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.     

Determination of the axial and circumferential mechanical properties of the skin tissue using experimental testing and constitutive modeling

The skin, being a multi-layered material, is responsible for protecting the human body from the mechanical, bacterial, and viral insults. The skin tissue may display different mechanical properties according to the anatomical locations of a body. However, these mechanical properties in different anatomical regions and at different loading directions (axial and circumferential) of the mice body to date have not been determined. In this study, the axial and circumferential loads were imposed on the mice skin samples. The elastic modulus and maximum stress of the skin tissues were measured before the failure occurred. The nonlinear mechanical behavior of the skin tissues was also computationally investigated through a suitable constitutive equation. Hyperelastic material model was calibrated using the experimental data. Regardless of the anatomic locations of the mice body, the results revealed significantly different mechanical properties in the axial and circumferential directions and, consequently, the mice skin tissue behaves like a pure anisotropic material. The highest elastic modulus was observed in the back skin under the circumferential direction (6.67 MPa), while the lowest one was seen in the abdomen skin under circumferential loading (0.80 MPa). The Ogden material model was narrowly captured the nonlinear mechanical response of the skin at different loading directions. The results help to understand the isotropic/anisotropic mechanical behavior of the skin tissue at different anatomical locations. They also have implications for a diversity of disciplines, i.e., dermatology, cosmetics industry, clinical decision making, and clinical intervention.     

Microplane constitutive model and computational framework for blood vessel tissue

This paper presents a nonlinearly elastic anisotropic microplane formulation in 3D for computational constitutive modeling of arterial soft tissue in the passive regime. The constitutive modeling of arterial (and other biological) soft tissue is crucial for accurate finite element calculations, which in turn are essential for design of implants, surgical procedures, bioartificial tissue, as well as determination of effect of progressive diseases on tissues and implants. The model presented is defined at a lower scale (mesoscale) than the conventional macroscale and it incorporates the effect of all the (collagen) fibers which are anisotropic structural components distributed in all directions within the tissue material in addition to that of isotropic bulk tissue. It is shown that the proposed model not only reproduces Holzapfel's recent model but also improves on it by accounting for the actual three-dimensional distribution of fiber orientation in the arterial wall, which endows the model with advanced capabilities in simulation of remodeling of soft tissue. The formulation is flexible so that its parameters could be adjusted to represent the arterial wall either as a single material or a material composed of several layers in finite element analyses of arteries. Explicit algorithms for both the material subroutine and the explicit integration with dynamic relaxation of equations of motion using finite element method are given. To circumvent the slow convergence of the standard dynamic relaxation and small time steps dictated by the stability of the explicit integrator, an adaptive dynamic relaxation technique that ensures stability and fastest possible convergence rates is developed. Incompressibility is enforced using penalty method with an updated penalty parameter. The model is used to simulate experimental data from the literature demonstrating that the model response is in excellent agreement with the data. An experimental procedure to determine the distribution of fiber directions in 3D for biological soft tissue is suggested in accordance with the microplane concept. It is also argued that this microplane formulation could be modified or extended to model many other phenomena of interest in biomechanics.     

Analysis of indentation: implications for measuring mechanical properties with atomic force microscopy.

Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law--a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.     

Mechanical characterization of anisotropic planar biological soft tissues using finite indentation: experimental feasibility

Heart valve tissue engineering offers a promising alternative for current treatment and replacement strategies, e.g., synthetic or bioprosthetic heart valves. In vitro mechanical conditioning is an important tool for engineering strong, implantable heart valves. Detailed knowledge of the mechanical properties of the native tissue as well as the developing tissue construct is vital for a better understanding and control of the remodeling processes induced by mechanical conditioning. The nonlinear, anisotropic and inhomogeneous mechanical behavior of heart valve tissue necessitates a mechanical characterization method that is capable of dealing with these complexities. In a recent computational study we showed that one single indentation test, combining force and deformation gradient data, provides sufficient information for local characterization of nonlinear soft anisotropic tissue properties. In the current study this approach is validated in two steps. First, indentation tests with varying indenter sizes are performed on linear elastic PDMS rubbers and compared to tensile tests on the same specimen. For the second step, tissue constructs are engineered using uniaxial or equibiaxial static constrained culture conditions. Digital image correlation (DIC) is used to quantify the anisotropy in the tissue constructs. For both validation steps, material parameters are estimated by inverse fitting of a computational model to the experimental results.     

