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.     

Changes in the rheological properties of blood vessel tissue remodeling in the course of development of diabetes.

Rheological properties of blood vessels are expected to change in disease process if the structure of the vessel wall changes. This is illustrated in diabetes, which can be induced in rat by a single injection of Streptozocin. One of the rheological properties of the blood vessel is the stress-strain relationship. The nonlinear stress-strain relationship of arteries is best expressed as derivations of a strain-energy function. In this paper, the stress-strain relations are measured and the coefficients in the strain energy function of arteries are determined for diabetic and control rats. The meaning of these coefficients are explained. The influence of diabetes on the elastic property of the arteries is expressed by the changes of these coefficients. A point of departure of the present paper from all other blood vessel papers published so far is that all strains used here are referred to the zero-stress state of the arteries, whereas all other papers refer strains to the no-load state. The existence of a large difference between the zero-stress state and no-load state of arteries is one of our recent findings. We have explained that the use of zero-stress state as a basis of strain measurements reveals that the in vivo circumferential stress distribution is quite uniform in the vessel wall at the homeostatic condition. It also makes the strain energy function much more accurate than those in which the residual stress is ignored. Using these new results, the stress and strain distribution in normal and diabetic arteries are presented.     

Viscoelastic material properties of the myocardium and cardiac jelly in the looping chick heart

Accurate material properties of developing embryonic tissues are a crucial factor in studies of the mechanics of morphogenesis. In the present work, we characterize the viscoelastic material properties of the looping heart tube in the chick embryo through nonlinear finite element modeling and microindentation experiments. Both hysteresis and ramp-hold experiments were performed on the intact heart and isolated cardiac jelly (extracellular matrix). An inverse computational method was used to determine the constitutive relations for the myocardium and cardiac jelly. With both layers assumed to be quasilinear viscoelastic, material coefficients for an Ogden type strain-energy density function combined with Prony series of two terms or less were determined by fitting numerical results from a simplified model of a heart segment to experimental data. The experimental and modeling techniques can be applied generally for determining viscoelastic material properties of embryonic tissues.     

Material properties and residual stress in the stage 12 chick heart during cardiac looping

During the morphogenetic process of cardiac looping, the initially straight cardiac tube bends and twists into a curved tube. The biophysical mechanisms that drive looping remain unknown, but the process clearly involves mechanical forces. Hence, it is important to determine mechanical properties of the early heart, which is a muscle-wrapped tube consisting primarily of a thin outer layer of myocardium surrounding a thick extracellular matrix compartment known as cardiac jelly. In this work, we used microindentation experiments and finite element modeling, combined with an inverse computational method, to determine constitutive relations for the myocardium and cardiac jelly at the outer curvature of stage 12 chick hearts. Material coefficients for exponential strain-energy density functions were found by fitting force-displacement and surface displacement data near the indenter Residual stress in the myocardium also was estimated. These results should be useful for computational models of the looping heart.     

Limiting models for calcification in fibrous tissues adjacent to orthopedic implants: Variational indicator functions and influences of implant stiffness

Calcification and eventual integration of orthopedic implants into bone is important to many load-bearing devices, and the influence of load and implant stiffness on this process are assessed in this mathematical modelling study. Three research questions are posed in this study. First, can limiting material models provide useful information on the overall behavior of the tissue adjacent to a loaded orthopedic implant? Second, can the limiting models lead to optimization criteria? Third, can an optimization approach be used to differentiate between the four prospective remodeling rate equations which are proposed? The answers are yes, yes, and no, respectively. A two degree of freedom lumped parameter model for axial loading of an intramedullary implant is considered. Two limiting composite material models are used, and the strain energy density in the calcified and non-calcified phases are assessed as stimuli for calcification. The rate equations posed here assume that the calcified material volume fraction decreases at high strain-energy densities, and increases at small strain-energy densities. In all four cases (both models, both phases) the steady states for these rate equations find equilibrium points of indicator functions which are a weighted sum of total strain energy and the mass of calcified tissue in the layer considered. The weights on strain-energy density and mass differ in each case. This shows that for appropriate choices of parameters, all four models can yield the same results, and it also shows that an optimization approach does not uniquely determine the appropriate rate equation in these cases. The rate equations showed complicated dynamic behavior and a phase-plane analysis was used which led to upper bounds on load, which depended on implant stiffness and distal support. The predictions of the four cases studied are compared.     

