite element models of trabecular bone failure could be used to augment current techniques for measuring damage in trabecular bone. However, the sensitivity of such models to the assumed tissue yield properties and apparent loading conditions is unknown. The goal of this study was to assess the sensitivity of the amount and mode (tension vs. compression) of tissue level yielding in trabecular bone to these factors. Linear elastic, high-resolution finite element models of nine bovine tibial trabecular bone specimens were used to calculate the fraction of the total tissue volume that exceeded each criterion for apparent level loading to the reported elastic limit in both on-axis and transverse compression and tension, and in shear. Four candidate yield criteria were studied, based on values suggested in the literature. Both the amount and the failure mode of yielded tissue were sensitive to the magnitudes of the tissue yield strains, the degree of tension-compression asymmetry of the yield criterion, and the applied apparent loads. The amount of yielded tissue was most sensitive to the orientation of the applied apparent loading, with the most tissue yielding for loading along the principal trabecular orientation and the least for loading perpendicular to it, regardless of the assumed tissue level yield criterion. Small changes in the magnitudes and the degree of asymmetry of the tissue yield criterion resulted in much larger changes in the amount of yielded tissue in the model. The results indicate that damage predictions based on high-resolution finite element models are highly sensitive to the assumed tissue yield properties. As such, good estimates of these values are needed before high-resolution finite element models can be applied to the study of trabecular bone damage. Regardless of the assumed tissue yield properties, the amount and type of damage that occurs in trabecular bone depends on the relative orientations of the applied apparent loads to the trabecular architecture, and this parameter should be controlled for both experimental and computational damage studies.     

Method for characterizing viscoelasticity of human gluteal tissue

Characterizing compressive transient large deformation properties of biological tissue is becoming increasingly important in impact biomechanics and rehabilitation engineering, which includes devices interfacing with the human body and virtual surgical guidance simulation. Individual mechanical in vivo behaviour, specifically of human gluteal adipose and passive skeletal muscle tissue compressed with finite strain, has, however, been sparsely characterised. Employing a combined experimental and numerical approach, a method is presented to investigate the time-dependent properties of in vivo gluteal adipose and passive skeletal muscle tissue. Specifically, displacement-controlled ramp-and-hold indentation relaxation tests were performed and documented with magnetic resonance imaging. A time domain quasi-linear viscoelasticity (QLV) formulation with Prony series valid for finite strains was used in conjunction with a hyperelastic model formulation for soft tissue constitutive model parameter identification and calibration of the relaxation test data. A finite element model of the indentation region was employed. Strong non-linear elastic but linear viscoelastic tissue material behaviour at finite strains was apparent for both adipose and passive skeletal muscle mechanical properties with orthogonal skin and transversal muscle fibre loading. Using a force-equilibrium assumption, the employed material model was well suited to fit the experimental data and derive viscoelastic model parameters by inverse finite element parameter estimation. An individual characterisation of in vivo gluteal adipose and muscle tissue could thus be established. Initial shear moduli were calculated from the long-term parameters for human gluteal skin/fat: G(∞,S/F)=1850 Pa and for cross-fibre gluteal muscle tissue: G(∞,M)=881 Pa. Instantaneous shear moduli were found at the employed ramp speed: G(0,S/F)=1920 Pa and G(0,M)=1032 Pa.     

Passive movement of human soft palate during respiration: A simulation of 3D fluid/structure interaction

This study reconstructed a three dimensional fluid/structure interaction (FSI) model to investigate the compliance of human soft palate during calm respiration. Magnetic resonance imaging scans of a healthy male subject were obtained for model reconstruction of the upper airway and the soft palate. The fluid domain consists of nasal cavity, nasopharynx and oropharynx. The airflow in upper airway was assumed as laminar and incompressible. The soft palate was assumed as linear elastic. The interface between airway and soft palate was the FSI interface. Sinusoidal variation of velocity magnitude was applied at the oropharynx corresponding to ventilation rate of 7.5L/min. Simulations of fluid model in upper airway, FSI models with palatal Young's modulus of 7539Pa and 3000Pa were carried out for two cycles of respiration. The results showed that the integrated shear forces over the FSI interface were much smaller than integrated pressure forces in all the three directions (axial, coronal and sagittal). The total integrated force in sagittal direction was much smaller than that of coronal and axial directions. The soft palate was almost static during inspiration but moved towards the posterior pharyngeal wall during expiration. In conclusion, the displacement of human soft palate during respiration was mainly driven by air pressure around the surface of the soft palate with minimal contribution of shear stress of the upper airway flow. Despite inspirational negative pressure, expiratory posterior movement of soft palate could be another factor for the induction of airway collapse.     

Nonlocal shear deformable shell model for postbuckling of axially compressed microtubules embedded in an elastic medium

Buckling and postbuckling analysis is presented for axially compressed microtubules (MTs) embedded in an elastic matrix of cytoplasm. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The surrounding elastic medium is modeled as a Pasternak foundation. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. The small scale parameter e ₀ a is estimated by matching the buckling load from their vibrational behavior of MTs with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling load and postbuckling behavior of MTs are very sensitive to the small scale parameter e ₀ a. The results reveal that the MTs under axial compressive loading condition have an unstable postbuckling path, and the lateral constraint has a significant effect on the postbuckling response of a microtubule when the foundation stiffness is sufficiently large.     