Identifying heterogeneous anisotropic properties in cerebral aneurysms: a pointwise approach

The traditional approaches of estimating heterogeneous properties in a soft tissue structure using optimization-based inverse methods often face difficulties because of the large number of unknowns to be simultaneously determined. This article proposes a new method for identifying the heterogeneous anisotropic nonlinear elastic properties in cerebral aneurysms. In this method, the local properties are determined directly from the pointwise stress–strain data, thus avoiding the need for simultaneously optimizing for the property values at all points/regions in the aneurysm. The stress distributions needed for a pointwise identification are computed using an inverse elastostatic method without invoking the material properties in question. This paradigm is tested numerically through simulated inflation tests on an image-based cerebral aneurysm sac. The wall tissue is modeled as an eight-ply laminate whose constitutive behavior is described by an anisotropic hyperelastic strain energy function containing four parameters. The parameters are assumed to vary continuously in the sac. Deformed configurations generated from forward finite element analysis are taken as input to inversely establish the parameter distributions. The delineated and the assigned distributions are in excellent agreement. A forward verification is conducted by comparing the displacement solutions obtained from the delineated and the assigned material parameters at a different pressure. The deviations in nodal displacements are found to be within 0.2% in most part of the sac. The study highlights some distinct features of the proposed method, and demonstrates the feasibility of organ level identification of the distributive anisotropic nonlinear properties in cerebral aneurysms.     

On the anisotropy and inhomogeneity of permeability in articular cartilage

Articular cartilage is known to be anisotropic and inhomogeneous because of its microstructure. In particular, its elastic properties are influenced by the arrangement of the collagen fibres, which are orthogonal to the bone-cartilage interface in the deep zone, randomly oriented in the middle zone, and parallel to the surface in the superficial zone. In past studies, cartilage permeability has been related directly to the orientation of the glycosaminoglycan chains attached to the proteoglycans which constitute the tissue matrix. These studies predicted permeability to be isotropic in the undeformed configuration, and anisotropic under compression. They neglected tissue anisotropy caused by the collagen network. However, magnetic resonance studies suggest that fluid flow is "directed" by collagen fibres in biological tissues. Therefore, the aim of this study was to express the permeability of cartilage accounting for the microstructural anisotropy and inhomogeneity caused by the collagen fibres. Permeability is predicted to be anisotropic and inhomogeneous, independent of the state of strain, which is consistent with the morphology of the tissue. Looking at the local anisotropy of permeability, we may infer that the arrangement of the collagen fibre network plays an important role in directing fluid flow to optimise tissue functioning.     

Elastic modulus variation in mandibular bone: a microindentation study of Macaca fascicularis

We characterized the heterogeneous anisotropic elastic properties of mandibular bone in an adult female specimen of Macaca fascicularis using the technique of microindentation. This approach used an indenter of known mass and geometry to sample bone hardness at a spatial resolution in the order of 100 mum. Hardness values were converted to elastic modulus using empirically derived regression. We determined properties in alveolar, midcorpus, and basal regions of coronal and transverse sections taken from multiple locations along the corpus and ramus. Within sections, we determined properties from endosteal, midcortical, and periosteal regions. We found regional variations in bone structure, including bands of orthotropic circumferential lamellar bone at the endosteal and periosteal corpus base, angular region, and ramus. Transversely isotropic osteonal bone characterizes the midcortices of alveolar and basal regions, with many resorption spaces in alveolar regions restricting sampling opportunities. Regional variations in elasticity include relatively compliant bone in the anterior corpus and ramus. Basal cortical bone is stiffer longitudinally than transversely or superoinferiorly, while the evidence for directional dependence in alveolar bone is equivocal. Alveolar bone appears to be relatively compliant with respect to bone found in midcorpus or basal regions. Considerable variation exists in structure and material properties on a highly localized scale, more so than is discernible through conventional approaches for determining material property variation.     

Finite element study of a tissue-engineered cartilage transplant in human tibiofemoral joint

Most tissue-engineered cartilage constructs are more compliant than native articular cartilage (AC) and are poorly integrated to the surrounding tissue. To investigate the effect of an implanted tissue-engineered construct (TEC) with these inferior properties on the mechanical environment of both the engineered and adjacent native tissues, a finite element study was conducted. Biphasic swelling was used to model tibial cartilage and an implanted TEC with the material properties of either native tissue or a decreased elastic modulus and fixed charged density. Creep loading was applied with a rigid impermeable indenter that represented the femur. In comparison with an intact joint, compressive strains in the transplant, surface contact stress in the adjacent native AC and load partitioning between different phases of cartilage were affected by inferior properties of TEC. Results of this study may lead to a better understanding of the complex mechanical environment of an implanted TEC.     

