Myocardial material parameter estimation

The passive material properties of myocardium play a major role in diastolic performance of the heart. In particular, the shear behaviour is thought to play an important mechanical role due to the laminar architecture of myocardium. We have previously compared a number of myocardial constitutive relations with the aim to extract their suitability for inverse material parameter estimation. The previous study assumed a homogeneous deformation. In the present study we relaxed the homogeneous assumption by implementing these laws into a finite element environment in order to obtain more realistic measures for the suitability of these laws in both their ability to fit a given set of experimental data, as well as their stability in the finite element environment. In particular, we examined five constitutive laws and compare them on the basis of (i) "goodness of fit": how well they fit a set of six shear deformation tests, (ii) "determinability": how well determined the objective function is at the optimal parameter fit, and (iii) "variability": how well determined the material parameters are over the range of experiments. Furthermore, we compared the FE results with those from the previous study.It was found that the same material law as in the previous study, the orthotropic Fung-type "Costa-Law", was the most suitable for inverse material parameter estimation for myocardium in simple shear.     

Myocardial material parameter estimation: a comparison of invariant based orthotropic constitutive equations

This study investigated a number of invariant based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue.

A number of orthotropic laws based on Green strain components and one microstructurally based law have previously been investigated to fit experimental measurements of stress-strain behaviour. Here we extend this investigation to include several recently proposed functional forms, i.e. invariant based orthotropic and transversely isotropic constitutive relations.

These laws were compared on the basis of (i) ‘goodness of fit’: how well they fit a set of six shear deformation tests, (ii) ‘variability’: how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive laws.

It was found that a specific form of the polyconvex type as well as the exponential Fung-type law from the previous study were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.     

A finite element study of invariant-based orthotropic constitutive equations in the context of myocardial material parameter estimation

A previous study investigated a number of invariant-based orthotropic and transversely isotropic constitutive equations for their suitability to fit three-dimensional simple shear mechanics data of passive myocardial tissue. The study was based on the assumption of a homogeneous deformation. Here, we extend the previous study by performing an inverse finite element material parameter estimation. This ensures a more realistic deformation state and material parameter estimates. The constitutive relations were compared on the basis of (i) ‘goodness of fit’: how well they fit a set of six shear deformation tests and (ii) ‘variability’: how well determined the material parameters are over the range of experiments. These criteria were utilised to discuss the advantages and disadvantages of the constitutive relations. It was found that a specific form of the polyconvex type as well as the exponential Fung-type equations were most suitable for modelling the orthotropic behaviour of myocardium under simple shear.     

Mechanical properties and microstructure of intraluminal thrombus from abdominal aortic aneurysm.

Accurate estimation of the wall stress distribution in an abdominal aortic aneurysm (AAA) may prove clinically useful by predicting when a particular aneurysm will rupture. Appropriate constitutive models for both the wall and the intraluminal thrombus (ILT) found in most AAA are necessary for this task. The purpose of this work was to determine the mechanical properties of ILT within AAA and to derive a more suitable constitutive model for this material. Uniaxial tensile testing was carried out on 50 specimens, including 14 longitudinally oriented and 14 circumferentially oriented specimens from the luminal region of the ILT, and 11 longitudinally oriented and 11 circumferentially oriented specimens from the medial region. A two-parameter, large-strain, hyperelastic constitutive model was developed and used to fit the uniaxial tensile testing data for determination of the material parameters. Maximum stiffness and strength were also determined from the data for each specimen. Scanning electron microscopy (SEM) was conducted to study the regional microstructural difference. Our results indicate that the microstructure of ILT differs between the luminal, medial, and abluminal regions, with the luminal region stronger and stiffer than the medial region. In all cases, the constitutive model fit the experimental data very well (R2>0.98). No significant difference was found for either of the two material parameters between longitudinal and circumferential directions, but a significant difference in material parameters, stiffness, and strength between the laminal and medial regions was determined (p<0.01). Therefore, our results suggest that ILT is an inhomogeneous and possibly isotropic material. The two-parameter, hyperelastic, isotropic, incompressible material model derived here for ILT can be easily incorporated into finite element models for simulation of wall stress distribution in AAA.     

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.     

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 behavior of fetal dura mater under large deformation biaxial tension

The mechanical behavior of fetal dura mater was investigated by means of a biaxial tension test designed to simulate the constraints imposed on the membrane by the cranial bones. The experimental results are compared with the theoretical results obtained by using two published strain energy functions: one defined by Mooney and Rivlin (MR) and the other by Skalak, Tozeren, Zarda and Chien (STZC). The latter constitutive relations fit the experimental results consistently well. The STZC stiffness values from this series of tests are compared with those from membrane inflation tests performed previously and reported elsewhere by the authors.     

