Continuous Softening as a State of Hyperelasticity: Examples of Application to the Softening Behavior of the Brain Tissue

The continuous softening behavior of the brain tissue, i.e., the softening in the primary loading path with an increase in deformation, is modeled in this work as a state of hyperelasticity up to the onset of failure. That is, the softening behavior is captured via a core hyperelastic model without the addition of damage variables and/or functions. Examples of the application of the model will be provided to extant datasets of uniaxial tension and simple shear deformations, demonstrating the capability of the model to capture the whole-range deformation of the brain tissue specimens, including their softening behavior. Quantitative and qualitative comparisons with other models within the brain biomechanics literature will also be presented, showing the clear advantages of the current approach. The application of the model is then extended to capturing the rate-dependent softening behavior of the tissue by allowing the parameters of the core hyperelastic model to evolve, i.e., vary, with the deformation rate. It is shown that the model captures the rate-dependent and softening behaviors of the specimens favorably and also predicts the behavior at other rates. These results offer a clear set of advantages in favor of the considered modeling approach here for capturing the quasi-static and rate-dependent mechanical properties of the brain tissue, including its softening behavior, over the existing models in the literature, which at best may purport to capture only a reduced set of the foregoing behaviors, and with ill-posed effects.

Modelling the rate-dependent mechanical behaviour of the brain tissue

A new modelling approach is employed in this work for application to the rate-dependent mechanical behaviour of the brain tissue, as an incompressible isotropic material. Extant datasets encompassing single- and multi-mode compression, tension and simple shear deformation(s) are considered, across a wide range of deformation rates from quasi-static to rates akin to blast loading conditions, in the order of 1000 s-1 . With a simple functional form and a reduced number of parameters, the model is shown to capture the considered rate-dependent behaviours favourably, including in both single- and multi-mode deformation fits, and over all range of deformation rates. The provided modelling results here are obtained from either first fitting the model to the quasi-static data, or/and predicting the behaviour at a different rate than those used for calibrating the model parameters. Given its simplicity, versatility, predictive capability and accuracy, the application of the utilised modelling framework in this work to the rate-dependent mechanical behaviour of the brain tissue is proposed.

A New Dissipation Function To Model The Rate-Dependent Mechanical Behaviour Of Semilunar Valve Leaflets

A new dissipation function Wv is devised and presented to capture the rate-dependent mechanical behaviour of the semilunar heart valves. Following the experimentally-guided framework introduced in our previous work (J. Mech. Behav. Biomed. Mater. (2022), https://doi.org/10.1016/j.jmbbm.2022.105341), we derive our proposed Wv function from the experimental data pertaining to the biaxial deformation of the aortic and pulmonary valve specimens across a 10,000-fold range of deformation rate, exhibiting two distinct rate-dependent features: (i) the stiffening effect in s-? curves with increase in rate; and (ii) the asymptotic effect of rate on stress levels at higher rates. The devised Wv function is then used in conjunction with a hyperelastic strain energy function We to model the rate-dependent behaviour of the valves, incorporating the rate of deformation as an explicit variable. It is shown that the devised function favourably captures the observed rate-dependent features, and the model provides excellent fits to the experimentally obtained σ-λ curves. The proposed function is thereby recommended for application to the rate-dependent mechanical behaviour of heart valves, as well as other soft tissues that exhibit a similar rate-dependent behaviour.

Modelling brain tissue elasticity with the Ogden model and an alternative family of constitutive models†

The Ogden model is often considered as a standard model in the literature for application to the deformation of brain tissue. Here, we show that, in some of those applications, the use of the Ogden model leads to the non-convexity of the strain-energy function and mis-prediction of the correct concavity of the experimental stress-stretch curves over a range of the deformation domain. By contrast, we propose a family of models which provides a favourable fit to the considered datasets while remaining free from the highlighted shortcomings of the Ogden model. While, as we discuss, those shortcomings might be due to the artefacts of the testing protocols, the proposed family of models proves impervious to such artefacts. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

New results in the theory of plane strain flexure of incompressible isotropic hyperelastic materials