Finite element study of a tissue-engineered cartilage transplant in human tibiofemoral joint

Most tissue-engineered cartilage constructs are more compliant than native articular cartilage (AC) and are poorly integrated to the surrounding tissue. To investigate the effect of an implanted tissue-engineered construct (TEC) with these inferior properties on the mechanical environment of both the engineered and adjacent native tissues, a finite element study was conducted. Biphasic swelling was used to model tibial cartilage and an implanted TEC with the material properties of either native tissue or a decreased elastic modulus and fixed charged density. Creep loading was applied with a rigid impermeable indenter that represented the femur. In comparison with an intact joint, compressive strains in the transplant, surface contact stress in the adjacent native AC and load partitioning between different phases of cartilage were affected by inferior properties of TEC. Results of this study may lead to a better understanding of the complex mechanical environment of an implanted TEC.     

A dual optimization method for the material parameter identification of a biphasic poroviscoelastic hydrogel: Potential application to hypercompliant soft tissues

A dual-indentation creep and stress relaxation methodology was developed and validated for the material characterization of very soft biological tissue within the framework of the biphasic poroviscoelastic (BPVE) constitutive model. Agarose hydrogel, a generic porous medium with mobile fluid, served as a mechanical tissue analogue for validation of the experimental procedure. Indentation creep and stress relaxation tests with a solid plane-ended cylindrical indenter were performed at identical sites on a gel sample with dimensions large enough with respect to indenter size in order to satisfy an infinite layer assumption. A finite element (FE) formulation coupled to a global optimization algorithm was utilized to simultaneously curve-fit the creep and stress relaxation data and extract the BPVE model parameters for the agarose gel. A numerical analysis with artificial data was conducted to validate the uniqueness of the computational procedure. The BPVE model was able to successfully cross-predict both creep and stress relaxation behavior for each pair of experiments with a single unique set of material parameters. Optimized elastic moduli were consistent with those reported in the literature for agarose gel. With the incorporation of appropriately-sized indenters to satisfy more stringent geometric constraints, this simple yet powerful indentation methodology can provide a straightforward means by which to obtain the BPVE model parameters of biological soft tissues that are difficult to manipulate (such as brain and adipose) while maintaining a realistic in situ loading environment.     

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.     

Nonlinear anisotropic stress analysis of anatomically realistic cerebral aneurysms

BACKGROUND: Static deformation analysis and estimation of wall stress distribution of patient-specific cerebral aneurysms can provide useful insights into the disease process and rupture. METHOD OF APPROACH: The three-dimensional geometry of saccular cerebral aneurysms from 27 patients (18 unruptured and nine ruptured) was reconstructed based on computer tomography angiography images. The aneurysm wall tissue was modeled using a nonlinear, anisotropic, hyperelastic material model (Fung-type) which was incorporated in a user subroutine in ABAQUS. Effective material fiber orientations were assumed to align with principal surface curvatures. Static deformation of the aneurysm models were simulated assuming uniform wall thickness and internal pressure load of 100 mm Hg. RESULTS: The numerical analysis technique was validated by quantitative comparisons to results in the literature. For the patient-specific models, in-plane stresses in the aneurysm wall along both the stiff and weak fiber directions showed significant regional variations with the former being higher. The spatial maximum of stress ranged from as low as 0.30 MPa in a small aneurysm to as high as 1.06 MPa in a giant aneurysm. The patterns of distribution of stress, strain, and surface curvature were found to be similar. Sensitivity analyses showed that the computed stress is mesh independent and not very sensitive to reasonable perturbations in model parameters, and the curvature-based criteria for fiber orientations tend to minimize the total elastic strain energy in the aneurysms wall. Within this small study population, there were no statistically significant differences in the spatial means and maximums of stress and strain values between the ruptured and unruptured groups. However, the ratios between the stress components in the stiff and weak fiber directions were significantly higher in the ruptured group than those in the unruptured group. CONCLUSIONS: A methodology for nonlinear, anisotropic static deformation analysis of geometrically realistic aneurysms was developed, which can be used for a more accurate estimation of the stresses and strains than previous methods and to facilitate prospective studies on the role of stress in aneurysm rupture.     

