On parameter estimation for biaxial mechanical behavior of arteries

This article considers the parameter estimation of multi-fiber family models for biaxial mechanical behavior of passive arteries in the presence of the measurement errors. First, the uncertainty propagation due to the errors in variables has been carefully characterized using the constitutive model. Then, the parameter estimation of the artery model has been formulated into nonlinear least squares optimization with an appropriately chosen weight from the uncertainty model. The proposed technique is evaluated using multiple sets of synthesized data with fictitious measurement noises. The results of the estimation are compared with those of the conventional nonlinear least squares optimization without a proper weight factor. The proposed method significantly improves the quality of parameter estimation as the amplitude of the errors in variables becomes larger. We also investigate model selection criteria to decide the optimal number of fiber families in the multi-fiber family model with respect to the experimental data balancing between variance and bias errors.     

Simple shear testing of parallel-fibered planar soft tissues

The simple shear test may provide unique information regarding the material response of parallel-fibered soft tissues because it allows the elimination of the dominant fiber material response from the overall stresses. However, inhomogeneities in the strain field due to clamping and free edge effects have not been documented. The finite element method was used to study finite simple shear of simulated ligament material parallel to the fiber direction. The effects of aspect ratio, clamping prestrain, and bulk modulus were assessed using a transversely isotropic, hyperelastic material model. For certain geometries, there was a central area of uniform strain. An aspect ratio of 1:2 for the fiber to cross-fiber directions provided the largest region of uniform strain. The deformation was nearly isochoric for all bulk moduli indicating this test may be useful for isolating solid viscoelasticity from interstitial flow effects. Results suggest this test can be used to characterize the matrix properties for the type of materials examined in this study, and that planar measurements will suffice to characterize the strain. The test configuration may be useful for the study of matrix, fiber-matrix, and fiber-fiber material response in other types of parallel-fibered transversely isotropic soft tissues.     

Internal mechanical conditions in the soft tissues of a residual limb of a trans-tibial amputee

Most trans-tibial amputation (TTA) patients use a prosthesis to retain upright mobility capabilities. Unfortunately, interaction between the residual limb and the prosthetic socket causes elevated internal strains and stresses in the muscle and fat tissues in the residual limb, which may lead to deep tissue injury (DTI) and other complications. Presently, there is paucity of information on the mechanical conditions in the TTA residual limb during load-bearing. Accordingly, our aim was to characterize the mechanical conditions in the muscle flap of the residual limb of a TTA patient after donning the prosthetic socket and during load-bearing. Knowledge of internal mechanical conditions in the muscle flap can be used to identify the risk for DTI and improve the fitting of the prosthesis. We used a patient-specific modelling approach which involved an MRI scan, interface pressure measurements between the residual limb and the socket of the prosthesis and three-dimensional non-linear large-deformation finite-element (FE) modelling to quantify internal soft tissue strains and stresses in a female TTA patient during static load-bearing. Movement of the truncated tibia and fibula during load-bearing was measured by means of MRI and used as displacement boundary conditions for the FE model. Subsequently, we calculated the internal strains, strain energy density (SED) and stresses in the muscle flap under the truncated bones. Internal strains under the tibia peaked at 85%, 129% and 106% for compression, tension and shear strains, respectively. Internal strains under the fibula peaked at substantially lower values, that is, 19%, 22% and 19% for compression, tension and shear strains, respectively. Strain energy density peaked at the tibial end (104kJ/m(3)). The von Mises stresses peaked at 215kPa around the distal end of the tibia. Stresses under the fibula were at least one order of magnitude lower than the stresses under the tibia. We surmise that our present patient-specific modelling method is an important tool in understanding the etiology of DTI in the residual limbs of TTA patients.     

Inexpensive, semi-automated system for measuring mechanical properties of soft tissues.

Stiffness and strength are important properties of many tissues, but standard material-testing equipment is expensive, often ill-suited for testing soft tissues, and rarely accessible to biologists. We describe a system built around a microcomputer and an electronic balance which is particularly well-suited for measuring stress and strain in small samples of soft tissue. We use a discarded floppy disk drive as a linear actuator to strain the sample, while an electronic balance measures the tension (used to calculate stress). We give an algorithm for a program to drive a microcomputer which controls the floppy disk drive via its parallel port and records the balance measurements via its serial port. We used this system to obtain stress-strain curves from a sample of latex rubber and a sample of soft insect cuticle. Three tests of the rubber sample gave nearly identical results, with smooth, J-shaped stress-strain curves. The stress-strain curves gave a modulus elasticity value of 1.72 Mpa over the steep, straight region, well within the range for natural latex rubber. We also tested a sample of abdominal cuticle from a caterpillar (Manduca sexta). The caterpillar cuticle had a J-shaped stress-strain curve with a modulus of elasticity of 2.11 Mpa over the steep part of the curve. J. Exp. Zool. 284:374-378, 1999.     

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.     

Mechanical properties of soft biological tissues

Odd published data concerning the shear mechanical properties of some soft tissues in norm and pathology are reviewed.     