Deficiencies in numerical models of anisotropic nonlinearly elastic materials

Incompressible nonlinearly hyperelastic materials are rarely simulated in finite element numerical experiments as being perfectly incompressible because of the numerical difficulties associated with globally satisfying this constraint. Most commercial finite element packages therefore assume that the material is slightly compressible. It is then further assumed that the corresponding strain-energy function can be decomposed additively into volumetric and deviatoric parts. We show that this decomposition is not physically realistic, especially for anisotropic materials, which are of particular interest for simulating the mechanical response of biological soft tissue. The most striking illustration of the shortcoming is that with this decomposition, an anisotropic cube under hydrostatic tension deforms into another cube instead of a hexahedron with non-parallel faces. Furthermore, commercial numerical codes require the specification of a ‘compressibility parameter’ (or ‘penalty factor’), which arises naturally from the flawed additive decomposition of the strain-energy function. This parameter is often linked to a ‘bulk modulus’, although this notion makes no sense for anisotropic solids; we show that it is essentially an arbitrary parameter and that infinitesimal changes to it result in significant changes in the predicted stress response. This is illustrated with numerical simulations for biaxial tension experiments of arteries, where the magnitude of the stress response is found to change by several orders of magnitude when infinitesimal changes in ‘Poisson’s ratio’ close to the perfect incompressibility limit of 1/2 are made.     

Dynamic finite element implementation of nonlinear,anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.     

Collective human mobility pattern from taxi trips in urban area

We analyze the passengers' traffic pattern for 1.58 million taxi trips of Shanghai, China. By employing the non-negative matrix factorization and optimization methods, we find that, people travel on workdays mainly for three purposes: commuting between home and workplace, traveling from workplace to workplace, and others such as leisure activities. Therefore, traffic flow in one area or between any pair of locations can be approximated by a linear combination of three basis flows, corresponding to the three purposes respectively. We name the coefficients in the linear combination as traffic powers, each of which indicates the strength of each basis flow. The traffic powers on different days are typically different even for the same location, due to the uncertainty of the human motion. Therefore, we provide a probability distribution function for the relative deviation of the traffic power. This distribution function is in terms of a series of functions for normalized binomial distributions. It can be well explained by statistical theories and is verified by empirical data. These findings are applicable in predicting the road traffic, tracing the traffic pattern and diagnosing the traffic related abnormal events. These results can also be used to infer land uses of urban area quite parsimoniously.     

Mechanical stresses in abdominal aortic aneurysms: influence of diameter, asymmetry, and material anisotropy

Biomechanical studies suggest that one determinant of abdominal aortic aneurysm (AAA) rupture is related to the stress in the wall. In this regard, a reliable and accurate stress analysis of an in vivo AAA requires a suitable 3D constitutive model. To date, stress analysis conducted on AAA is mainly driven by isotropic tissue models. However, recent biaxial tensile tests performed on AAA tissue samples demonstrate the anisotropic nature of this tissue. The purpose of this work is to study the influence of geometry and material anisotropy on the magnitude and distribution of the peak wall stress in AAAs. Three-dimensional computer models of symmetric and asymmetric AAAs were generated in which the maximum diameter and length of the aneurysm were individually controlled. A five parameter exponential type structural strain-energy function was used to model the anisotropic behavior of the AAA tissue. The anisotropy is determined by the orientation of the collagen fibers (one parameter of the model). The results suggest that shorter aneurysms are more critical when asymmetries are present. They show a strong influence of the material anisotropy on the magnitude and distribution of the peak stress. Results confirm that the relative aneurysm length and the degree of aneurysmal asymmetry should be considered in a rupture risk decision criterion for AAAs.     

In situ intercellular mechanics of the bovine outer annulus fibrosus subjected to biaxial strains

In situ intercellular strains in the outer annulus fibrosus of bovine caudal discs were determined under two states of biaxial strain. Confocal microscopy was used to track and capture images of fluorescently labelled nuclei at applied Lagrangian strains in the axial direction (E(A)(S)) of 0%, 7.5% and 15% while the circumferential direction (E(C)(S)) was constrained to either 0% or -2.5%. The position of the nuclear centroids were calculated in each image and used to investigate the in situ intercellular mechanics of both lamellar and interlamellar cells. The intercellular Lagrangian strains measured in situ were non-uniform and did not correspond with the biaxial Lagrangian strains applied to the tissue. A row-oriented analysis of intercellular unit displacements within the lamellar layers found that the magnitudes of unit displacements between cells along a row (delta;(II)) were small (|delta;(IIavg)|=1.6% at E(C)(S)=0%, E(A)(S)=15%; |delta;(IIavg)|=3.0% at E(C)(S)=-2.5%, E(A)(S)=15%) with negative unit displacements occurring greater than one-third of the time. Evidence of interlamellar shear and increased intercellular Lagrangian strains among the cells within the interlamellar septa suggested that their in situ mechanical environment may be more complex. The in situ intercellular strains of annular cells were strongly dependent upon the local structure and behaviour of the extracellular matrix and did not correspond with applied tissue strains. This knowledge has immediate relevance for in vitro investigations of disc mechanobiology, and will also provide a base to investigate the mechanical implications of disc degeneration at the cellular level.     