Mechanical characterization of anisotropic planar biological soft tissues using large indentation: a computational feasibility study

Traditionally, the complex mechanical behavior of planar soft biological tissues is characterized by (multi)axial tensile testing. While uniaxial tests do not provide sufficient information for a full characterization of the material anisotropy, biaxial tensile tests are difficult to perform and tethering effects limit the analyses to a small central portion of the test sample. In both cases, determination of local mechanical properties is not trivial. Local mechanical characterization may be performed by indentation testing. Conventional indentation tests, however, often assume linear elastic and isotropic material properties, and therefore these tests are of limited use in characterizing the nonlinear, anisotropic material behavior typical for planar soft biological tissues. In this study, a spherical indentation experiment assuming large deformations is proposed. A finite element model of the aortic valve leaflet demonstrates that combining force and deformation gradient data, one single indentation test provides sufficient information to characterize the local material behavior. Parameter estimation is used to fit the computational model to simulated experimental data. The aortic valve leaflet is chosen as a typical example. However, the proposed method is expected to apply for the mechanical characterization of planar soft biological materials in general.     

Dehomogenized Elastic Properties of Heterogeneous Layered Materials in AFM Indentation Experiments

Atomic force microscopy (AFM) is used to study mechanical properties of biological materials at submicron length scales. However, such samples are often structurally heterogeneous even at the local level, with different regions having distinct mechanical properties. Physical or chemical disruption can isolate individual structural elements but may alter the properties being measured. Therefore, to determine the micromechanical properties of intact heterogeneous multilayered samples indented by AFM, we propose the Hybrid Eshelby Decomposition (HED) analysis, which combines a modified homogenization theory and finite element modeling to extract layer-specific elastic moduli of composite structures from single indentations, utilizing knowledge of the component distribution to achieve solution uniqueness. Using finite element model-simulated indentation of layered samples with micron-scale thickness dimensions, biologically relevant elastic properties for incompressible soft tissues, and layer-specific heterogeneity of an order of magnitude or less, HED analysis recovered the prescribed modulus values typically within 10% error. Experimental validation using bilayer spin-coated polydimethylsiloxane samples also yielded self-consistent layer-specific modulus values whether arranged as stiff layer on soft substrate or soft layer on stiff substrate. We further examined a biophysical application by characterizing layer-specific microelastic properties of full-thickness mouse aortic wall tissue, demonstrating that the HED-extracted modulus of the tunica media was more than fivefold stiffer than the intima and not significantly different from direct indentation of exposed media tissue. Our results show that the elastic properties of surface and subsurface layers of microscale synthetic and biological samples can be simultaneously extracted from the composite material response to AFM indentation. HED analysis offers a robust approach to studying regional micromechanics of heterogeneous multilayered samples without destructively separating individual components before testing.     

A method for planar biaxial mechanical testing that includes in-plane shear.

A limitation in virtually all planar biaxial studies of soft tissues has been the inability to include the effects of in-plane shear. This is due to the inability of current mechanical testing devices to induce a state of in-plane shear, due to the added cost and complexity. In the current study, a straightforward method is presented for planar biaxial testing that induces a combined state of in-plane shear and normal strains. The method relies on rotation of the test specimen's material axes with respect to the device axes and on rotating carriages to allow the specimen to undergo in-plane shear freely. To demonstrate the method, five glutaraldehyde treated bovine pericardium specimens were prepared with their preferred fiber directions (defining the material axes) oriented at 45 deg to the device axes to induce a maximum shear state. The test protocol included a wide range of biaxial strain states, and the resulting biaxial data re-expressed in material axes coordinate system. The resulting biaxial data was then fit to the following strain energy function W: [equation: see text] where E'ij is the Green's strain tensor in the material axes coordinate system and c and Ai are constants. While W was able to fit the data very well, the constants A5 and A6 were found not to contribute significantly to the fit and were considered unnecessary to model the shear strain response. In conclusion, while not able to control the amount of shear strain independently or induce a state of pure shear, the method presented readily produces a state of simultaneous in-plane shear and normal strains. Further, the method is very general and can be applied to any anisotropic planar tissue that has identifiable material axes.     