New results on the classical problem of bending by end moments for incompressible isotropic hyperelastic materials within the framework of nonlinear elasticity are investigated and presented in this paper. The particular results of concern here include (i) the adaptation of Rivlin's standard analysis to the case where one end of the beam is fixed and the other end is subjected to a bending moment; and (ii) results on the finite bending of (infinitesimally) thin isotropic hyperelastic plates which are valid for large deformations , extending the classical results from the linear elasticity theory which are restricted to small deformations. An interesting feature observed in this context is that a flexed thin plate develops an oscillatory surface along the circular arc near the free end, due to local (small) deviations of the radius of curvature. A potential application to the bending of a biological soft tissue, namely the aortic valve leaflet, is briefly described by way of an example. Finally, some new results are obtained for finite bending of hyperelastic materials that exhibit limiting chain extensibility at the molecular level and involve constraints on the deformation. The amount of bending that such materials can sustain is limited by the constraint. On using a limiting chain extensibility model, closed-form solutions for the Cauchy stress components, the bending moment and the normal out-of-plane force required to sustain the bending deformation are derived.

ASSESSMENT OF A NEW ISOTROPIC HYPERELASTIC CONSTITUTIVE MODEL FOR A RANGE OF RUBBERLIKE MATERIALS AND DEFORMATIONS

The choice of an appropriate strain energy function W is key to accurate modeling and computational finite element analysis of the mechanical behavior of unfilled non-crystalizing rubberlike materials. Despite the existing variety of models, finding a suitable model that can capture many deformation modes of a rubber specimen with a single set of parameter values and satisfy the a priori mathematical and structural requirements remains a formidable task. Previous work proposed a new generalized neo-Hookean W (I1) function, showing a promising fitting capability and enjoying a structural basis. We now use two extended forms of that model that include an I1 term adjunct, W (I1, I2), for application to various boundary value problems commonly encountered in rubber mechanics applications. Specifically, two functional forms of the I2 invariant are considered: a linear function and a logarithmic function. The boundary value problems of interest include the in-plane uniaxial, equi-biaxial, and pure shear deformations and simple shear, inflation, and nonhomogeneous deformations such as torsion. By simultaneous fitting of each model to various deformation modes of rubber specimens, it is demonstrated that a single set of model parameter values favorably captures the mechanical response for all the considered deformations of each specimen. It is further shown that the model with a logarithmic I2 function provides better fits than the linear function. Given the functional simplicity of the considered W (I1, I2) models, the low number of model parameters (three in total), the structurally motivated bases of the models, and their capability to capture the mechanical response for various deformations of rubber specimens, the considered models are recommended as a powerful tool for practical applications and analysis of rubber elasticity.

Rate-dependent mechanical behaviour of semilunar valves under biaxial deformation: From quasi-static to physiological loading rates

In this study we investigate the rate-dependency of the mechanical behaviour of semilunar heart valves under biaxial deformation, from quasi-static to physiological loading rates. This work extends and complements our previous undertaking, where the rate-dependency in the mechanical behaviour of semilunar valve specimens was documented in sub-physiological rate domains (Acta Biomater. 2019; https://doi.org/10.1016/j.actbio.2019.02.008). For the first time we demonstrate herein that the stress-stretch curves obtained from specimens under physiological rates too are markedly different to those at sufficiently lower rates and at quasi-static conditions. The results importantly underline that the mechanical behaviour of semilunar heart valves is rate dependent, and the physiological mechanical behaviour of the valves may not be correctly obtained via material characterisation tests at arbitrary low deformation rates. Presented results in this work provide an inclusive dataset for material characterisation and modelling of semilunar heart valves across a 10,000 fold deformation rate, both under equi-biaxial and 1:3 ratio deformation rates. The important application of these results is to inform the development of appropriate mechanical testing protocols, as well as devising new models, for suitable determination of the rate-dependent constitutive mechanical behaviour of the semilunar valves.

A transverse isotropic constitutive model for the aortic valve tissue incorporating rate-dependency and fibre dispersion: Application to biaxial deformation

This paper presents a continuum-based transverse isotropic model incorporating rate-dependency and fibre dispersion, applied to the planar biaxial deformation of aortic valve (AV) specimens under various stretch rates. The rate dependency of the mechanical behaviour of the AV tissue under biaxial deformation, the (pseudo-) invariants of the right Cauchy-Green deformation-rate tensor Ċ associated with fibre dispersion, and a new fibre orientation density function motivated by fibre kinematics are presented for the first time. It is shown that the model captures the experimentally observed deformation of the specimens, and characterises a shear-thinning behaviour associated with the dissipative (viscous) kinematics of the matrix and the fibres. The application of the model for predicting the deformation behaviour of the AV under physiological rates is illustrated and an example of the predicted σ-λ curves is presented. While the development of the model was principally motivated by the AV biomechanics requisites, the comprehensive theoretical approach employed in the study renders the model suitable for application to other fibrous soft tissues that possess similar rate-dependent and structural attributes.