Development of fibroblast-seeded collagen gels under planar biaxial mechanical constraints: a biomechanical study

Prior studies indicated that mechanical loading influences cell turnover and matrix remodeling in tissues, suggesting that mechanical stimuli can play an active role in engineering artificial tissues. While most tissue culture studies focus on influence of uniaxial loading or constraints, effects of multi-axial loading or constraints on tissue development are far from clear. In this study, we examined the biaxial mechanical properties of fibroblast-seeded collagen gels cultured under four different mechanical constraints for 6 days: free-floating, equibiaxial stretching (with three different stretch ratios), strip-biaxial stretching, and uniaxial stretching. Passive mechanical behavior of the cell-seeded gels was also examined after decellularization. A continuum-based two-dimensional Fung model was used to quantify the mechanical behavior of the gel. Based on the model, the value of stored strain energy and the ratio of stiffness in the stretching directions were calculated at prescribed strains for each gel, and statistical comparisons were made among the gels cultured under the various mechanical constraints. Results showed that gels cultured under the free-floating and equibiaxial stretching conditions exhibited a nearly isotropic mechanical behavior, while gels cultured under the strip-biaxial and uniaxial stretching conditions developed a significant degree of mechanical anisotropy. In particular, gels cultured under the equibiaxial stretching condition with a greater stretch ratio appeared to be stiffer than those with a smaller stretch ratio. Also, a decellularized gel was stiffer than its non-decellularized counterpart. Finally, the retained mechanical anisotropy in gels cultured under the strip-biaxial stretching and uniaxial stretching conditions after cell removal reflected an irreversible matrix remodeling.     

Evolving biaxial mechanical properties of mouse carotid arteries in hypertension

Quantifying the time course of load-induced changes in arterial wall geometry, microstructure, and properties is fundamental to developing mathematical models of growth and remodeling. Arteries adapt to altered pressure and flow by modifying wall thickness, inner diameter, and axial length via marked cell and matrix turnover. To estimate particular biomaterial implications of such adaptations, we used a 4-fiber family constitutive relation to quantify passive biaxial mechanical behaviors of mouse carotid arteries 0 (control), 7-10, 10-14, or 35-56 days after an aortic arch banding surgery that increased pulse pressure and pulsatile flow in the right carotid artery. In vivo circumferential and axial stretches at mean arterial pressure were, for example, 11% and 26% lower, respectively, in hypertensive carotids 35-56 days after banding than in normotensive controls; this finding is consistent with observations that hypertension decreases distensibility. Interestingly, the strain energy W stored in the carotids at individual in vivo conditions was also less in hypertensive compared with normotensive carotids. For example, at 35-56 days after banding, W was 24%, 39%, and 47% of normal values at diastolic, mean, and systolic pressures, respectively. The energy stored during the cardiac cycle, W(sys)-W(dias), also tended to be less, but this reduction did not reach significance. When computed at normal in vivo values of biaxial stretch, however, W was well above normal for the hypertensive carotids. This net increase resulted from an overall increase in the collagen-related anisotropic contribution to W despite a decrease in the elastin-related isotropic contribution. The latter was consistent with observed decreases in the mass fraction of elastin.     

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

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

On the biaxial mechanical properties of the layers of the aortic valve leaflet

All existing constitutive models for heart valve leaflet tissues either assume a uniform transmural stress distribution or utilize a membrane tension formulation. Both approaches ignore layer specific mechanical contributions and the implicit nonuniformity of the transmural stress distribution. To begin to address these limitations, we conducted novel studies to quantify the biaxial mechanical behavior of the two structurally distinct, load bearing aortic valve (AV) leaflet layers: the fibrosa and ventricularis. Strip biaxial tests, with extremely sensitive force sensing capabilities, were further utilized to determine the mechanical behavior of the separated ventricularis layer at very low stress levels. Results indicated that both layers exhibited very different nonlinear, highly anisotropic mechanical behaviors. While the leaflet tissue mechanical response was dominated by the fibrosa layer, the ventricularis contributed double the amount of the fibrosa to the total radial tension and experienced four times the stress level. The strip biaxial test results further indicated that the ventricularis exhibited substantial anisotropic mechanical properties at very low stress levels. This result suggested that for all strain levels, the ventricularis layer is dominated by circumferentially oriented collagen fibers, and the initial loading phase of this layer cannot be modeled as an isotropic material. Histological-based thickness measurements indicated that the fibrosa and ventricularis constitute 41% and 29% of the total layer thickness, respectively. Moreover, the extensive network of interlayer connections and identical strains under biaxial loading in the intact state suggests that these layers are tightly bonded. In addition to advancing our knowledge of the subtle but important mechanical properties of the AV leaflet, this study provided a comprehensive database required for the development of a true 3D stress constitutive model for the native AV leaflet.     