A model of growth and rupture of abdominal aortic aneurysm

We present here a coupled mathematical model of growth and failure of the abdominal aortic aneurysm (AAA). The failure portion of the model is based on the constitutive theory of softening hyperelasticity where the classical hyperelastic law is enhanced with a new constant indicating the maximum energy that an infinitesimal material volume can accumulate without failure. The new constant controls material failure and it can be interpreted as the average energy of molecular bonds from the microstructural standpoint. The constitutive model is compared to the data from uniaxial tension tests providing an excellent fit to the experiment. The AAA failure model is coupled with a phenomenological theory of soft tissue growth. The unified theory includes both momentum and mass balance laws coupled with the help of the constitutive equations. The microstructural alterations in the production of elastin and remodeling of collagen are reflected in the changing macroscopic parameters characterizing tissue stiffness, strength and density. The coupled theory is used to simulate growth and rupture of an idealized spherical AAA. The results of the simulation showing possible AAA ruptures in growth are reasonable qualitatively while the quantitative calibration of the model will require further clinical observations and in vitro tests. The presented model is the first where growth and rupture are coupled.     

All human Na(+)-K(+)-ATPase alpha-subunit isoforms have a similar affinity for cardiac glycosides

Three-subunit isoforms of the sodium pump, which is the receptor forcardiac glycosides, are expressed in human heart. The aim of this studywas to determine whether these isoforms have distinct affinities forthe cardiac glycoside ouabain. Equilibrium ouabain binding to membranesfrom a panel of different human tissues and cell lines derived fromhuman tissues was compared by an F statistic to determinewhether a single population of binding sites or two populations ofsites with different affinities would better fit the data. For alltissues, the single-site model fit the data as well as the two-sitemodel. The mean equilibrium dissociation constant(K_d) for all samples calculated using thesingle-site model was 18 ± 6 nM (mean ± SD). No differencein K_d was found between nonfailing and failinghuman heart samples, although the maximum number of binding sites infailing heart was only ~50% of the number of sites in nonfailingheart. Measurement of association rate constants and dissociation rateconstants confirmed that the binding affinities of the different human-isoforms are similar to each other, although calculatedK_d values were lower than those determined byequilibrium binding. These results indicate both that the affinity ofall human -subunit isoforms for ouabain is similar and that theincreased sensitivity of failing human heart to cardiac glycosides isprobably due to a reduction in the number of pumps in the heart ratherthan to a selective inhibition of a subset of pumps with differentaffinities for the drugs.

     

A porous-medium approach for modeling heart mechanics. I. theory

A method is developed for evaluating the distribution of the blood pressure, stresses and strains of the muscle fibers, and motion of the cardiac wall due to the cyclic contractions of the heart. The cardiac system is subdivided into two media: the chambers and the wall; the latter is enclosed by two impermeable surfaces (with one interface separating the two media and the other confining the wall). The momentum balance equation for the blood in the (two) cardiac ventricles is averaged, yielding a modification of Forchheimer's law, namely inclusion of the time derivative of the flux. The contracting muscle provides the driving force for the blood flow, and the endocardial velocity is thus taken as identical to that of the blood in the cavity next to the wall. Translation of the endocardium is governed by the blood pressure gradients in the ventricles. The blood pressure and stress-strain pattern in the myocardium are analyzed by applying concepts of the theory of mixtures to the blood and to the saturated solid matrix. With the blood pressure simulated by a modified Darcy law with relative fluid-solid velocity, fiber stresses and strains can be assessed with the aid of appropriate constitutive and compatibility laws.     

Oscillatory shear loading of bovine periodontal ligament--a methodological study

This study examined the stress response of bovine periodontal ligament (PDL) under sinusoidal straining. The principle of the test consisted in subjecting transverse tooth, PDL and bone sections of known geometries to controlled oscillatory force application. The samples were secured to the actuator by support plates fabricated using a laser sintering technique to fit their contours to the tooth and the alveolar bone. The actuator was attached to the root slices located in the specimen's center. Hence the machine was able to push or pull the root relative to its surrounding alveolar bone. After determining an optimal distraction amplitude, the samples were cyclically loaded first in ramps and then in sinusoidal oscillations at frequencies ranging from 0.2 to 5 Hz. In the present study the following observations were made: (1) Imaging and the laser sintering technique can be used successfully to fabricate custom-made support plates for cross-sectional root-PDL-bone sections using a laser sintering technique, (2) the load-response curves were symmetric in the apical and the coronal directions, (3) both the stress response versus phase angle and the stress response versus. strain curves tended to "straighten" with increasing frequency, and (4) the phase lag between applied strain and resulting stress was small and did not differ in the intrusive and the extrusive directions. As no mechanical or time-dependent anisotropy was demonstrable in the intrusive and extrusive directions, such results may considerably simplify the development of constitutive laws for the PDL.     

