Mechanical properties of passive myocardium: experiment and mathematical model

We present here a 3D mathematical model of rheological properties of a morphofunctional unit of myocardium as example of biological tissue. The model consists of longitudinal and transverse elastic elements and inclined viscoelastic elements connected pivotally without friction. The parameters of viscosity and elasticity of the structural elements of the model do not depend on the magnitude of deformation. The model adequately describes nonlinear viscoelastic behavior of isolated samples of passive myocardium both in static condition and under dynamic loading. The simulation data fit the experiment very well both for intact rat papillary muscle and for a decellularized specimen.     

Viscoelastic properties of relaxed papillary muscle under physiological hypertrophy

Viscoelastic properties of relaxed rat papillary muscles in physiological hypertrophy (intensive swimming for 5 weeks) have been obtained. It has been ascertained that viscoelastic properties of hypertrophied muscles are not significantly distinguished from those of control papillary muscles. A three-dimensional model of myocardial fascicle has been verified in compliance with experimental data of biomechanical tests of hypertrophied muscles. Elastic and viscous parameters of structural elements of the model negligibly differ from the parameters of the model of a control muscle. It is shown that physiological hypertrophy has a slight influence on viscoelastic properties of papillary muscles.     

Trabeculated embryonic myocardium shows rapid stress relaxation and non-quasi-linear viscoelastic behavior

Passive viscoelastic behavior is important in embryonic cardiovascular function, influencing the rate and magnitude of contraction and relaxation. We hypothesized that if viscoelastic behavior is influenced by interstitial fluid flow, then the stage-21 (312d) and stage-24 (4d) chick myocardium with large intertrabecular spaces will exhibit much different viscoelastic behavior than stage-16 (212d) and stage-18 (3d) compact myocardium and a non-quasi-linear response. Excised left ventricular sections were tested with ramp-and-hold stress relaxation tests at axial stretch ratios of 1.05:1.1:1.2:1.3. The measured stress relaxation was much more rapid than previously observed in the compact, non-trabeculated myocardium. The reduced relaxation curves depended significantly on the stretch level. A continuous-spectrum quasi-linear relaxation function described their shape well but the model-fit parameters also depended on the stretch level. Sinusoidal stretching of ventricular sections at rates from 0.2 to 25Hz showed that the steepening of stress-strain curves with increasing strain rate was half as much as predicted by a quasi-linear model. Hysteresis ranged from 25-35%, varied little with loading rate from 0.2 to 8Hz, and was twice that predicted from a quasi-linear model. Doubling the viscosity of the perfusate in stress-relaxation tests produced increased stiffness and decreased relaxation rate. These results demonstrate that the passive viscoelastic behavior of the trabeculated embryonic myocardium is markedly different from that of younger, compact myocardium and is not quasi-linear.     

An orthotropic viscoelastic model for the passive myocardium: continuum basis and numerical treatment

This study deals with the viscoelastic constitutive modeling and the respective computational analysis of the human passive myocardium. We start by recapitulating the locally orthotropic inner structure of the human myocardial tissue and model the mechanical response through invariants and structure tensors associated with three orthonormal basis vectors. In accordance with recent experimental findings the ventricular myocardial tissue is assumed to be incompressible, thick-walled, orthotropic and viscoelastic. In particular, one spring element coupled with Maxwell elements in parallel endows the model with viscoelastic features such that four dashpots describe the viscous response due to matrix, fiber, sheet and fiber-sheet fragments. In order to alleviate the numerical obstacles, the strictly incompressible model is altered by decomposing the free-energy function into volumetric-isochoric elastic and isochoric-viscoelastic parts along with the multiplicative split of the deformation gradient which enables the three-field mixed finite element method. The crucial aspect of the viscoelastic formulation is linked to the rate equations of the viscous overstresses resulting from a 3-D analogy of a generalized 1-D Maxwell model. We provide algorithmic updates for second Piola–Kirchhoff stress and elasticity tensors. In the sequel, we address some numerical aspects of the constitutive model by applying it to elastic, cyclic and relaxation test data obtained from biaxial extension and triaxial shear tests whereby we assess the fitting capacity of the model. With the tissue parameters identified, we conduct (elastic and viscoelastic) finite element simulations for an ellipsoidal geometry retrieved from a human specimen.     