Review: total doses in fractionated radiotherapy--implications of new radiobiological data

The total dose in radiotherapy has been adjusted in the past for different fractionation schedules by the use of empirical formulae such as NSD, TDF and CRE. It is now appropriate to consider fractionation factors which include more biological insight in their formulation than was possible earlier. It has become clear, from both clinical and experimental animal data, that the total dose in multi-fraction irradiations depends more critically on size of dose-per-fraction for late than for early damage to normal tissues. This difference has been interpreted as due to different shapes of the underlying dose-response curves. The late reactions respond with more curvature in the dose-response curve, i.e. with more repair capability at very low doses per fraction, than the early tissue reactions. A linear-quadratic relationship for the dose-response curves has been found to fit experimental data well, with few exceptions. This paper reviews this interpretation and explores some of its implications for radiotherapy and for radiobiology applied to therapy. Of many repair factors that have been suggested, the ratio alpha/beta (of the linear to the quadratic coefficients) is one that should be independent of the level of damage assayed. Values of alpha/beta of about 10 Gy have been reported for a number of early tissue responses but a range of values from about 1 to 5 or 6 Gy for late responses. It is a current challenge to radiobiology to explain why this difference occurs. Once such values are known for different tissues--and the dangers of premature assumptions are emphasized--calculations are possible which might be useful in radiotherapy as an alternative to NSD, TDF, CRE etc. Some data are presented on the magnitude of differences from these previously used empirical formulae, with a discussion about how easily detected the discrepancies might be in clinical practice. Applications to hypofractionation, hyperfractionation and accelerated fractionation are illustrated.     

Determination of material models for arterial walls from uniaxial extension tests and histological structure

An approach is proposed that allows the determination of material models from uniaxial tests and histostructural data including fiber orientation of the tissue. A combination of neo-Hookean and Fung-type strain-energy functions is utilized, and inequality constraints imposed on the constitutive parameters are derived providing strict local convexity and preferred fiber orientations. It is shown how the Fung-type model gets a pseudo-structural aspect inherent in the phenomenological model; a correlation between the fiber structure and the parameters of the Fung-type model is explicitly provided. In order to apply the proposed approach, quasi-static uniaxial extension tests of preconditioned prepared strips from the intima, media and adventitia of a human aorta with non-atherosclerotic intimal thickening are acquired in axial and circumferential directions; structural information from histological analyses for each aortic tissue are documented. Data reveal a remarkable thickness, load-bearing capacity and stiffness of the intimal samples in comparison with the media and adventitia. Constitutive parameters for each aortic tissue layer are determined by solving the constrained problem using a penalty function method; a new approach for the estimation of appropriate start values is proposed. Finally, the predictivity and efficacy of the material models is shown by comparing model data with data from the uniaxial extension tests and histological image analyses.     

A mathematical model to detect inspiratory flow limitation during sleep.

The physiological significance of inspiratory flow limitation (IFL) has recently been recognized, but methods of detecting IFL can be subjective. We sought to develop a mathematical model of the upper airway pressure-flow relationship that would objectively detect flow limitation. We present a theoretical discussion that predicts that a polynomial function [F(P) = AP(3) + BP(2) + CP + D, where F(P) is flow and P is supraglottic pressure] best characterizes the pressure-flow relationship and allows for the objective detection of IFL. In protocol 1, step 1, we performed curve-fitting of the pressure-flow relationship of 20 breaths to 5 mathematical functions and found that highest correlation coefficients (R(2)) for quadratic (0.88 +/- 0.10) and polynomial (0.91 +/- 0.05; P < 0.05 for both compared with the other functions) functions. In step 2, we performed error-fit calculations on 50 breaths by comparing the quadratic and polynomial functions and found that the error fit was lowest for the polynomial function (3.3 +/- 0.06 vs. 21.1 +/- 19.0%; P < 0.001). In protocol 2, we performed sensitivity/specificity analysis on two sets of breaths (50 and 544 breaths) by comparing the mathematical determination of IFL to manual determination. Mathematical determination of IFL had high sensitivity and specificity and a positive predictive value (>99% for each). We conclude that a polynomial function can be used to predict the relationship between pressure and flow in the upper airway and objectively determine the presence of IFL.     