A first principles derivation of animal group size distributions

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.     

Effect of Matrix on Cardiomyocyte Viscoelastic Properties in 2D Culture

Cardiomyocyte phenotype changes significantly in 2D culture systems depending on the substrate composition and organization. Given the variety of substrates that are used both for basic cardiac cell culture studies and for regenerative medicine applications, there is a critical need to understand how the different matrices influence cardiac cell mechanics. In the current study, the mechanical properties of neonatal rat cardiomyocytes cultured in a subconfluent layer upon aligned and unaligned collagen and fibronectin matrices were assessed over a two week period using atomic force microscopy. The elastic modulus was estimated by fitting the Hertz model to force curve data and the percent relaxation was determined from stress relaxation curves. The Quasilinear Viscoelastic (QLV) and Standard Linear Solid (SLS) models were fit to the stress relaxation data. Cardiomyocyte cellular mechanical properties were found to be highly dependent on matrix composition and organization as well as time in culture. It was observed that the cells stiffened and relaxed less over the first 3 to 5 days in culture before reaching a plateau in their mechanical properties. After day 5, cells on aligned matrices were stiffer than cells on unaligned matrices and cells on fibronectin matrices were stiffer than cells on collagen matrices. No such significant trends in percent relaxation measurements were observed but the QLV model fit the data very well. These results were correlated with observed changes in cellular structure associated with culture on the different substrates and analyzed for cell-to-cell variability.     

A model of prebiotic replication: survival of the fittest versus extinction of the unfittest.

A model of prebiotic replication discussed by Szathmary and Maynard Smith (1997) is refined in accordance with Lifson's (1997) theory on the crucial stages in the origin of animate matter. The refined model accounts for two processes associated with replication: (1) Replicators always decompose at some rate. (2) Replicators deplete their reactants at a rate equal to the rate of their own replication. Consequently, replicators and their reactants are linked by a non-linear feedback process that keeps them within limits and leads them toward a steady state. The model suggests that: (a) All competing replicators, including those which replicate at sub-exponential "parabolic" and super-exponential "hyperbolic" rates, are subject to natural selection. (b) Survival/extinction is determined by positive/negative net-replication irrespective of the mechanism of replication. (c) Being fit/unfit is the consequence of survival/extinction rather than its cause. In other words, natural selection and survival of the fit is the outcome of continual extinction of the unfit.     

Predicting maintenance or achievement of healthy weight in children: the impact of changes in physical fitness

Physical fitness is often inversely associated with adiposity in children cross-sectionally, but the effect of becoming fit or maintaining fitness over time on changes in weight status has not been well studied in children. We investigated the impact of changes in fitness over 1-4 years of follow-up on the maintenance or achievement of healthy weight among 2,793 schoolchildren who were first measured as 1st to 7th graders. Students were classified as "fit" or "underfit" according to age- and gender-specific norms in five fitness domains: endurance, agility, flexibility, upper body strength, and abdominal strength. Weight status was dichotomized by BMI percentile: "healthy weight" (<85th percentile) or "overweight/obese" (≥85th percentile). At baseline, of the 38.3% overweight/obese children, 81.9% (N = 875) were underfit. Underfit overweight students were more likely to achieve healthy weight if they achieved fitness (boys: odds ratio (OR) = 2.68, 95% confidence interval (CI) = 1.24-5.77; girls: OR = 4.67, 95%CI = 2.09-10.45). Initially fit overweight children (N = 194) were more likely to achieve healthy weight if they maintained fitness (boys: OR = 11.99, 95%CI = 2.18-65.89; girls: OR = 2.46, 95%CI = 1.04-5.83). Similarly, initially fit healthy-weight children (N = 717) were more likely to maintain healthy weight if they maintained fitness (boys: OR 3.70, 95%CI = 1.40-9.78; girls: OR = 4.14, 95%CI = 1.95-8.78). Overweight schoolchildren who achieve or maintain physical fitness are more likely to achieve healthy weight, and healthy-weight children who maintain fitness are more likely to maintain healthy weight. School-based policies/practices that support physical fitness may contribute to obesity reduction and maintenance of healthy weight among schoolchildren.