Micro-Mechanical Characterization of Lung Tissue Using Atomic Force Microscopy

Matrix stiffness strongly influences growth, differentiation and function of adherent cells^1-3. On the macro scale the stiffness of tissues and organs within the human body span several orders of magnitude⁴. Much less is known about how stiffness varies spatially within tissues, and what the scope and spatial scale of stiffness changes are in disease processes that result in tissue remodeling. To better understand how changes in matrix stiffness contribute to cellular physiology in health and disease, measurements of tissue stiffness obtained at a spatial scale relevant to resident cells are needed. This is particularly true for the lung, a highly compliant and elastic tissue in which matrix remodeling is a prominent feature in diseases such as asthma, emphysema, hypertension and fibrosis. To characterize the local mechanical environment of lung parenchyma at a spatial scale relevant to resident cells, we have developed methods to directly measure the local elastic properties of fresh murine lung tissue using atomic force microscopy (AFM) microindentation. With appropriate choice of AFM indentor, cantilever, and indentation depth, these methods allow measurements of local tissue shear modulus in parallel with phase contrast and fluorescence imaging of the region of interest. Systematic sampling of tissue strips provides maps of tissue mechanical properties that reveal local spatial variations in shear modulus. Correlations between mechanical properties and underlying anatomical and pathological features illustrate how stiffness varies with matrix deposition in fibrosis. These methods can be extended to other soft tissues and disease processes to reveal how local tissue mechanical properties vary across space and disease progression.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Extracellular matrix and vascular ageing

The extracellular matrix provides a structural framework essential for the functional properties of tissues. In each tissue, the three-dimensional organisation of the extracellular matrix molecules--elastin, collagens, proteoglycans and structural glycoproteins--synthesized during development and growth is optimal for these functions. In adult tissues, proteases are constitutively expressed but have a very low activity and the turn-over of elastic and collagen fibers is very low. During ageing, the interaction of environmental factors (glucose, lipids, calcium...) and modifications of the biosynthesis and degradation processes lead to modifications of extracellular matrix homeostasis and consequently to alterations of tissue functionality. These alterations are increased during pathological processes such as cardiovascular diseases.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Interaction of stress and growth in a fibrous tissue

The influence of stress on the growth and remodeling of a soft biological tissue is considered. For this purpose, the soft tissue is idealized as a fiber network. The stress-free lengths of the fibers composing the network are not fixed as in an inert elastic solid, but are assumed to evolve as a result of growth and stress adaptation. Similarly, the topology of the fiber network may also evolve under the application of stress. A set of constitutive equations are proposed which relate the tissue stress to the deformation of the tissue as well as to its growth and microstructure. It is shown that distinctly different growth patterns which may arise during initial growth or wound healing can be modeled by the proposed mathematical analysis.     

A two-parameter model of the effective elastic tensor for cortical bone

Multiscale models of cortical bone elasticity require a large number of parameters to describe the organization and composition of the tissue. We hypothesize that the macro-scale anisotropic elastic properties of different bones can be modeled retaining only two variable parameters, and setting the others to universal values identical for all bones. Cortical bone is regarded as a two-phase composite material: a dense mineralized matrix (ultrastructure) and a soft phase (pores). The ultrastructure is assumed to be a homogeneous and transversely isotropic tissue whose elastic properties in different directions are mutually dependent and can be scaled with a single parameter driving the overall rigidity. This parameter is taken to be the volume fraction of mineral f(ha). The pore network is modeled as an ensemble of water-filled cylinders and described only by the porosity p. The effective macroscopic elasticity tensor C(ij)(f(ha),p) is calculated with a multiscale micromechanics approach starting from existing models. The modeled stiffness coefficients compare favorably to four literature datasets which were chosen because they provide the full stiffness tensors of groups of human samples. Since the physical counterparts of f(ha) and p were unknown for the datasets, their values which allow the best fit of experimental tensors by the modeled ones were determined by optimization. Optimum values of f(ha) and p are found to be unique and realistic. These results suggest that a two-parameter model may be sufficient to model the elasticity of different samples of human femora and tibiae. Such a model would in particular be useful in large-scale parametric studies of bone mechanical response.     

Unconfined compression of hydrated viscoelastic tissues: a biphasic poroviscoelastic analysis

A biphasic poroviscoelastic theory was used to analyze the unconfined compression creep and stress relaxation of a hydrated viscoelastic tissue. The intrinsic shear properties of the tissue matrix was described by an integral-type viscoelastic constitutive law while the intrinsic bulk property of the matrix was assumed to be linearly elastic. Parametric data were presented to show how the two major energy dissipative mechanisms, namely the interstitial fluid flow and the intrinsic matrix viscoelasticity, may each contribute to the apparent viscoelastic behavior of the whole tissue under unconfined compression. The hydraulic permeability of the tissue was found to enter in only as a scaling factor for time.