A versatile biaxial testing platform for soft tissues.

Uniaxial testing remains the most common modality of mechanical analysis for biological and other soft materials; however, biaxial testing enables a more comprehensive understanding of these materials' mechanical behavior. In recent years, a number of commercially available biaxial testing systems designed for biological materials have been produced; however, there are common limitations that are often associated with using these systems. For example, the range of allowable sample geometries are relatively constrained, the clamping systems are relatively limited with respect to allowable configurations, the load and displacement ranges are relatively small, and the software and control elements offer relatively limited options. Due to these constraints, there are significant benefits associated with designing custom biaxial testing systems that meet the technical requirements for testing a broad range of materials. Herein we present a design for a biaxial testing system with capabilities that extend beyond those associated with typical commercially available systems. Our design is capable of performing uniaxial tests, traditional biaxial tests, and double lap shear (simple shear) tests, in either a displacement or load control mode. Testing protocols have been developed and proof-of-concept experiments have been performed on commercially available silicone membranes and rat abdominal skin samples.

Characterizing the non-linear mechanical behavior of native and biomimetic engineered tissues in 1D with physically meaningful parameters.

It is common practice to evaluate the mechanical performance of a scaffold for tissue engineering using concepts from linear elasticity theory (i.e. Young's modulus), or variations thereof, and uniaxial testing data. In some cases the non-linear nature of tissue stress-strain behavior has prompted development of empirical approaches to obtain a more comprehensive description of the observed mechanical behavior. Such approaches constitute improvements over singular stiffness measures but the lack of an appropriate non-linear theoretical foundation renders them somewhat arbitrary and potentially incomplete. Recently, a constitutive model for non-linear tissues was developed based on first principles in physics. The Freed-Rajagopal 1-D Fiber Model incorporates physically meaningful parameters that provide a unique and comprehensive characterization of non-linear tissue behavior for the class of tissues with strain limiting behavior in 1D. The physical interpretation that these parameters provide suggests they may serve as useful design targets for tissue engineering applications. In this study, the Freed-Rajagopal model is employed with conventional uniaxial mechanical testing data obtained from experiments with collagen scaffolds for hernia repair grafts and the healthy native tissue counterpart. Results from the Freed-Rajagopal analysis revealed that tissue-engineered constructs that qualify as "biomimetic" according to linear elasticity theory, or variations thereof, are not truly biomimetic, as they do not mimic the non-linear mechanical behaviors observed in their native tissue counterparts. Most importantly, the Freed-Rajagopal model was easy to employ (it can be done using a standard uniaxial testing system, with minimal additional effort) and revealed specific design improvements that could be targeted to improve the biofidelity of these constructs. A performance comparison with conventional non-linear models (including Fung's 1D Law and a one-dimensionalized version of the Holzapfel, Gasser, Ogden model), was then conducted and revealed the Freed-Rajagopal model produced results that correlated exceptionally well with experimental data and better describes material behavior at low strains as compared to competing models.

An Implicit Elastic Theory for Lung Parenchyma.

The airways and parenchyma of lung experience large deformations during normal respiration. Spatially accurate predictions of airflow patterns and aerosol transport therefore require respiration to be modeled as a fluid-structure interaction problem. Such computational models in turn require constitutive models for the parencyhma that are both accurate and efficient. Herein, an implicit theory of elasticity is derived from thermodynamics to meet this need, leading to a generic template for strain-energy that is shown to be an exact analogue for the well-known Fung model that is the root of modern constitutive theory of tissues. To support this theory, we also propose a novel definition of Lagrangian strain rate. Unlike the classic definition of Lagrangian strain rate, this new definition is separable into volumetric and deviatoric terms, a separation that is both mathematically and physically justified. Within this framework, a novel material model capable of describing the elastic contribution of the nonlinear response of parenchyma is constructed and characterized against published data.

Hypo-elastic model for lung parenchyma.

A simple, isotropic, elastic constitutive model for the spongy tissue in lung is formulated from the theory of hypo-elasticity. The model is shown to exhibit a pressure dependent behavior that has been interpreted in the literature as indicating extensional anisotropy. In contrast, we show that this behavior arises naturally from an analysis of isotropic hypo-elastic invariants and is a result of non-linearity, not anisotropy. The response of the model is determined analytically for several boundary value problems used for material characterization. These responses give insight into both the material behavior as well as admissible bounds on parameters. The model predictions are compared with published experimental data for dog lung.

Hypoelastic Soft Tissues: Part II: In-Plane Biaxial Experiments.

In Part I, a novel hypoelastic framework for soft-tissues was presented. One of the hallmarks of this new theory is that the well-known exponential behavior of soft-tissues arises consistently and spontaneously from the integration of a rate based formulation. In Part II, we examine the application of this framework to the problem of biaxial kinematics, which are common in experimental soft-tissue characterization. We confine our attention to an isotropic formulation in order to highlight the distinction between non-linearity and anisotropy. In order to provide a sound foundation for the membrane extension of our earlier hypoelastic framework, the kinematics and kinetics of in-plane biaxial extension are revisited, and some enhancements are provided. Specifically, the conventional stress-to-traction mapping for this boundary value problem is shown to violate the conservation of angular momentum. In response, we provide a corrected mapping. In addition, a novel means for applying loads to in-plane biaxial experiments is proposed. An isotropic, isochoric, hypoelastic, constitutive model is applied to an in-plane biaxial experiment done on glutaraldehyde treated bovine pericardium. The experiment is comprised of eight protocols that radially probe the biaxial plane. Considering its simplicity (two adjustable parameters) the model does a reasonably good job of describing the non-linear normal responses observed in these experimental data, which are more prevalent than are the anisotropic responses exhibited by this tissue.