ULTRASTRUCTURAL ALTERATIONS PRODUCED IN MAMMALIAN MYOCARDIUM BY VARIATION IN PERFUSATE IONIC COMPOSITION

This study describes the changes produced in the subcellular morphology of mammalian myocardium when perfusate sodium, calcium, and chloride concentrations are varied. By means of a recently developed perfusion technique, functioning dog papillary muscles were perfused with isotonic solutions of varying ionic compositions. Examination of the tissue in the electron microscope revealed that control muscles showed satisfactory preservation of ultrastructure, demonstrating that the protocol itself did not create significant morphological artefact. Low sodium chloride perfusion produced dilatation of both transverse tubules and longitudinal sarcoplasmic reticulum elements. Low sodium or high calcium concentrations produced dilation of tubular elements of the longitudinal sarcoplasmic reticulum while leaving transverse tubules intact. High calcium perfusion produced mitochondrial swelling and vacuolization. Mitochondrial precipitate, both crystalline and amorphous in form, was observed and presumed to be calcium phosphate, either alone or mixed with calcium carbonate. The possibility that the morphological changes observed might indicate subcellular loci of specific ion permeability is discussed. A correlation of the known kinetic behavior of sodium and calcium ions in mammalian myocardium with the ultrastructural alterations produced is suggested.     

A fibre optic catheter for simultaneous measurement of longitudinal and circumferential muscular activity in the gastrointestinal tract

Diagnostic catheters based on fibre Bragg gratings (FBG's) are proving to be highly effective for measurement of the muscular activity associated with motility in the human gut. While the primary muscular contractions that generate peristalsis are circumferential in nature, it has long been known that there is also a component of longitudinal contractility present, acting in harmony with the circumferential component to improve the overall efficiency of material movement. We report the detection of longitudinal motion in mammalian intestine using an FBG technique that should be viable for similar detection in humans. The longitudinal sensors have been combined with our previously reported FBG pressure sensing elements to form a composite catheter that allows the relative phase between the two components to be detected. The catheter output has been validated using video mapping in an ex‐vivo rabbit ileum preparation. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shear properties of passive ventricular myocardium

We examined the shear properties of passive ventricular myocardium in six pig hearts. Samples (3 x 3 x 3 mm) were cut from adjacent regions of the lateral left ventricular midwall, with sides aligned with the principal material axes. Four cycles of sinusoidal simple shear (maximum shear displacements of 0.1-0.5) were applied separately to each specimen in two orthogonal directions. Resulting forces along the three axes were measured. Three specimens from each heart were tested in different orientations to cover all six modes of simple shear deformation. Passive myocardium has nonlinear viscoelastic shear properties with reproducible, directionally dependent softening as strain is increased. Shear properties were clearly anisotropic with respect to the three principal material directions: passive ventricular myocardium is least resistant to simple shear displacements imposed in the plane of the myocardial layers and most resistant to shear deformations that produce extension of the myocyte axis. Comparison of results for the six different shear modes suggests that simple shear deformation is resisted by elastic elements aligned with the microstructural axes of the tissue.     

Wave Propagation through a Newtonian Fluid Contained within a Thick-Walled,Viscoelastic Tube

The propagation of harmonic pressure waves through a Newtonian fluid contained within a thick-walled, viscoelastic tube is considered as a model of arterial blood flow. The fluid is assumed to be homogeneous and Newtonian, and its motion to be laminar and axisymmetric. The wall is assumed to be isotropic, incompressible, linear, and viscoelastic. It is also assumed that the motion is such that the convective acceleration is negligible. The motion of the fluid is described by the linearized form of the Navier-Stokes equations and the motion of the wall by classical elasticity theory. The frequency dependence of the wall mechanical properties are represented by a three parameter, relaxation-type model. Using boundary conditions describing the continuity of stress and velocity components in the fluid and the wall, explicit solutions for the system of equations of the model have been obtained. The longitudinal fluid impedance has been expressed in terms of frequency and the system parameters. The frequency equation has been solved and the propagation constant also expressed in terms of frequency and system parameters. The results indicate that the fluid impedance is smaller than predicted by the rigid tube model or by Womersley''s constrained elastic tube model. Also, the velocity of propagation is generally slower and the transmission per wavelength less than predicted by Womersley''s elastic tube model. The propagation constant is very sensitive to changes in the degree of wall viscoelasticity.     

