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.     

Identification of regional mechanical anisotropy in soft tissue analogs

In a previous work (Raghupathy and Barocas, 2010, "Generalized Anisotropic Inverse Mechanics for Soft Tissues,"J. Biomech. Eng., 132(8), pp. 081006), a generalized anisotropic inverse mechanics method applicable to soft tissues was presented and tested against simulated data. Here we demonstrate the ability of the method to identify regional differences in anisotropy from full-field displacements and boundary forces obtained from biaxial extension tests on soft tissue analogs. Tissue heterogeneity was evaluated by partitioning the domain into homogeneous subdomains. Tests on elastomer samples demonstrated the performance of the method on isotropic materials with uniform and nonuniform properties. Tests on fibroblast-remodeled collagen cruciforms indicated a strong correlation between local structural anisotropy (measured by polarized light microscopy) and the evaluated local mechanical anisotropy. The results demonstrate the potential to quantify regional anisotropic material behavior on an intact tissue sample.     

Biaxial mechanical properties of the natural and glutaraldehyde treated aortic valve cusp--Part I: Experimental results

To date, there are no constitutive models for either the natural or bioprosthetic aortic valve (AV), in part due to experimental complications related to the AV's small size and heterogeneous fibrous structure. In this study, we developed specialized biaxial testing techniques for the AV cusp, including a method to determine the local structure-strain relationship to assess the effects of boundary tethering forces. Natural and glutaraldehyde (GL) treated cusps were subjected to an extensive biaxial testing protocol in which the ratios of the axial tensions were held at constant values. Results indicated that the local fiber architecture clearly dominated cuspal deformation, and that the tethering effects at the specimen boundaries were negligible. Due to unique aspects of cuspal fiber architecture, the most uniform region of deformation was found at the lower portion as opposed to the center of the cuspal specimen. In general, the circumferential strains were much smaller than the radial strains, indicating a profound degree of mechanical anisotropy, and that natural cusps were significantly more extensible than the GL treated cusps. Strong mechanical coupling between biaxial stretch axes produced negative circumferential strains under equibiaxial tension. Further, the large radial strains observed could not be explained by uncrimping of the collagen fibers, but may be due to large rotations of the highly aligned, circumferential-oriented collagen fibers in the fibrosa. In conclusion, this study provides new insights into the AV cusp's structure-function relationship in addition to requisite data for constitutive modeling.     

Inertia effects on characterization of dynamic response of brain tissue

Modeling and simulation of traumatic brain injury (TBI) resulted from collision or blast loading requires characterization of mechanical response over a wide range of loading rates under valid testing conditions. In this study, mechanical response of fresh bovine brain tissue was studied using the two modified Kolsky bar techniques. Radial deformation behavior of annular specimens, which are typically used to characterize the dynamic uniaxial compressive response of biological tissues, was examined using a modified Kolsky bar and a high speed camera to collect images while the specimen deforms at an axial strain rate of 2000s(-1). The high-speed images revealed inhomogeneous specimen deformation possibly brought about by radial inertia and causing a multi-axial stress state. To acquire valid stress-strain results that can be used to produce constitutive behavior of the soft materials, a novel torsion technique was developed to obtain pure shear response at dynamic loading rates. Experimental results show clear differences in the material response using the two methods. These results indicate that the previously demonstrated annular specimen geometry aimed at reducing inertia induced stress components for high rate soft materials uniaxial-compressive testing may still possess a significant component of radial inertia induced radial stress which consequently caused the observed inhomogeneous deformation in brain tissue test samples.     

A constitutive model for two-dimensional soft tissues and its application to experimental data

A constitutive relation proposed by Shoemaker (Ph.D. dissertation, 1984) to model the mechanical behavior of membraneous or two-dimensional soft tissues is described. Experiments by Schneider (Ph.D. dissertation, 1982) on human skin and Lee et al. (Am. J. Physiol., 249, H222-H230, 1985) on canine pericardium, and the application of the constitutive model to biaxial stress-strain data from these experiments, are discussed. Some experimental data and predictions of the model obtained by curvefitting are presented for comparison. Values of material parameters are also presented. It is concluded that the constitutive model is well able to fit results of individual tests, and that its generality (judged by consistency of parameters from test to test of the same specimen), though not complete, does compare favorably with some other results presented in the literature.     

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.     

Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading-unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah-Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.     

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.     