A constitutive model for the mechanical behaviour of soft connective tissues

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

Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading

Detailed understanding of cardiac mechanics depends upon accurate and complete characterization of the three-dimensional properties of both normal and diseased myocardial tissue. This, however, can only be obtained by performing multiaxial tests on cardiac tissue. In this study we subjected thin sheets of passive canine left ventricular myocardium to various combinations of simultaneous biaxial stretching. During each stretch the ratio of the orthogonal strains was kept constant and the corresponding stresses remained proportional. We fitted the biaxial stress-strain data both with exponential strain-energy functions with quadratic powers of strains as well as with an alternative function with nonintegral powers of strains. We used our recently developed nonparametric method to assess the reliability of the coefficients for each of these functions. The quadratic strain-energy functions resulted in wide intra- and interspecimen variability in the coefficients. Moreover, both their absolute and relative values demonstrated marked load history dependence such that interpretation of the direction of anisotropy was difficult. Fitting the data with the alternative nonintegral strain-energy function seemed to alleviate these problems. This alternative strain-energy function may provide more self-consistent results than the more commonly used quadratic strain-energy functions.     

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

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

Adaptation of a planar microbiaxial optomechanical device for the tubular biaxial microstructural and macroscopic characterization of small vascular tissues

Murine models of disease are a powerful tool for researchers to gain insight into disease formation, progression, and therapies. The biomechanical indicators of diseased tissue provide a unique insight into some of these murine models, since the biomechanical properties in scenarios such as aneurysm and Marfan syndrome can dictate tissue failure and mortality. Understanding the properties of the tissue on the macroscopic scale has been shown to be important, as one can then understand the tissue's ability to withstand the high stresses seen in the cardiac pulsatile cycle. Alterations in the biomechanical response can foreshadow prospective mechanical failure of the tissue. These alterations are often seen on the microstructural level, and obtaining detailed information on such changes can offer a better understanding of the phenomena seen on the macroscopic level. Unfortunately, mouse models present problems due to the size and delicate features in the mechanical testing of such tissues. In addition, some smaller arteries in large-animal studies (e.g., coronary and cerebral arteries) can present the same issues, and are sometimes unsuitable for planar biaxial testing. The purpose of this paper is to present a robust method for the investigation of the mechanical properties of small arteries and the classification of the microstructural orientation and degree of fiber alignment. This occurs through the cost-efficient modification of a planar biaxial tester that works in conjunction with a two-photon nonlinear microscope. This system provides a means to further investigate how microstructure and mechanical properties are modified in diseased transgenic animals where the tissue is in small tube form. Several other hard-to-test tubular specimens such as cerebral aneurysm arteries and atherosclerotic coronary arteries can also be tested using the described modular device.     

Studies of biaxial mechanical properties and nonlinear finite element modeling of skin

The objective of this research is to conduct mechanical property studies of skin from two individual but potentially connected aspects. One is to determine the mechanical properties of the skin experimentally by biaxial tests, and the other is to use the finite element method to model the skin properties. Dynamic biaxial tests were performed on 16 pieces of abdominal skin specimen from rats. Typical biaxial stress-strain responses show that skin possesses anisotropy, nonlinearity and hysteresis. To describe the stress-strain relationship in forms of strain energy function, the material constants of each specimen were obtained and the results show a high correlation between theory and experiments. Based on the experimental results, a finite element model of skin was built to model the skin's special properties including anisotropy and nonlinearity. This model was based on Arruda and Boyce's eight-chain model and Bischoff et al.'s finite element model of skin. The simulation results show that the isotropic, nonlinear eight-chain model could predict the skin's anisotropic and nonlinear responses to biaxial loading by the presence of an anisotropic prestress state.     

Finite element implementation of a generalized Fung-elastic constitutive model for planar soft tissues

Numerical simulations of the anisotropic mechanical properties of soft tissues and tissue-derived biomaterials using accurate constitutive models remain an important and challenging research area in biomechanics. While most constitutive modeling efforts have focused on the characterization of experimental data, only limited studies are available on the feasibility of utilizing those models in complex computational applications. An example is the widely utilized exponential constitutive model proposed by Fung. Although present in the biomechanics literature for several decades, implementation of this model into finite element (FE) simulations has been limited. A major reason for limited numerical implementations are problems associated with inherent numerical instability and convergence. To address this issue, we developed and applied two restrictions for a generalized Fung-elastic constitutive model necessary to achieve numerical stability. These are (1) convexity of the strain energy function, and (2) the condition number of material stiffness matrix set lower than a prescribed value. These constraints were implemented in the nonlinear regression used for constitutive model parameter estimation to the experimental biaxial mechanical data. We then implemented the generalized Fung-elastic model into a commercial FE code (ABAQUS, Pawtucket, RI, USA). Single element and multi-element planar biaxial test simulations were conducted to verify the accuracy and robustness of the implementation. Results indicated that numerical convergence and accurate FE implementation were consistently obtained. The present study thus presents an integrated framework for accurate and robust implementation of pseudo-elastic constitutive models for planar soft tissues. Moreover, since our approach is formulated within a general FE code, it can be straightforwardly adopted across multiple software platforms.