Modeling the Deformation of the Elastin Network in the Aortic Valve

This paper is concerned with proposing a suitable structurally motivated strain energy function, denoted by Weelastin network, for modeling the deformation of the elastin network within the aortic valve (AV) tissue. The AV elastin network is the main noncollagenous load-bearing component of the valve matrix, and therefore, in the context of continuum-based modeling of the AV, the Weelastin network strain energy function would essentially serve to model the contribution of the “isotropic matrix.” To date, such a function has mainly been considered as either a generic neo-Hookean term or a general exponential function. In this paper, we take advantage of the established structural analogy between the network of elastin chains and the freely jointed molecular chain networks to customize a structurally motivated Weelastin network function on this basis. The ensuing stress–strain (force-stretch) relationships are thus derived and fitted to the experimental data points reported by (Vesely, 1998, “The Role of Elastin in Aortic Valve Mechanics,” J. Biomech., 31, pp. 115–123) for intact AV elastin network specimens under uniaxial tension. The fitting results are then compared with those of the neo-Hookean and the general exponential models, as the frequently used models in the literature, as well as the “Arruda–Boyce” model as the gold standard of the network chain models. It is shown that our proposed Weelastin network function, together with the general exponential and the Arruda–Boyce models provide excellent fits to the data, with R2 values in excess of 0.98, while the neo-Hookean function is entirely inadequate for modeling the AV elastin network. However, the general exponential function may not be amenable to rigorous interpretation, as there is no structural meaning attached to the model. It is also shown that the parameters estimated by the Arruda–Boyce model are not mathematically and structurally valid, despite providing very good fits. We thus conclude that our proposed strain energy function Weelastin network is the preferred choice for modeling the behavior of the AV elastin network and thereby the isotropic matrix. This function may therefore be superimposed onto that of the anisotropic collagen fibers family in order to develop a structurally motivated continuum-based model for the AV.

A transverse isotropic viscoelastic constitutive model for aortic valve tissue

A new anisotropic viscoelastic model is developed for application to the aortic valve (AV). The directional dependency in the mechanical properties of the valve, arising from the predominantly circumferential alignment of collagen fibres, is accounted for in the form of transverse isotropy. The rate dependency of the valve's mechanical behaviour is considered to stem from the viscous (η) dissipative effects of the AV matrix, and is incorporated as an explicit function of the deformation rate ([Formula: see text]). Model (material) parameters were determined from uniaxial tensile deformation tests of porcine AV specimens at various deformation rates, by fitting the model to each experimental dataset. It is shown that the model provides an excellent fit to the experimental data across all different rates and satisfies the condition of strict local convexity. Based on the fitting results, a nonlinear relationship between η and [Formula: see text] is established, highlighting a 'shear-thinning' behaviour for the AV with increase in the deformation rate. Using the model and these outcomes, the stress-deformation curves of the AV tissue under physiological deformation rates in both the circumferential and radial directions are predicted and presented. To verify the predictive capabilities of the model, the stress-deformation curves of AV specimens at an intermediate deformation rate were estimated and validated against the experimental data at that rate, showing an excellent agreement. While the model is primarily developed for application to the AV, it may be applied without the loss of generality to other collagenous soft tissues possessing a similar structure, with a single preferred direction of embedded collagen fibres.