Biaxial mechanical response of bioprosthetic heart valve biomaterials to high in-plane shear

Utilization of novel biologically-derived biomaterials in bioprosthetic heart valves (BHV) requires robust constitutive models to predict the mechanical behavior under generalized loading states. Thus, it is necessary to perform rigorous experimentation involving all functional deformations to obtain both the form and material constants of a strain-energy density function. In this study, we generated a comprehensive experimental biaxial mechanical dataset that included high in-plane shear stresses using glutaraldehyde treated bovine pericardium (GLBP) as the representative BHV biomaterial. Compared to our previous study (Sacks, JBME, v.121, pp. 551-555, 1999), GLBP demonstrated a substantially different response under high shear strains. This finding was underscored by the inability of the standard Fung model, applied successfully in our previous GLBP study, to fit the high-shear data. To develop an appropriate constitutive model, we utilized an interpolation technique for the pseudo-elastic response to guide modification of the final model form. An eight parameter modified Fung model utilizing additional quartic terms was developed, which fitted the complete dataset well. Model parameters were also constrained to satisfy physical plausibility of the strain energy function. The results of this study underscore the limited predictive ability of current soft tissue models, and the need to collect experimental data for soft tissue simulations over the complete functional range.     

The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta

The biomechanical response of normal and pathologic human abdominal aortic tissue to uniaxial loading conditions is insufficient for the characterization of its three-dimensional (3D) mechanical behavior. Planar biaxial mechanical evaluation allows for 3D constitutive modeling of nearly incompressible tissues, as well as the investigation of the nature of mechanical anisotropy. In the current study, 26 abdominal aortic aneurysm (AAA) tissue samples and 8 age-matched (> 60 years of age) nonaneurysmal abdominal aortic (AA) tissue samples were obtained and tested using a tension-controlled biaxial testing protocol. Graphical response functions (Sun et al., 2003. J. Biomech. Eng. 125, 372-380) were used as a guide to describe the pseudo-elastic response of AA and AAA. Based on the observed pseudo-elastic response, a four-parameter exponential strain energy function developed by Vito (1990. J. Biomech. Eng. 112, 153-159) was used from which both an individual specimen and group material parameter sets were determined for both AA and AAA. Peak Green strain values in the circumferential (Ethetatheta,max) and longitudinal (ELL,max) directions under an equibiaxial tension of 120 N/m were also compared. The strain energy function fit all of the individual specimens well with an average R2 of 0.95 +/- 0.02 and 0.90 +/- 0.02 (mean +/- SEM) for the AA and AAA groups, respectively. The average Ethetatheta,max at 200 N/m equibiaxial tension was found to be significantly smaller for AAAs as compared to AAs (0.07 +/- 0.01 versus 0.13 +/- 0.03, respectively; p < 0.01). There was also a pronounced increase in the circumferential stiffness for AAA tissue as compared to AA tissue, indicating a larger degree of anisotropy for this tissue as compared to age-matched AA tissue. We also observed that the four-parameter Fung-elastic model was not able to fit the AAA tissue mechanical response using physically realistic material parameter values. It was concluded that aneurysmal degeneration of the abdominal aorta is associated with an increase in mechanical anisotropy, with preferential stiffening in the circumferential direction.     

Biaxial mechanical properties of the native and glutaraldehyde-treated aortic valve cusp: Part II--A structural constitutive model

We have formulated the first constitutive model to describe the complete measured planar biaxial stress-strain relationship of the native and glutaraldehyde-treated aortic valve cusp using a structurally guided approach. When applied to native, zero-pressure fixed, and low-pressure fixed cusps, only three parameters were needed to simulate fully the highly anisotropic, and nonlinear in-plane biaxial mechanical behavior. Differences in the behavior of the native and zero- and low-pressure fixed cusps were found to be primarily due to changes in the effective fiber stress-strain behavior. Further, the model was able to account for the effects of small (< 10 deg) misalignments in the cuspal specimens with respect to the biaxial test axes that increased the accuracy of the model material parameters. Although based upon a simplified cuspal structure, the model underscored the role of the angular orientation of the fibers that completely accounted for extreme mechanical anisotropy and pronounced axial coupling. Knowledge of the mechanics of the aortic cusp derived from this model may aid in the understanding of fatigue damage in bioprosthetic heart valves and, potentially, lay the groundwork for the design of tissue-engineered scaffolds for replacement heart valves.     