Anomalous rate dependence of the preconditioned response of soft tissue during load controlled deformation

It has been observed in load controlled laboratory tests of myocardium and skin that the tissues can exhibit a decrease in nonlinear stiffness with an increase in loading rate: the faster a test is performed, the more compliant is the preconditioned material behavior. This response seems to conflict with what is generally expected of soft tissues based on stretch or strain controlled tests, in which an increased rate of deformation results in a stiffer material response. It is hypothesized that this anomalous behavior has not been observed previously due to the small number of cyclic load controlled mechanical characterization tests that are geared specifically towards viscoelastic tissue response. The goal of this paper is to examine the preconditioned response of soft tissue to load controlled deformation using nonlinear viscoelastic material models including quasi-linear viscoelasticity, and to determine under what conditions this anomalous behavior becomes apparent. Results from this study suggest that this behavior is a true phenomenon unique to load controlled deformations that results from the interplay of nonlinear effects and creep behavior. These results call for increased attention to experimental parameters when testing and modeling nonlinear viscoelastic material behavior.     

Effect of drugs on the resistance of the myocardium to hypoxia

A method for the investigation of drug effects in the myocardium resistance to hypoxia has been suggested. It is based on the determination of drug effects on the performance of the isolated spontaneously contracting atrium (ISCA) of rats under hypoxic conditions. Hypoxia was induced by oxygen displacement from the nutritional solution by nitrogen. ISCA resistance to hypoxia was assessed by the mechanogram of the heart preparation (the duration and volume of ISCA performance being up to 50% of the initial amplitude). Using the inhibitor analysis, it has been demonstrated that the given model of myocardial hypoxia adequately reflects the role of energy cellular metabolism in the regulation of ISCA resistance to hypoxia and can be used in the search for myocardial antihypoxic agents.     

Rheological properties and wall structures of large veins

The static and dynamic viscoelastic properties were studied of longitudinal and circumferential strips excised from various large veins of dogs. The mechanical behavior in longitudinal direction could be regarded as elastic, while that in circumferential direction was highly viscoelastic. No distinct regionality was observed in either of the longitudinal and the circumferential groups. Noradrenaline and papaverine did not alter the elastic behavior of the longitudinal strips. In circumferential strips, however, noradrenaline caused a considerable decrease in stress relaxation and some steepening in the slope of the upper limb of hysteresis loop. Papaverine did not affect the circumferential characteristics. These findings suggest the dominant contribution of smooth muscle tone to the circumferential characteristics of venous walls. Pretreatment with formic acid abolished the occurrence of stress relaxation in circumferential direction but produced no change in the longitudinal behavior. This indicates that elastin fibers may be a principal determinant of the elastic behavior in longitudinal direction and that a residual tension observed in stress-relaxation tests of circumferential strips may be due to stretched elastin fibers. The elastic moduli of elastase pretreated venous walls were in the order of 10(8) dynes/cm2, about 1000 times higher than those of the control. Accordingly, collagen fibers seemed not to play any appreciable role in the rheological behavior of venous walls under physiological conditions. This inference was supported by histological observations of venous walls under unstretched and stretched states. Models were proposed in regard to the architecture of the fibrous elements in the venous walls.     

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.     

Morphologic aspects of local blood flow regulation in the myocardium

In 67 preparations of the human hearts at the first and second periods of mature age, spatial interrelations between blood vessels and cardiac muscle fibers in the ventricle myocardium have been studied. All the elements of the myocardial blood bed are oriented under a certain angle in relation to the cardiac muscle fibers. Regular arrangement of the arteries and sinusoid dilated veins under endocardium on the top of the papillary muscles and in the muscular trabecules is demonstrated. As proves the mathematical model, the slope orientation of the blood bed elements towards the cardiac muscle fibers ensures and adequate realization of the external influence of the contractile cardiomyocytes to the successive movement of blood along the intramural myocardial vessels. From morphological positions, a conclusion on the mechanism of the intracavitary pressure effect on blood movement along the intramural veins of the ventricular myocardium is argued. A conclusion is made on the leading role of the extravascular factors (intramyocardial and intercavitary pressure) in the local regulation of the blood stream in the myocardium and in development of working cardiac hyperemia.     

In situ multi-level analysis of viscoelastic deformation mechanisms in tendon collagen

Tendon is a hydrated multi-level fibre composite, in which time-dependent behaviour is well established. Studies indicate significant stress relaxation, considered important for optimising tissue stiffness. However, whilst this behaviour is well documented, the mechanisms associated with the response are largely unknown. This study investigates the sub-structural mechanisms occurring during stress relaxation at both the macro (fibre) and nano (fibril) levels of the tendon hierarchy. Stress relaxation followed a two-stage exponential behaviour, during which structural changes were visible at the fibre and fibril levels. Fibril relaxation and fibre sliding showed a double exponential response, while fibre sliding was clearly the largest contributor to relaxation. The amount of stress relaxation and sub-structural reorganisation increased with increasing load increments, but fibre sliding was consistently the largest contributor to stress relaxation. A simple model of tendon viscoelasticity at the fibril and fibre levels has been developed, capturing this behaviour by serially coupling a Voigt element (collagen fibril), with two Maxwell elements (non-collagenous matrix between fibrils and fibres). This multi-level analysis provides a first step towards understanding how sub-structural interactions contribute to viscoelastic behaviour. It indicates that nano- and micro-scale shearing are significant dissipative mechanisms, and the kinetics of relaxation follows a two-stage exponential decay, well fitted by serially coupled viscoelastic elements.     

Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularized Fractional Derivative Kernels with Application to the Human Calcaneal Fat Pad

A viscoelastic model of the K-BKZ (Kaye, Technical Report 134, College of Aeronautics, Cranfield 1962; Bernstein et al., Trans Soc Rheol 7: 391–410, 1963) type is developed for isotropic biological tissues and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. This elastic potential – via the K-BKZ hypothesis – also produces the tensorial structure of the viscoelastic model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including two functions whose origins lie in the fractional calculus. The Akaike information criterion is used to perform multi-model inference, enabling an objective selection to be made as to the best material function from within a candidate set.Dedicated to Prof. Ronald L. Bagley.     

Axial dispersion in packed bed reactors involving viscoinelastic and viscoelastic non-Newtonian fluids

Axial dispersion is an important parameter in the performance of packed bed reactors. A lot of fluids exhibit non-Newtonian behaviour but the effect of rheological parameters on axial dispersion is not available in literature. The effect of rheology on axial dispersion has been analysed for viscoinelastic and viscoelastic non-Newtonian fluids. Aqueous solutions of carboxymethyl cellulose and polyacrylamide have been chosen to represent viscoinelastic and viscoelastic liquid-phases. Axial dispersion has been measured in terms of Bo_L number. The single parameter axial dispersion model has been applied to analyse RTD response curve. The Bo_L numbers were observed to increase with increase in liquid flow rate and consistency index ‘K’ for viscoinelastic as well as viscoelastic fluids. Bodenstein correlation for Newtonian fluids proposed has been modified to account for the effect of fluid rheology. Further, Weissenberg number is introduced to quantify the effect of viscoelasticity.     

Lengthwise splitting of myocardial cells and nuclei]

A possibility of a peculiar method for division of myocardial cells and nuclei--longitudinal splitting--has been studied in hypertrophic hearts of 19 corpses. Morphological examination of the myocardium (separate weighing of the cardiac ventricles, measuring of myocardial components and muscular fibres in microphotos of transversal sections) has not revealed any signs of longitudinal fibres splitting: their average thickness either increase or remain the same with growth of muscular mass, and per cent of thin and thick fibres remains within limits of unreliable fluctuations. Analysis of serial transversal sections of muscular fibres and their micrometry confirm the reality of longitudinal splitting of the nuclei which is realized through several steps and depends on the degree of hypertrophy of muscular fibres.     

Stress Distribution in the Canine Left Ventricle during Diastole and Systole

A model is proposed for stress analysis of the left ventricular wall (LV wall) based on the realistic assumption that the myocardium is essentially composed of fiber elements which carry only axial tension and vary in orientation through the wall. Stress analysis based on such a model requires an extensive study of muscle fiber orientation and curvature through the myocardium. Accordingly, the principal curvatures were studied at a local site near the equator in ten dog hearts rapidly fixed in situ at end diastole and end systole; the fiber orientation for these hearts had already been established in a previous study. The principal radii of curvature were (a) measured by fitting templates to the endocardial and epicardial wall surfaces in the circumferential and longitudinal directions and (b) computed from measured lengths of semiaxes of ellipsoids of revolution representing the LV wall (“ellipsoid” data). The wall was regarded as a tethered set of nested shells, each having a unique fiber orientation. Results indicate the following. (a) Fiber curvature, k, is maximum at midwall at end systole; this peak shifts towards endocardium at end diastole. (b) The pressure or radial stress through the wall decreases more rapidly near the endocardium than near the epicardium at end diastole and at end systole when a constant tension is assumed for each fiber through the wall. (c) At end diastole the curve for the circumferential stress vs. wall thickness is convex with a maximum at midwall. In the longitudinal direction the stress distribution curve is concave with a minimum at midwall. Similar distributions are obtained at end systole when a constant tension is assumed for each fiber through the wall. (d) The curvature and stress distributions obtained by direct measurements at a selected local site agree well with those computed from “ellipsoid” data.