Evaluation of bioprosthetic heart valve failure using a matrix-fibril shear stress transfer approach

A matrix-fibril shear stress transfer approach is devised and developed in this paper to analyse the primary biomechanical factors which initiate the structural degeneration of the bioprosthetic heart valves (BHVs). Using this approach, the critical length of the collagen fibrils l c and the interface shear acting on the fibrils in both BHV and natural aortic valve (AV) tissues under physiological loading conditions are calculated and presented. It is shown that the required critical fibril length to provide effective reinforcement to the natural AV and the BHV tissue is l c  = 25.36 µm and l c  = 66.81 µm, respectively. Furthermore, the magnitude of the required shear force acting on fibril interface to break a cross-linked fibril in the BHV tissue is shown to be 38 µN, while the required interfacial force to break the bonds between the fibril and the surrounding extracellular matrix is 31 µN. Direct correlations are underpinned between these values and the ultimate failure strength and the failure mode of the BHV tissue compared with the natural AV, and are verified against the existing experimental data. The analyses presented in this paper explain the role of fibril interface shear and critical length in regulating the biomechanics of the structural failure of the BHVs, for the first time. This insight facilitates further understanding into the underlying causes of the structural degeneration of the BHVs in vivo.

Atherosclerotic plaques: is endothelial shear stress the only factor?

Initiation and development of atherosclerosis has largely been attributed to irregular shear stress patterns and values, in the current literature. Abnormalities such as low shear stress, reversing and oscillatory shear force patterns, as well as temporal variations of shear stress are the most cited factors. However, clinical findings have further indicated that plaques have still been formed and developed in arterial sites that possess relatively more steady and higher shear stresses than those observed in studies correlating low or oscillatory shear stresses with atherosclerosis. These data imply that deviations in shear stress from its normal physiological pattern alone may not be the only factor inducing atherosclerosis, and additional haemodynamics parameter other then shear stress may also contribute to the initiation and development of plaques. In this paper, we hypothesise that the combined effect of wall shear stress and circumferential stress waves, in the form of angular phase difference between the two waves at each cardiac cycle, may be a more accurate determinant of plaque formation and growth. Furthermore, arterial sites that possess more positive values of this angular phase difference may be more prone to plaque formation or development. If proved correct, this theory can transform our understanding of endothelial cell mechanotransduction and mechanobiology, and may lead to design and utilisation of new diagnostic procedures and equipment as predictive and preventive clinical tools for patients with abnormal arterial blood pressure.

Strain Transfer Through the Aortic Valve

The complex structural organization of the aortic valve (AV) extracellular matrix (ECM) enables large and highly nonlinear tissue level deformations. The collagen and elastin (elastic) fibers within the ECM form an interconnected fibrous network (FN) and are known to be the main load-bearing elements of the AV matrix. The role of the FN in enabling deformation has been investigated and documented. However, there is little data on the correlation between tissue level and FN-level strains. Investigating this correlation will help establish the mode of strain transfer (affine or nonaffine) through the AV tissue as a key feature in microstructural modeling and will also help characterize the local FN deformation across the AV sample in response to applied tissue level strains. In this study, the correlation between applied strains at tissue level, macrostrains across the tissue surface, and local FN strains were investigated. Results showed that the FN strain distribution across AV samples was inhomogeneous and nonuniform, as well as anisotropic. There was no direct transfer of the deformation applied at tissue level to the fibrous network. Loading modes induced in the FN are different than those applied at the tissue as a result of different local strains in the valve layers. This nonuniformity of local strains induced internal shearing within the FN of the AV, possibly exposing the aortic valve interstitial cells (AVICs) to shear strains and stresses.

A combined experimental and modelling approach to aortic valve viscoelasticity in tensile deformation

The quasi-static mechanical behaviour of the aortic valve (AV) is highly non-linear and anisotropic in nature and reflects the complex collagen fibre kinematics in response to applied loading. However, little is known about the viscoelastic behaviour of the AV. The aim of this study was to investigate porcine AV tissue under uniaxial tensile deformation, in order to establish the directional dependence of its viscoelastic behaviour. Rate dependency associated with different mechanical properties was investigated, and a new viscoelastic model incorporating rate effects developed, based on the Kelvin-Voigt model. Even at low applied loads, experimental results showed rate dependency in the stress-strain response, and also hysteresis and dissipation effects. Furthermore, corresponding values of each parameter depended on the loading direction. The model successfully predicted the experimental data and indicated a 'shear-thinning' behaviour. By extrapolating the experimental data to that at physiological strain rates, the model predicts viscous damping coefficients of 8.3 MPa s and 3.9 MPa s, in circumferential and radial directions, respectively. This implies that the native AV offers minimal resistance to internal shear forces induced by blood flow, a potentially critical design feature for substitute implants. These data suggest that the mechanical behaviour of the AV cannot be thoroughly characterised by elastic deformation and fibre recruitment assumptions alone.