Simulation of planar soft tissues using a structural constitutive model: Finite element implementation and validation

Computational implementation of physical and physiologically realistic constitutive models is critical for numerical simulation of soft biological tissues in a variety of biomedical applications. It is well established that the highly nonlinear and anisotropic mechanical behaviors of soft tissues are an emergent behavior of the underlying tissue microstructure. In the present study, we have implemented a structural constitutive model into a finite element framework specialized for membrane tissues. We noted that starting with a single element subjected to uniaxial tension, the non-fibrous tissue matrix must be present to prevent unrealistic tissue deformations. Flexural simulations were used to set the non-fibrous matrix modulus because fibers have little effects on tissue deformation under three-point bending. Multiple deformation modes were simulated, including strip biaxial, planar biaxial with two attachment methods, and membrane inflation. Detailed comparisons with experimental data were undertaken to insure faithful simulations of both the macro-level stress–strain insights into adaptations of the fiber architecture under stress, such as fiber reorientation and fiber recruitment. Results indicated a high degree of fidelity and demonstrated interesting microstructural adaptions to stress and the important role of the underlying tissue matrix. Moreover, we apparently resolve a discrepancy in our 1997 study (Billiar and Sacks, 1997. J. Biomech. 30 (7), 753–756) where we observed that under strip biaxial stretch the simulated fiber splay responses were not in good agreement with the experimental results, suggesting non-affine deformations may have occurred. However, by correctly accounting for the isotropic phase of the measured fiber splay, good agreement was obtained. While not the final word, these simulations suggest that affine fiber kinematics for planar collagenous tissues is a reasonable assumption at the macro level. Simulation tools such as these are imperative in the design and simulation of native and engineered tissues.     

Analysis of effects of friction on the deformation behavior of soft tissues in unconfined compression tests

Frictionless specimen/platen contact in unconfined compression tests has traditionally been assumed in determining material properties of soft tissues via an analytical solution. In the present study, the suitability of this assumption was examined using a finite element method. The effect of the specimen/platen friction on the mechanical characteristics of soft tissues in unconfined compression was analyzed based on the published experimental data of three different materials (pigskin, pig brain, and human calcaneal fat). The soft tissues were considered to be nonlinear and viscoelastic; the friction coefficient at the contact interface between the specimens and platens was assumed to vary from 0.0 to 0.5. Our numerical simulations show that the tissue specimens are, due to the specimen/platen friction, not compressed in a uniform stress/strain state, as has been traditionally assumed in analytical analysis. The stress of the specimens obtained with the specimen/platen friction can be greater than those with the frictionless specimen/platen contact by more than 50%, even in well-controlled test conditions.     

An integrated inverse model-experimental approach to determine soft tissue three-dimensional constitutive parameters: application to post-infarcted myocardium

Knowledge of the complete three-dimensional (3D) mechanical behavior of soft tissues is essential in understanding their pathophysiology and in developing novel therapies. Despite significant progress made in experimentation and modeling, a complete approach for the full characterization of soft tissue 3D behavior remains elusive. A major challenge is the complex architecture of soft tissues, such as myocardium, which endows them with strongly anisotropic and heterogeneous mechanical properties. Available experimental approaches for quantifying the 3D mechanical behavior of myocardium are limited to preselected planar biaxial and 3D cuboidal shear tests. These approaches fall short in pursuing a model-driven approach that operates over the full kinematic space. To address these limitations, we took the following approach. First, based on a kinematical analysis and using a given strain energy density function (SEDF), we obtained an optimal set of displacement paths based on the full 3D deformation gradient tensor. We then applied this optimal set to obtain novel experimental data from a 1-cm cube of post-infarcted left ventricular myocardium. Next, we developed an inverse finite element (FE) simulation of the experimental configuration embedded in a parameter optimization scheme for estimation of the SEDF parameters. Notable features of this approach include: (i) enhanced determinability and predictive capability of the estimated parameters following an optimal design of experiments, (ii) accurate simulation of the experimental setup and transmural variation of local fiber directions in the FE environment, and (iii) application of all displacement paths to a single specimen to minimize testing time so that tissue viability could be maintained. Our results indicated that, in contrast to the common approach of conducting preselected tests and choosing an SEDF a posteriori, the optimal design of experiments, integrated with a chosen SEDF and full 3D kinematics, leads to a more robust characterization of the mechanical behavior of myocardium and higher predictive capabilities of the SEDF. The methodology proposed and demonstrated herein will ultimately provide a means to reliably predict tissue-level behaviors, thus facilitating organ-level simulations for efficient diagnosis and evaluation of potential treatments. While applied to myocardium, such developments are also applicable to characterization of other types of soft tissues.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

