Mechanical properties of the passive myocardium: experiment and a mathematical model

A three-dimensional mathematical model of the rheological properties of a morphofunctional unit of the myocardium has been constructed, which consists of transversal and longitudinal elastic elements and tilted viscoelastic elements hinged without friction. The parameters of the viscosity and elasticity of structural elements of the model do not depend on the deformation value. The model makes it possible to adequately describe the features of the viscoelastic behavior of isolated samples of the passive myocardium and of the myocardium under longitudinal strain. A good agreement between the calculated and experimental data for an intact preparation of the rat myocardium and a preparation with removed cardiomyocytes has been shown.     

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.     

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.     

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.     

Internal viscoelastic loading in cat papillary muscle.

The passive mechanical properties of myocardium were defined by measuring force responses to rapid length ramps applied to unstimulated cat papillary muscles. The immediate force changes following these ramps recovered partially to their initial value, suggesting a series combination of viscous element and spring. Because the stretched muscle can bear force at rest, the viscous element must be in parallel with an additional spring. The instantaneous extension-force curves measured at different lengths were nonlinear, and could be made to superimpose by a simple horizontal shift. This finding suggests that the same spring was being measured at each length, and that this spring was in series with both the viscous element and its parallel spring (Voigt configuration), so that the parallel spring is held nearly rigid by the viscous element during rapid steps. The series spring in the passive muscle could account for most of the series elastic recoil in the active muscle, suggesting that the same spring is in series with both the contractile elements and the viscous element. It is postulated that the viscous element might be coupled to the contractile elements by a compliance, so that the load imposed on the contractile elements by the passive structures is viscoelastic rather than purely viscous. Such a viscoelastic load would give the muscle a length-independent, early diastolic restoring force. The possibility is discussed that the length-independent restoring force would allow some of the energy liberated during active shortening to be stored and released during relaxation.     

Application of the micropipette technique to the measurement of cultured porcine aortic endothelial cell viscoelastic properties

The viscoelastic deformation of porcine aortic endothelial cells grown under static culture conditions was measured using the micropipette technique. Experiments were conducted both for control cells (mechanically or trypsin detached from the substrate) and for cells in which cytoskeletal elements were disrupted by cytochalasin B or colchicine. The time course of the aspirated length into the pipette was measured after applying a stepwise increase in aspiration pressure. To analyze the data, a standard linear viscoelastic half-space model of the endothelial cell was used. The aspirated length was expressed as an exponential function of time. The actin microfilaments were found to be the major cytoskeletal component determining the viscoelastic response of endothelial cells grown in static culture.     

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.     

A recruitment-based rheological model for mechanical behavior of soft tissues

When lung tissue is subjected to finite deformations, phenomena appear that can only be described using nonlinear models. This paper considers the lung as a material composed of two elements, a continuous phase that acts uninterruptedly and a second phase composed of fiber elements that are recruited progressively into the mechanical process. Each individual fiber participates in the mechanical response of the set only when the deformation is above a certain value. A nine-parameter model was designed adopting standard viscoelastic elements both for the matrix and for each of the fibers. The mechanical behavior of the lung can be reproduced by a fitting process with standard numerical procedures in both dynamic-mechanical measurements and stress relaxation processes. Mechanical stress relaxation tests and dynamic-mechanical measurements have been carried out on subpleural parenchymal strips from rat lung. The model permits the reproduction of lung behavior in both types of measurements. The results show a recruitment ratio that decreases with deformation and the nonparticipation of the parallel matrix fraction in the lung's mechanical response so that a uniaxial transmission of force in the lung occurs via the recruited elements and the matrix series.     

Velocity transients and viscoelastic resistance to active shortening in cat papillary muscle.

