Structural three-dimensional constitutive law for the passive myocardium

A three-dimensional constitutive law is proposed for the myocardium. Its formulation is based on a structural approach in which the total strain energy of the tissue is the sum of the strain energies of its constituents: the muscle fibers, the collagen fibers and the fluid matrix which embeds them. The ensuing material law expresses the specific structural and mechanical properties of the tissue, namely, the spatial orientation of the comprising fibers, their waviness in the unstressed state and their stress-strain behavior when stretched. Having assumed specific functional forms for the distribution of the fibers spatial orientation and waviness, the results of biaxial mechanical tests serve for the estimation of the material constants appearing in the constitutive equations. A very good fit is obtained between the measured and the calculated stresses, indicating the suitability of the proposed model for describing the mechanical behavior of the passive myocardium. Moreover, the results provide general conclusions concerning the structural basis for the tissue overall mechanical properties, the main of which is that the collagen matrix, though comprising a relatively small fraction of the whole tissue volume, is the dominant component accounting for its stiffness.     

Development of an in vivo method for determining material properties of passive myocardium

Calculation of mechanical stresses and strains in the left ventricular (LV) myocardium by the finite element (FE) method relies on adequate knowledge of the material properties of myocardial tissue. In this paper, we present a model-based estimation procedure to characterize the stress-strain relationship in passive LV myocardium. A 3D FE model of the LV myocardium was used, which included morphological fiber and sheet structure and a nonlinear orthotropic constitutive law with different stiffness in the fiber, sheet, and sheet-normal directions. The estimation method was based on measured wall strains. We analyzed the method's ability to estimate the material parameters by generating a set of synthetic strain data by simulating the LV inflation phase with known material parameters. In this way we were able to verify the correctness of the solution and to analyze the effects of measurement and model error on the solution accuracy and stability. A sensitivity analysis was performed to investigate the observability of the material parameters and to determine which parameters to estimate. The results showed a high degree of coupling between the parameters governing the stiffness in each direction. Thus, only one parameter in each of the three directions was estimated. For the tested magnitudes of added noise and introduced model errors, the resulting estimated stress-strain characteristics in the fiber and sheet directions converged with good accuracy to the known relationship. The sheet-normal stress-strain relationship had a higher degree of uncertainty as more noise was added and model error was introduced.     

A modified Holzapfel-Ogden law for a residually stressed finite strain model of the human left ventricle in diastole

In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.     

Theoretical models in mechanics of the left ventricle

Different rheological concepts and theoretical studies have been recently presented using models of myocardial mechanics. Complex analysis of the mechanical behavior of the left ventricular wall have been developed in order to estimate the local stresses and deformations that occur during the heart cycle as well as the ventricular stroke volume and pressure. Theoretical models have taken into account non-linear and viscoelastic passive properties of the myocardium tissue, when subjected to large deformations, through given strain energy functions or stress-strain relations. Different prolate spheroid geometries have been considered for such thick shell cardiac structure. During the active state of the contraction, the rheological behavior of the fibers has been described using different muscle models and relationships between fiber tension and strain, and activation degree. A forthcoming approach for bridging the gap between the knowledge of the muscle fiber microrheological properties and the study of the mechanical behavior of the entire ventricle, consists in including anisotropic and inhomogeneous effects through fiber direction field.     

Multiaxial mechanical behavior of the porcine anterior lens capsule

The biomechanics of the lens capsule of the eye is important both in physiologic processes such as accommodation and clinical treatments such as cataract surgery. Although the lens capsule experiences multiaxial stresses in vivo, there have been no measurements of its multiaxial properties or possible regional heterogeneities. Rather all prior mechanical data have come from 1-D pressure–volume or uniaxial force-length tests. Here, we report a new experimental approach to study in situ the regional, multiaxial mechanical behavior of the lens capsule. Moreover, we report multiaxial data suggesting that the porcine anterior lens capsule exhibits a typical nonlinear pseudoelastic behavior over finite strains, that the in situ state is pre-stressed multiaxially, and that the meridional and circumferential directions are principal directions of strain, which is nearly equibiaxial at the pole but less so towards the equator. Such data are fundamental to much needed constitutive formulations.     

