A transverse isotropic viscoelastic constitutive model for aortic valve tissue

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 Bayesian constitutive model selection framework for biaxial mechanical testing of planar soft tissues: Application to porcine aortic valves

A variety of constitutive models have been developed for soft tissue mechanics. However, there is no established criterion to select a suitable model for a specific application. Although the model that best fits the experimental data can be deemed the most suitable model, this practice often can be insufficient given the inter-sample variability of experimental observations. Herein, we present a Bayesian approach to calculate the relative probabilities of constitutive models based on biaxial mechanical testing of tissue samples. Forty-six samples of porcine aortic valve tissue were tested using a biaxial stretching setup. For each sample, seven ratios of stresses along and perpendicular to the fiber direction were applied. The probabilities of eight invariant-based constitutive models were calculated based on the experimental data using the proposed model selection framework. The calculated probabilities showed that, out of the considered models and based on the information available through the utilized experimental dataset, the May-Newman model was the most probable model for the porcine aortic valve data. When the samples were further grouped into different cusp types, the May-Newman model remained the most probable for the left- and right-coronary cusps, whereas for non-coronary cusps two models were found to be equally probable: the Lee-Sacks model and the May-Newman model. This difference between cusp types was found to be associated with the first principal component analysis (PCA) mode, where this mode's amplitudes of the non-coronary and right-coronary cusps were found to be significantly different. Our results show that a PCA-based statistical model can capture significant variations in the mechanical properties of soft tissues. The presented framework is applicable to other tissue types, and has the potential to provide a structured and rational way of making simulations population-based.

Different Passive Viscoelastic Properties Between the Left and Right Ventricles in Healthy Adult Ovine

Ventricle dysfunction is the most common cause of heart failure, which leads to high mortality and morbidity. The mechanical behavior of the ventricle is critical to its physiological function. It is known that the ventricle is anisotropic and viscoelastic. However, the understanding of ventricular viscoelasticity is much less than that of its elasticity. Moreover, the left and right ventricles (LV&RV) are different in embryologic origin, anatomy, and function, but whether they distinguish in viscoelastic properties is unclear. We hypothesized that passive viscoelasticity is different between healthy LVs and RVs. Ex vivo cyclic biaxial tensile mechanical tests (1, 0.1, 0.01 Hz) and stress relaxation (strain of 3, 6, 9, 12, 15%) were performed for ventricles from healthy adult sheep. Outflow track direction was defined as the longitudinal direction. Hysteresis stress-strain loops and stress relaxation curves were obtained to quantify the viscoelastic properties. We found that the RV had more pronounced frequency-dependent viscoelastic changes than the LV. Under the physiological frequency (1 Hz), the LV was more anisotropic in the elasticity and stiffer than the RV in both directions, whereas the RV was more anisotropic in the viscosity and more viscous than the LV in the longitudinal direction. The LV was quasi-linear viscoelastic in the longitudinal but not circumferential direction, and the RV was nonlinear viscoelastic in both directions. This study is the first to investigate passive viscoelastic differences in healthy LVs and RVs, and the findings will deepen the understanding of biomechanical mechanisms of ventricular function.

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.

Mechanical considerations for polymeric heart valve development: Biomechanics, materials, design and manufacturing

The native human heart valve leaflet contains a layered microstructure comprising a hierarchical arrangement of collagen, elastin, proteoglycans and various cell types. Here, we review the various experimental methods that have been employed to probe this intricate microstructure and which attempt to elucidate the mechanisms that govern the leaflet's mechanical properties. These methods include uniaxial, biaxial, and flexural tests, coupled with microstructural characterization techniques such as small angle X-ray scattering (SAXS), small angle light scattering (SALS), and polarized light microscopy. These experiments have revealed complex elastic and viscoelastic mechanisms that are highly directional and dependent upon loading conditions and biochemistry. Of all engineering materials, polymers and polymer-based composites are best able to mimic the tissue-level mechanical behavior of the native leaflet. This similarity to native tissue permits the fabrication of polymeric valves with physiological flow patterns, reducing the risk of thrombosis compared to mechanical valves and in some cases surpassing the in vivo durability of bioprosthetic valves. Earlier work on polymeric valves simply assumed the mechanical properties of the polymer material to be linear elastic, while more recent studies have considered the full hyperelastic stress-strain response. These material models have been incorporated into computational models for the optimization of valve geometry, with the goal of minimizing internal stresses and improving durability. The latter portion of this review recounts these developments in polymeric heart valves, with a focus on mechanical testing of polymers, valve geometry, and manufacturing methods.