An anisotropic constitutive model for immersogeometric fluid–structure interaction analysis of bioprosthetic heart valves

This paper considers an anisotropic hyperelastic soft tissue model, originally proposed for native valve tissue and referred to herein as the Lee–Sacks model, in an isogeometric thin shell analysis framework that can be readily combined with immersogeometric fluid–structure interaction (FSI) analysis for high-fidelity simulations of bioprosthetic heart valves (BHVs) interacting with blood flow. We find that the Lee–Sacks model is well-suited to reproduce the anisotropic stress–strain behavior of the cross-linked bovine pericardial tissues that are commonly used in BHVs. An automated procedure for parameter selection leads to an instance of the Lee–Sacks model that matches biaxial stress–strain data from the literature more closely, over a wider range of strains, than other soft tissue models. The relative simplicity of the Lee–Sacks model is attractive for computationally-demanding applications such as FSI analysis and we use the model to demonstrate how the presence and direction of material anisotropy affect the FSI dynamics of BHV leaflets.     

TRs have common and isoform-specific functions in regulation of the cardiac myosin heavy chain genes.

TRalpha1 and TRbeta mediate the regulatory effects of T3 and have profound effects on the cardiovascular system. We have analyzed the expression of the cardiac myosin heavy chain (MyHC) genes alpha and beta in mouse strains deficient for one or several TR genes to identify specific regulatory functions of TRalpha1 and TRbeta. The results show that TRalpha1 deficiency, which slows the heart rate, causes chronic overexpression of MyHCbeta. However, MyHCbeta was still suppressible by T3 in both TRalpha1- and TRbeta-deficient mice, indicating that either receptor can mediate repression of MyHCbeta. T3-dependent induction of the positively regulated MyHCalpha gene was similar in both TRalpha1- and TRbeta-deficient mice. The data identify a specific role for TRalpha1 in the negative regulation of MyHCbeta, whereas TRalpha1 and TRbeta appear interchangeable for hormone-dependent induction of MyHCalpha. This suggests that TR isoforms exhibit distinct specificities in the genes that they regulate within a given tissue type. Thus, dysregulation of MyHCbeta is likely to contribute to the critical role of TRalpha1 in cardiac function.     

On constitutive relations and finite deformations of passive cardiac tissue: I. A pseudostrain-energy function

A three-dimensional constitutive relation for passive cardiac tissue is formulated in terms of a structurally motivated pseudostrain-energy function, W, while the mathematical simplicity of phenomenological approaches is preserved. A specific functional form of W is proposed on the basis of limited structural information and multiaxial experimental data. The material parameters are determined in a least-squared sense from both uniaxial and biaxial data. Our results suggest that (1) multiaxially-loaded cardiac tissue is nearly transversely-isotropic with respect to local muscle fiber directions, at least for a limited range of strain histories, (2) material parameters determined from uniaxial papillary muscle data result in gross underestimates of the stresses in multiaxially-loaded specimens, and (3) material parameters determined from equibiaxial tests predict the behavior of the tissue under various nonequibiaxial stretching protocols reasonably well.     

Sensitivity of stress and strain calculations to passive material parameters in cardiac mechanical models using unloaded geometries

Cardiac stress (load) and strain (stretch) are widely studied indicators of cardiac function and outcome, but are difficult or impossible to directly measure in relation to the cardiac microstructure. An alternative approach is to estimate these states using computer methods and image-based measurements, but this still requires knowledge of the tissue material properties and the unloaded state, both of which are difficult to determine. In this work, we tested the sensitivity of these two interdependent unknowns (reference geometry and material parameters) on stress and strain calculations in cardiac tissue. Our study used a finite element model of the human ventricle, with a hyperelastic passive material model, and was driven by a cell model mediated active contraction. We evaluated 21 different published parameter sets for the five parameters of the passive material model, and for each set we optimised the corresponding unloaded geometry and contractility parameter to model a single pressure-volume loop. The resulting mechanics were compared, and calculated systolic stresses were largely insensitive to the chosen parameter set when an unloading algorithm was used. Meanwhile, material strain calculations varied substantially depending on the choice of material parameters. These results indicate that determining the correct material and unloaded configuration may be highly important to understand strain driven processes, but less so for calculating stress estimates.     

A nonlinear anisotropic inverse method for computational dissection of inhomogeneous planar tissues

Quantification of the mechanical behavior of soft tissues is challenging due to their anisotropic, heterogeneous, and nonlinear nature. We present a method for the ‘computational dissection’ of a tissue, by which we mean the use of computational tools both to identify and to analyze regions within a tissue sample that have different mechanical properties. The approach employs an inverse technique applied to a series of planar biaxial experimental protocols. The aggregated data from multiple protocols provide the basis for (1) segmentation of the tissue into regions of similar properties, (2) linear analysis for the small-strain behavior, assuming uniform, linear, anisotropic behavior within each region, (3) subsequent nonlinear analysis following each individual experimental protocol path and using local linear properties, and (4) construction of a strain energy data set W(E) at every point in the material by integrating the differential stress–strain functions along each strain path. The approach has been applied to simulated data and captures not only the general nonlinear behavior but also the regional differences introduced into the simulated tissue sample.