Akinetic myocardial infarcts must contain contracting myocytes: finite-element model study

Infarcted segments of myocardium demonstrate functional impairment ranging in severity from hypokinesis to dyskinesis. We sought to better define the contributions of passive material properties (stiffness) and active properties (contracting myocytes) to infarct thickening. Using a finite-element (FE) model, we tested the hypothesis that infarcted myocardium must contain contracting myocytes to be akinetic and not dyskinetic. A three-dimensional FE mesh of the left ventricle was developed with echocardiographs from a reperfused ovine anteroapical infarct. The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction, and an elastance model for active fiber stress was incorporated. The diastolic stiffness (C) and systolic material property (isometric tension at longest sarcomere length and peak intracellular calcium concentration, T(max)) of the uninfarcted remote myocardium were assumed to be normal (C = 0.876 kPa, T(max) = 135.7 kPa). Diastolic and systolic properties of the infarct necessary to produce akinesis, defined as an average radial strain between -0.01 and 0.01, were determined by assigning a range of diastolic stiffnesses and scaling infarct T(max) to represent the percentage of contracting myocytes between 0% and 100%. As C was increased to 11 times normal (C = 10 kPa) the percentage of T(max) necessary for akinesis increased from 20% to 50%. Without contracting myocytes, C = 250 kPa was necessary to achieve akinesis. If infarct stiffness is <285 times normal, contracting myocytes are required to prevent dyskinetic infarct wall motion.     

Influence of dispersion in myosin filament orientation and anisotropic filament contractions in smooth muscle

A new constitutive model for the biomechanical behaviour of smooth muscle tissue is proposed. The active muscle contraction is accomplished by the relative sliding between actin and myosin filaments, comprising contractile units in the smooth muscle cells. The orientation of the myosin filaments, and thereby the contractile units, are taken to exhibit a statistical dispersion around a preferred direction. The number of activated cross-bridges between the actin and myosin filaments governs the contractile force generated by the muscle and also the contraction speed. A strain-energy function is used to describe the mechanical behaviour of the smooth muscle tissue. Besides the active contractile apparatus, the mechanical model also incorporates a passive elastic part. The constitutive model was compared to histological and isometric tensile test results for smooth muscle tissue from swine carotid artery. In order to be able to predict the active stress at different muscle lengths, a filament dispersion significantly larger than the one observed experimentally was required. Furthermore, a comparison of the predicted active stress for a case of uniaxially oriented myosin filaments and a case of filaments with a dispersion based on the experimental histological data shows that the difference in generated stress is noticeable but limited. Thus, the results suggest that myosin filament dispersion alone cannot explain the increase in active muscle stress with increasing muscle stretch.     

A model of the structural and functional development of the normal human fetal left ventricle based on a global growth law

The purpose of this research is to study the growth of the normal human left ventricle (LV) during the fetal period from 14 to 40 weeks of gestation. A new constitutive law for the active myocardium describing the mechanical properties of the active muscle during the whole cardiac cycle has been proposed. The LV model is a thick-walled, incompressible, hyperelastic cylinder, with families of helicoidal fibers running on cylindrical surfaces [1]. Based on the works of Lin and Taber [2] done on the embryonic chick heart, we use for the human fetal heart a growth law in which the growth rate depends on the wall stresses. The parameters of the growth law are adapted to agree with sizes and volumes inferred from two dimensional ultrasound measurements performed on 18 human fetuses.Then calculations are performed to extrapolate the cardiac performance during normal growth of the fetal LV. The results presented support the idea that a growth law in which the growth rate depends linearly on the mean wall stresses averaged through the space and during whole cardiac cycle, is adapted to the normal human fetal LV development.     

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.     

Mechanical feedback in morphogenesis and cell differentiation

Studies of mechanical stresses and mechanical feedback at the cell level are reviewed. It is shown that cells and embryonic tissues respond to external mechanical stresses and can generate such stresses themselves. Regular feedback loops between external (passive) and internal (active) mechanical stresses have been established. They are essential for cell survival, determination of the direction of their differentiation, and selforganization of morphogenetic processes. Relevant experimental data are presented, and models of mechanical feedback loops are discussed.     

A Model of the Structural and Functional Development of the Normal Human Fetal Left Ventricle Based on a Global Growth Law

The purpose of this research is to study the growth of the normal human left ventricle (LV) during the fetal period from 14 to 40 weeks of gestation. A new constitutive law for the active myocardium describing the mechanical properties of the active muscle during the whole cardiac cycle has been proposed. The LV model is a thick-walled, incompressible, hyperelastic cylinder, with families of helicoidal fibers running on cylindrical surfaces [1] . Based on the works of Lin and Taber [2] done on the embryonic chick heart, we use for the human fetal heart a growth law in which the growth rate depends on the wall stresses. The parameters of the growth law are adapted to agree with sizes and volumes inferred from two dimensional ultrasound measurements performed on 18 human fetuses. Then calculations are performed to extrapolate the cardiac performance during normal growth of the fetal LV. The results presented support the idea that a growth law in which the growth rate depends linearly on the mean wall stresses averaged through the space and during whole cardiac cycle, is adapted to the normal human fetal LV development.     