When isotonic force steps were applied to activated papillary muscles, the velocity was almost never constant. Early rapid shortening associated with the step persisted for 2-7 ms after the step ends. The early rapid shortening is attributed to lightly damped series elastic recoil and velocity transients of the contractile elements. In most steps, the subsequent velocity declines progressively with shortening, and most of the decline in velocity can be accounted for by compression of a viscoelastic element in parallel with the contractile elements. To demonstrate this, the time course of isotonic velocity was compared with a model in which the force-velocity characteristics of the contractile element were assumed to be constant, and the decline in velocity was due to increasing compression of the viscoelastic element. This model predicted the observed results except that the predicted velocities rose progressively above the measured values for steps to light loads applied late in the twitch, and fell below the velocity trace for heavy loads applied early in the twitch. These deviations would occur if rapid shortening caused deactivation late in the twitch, and if activation were rising early in the twitch. A conditioning step applied to the muscle during the rise of force depressed both isometric force and maximum velocity measured at the peak of force; isometric force was more depressed with later conditioning steps than with earlier steps, while maximum velocity was depressed by about the same extent with both early and late steps. This difference between the effects on isometric force and maximum velocity are explained by a combination of deactivation and viscoelastic load.     

A viscoelastic model of mechanically induced and spontaneous contractions of vascular smooth muscle

A new viscoelastic model was developed for the mathematical characterisation of mechanically induced and intrinsic contractional responses of the vascular smooth muscle. To this end, the elastic and viscous analogue elements were supplemented with a new active element generating stress proportional to its momentary elongation. The four-element model consisting of an active element, a parallel viscous element and both series and parallel elastic elements predicted biphasic or damped oscillatory stress relaxation and creep responses which were similar to that found experimentally earlier. Above a certain exciting frequency the model exhibited dissipative and below energy producing behaviour, as indicated by the sign change of the hysteresis loop area. In the case of sinusoidal modulation of the stress generation parameter the model showed parametric resonance, which was regarded as a simulation of intrinsic oscillation of the smooth muscle.     

A computational model for dynamic analysis of the human gait

Biomechanical models are important tools in the study of human motion. This work proposes a computational model to analyse the dynamics of lower limb motion using a kinematic chain to represent the body segments and rotational joints linked by viscoelastic elements. The model uses anthropometric parameters, ground reaction forces and joint Cardan angles from subjects to analyse lower limb motion during the gait. The model allows evaluating these data in each body plane. Six healthy subjects walked on a treadmill to record the kinematic and kinetic data. In addition, anthropometric parameters were recorded to construct the model. The viscoelastic parameter values were fitted for the model joints (hip, knee and ankle). The proposed model demonstrated that manipulating the viscoelastic parameters between the body segments could fit the amplitudes and frequencies of motion. The data collected in this work have viscoelastic parameter values that follow a normal distribution, indicating that these values are directly related to the gait pattern. To validate the model, we used the values of the joint angles to perform a comparison between the model results and previously published data. The model results show a same pattern and range of values found in the literature for the human gait motion.     

A cellular tensegrity model to analyse the structural viscoelasticity of the cytoskeleton

This study describes the viscoelastic properties of a refined cellular-tensegrity model composed of six rigid bars connected to a continuous network of 24 viscoelastic pre-stretched cables (Voigt bodies) in order to analyse the role of the cytoskeleton spatial rearrangement on the viscoelastic response of living adherent cells. This structural contribution was determined from the relationships between the global viscoelastic properties of the tensegrity model, i.e., normalized viscosity modulus (eta(*)), normalized elasticity modulus (E(*)), and the physical properties of the constitutive elements, i.e., their normalized length (L(*)) and normalized initial internal tension (T(*)). We used a numerical method to simulate the deformation of the structure in response to different types of loading, while varying by several orders of magnitude L(*) and T(*). The numerical results obtained reveal that eta(*) remains almost independent of changes in T(*) (eta(*) proportional, variant T(*+0.1)), whereas E(*) increases with approximately the square root of the internal tension T(*) (from E(*) proportional, variant T(*+0.3) to E(*) proportional, variant T(*+0.7)). Moreover, structural viscosity eta(*) and elasticity E(*) are both inversely proportional to the square of the size of the structure (eta(*) proportional, variant L(*-2) and E(*) proportional, variant L(*-2)). These structural properties appear consistent with cytoskeleton (CSK) mechanical properties measured experimentally by various methods which are specific to the CSK micromanipulation in living adherent cells. Present results suggest, for the first time, that the effect of structural rearrangement of CSK elements on global CSK behavior is characterized by a faster cellular mechanical response relatively to the CSK element response, which thus contributes to the solidification process observed in adherent cells. In extending to the viscoelastic properties the analysis of the mechanical response of the cellular 30-element tensegrity model, the present study contributes to the understanding of recent results on the cellular-dynamic response and allows to reunify the scattered data reported for the viscoelastic properties of living adherent cells.     