Simulation of personalised haemodynamics by various mounting positions of a prosthetic valve using computational fluid dynamics

Diseases of the cardiovascular system account for nearly 42% of all deaths in the European Union. In Germany, approximately 12,000 patients receive surgical replacement of the aortic valve due to heart valve disease alone each year. A three-dimensional (3D) numerical model based on patient-specific anatomy derived from four-dimensional (4D) magnetic resonance imaging (MRI) data was developed to investigate preoperatively the flow-induced impact of mounting positions of aortic prosthetic valves to select the best orientation for individual patients. Systematic steady-state analysis of blood flow for different rotational mounting positions of the valve is only possible using a virtual patient model. A maximum velocity of 1 m/s was used as an inlet boundary condition, because the opening angle of the valve is at its largest at this velocity. For a comparative serial examination, it is important to define the standardised general requirements to avoid impacts other than the rotated implantation of the prosthetic aortic valve. In this study, a uniform velocity profile at the inlet for the inflow of the aortic valve and the real aortic anatomy were chosen for all simulations. An iterative process, with the weighted parameters flow resistance (1), shear stress (2) and velocity (3), was necessary to determine the best rotated orientation. Blood flow was optimal at a 45° rotation from the standard implantation orientation, which will offer a supply to the coronary arteries.

Elastin-Dependent Aortic Heart Valve Leaflet Curvature Changes During Cyclic Flexure

The progression of calcific aortic valve disease (CAVD) is characterized by extracellular matrix (ECM) remodeling, leading to structural abnormalities and improper valve function. The focus of the present study was to relate aortic valve leaflet axial curvature changes as a function of elastin degradation, which has been associated with CAVD. Circumferential rectangular strips (L × W = 10 × 2.5 mm) of normal and elastin-degraded (via enzymatic digestion) porcine AV leaflets were subjected to cyclic flexure (1 Hz). A significant increase in mean curvature (p < 0.05) was found in elastin-degraded leaflet specimens in comparison to un-degraded controls at both the semi-constrained (50% of maximum flexed state during specimen bending and straightening events) and fully-constrained (maximally-flexed) states. This significance did not occur in all three flexed configurations when measurements were performed using either minimum or maximum curvature. Moreover, the mean curvature increase in the elastin-degraded leaflets was most pronounced at the instance of maximum flexure, compared to un-degraded controls. We conclude that the mean axial curvature metric can detect distinct spatial changes in aortic valve ECM arising from the loss in bulk content and/or structure of elastin, particularly when there is a high degree of tissue bending. Therefore, the instance of maximum leaflet flexure during the cardiac cycle could be targeted for mean curvature measurements and serve as a potential biomarker for elastin degradation in early CAVD remodeling.

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 study of the damage behaviour of porcine intervertebral discs in a bioreactor environment

The intervertebral discs are cartilaginous, articulating structures that lie between vertebral bodies, allowing flexibility, transmission, modification, and also distribution of the forces to the spinal column. Disc degeneration is characterised by progressive loss of disc height and exaggerated radial bulging. Therefore, the spine becomes shorter, stiffer, and less mobile. In the last several decades, there is a strong need for a tissue engineering strategy that alleviates pain and restores spine function by directly addressing the underlying biological causes of disc degeneration. Numerous studies that are currently showing potential have been conducted on developing regenerative and reparative strategies for treating this condition. In this study, to numerically describe the anisotropic mechanical damage behaviour of discs, the pseudo-elastic damage model was applied. To experimentally picture the biomechanical response of discs and to study the damage mechanisms as well as the spinal disc herniation, a special bioreactor was evolved. The specimens were obtained from pigs aged six months. A total of eight functional spine units were taken from porcine lumbar spines (L1-L2). Firstly, the experiments were performed by using long-term cyclic uniaxial compression tests. Secondly, the mean value of experimental results with consideration of the different shapes and sizes of the samples was calculated. Afterwards, the experimental results were compared with outcomes of numerical simulations.