Biaxial vasoactivity of porcine coronary artery

The passive mechanical properties of blood vessel mainly stem from the interaction of collagen and elastin fibers, but vessel constriction is attributed to smooth muscle cell (SMC) contraction. Although the passive properties of coronary arteries have been well characterized, the active biaxial stress-strain relationship is not known. Here, we carry out biaxial (inflation and axial extension) mechanical tests in right coronary arteries that provide the active coronary stress-strain relationship in circumferential and axial directions. Based on the measurements, a biaxial active strain energy function is proposed to quantify the constitutive stress-strain relationship in the physiological range of loading. The strain energy is expressed as a Gauss error function in the physiological pressure range. In K(+)-induced vasoconstriction, the mean ± SE values of outer diameters at transmural pressure of 80 mmHg were 3.41 ± 0.17 and 3.28 ± 0.24 mm at axial stretch ratios of 1.3 and 1.5, respectively, which were significantly smaller than those in Ca(2+)-free-induced vasodilated state (i.e., 4.01 ± 0.16 and 3.75 ± 0.20 mm, respectively). The mean ± SE values of the inner and outer diameters in no-load state and the opening angles in zero-stress state were 1.69 ± 0.04 mm and 2.25 ± 0.08 mm and 126 ± 22°, respectively. The active stresses have a maximal value at the passive pressure of 80-100 mmHg and at the active pressure of 140-160 mmHg. Moreover, a mechanical analysis shows a significant reduction of mean stress and strain (averaged through the vessel wall). These findings have important implications for understanding SMC mechanics.     

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.     

Effects of C-reactive protein on adipokines genes expression in 3T3-L1 adipocytes

Transplantation of skeletal myoblasts (SMs) has been investigated as a potential cardiac cell therapy approach. SM are available autologously, predetermined for muscular differentiation and resistant to ischemia. Major hurdles for their clinical application are limitations in purity and yield during cell isolation as well as the absence of gap junction expression after differentiation into myotubes. Furthermore, transplanted SMs do not functionally or electrically integrate with the host myocardium. Here, we describe an efficient method for isolating homogeneous SM populations from neonatal mice and demonstrate persistent gap junction expression in an engineered tissue. This method resulted in a yield of 1.4 × 10(8) high-purity SMs (>99% desmin positive) after 10 days in culture from 162.12 ± 11.85 mg muscle tissue. Serum starvation conditions efficiently induced differentiation into spontaneously contracting myotubes that coincided with loss of gap junction expression. For mechanical conditioning, cells were integrated into engineered tissue constructs. SMs within tissue constructs exhibited long term survival, ordered alignment, and a preserved ability to differentiate into contractile myotubes. When the tissue constructs were subjected to passive longitudinal tensile stress, the expression of gap junction and cell adherence proteins was maintained or increased throughout differentiation. Our studies demonstrate that mechanical loading of SMs may provide for improved electromechanical integration within the myocardium, which could lead to more therapeutic opportunities.     

Elastic constitutive laws for cow teat tissue undergoing finite deformations

Five constitutive laws are investigated to model the effect of machine milking. A nonlinear least squares procedure is employed to estimate material constants from in vivo teat inflation data. An exponential form is found to be statistically adequate as a constitutive law, and is used to determine the mechanical stresses in teat tissue during finite deformations.     

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.     

Assessing the mechanical energy costs of various tissue reshaping mechanisms

Early-stage embryos must reshape the tissues of which they are made into organs and other life-sustaining structures; and if non-mammalian embryos fail to complete these tasks before the energy provided by their yolk runs out, they die. The aim of this study is to use a cell-level computational model to investigate the energetic cost of a variety of mechanisms that can drive an in-plane reshaping pattern known as convergent extension—a motif in which a tissue narrows in one in-plane direction and expands in another. Mechanisms considered include oriented lamellipodia, directed mitosis, stress fibers, and anisotropic external tension. Both isolated patches of tissue and actively contracting tissues that deform adjacent passive areas are considered. The cell-level finite element model used here assumes that the cell membrane and its associated proteins generate a net tension γ along each cell–cell interface and that the cytoplasm and its embedded networks and structures have an effective viscosity?μ. Work costs are based exclusively on mechanical considerations such as edge lengths and tensions, and because a traditional mechanical efficiency cannot be calculated, mechanisms are compared on the basis of the work they must do to the tissue to cause a specified rate of in-plane reshaping. Although the model contains a number of simplifications compared to real embryonic tissues, it is able to show that the work requirements for tissue reshaping by mitoses and by lamellipodia are of the same order. Lamellipodia are energetically most effective when their tensions are approximately twice as large as the interfacial tensions in the surrounding cells. The model also shows that stress fibers or other direct stretch or compression mechanisms are at least five times more efficient for tissue reshaping than are mitoses or lamellipodia and that the work needed to deform a typical cellular tissue is more than thirty times greater than if it did not contain cell boundaries. Collectively, these findings indicate that common tissue reshaping mechanisms have mechanical efficiencies of less than one percent and that mechanical efficiency is not the primary determinant of which mechanism(s) an embryo uses to reshape its tissues.