Biphasic poroviscoelastic simulation of the unconfined compression of articular cartilage: I--Simultaneous prediction of reaction force and lateral displacement

This study investigated the ability of the linear biphasic poroelastic (BPE) model and the linear biphasic poroviscoelastic (BPVE) model to simultaneously predict the reaction force and lateral displacement exhibited by articular cartilage during stress relaxation in unconfined compression. Both models consider articular cartilage as a binary mixture of a porous incompressible solid phase and an incompressible inviscid fluid phase. The BPE model assumes the solid phase is elastic, while the BPVE model assumes the solid phase is viscoelastic. In addition, the efficacy of two additional models was also examined, i.e., the transversely isotropic BPE (TIBPE) model, which considers transverse isotropy of the solid matrix within the framework of the linear BPE model assumptions, and a linear viscoelastic solid (LVE) model, which assumes that the viscoelastic behavior of articular cartilage is solely governed by the intrinsic viscoelastic nature of the solid matrix, independent of the interstitial fluid flow. It was found that the BPE model was able to accurately account for the lateral displacement, but unable to fit the short-term reaction force data of all specimens tested. The TIBPE model was able to account for either the lateral displacement or the reaction force, but not both simultaneously. The LVE model was able to account for the complete reaction force, but unable to fit the lateral displacement measured experimentally. The BPVE model was able to completely account for both lateral displacement and reaction force for all specimens tested. These results suggest that both the fluid flow-dependent and fluid flow-independent viscoelastic mechanisms are essential for a complete simulation of the viscoelastic phenomena of articular cartilage.     

Models of skeletal muscle to explain the increase in passive stiffness in desmin knockout muscle

Absence of desmin in skeletal muscle was found to induce an increase in passive stiffness. The present study aimed at developing rheological models of passive muscle to explain this stiffening. Models were elaborated by using experimental data depicting muscle viscoelastic behaviour. The experimental protocol included stepwise extension tests applied on control and desmin knockout soleus muscles from mice. Linear and non-linear models were composed of elastic and viscous elements. They were constructed with the aim at taking the presence or absence of desmin into account by simulating desmin as an elastic element. Furthermore, associated adaptation of connective tissues in absence of desmin was modelled as an additional elastic element. Differences in passive behaviour induced by absence of desmin were predicted by using a linear model and a non-linear one. The non-linear model was selected because: (1) it is able to predict experimental viscoelastic kinetics accounting for the increase in passive stiffness in muscles lacking desmin, (2) its design is consistent with morphological data, and (3) stiffness characteristics of its elements are in accordance with the literature. Finally, this modelling approach demonstrates that both absence of desmin and adaptation of connective tissue are required to explain the increase in passive stiffness in desmin knockout muscles.     

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.     

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.     