Biaxial tensile testing and constitutive modeling of human supraspinatus tendon

The heterogeneous composition and mechanical properties of the supraspinatus tendon offer an opportunity for studying the structure-function relationships of fibrous musculoskeletal connective tissues. Previous uniaxial testing has demonstrated a correlation between the collagen fiber angle distribution and tendon mechanics in response to tensile loading both parallel and transverse to the tendon longitudinal axis. However, the planar mechanics of the supraspinatus tendon may be more appropriately characterized through biaxial tensile testing, which avoids the limitation of nonphysiologic traction-free boundary conditions present during uniaxial testing. Combined with a structural constitutive model, biaxial testing can help identify the specific structural mechanisms underlying the tendon's two-dimensional mechanical behavior. Therefore, the objective of this study was to evaluate the contribution of collagen fiber organization to the planar tensile mechanics of the human supraspinatus tendon by fitting biaxial tensile data with a structural constitutive model that incorporates a sample-specific angular distribution of nonlinear fibers. Regional samples were tested under several biaxial boundary conditions while simultaneously measuring the collagen fiber orientations via polarized light imaging. The histograms of fiber angles were fit with a von Mises probability distribution and input into a hyperelastic constitutive model incorporating the contributions of the uncrimped fibers. Samples with a wide fiber angle distribution produced greater transverse stresses than more highly aligned samples. The structural model fit the longitudinal stresses well (median R(2) ≥ 0.96) and was validated by successfully predicting the stress response to a mechanical protocol not used for parameter estimation. The transverse stresses were fit less well with greater errors observed for less aligned samples. Sensitivity analyses and relatively affine fiber kinematics suggest that these errors are not due to inaccuracies in measuring the collagen fiber organization. More likely, additional strain energy terms representing fiber-fiber interactions are necessary to provide a closer approximation of the transverse stresses. Nevertheless, this approach demonstrated that the longitudinal tensile mechanics of the supraspinatus tendon are primarily dependent on the moduli, crimp, and angular distribution of its collagen fibers. These results add to the existing knowledge of structure-function relationships in fibrous musculoskeletal tissue, which is valuable for understanding the etiology of degenerative disease, developing effective tissue engineering design strategies, and predicting outcomes of tissue repair.     

The relation between collagen fibril kinematics and mechanical properties in the mitral valve anterior leaflet

We have recently demonstrated that the mitral valve anterior leaflet (MVAL) exhibited minimal hysteresis, no strain rate sensitivity, stress relaxation but not creep (Grashow et al., 2006, Ann Biomed Eng., 34(2), pp. 315-325; Grashow et al., 2006, Ann Biomed. Eng., 34(10), pp. 1509-1518). However, the underlying structural basis for this unique quasi-elastic mechanical behavior is presently unknown. As collagen is the major structural component of the MVAL, we investigated the relation between collagen fibril kinematics (rotation and stretch) and tissue-level mechanical properties in the MVAL under biaxial loading using small angle X-ray scattering. A novel device was developed and utilized to perform simultaneous measurements of tissue level forces and strain under a planar biaxial loading state. Collagen fibril D-period strain (epsilonD) and the fibrillar angular distribution were measured under equibiaxial tension, creep, and stress relaxation to a peak tension of 90 N/m. Results indicated that, under equibiaxial tension, collagen fibril straining did not initiate until the end of the nonlinear region of the tissue-level stress-strain curve. At higher tissue tension levels, epsilonD increased linearly with increasing tension. Changes in the angular distribution of the collagen fibrils mainly occurred in the tissue toe region. Using epsilonD, the tangent modulus of collagen fibrils was estimated to be 95.5+/-25.5 MPa, which was approximately 27 times higher than the tissue tensile tangent modulus of 3.58+/-1.83 MPa. In creep tests performed at 90 N/m equibiaxial tension for 60 min, both tissue strain and epsilonD remained constant with no observable changes over the test length. In contrast, in stress relaxation tests performed for 90 min epsilonD was found to rapidly decrease in the first 10 min followed by a slower decay rate for the remainder of the test. Using a single exponential model, the time constant for the reduction in collagen fibril strain was 8.3 min, which was smaller than the tissue-level stress relaxation time constants of 22.0 and 16.9 min in the circumferential and radial directions, respectively. Moreover, there was no change in the fibril angular distribution under both creep and stress relaxation over the test period. Our results suggest that (1) the MVAL collagen fibrils do not exhibit intrinsic viscoelastic behavior, (2) tissue relaxation results from the removal of stress from the fibrils, possibly by a slipping mechanism modulated by noncollagenous components (e.g. proteoglycans), and (3) the lack of creep but the occurrence of stress relaxation suggests a "load-locking" behavior under maintained loading conditions. These unique mechanical characteristics are likely necessary for normal valvular function.     