Determination of dynamically adapting anisotropic material properties of bone under cyclic loading

Because bone tissue adapts to loading conditions, finite element simulations of remodelling bone require a precise prediction of dynamically changing anisotropic elastic parameters. We present a phenomenological theory that refers to the tissue in terms of the tendency of the structure to align with principal stress directions. We describe the material parameters of remodelling bone. This work follows findings by the same research group and independently by Danilov (1971) in the field of plasticity, where the dependencies of the components of the stiffness tensor in terms of time are based on Hill's anisotropy. We modify such an approach in this novel theory that addresses bone tissue that can regenerate. The computational assumption of the theory is that bone trabeculae have the tendency to orient along one of the principal stress directions but during remodelling the principal stresses change continuously and the resulting orientation of the trabeculae can differ from the principal stress direction at any given time. The novelty of this work consists in the limited number of parameters needed to compute the twenty-one anisotropic material parameters at any given location in the bone tissue. In addition to the theory, we present here two cases of simplified geometry, loading and boundary conditions to show the effect of (1) time on the material properties; and (2) change of loading conditions on the anisotropic parameters. The long term goal is to experimentally verify that the predictions generated by theory provide a reliable simulation of cancellous bone properties.     

Micropipette aspiration method for characterizing biological materials with surface energy

Many soft biological tissues possess a considerable surface stress, which plays a significant role in their biophysical functions, but most previous methods for characterizing their mechanical properties have neglected the effects of surface stress. In this work, we investigate the micropipette aspiration method to measure the mechanical properties of soft tissues and cells with surface effects. The neo-Hookean constitutive model is adopted to describe the hyperelasticity of the measured biological material, and the surface effect is taken into account by the finite element method. It is found that when the pipette radius or aspiration length is comparable to the elastocapillary length, surface energy may distinctly alter the aspiration response. Generally, both the aspiration length and the bulk normal stress decrease with increasing surface energy, and thus neglecting the surface energy would lead to an overestimation of elastic modulus. Through dimensional analysis and numerical simulations, we provide an explicit relation between the imposed pressure and the aspiration length. This method can be applied to determine the mechanical properties of soft biological tissues and organs, e.g., livers, tumors and embryos.     

Transversely isotropic elasticity imaging of cancellous bone

To measure spatial variations in mechanical properties of biological materials, prior studies have typically performed mechanical tests on excised specimens of tissue. Less invasive measurements, however, are preferable in many applications, such as patient-specific modeling, disease diagnosis, and tracking of age- or damage-related degradation of mechanical properties. Elasticity imaging (elastography) is a nondestructive imaging method in which the distribution of elastic properties throughout a specimen can be reconstructed from measured strain or displacement fields. To date, most work in elasticity imaging has concerned incompressible, isotropic materials. This study presents an extension of elasticity imaging to three-dimensional, compressible, transversely isotropic materials. The formulation and solution of an inverse problem for an anisotropic tissue subjected to a combination of quasi-static loads is described, and an optimization and regularization strategy that indirectly obtains the solution to the inverse problem is presented. Several applications of transversely isotropic elasticity imaging to cancellous bone from the human vertebra are then considered. The feasibility of using isotropic elasticity imaging to obtain meaningful reconstructions of the distribution of material properties for vertebral cancellous bone from experiment is established. However, using simulation, it is shown that an isotropic reconstruction is not appropriate for anisotropic materials. It is further shown that the transversely isotropic method identifies a solution that predicts the measured displacements, reveals regions of low stiffness, and recovers all five elastic parameters with approximately 10% error. The recovery of a given elastic parameter is found to require the presence of its corresponding strain (e.g., a deformation that generates ??? is necessary to reconstruct C????), and the application of regularization is shown to improve accuracy. Finally, the effects of noise on reconstruction quality is demonstrated and a signal-to-noise ratio (SNR) of 40 dB is identified as a reasonable threshold for obtaining accurate reconstructions from experimental data. This study demonstrates that given an appropriate set of displacement fields, level of regularization, and signal strength, the transversely isotropic method can recover the relative magnitudes of all five elastic parameters without an independent measurement of stress. The quality of the reconstructions improves with increasing contrast, magnitude of deformation, and asymmetry in the distributions of material properties, indicating that elasticity imaging of cancellous bone could be a useful tool in laboratory studies to monitor the progression of damage and disease in this tissue.