Viscoelastic behaviour aids extrusion from and reabsorption of the liquid phase into the digesta plug: creep rheometry of hindgut digesta in the common brushtail possum Trichosurus vulpecula

Temporal variation in the rheometric properties of the proximal and distal colonic digesta of an arboreal marsupial folivore, the common brushtail possum, was examined to assess flow behaviour during peristalsis, segmentation and other aspects of intestinal motility. The time-dependent rheometric characteristics on application of a constant shear stress within the physiological range showed an initial elastic and subsequent viscoelastic phase, which fitted Burger’s model of creep compliance. Similarly, the time-dependent rheometric characteristics on recovery from shear stress fitted with a generalised two-component Maxwell model of elastic and viscoelastic components for creep recovery. Differences in the relative magnitudes of the viscoelastic components during recovery from those during shear indicated that the physical properties of the digesta plug changed with sustained shear stress, a phenomenon, which is likely to result from extrusion of the liquid phase from the solid elements of the digesta plug. There was significant viscoelastic recovery during the initial 4 s following cessation of stress, which would allow for prompt concomitant reabsorption of the liquid phase into the digesta plug. This supports a hypothesis of alternate extrusion and reabsorption of the liquid phase of the digesta plug. This would promote both nutrient absorption across the intestinal wall (from liquid extrusion) and enzyme permeation and digestion (from liquid absorption into the plug). However, the presence of a slower component of viscoelastic recovery indicates that liquid phase reabsorption into the digesta plug is incomplete if the interval before a subsequent contraction is less than 150 s, in which case unreabsorbed liquid may be driven either orally or aborally. This would at least partly account for differences in retention times of liquid and solid phase digesta markers reported for the gastrointestinal tracts of numerous vertebrate species.     

The possible role of poroelasticity in the apparent viscoelastic behavior of passive cardiac muscle

This paper investigates the contribution of extracellular fluid flow to the apparent viscoelastic behavior of passive cardiac muscle. The muscle is modeled as an incompressible, isotropic, poroelastic solid saturated by an incompressible viscous fluid. Based on Biot's linear and nonlinear consolidation theories, solutions are presented for general time-dependent uniaxial loading of unconfined cylindrical muscle specimens. The nonlinear analysis includes the effects of large strain, material nonlinearity, and strain-dependent permeability. The computed results show that, for axial stretch ratios greater than 1.1, the changing permeability and the loading rate strongly affect the total stress relaxation and the short-time relaxation rate. Comparisons of theoretical and published experimental results show that extracellular fluid flow can account for several observed biomechanical features of passive myocardium, including the insensitivity of stress-strain curves to loading rate and of stress-relaxation curves to the amount of stretch. Theoretical hysteresis loops, however, are too small. Thus, both poroelastic and tissue viscoelastic effects must be considered in studies of passive cardiac muscle.     

Influence of viscosity on myocardium mechanical activity: a mathematical model

We have previously proposed and validated a mathematical model of myocardium contraction-relaxation cycle based on current knowledge of regulatory role of Ca2+ and cross-bridge kinetics in cardiac cell. That model did not include viscous elements. Here we propose a modification of the model, in which two viscous elements are added, one in parallel to the contractile element, and one more in parallel to the series elastic element. The modified model allowed us to simulate and explain some subtle experimental data on relaxation velocity in isotonic twitches and on a mismatch between the time course of sarcomere shortening/lengthening and the time course of active force generation in isometric twitches. Model results were compared with experimental data obtained from 28 rat LV papillary muscles contracting and relaxing against various loads. Additional model analysis suggested contribution of viscosity to main inotropic and lusitropic characteristics of myocardium performance.     

Viscoelastic fluid description of bacterial biofilm material properties

A mathematical model describing the constitutive properties of biofilms is required for predicting biofilm deformation, failure, and detachment in response to mechanical forces. Laboratory observations indicate that biofilms are viscoelastic materials. Likewise, current knowledge of biofilm internal structure suggests modeling biofilms as associated polymer viscoelastic systems. Supporting experimental results and a system of viscoelastic fluid equations with a linear Jeffreys viscoelastic stress-strain law are presented here. This system of equations is based on elements of associated polymer physics and is also consistent with presented and previous experimental results. A number of predictions can be made. One particularly interesting result is the prediction of an elastic relaxation time on the order of a few minutes-biofilm disturbances on shorter time scales produce an elastic response, biofilm disturbances on longer time scales result in viscous flow, i.e., nonreversible biofilm deformation. Although not previously recognized, evidence of this phenomenon is in fact present in recent experimental results.