Maturational and adaptive modulation of left ventricular torsional biomechanics: Doppler tissue imaging observation from infancy to adulthood.

BACKGROUND: Left ventricular (LV) torsional deformation, based in part on the helical myocardial fiber architecture, is an important component of LV systolic and diastolic performance. However, there is no comprehensive study describing its normal development during childhood and adult life. METHODS AND RESULTS: Forty-five normal subjects (25 children and 20 adults; aged 9 days to 49 years; divided into 5 groups: infants, children, adolescents, and young and middle-age adults) underwent assessment of LV torsion and untwisting rate by Doppler tissue imaging. LV torsion increased with age, primarily owing to augmentation in basal clockwise rotation during childhood and apical counterclockwise rotation during adulthood. Although LV torsion and untwisting overall showed age-related increases, when normalized by LV length, they showed higher values in infancy and middle age. The proportion of untwisting during isovolumic relaxation was lowest in infancy, increased during childhood, and leveled off thereafter, whereas peak untwisting performance (peak untwisting velocity normalized by peak LV torsion) showed a decrease during adulthood. CONCLUSIONS: We have shown the maturational process of LV torsion in normal subjects. Net LV torsion increases gradually from infancy to adulthood, but the determinants of this were different in the 2 age groups. The smaller LV isovolumic untwisting recoil during infancy and its decline in adulthood may suggest mechanisms for alterations in diastolic function.

Invariant formulation for dispersed transverse isotropy in aortic heart valves: an efficient means for modeling fiber splay.

Most soft tissues possess an oriented architecture of collagen fiber bundles, conferring both anisotropy and nonlinearity to their elastic behavior. Transverse isotropy has often been assumed for a subset of these tissues that have a single macroscopically-identifiable preferred fiber direction. Micro-structural studies, however, suggest that, in some tissues, collagen fibers are approximately normally distributed about a mean preferred fiber direction. Structural constitutive equations that account for this dispersion of fibers have been shown to capture the mechanical complexity of these tissues quite well. Such descriptions, however, are computationally cumbersome for two-dimensional (2D) fiber distributions, let alone for fully three-dimensional (3D) fiber populations. In this paper, we develop a new constitutive law for such tissues, based on a novel invariant theory for dispersed transverse isotropy. The invariant theory is derived from a novel closed-form 'splay invariant' that can easily handle 3D fiber populations, and that only requires a single parameter in the 2D case. The model fits biaxial data for aortic valve tissue as accurately as the standard structural model. Modification of the fiber stress-strain law requires no reformulation of the constitutive tangent matrix, making the model flexible for different types of soft tissues. Most importantly, the model is computationally expedient in a finite-element analysis, demonstrated by modeling a bioprosthetic heart valve.

Elastic model for crimped collagen fibrils.

A physiologic constitutive expression is presented in algorithmic format for the nonlinear elastic response of wavy collagen fibrils found in soft connective tissues. The model is based on the observation that crimped fibrils in a fascicle have a three-dimensional structure at the micron scale that we approximate as a helical spring. The symmetry of this wave form allows the force/displacement relationship derived from Castigliano's theorem to be solved in closed form: all integrals become analytic. Model predictions are in good agreement with experimental observations for mitral-valve chordae tendinece.

Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity.

BACKGROUND: Quasilinear viscoelasticity (QLV) theory has been widely and successfully used to describe the time-dependent response of connective tissues. Difficulties remain, however, particularly in material parameter estimation and sensitivities. In this study, we introduce a new alternative: the fractional order viscoelasticity (FOV) theory, which uses a fractional order integral to describe the relaxation response. FOV implies a fractal-like tissue structure, reflecting the hierarchical arrangement of collagenous tissues. METHOD OF APPROACH: A one-dimensional (I-D) FOV reduced relaxation function was developed, replacing the QLV "box-spectrum" function with a fractional relaxation function. A direct-fit, global optimization method was used to estimate material parameters from stress relaxation tests on aortic valve tissue. RESULTS: We found that for the aortic heart valve, FOV had similar accuracy and better parameter sensitivity than QLV, particularly for the long time constant (tau2). The mean (n = 5) fractional order was 0.29, indicating that the viscoelastic response of the tissue was strongly fractal-like. RESULTS SUMMARY: mean QLV parameters were C = 0.079, tau1 = 0.004, tau2 = 76, and mean FOV parameters were beta = 0.29, tau = 0.076, and rho = 1.84. CONCLUSIONS: FOV can provide valuable new insights into tissue viscoelastic behavior Determining the fractional order can provide a new and sensitive quantitative measure for tissue comparison.

Inverse parameter fitting of biological tissues: a response surface approach.

In this paper, we present the application of a semi-global inverse method for determining material parameters of biological tissues. The approach is based on the successive response surface method, and is illustrated by fitting constitutive parameters to two nonlinear anisotropic constitutive equations, one for aortic sinus and aortic wall, the other for aortic valve tissue. Material test data for the aortic sinus consisted of two independent orthogonal uniaxial tests. Material test data for the aortic valve was obtained from a dynamic inflation test. In each case, a numerical simulation of the experiment was performed and predictions were compared to the real data. For the uniaxial test simulation, the experimental targets were force at a measured displacement. For the inflation test, the experimental targets were the three-dimensional coordinates of material markers at a given pressure. For both sets of tissues, predictions with converged parameters showed excellent agreement with the data, and we found that the method was able to consistently identify model parameters. We believe the method will find wide application in biomedical material characterization and in diagnostic imaging.