Biaxial mechanical testing of posterior sclera using high-resolution ultrasound speckle tracking for strain measurements

This study aimed to characterize the mechanical responses of the sclera, the white outer coat of the eye, under equal-biaxial loading with unrestricted shear. An ultrasound speckle tracking technique was used to measure tissue deformation through sample thickness, expanding the capabilities of surface strain techniques. Eight porcine scleral samples were tested within 72 h postmortem  . High resolution ultrasound scans of scleral cross-sections along the two loading axes were acquired at 25 consecutive biaxial load levels. An additional repeat of the biaxial loading cycle was performed to measure a third normal strain emulating a strain gage rosette for calculating the in-plane shear. The repeatability of the strain measurements during identical biaxial ramps was evaluated. A correlation-based ultrasound speckle tracking algorithm was used to compute the displacement field and determine the distributive strains in the sample cross-sections. A Fung type constitutive model including a shear term was used to determine the material constants of each individual specimen by fitting the model parameters to the experimental stress–strain data. A non-linear stress–strain response was observed in all samples. The meridian direction had significantly larger strains than that of the circumferential direction during equal-biaxial loadings (P's<0.05$P\text{'}s<0.05$). The stiffness along the two directions was also significantly different (P=0.02) but highly correlated (R²=0.8). These results showed that the mechanical properties of the porcine sclera were nonlinear and anisotropic under biaxial loading. This work has also demonstrated the feasibility of using ultrasound speckle tracking for strain measurements during mechanical testing.     

A planar biaxial constitutive relation for the luminal layer of intra-luminal thrombus in abdominal aortic aneurysms

The rupture risk of abdominal aortic aneurysms (AAAs) is thought to be associated with increased levels of wall stress. Finite element analysis (FEA) allows the prediction of wall stresses in a patient-specific, non-invasive manner. We have recently shown that it is important to include the intra-luminal thrombus (ILT), present in approximately 70% of AAA, into FEA simulations of AAA. All FEA simulations to date assume an isotropic, homogeneous material behavior for this material. The purpose of this work was to investigate the multi-axial biomechanical behavior of ILT and to derive an appropriate constitutive relation. We performed planar biaxial testing on the luminal layer of nine ILT specimens obtained fresh in the operating room (9 patients, mean age 71+/-4.5 years, mean diameter 5.9+/-0.4 cm), and a constitutive relation was derived from this data. Peak stretch and maximum tangential modulus (MTM) values were recorded for the equibiaxial protocol in both the circumferential (theta) and longitudinal (L) directions. Stress contour plots were used to investigate the presence of mechanical anisotropy, after which an appropriate strain energy function was fit to each of the specimen datasets. The peak stretch values for the luminal layer of the ILT were (mean+/-SEM) 1.18+/-0.02 and 1.13+/-0.02 in the theta and L directions, respectively (p=0.14). The MTM values were 20+/-2 and 23+/-3N/cm(2) in the theta and L directions, respectively (p=0.37). From these results and our observation of the symmetry of the stress contour plots for each specimen, we concluded that the use of an isotropic strain energy function for ILT is appropriate. Each specimen data set was then fit to a second-order polynomial strain energy function of the first invariant of the left Cauchy-Green strain tensor, resulting in an accurate fit (average R(2)=0.92+/-0.02; range 0.80-0.99). Comparison of our previously reported, uniaxially derived constitutive relation with the biaxially derived relation derived here shows large differences in the predicted mechanical response, underscoring the importance of the appropriate experimental methods used to derive constitutive relations. Further work is merited in an effort to produce more accurate predictions of wall stresses in patient-specific AAA, and viscoelastic behaviors of the ILT.