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 new method for measuring passive length-tension properties of human gastrocnemius muscle in vivo

The study of muscle growth and muscle length adaptations requires measurement of passive length-tension properties of individual muscles, but until now such measurements have only been made in animal muscles. We describe a new method for measuring passive length-tension properties of human gastrocnemius muscles in vivo. Passive ankle torque and ankle angle data were obtained as the ankle was rotated through its full range with the knee in a range of positions. To extract gastrocnemius passive length-tension curves from passive torque-angle data it was assumed that passive ankle torque was the sum of torque due to structures which crossed only the ankle joint (this torque was a 6-parameter function of ankle joint angle) and a torque due to the gastrocnemius muscle (a 3-parameter function of knee and ankle angle). Parameter values were estimated with non-linear regression and used to reconstruct passive length-tension curves of the gastrocnemius. The reliability of the method was examined in 11 subjects by comparing three sets of measurements: two on the same day and the other at least a week later. Length-tension curves were reproducible: the average root mean square error was 5.1+/-1.1 N for pairs of measurements taken within a day and 7.3+/-1.2 N for pairs of measurements taken at least a week apart (about 3% and 6% of maximal passive tension, respectively). Length-tension curves were sensitive to mis-specification of moment arms, but changes in length-tension curves were not. The new method enables reliable measurement of passive length-tension properties of human gastrocnemius in vivo, and is likely to be useful for investigation of changes in length-tension curves over time.     

An orthotropic viscoelastic material model for passive myocardium: theory and algorithmic treatment

This contribution presents a novel constitutive model in order to simulate an orthotropic rate-dependent behaviour of the passive myocardium at finite strains. The motivation for the consideration of orthotropic viscous effects in a constitutive level lies in the disagreement between theoretical predictions and experimentally observed results. In view of experimental observations, the material is deemed as nearly incompressible, hyperelastic, orthotropic and viscous. The viscoelastic response is formulated by means of a rheological model consisting of a spring coupled with a Maxwell element in parallel. In this context, the isochoric free energy function is decomposed into elastic equilibrium and viscous non-equilibrium parts. The baseline elastic response is modelled by the orthotropic model of Holzapfel and Ogden [Holzapfel GA, Ogden RW. 2009. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans Roy Soc A Math Phys Eng Sci. 367:3445–3475]. The essential aspect of the proposed model is the account of distinct relaxation mechanisms for each orientation direction. To this end, the non-equilibrium response of the free energy function is constructed in the logarithmic strain space and additively decomposed into three anisotropic parts, denoting fibre, sheet and normal directions each accompanied by a distinct dissipation potential governing the evolution of viscous strains associated with each orientation direction. The evolution equations governing the viscous flow have an energy-activated nonlinear form. The energy storage in the Maxwell branches has a quadratic form leading to a linear stress–strain response in the logarithmic strain space. On the numerical side, the algorithmic aspects suitable for the implicit finite element method are discussed in a Lagrangian setting. The model shows excellent agreement compared to experimental data obtained from the literature. Furthermore, the finite element simulations of a heart cycle carried out with the proposed model show significant deviations in the strain field relative to the elastic solution.     

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.     

Muscle material properties in passive and active stroke-impaired muscle

Stroke survivors routinely experience long-term motor and sensory impairments. In parallel with neurological changes, material properties of muscles in the impaired limbs, such as muscle stiffness, may also change progressively. However, these stiffness measures are routinely derived from individual joint stiffness, representing whole muscle groups. Here, we use shear wave (SW) ultrasound elastography to measure SW velocity, as a surrogate measure of stiffness, to quantify material properties in individual muscles. Accordingly, the purpose of this study was to compare muscle material properties of the bicep brachii in stroke survivors and in age-matched control subjects by measuring SW velocity at rest and different voluntary activation levels. Our main findings show that at rest, the SW velocity was on average 41% greater in the paretic muscle compared the contralateral non-paretic muscle. The mean passive SW velocity across all subjects were 2.34 ± 0.41 m/s for the non-paretic side, 3.30 ± 1.20 m/s for the paretic side, and 2.24 ± 0.18 for controls. SW velocity was significantly different in muscles of the paretic and non-paretic side (p < 0.001), but not between muscles of the non-paretic and controls (p = 0.47). As voluntary activation increased, SW velocity increased non-linearly, with an average power fit of r² = 0.83 ± 0.09 for the non-paretic side, r² = 0.61 ± 0.24 for the paretic side, and r² = 0.24 ± 0.15 for the healthy age-matched controls. In active muscle (10, 25, 50, 75, 100% maximum voluntary contraction), there was no significant difference in SW velocity between the non-paretic, paretic, and control muscles.These findings suggest that stroke-impaired muscles have potentially altered muscle material properties, specifically stiffness, and that passive and active stiffness may contribute differently to total muscle stiffness.     

A novel method for determining articular cartilage chondrocyte mechanics in vivo

Work relating the mechanical states of articular cartilage chondrocytes to their biosynthetic responses is based on measurements in isolated cells or cells in explant samples removed from their natural in situ environment. Neither the mechanics nor the associated biological responses of chondrocytes have ever been studied in cartilage within a joint of a live animal, and no such measurements have ever been performed using physiologically relevant joint loading through muscular contractions. The purpose of this study was to design and apply a method to study the mechanics of chondrocytes in the exposed but fully intact knee of live animals, which was loaded near-physiologically through muscular contraction. In order to achieve this purpose, we developed an accurate and reliable method based on two-photon laser excitation microscopy. Near-physiological knee joint loading was achieved through controlled electrical activation of the knee extensor muscles that compress the articulating surfaces of the femur, tibia and patella. Accuracy of the system was assessed by inserting micro-beads of known dimensions into the articular cartilage of the mouse knee and comparing the measured volumes and diameters in the principal directions with known values of the beads. Accuracy was best in the plane perpendicular to the optical axis (average error = 1%) while it was slightly worse, but still excellent, along the optical axis (average error = 3%). Reliability of cell volume and shape measurements was 0.5% on average, and 2.9% in the worst-case-scenario. Pilot measurements of chondrocyte deformations upon sub-maximal muscular loading causing a mean articular contact pressure of 1.9 ± 0.2 MPa showed an "instantaneous" decrease in cell height (17 ± 4.5%) and loss of cell volume (22.3 ± 2.4%) that took minutes to recover upon deactivation of the knee extensor muscles.     

The passive permeability properties of in vivo perfused rat jejunum

The polarity of the microvillus membrane of rat and rabbit jejunum, determined in vitro by the incremental free energy changes associated with the addition of -CH₂ groups to fatty acids or -OH groups to bile acids, has been found to be more polar than other biological membranes. The reason for the difference in polarity is unexplained but could be due to artifacts caused by the in vitro conditions. In the current studies the apparent permeability coefficients were determined for a homologous series of fatty acids and bile acids in in vivo perfused rat jejunum. The true permeability coefficients were derived by correction for diffusion barrier resistance. The incremental free energy changes associated with the addition of a -CH₂ group to a fatty acid and a -OH group to a bile acid were −619 and +2069 cal/mol, respectively. These values correspond to values determined in other biological membranes such as erythrocytes and adipocytes. Thus, the rat microvillus membrane is more nonpolar than previously observed in vitro. The reason for the discrepancy between the in vivo and in vitro results is most likely due to an underestimation of the permeability coefficients in the in vitro studies.     

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.     

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.     

Physical invariant strain energy function for passive myocardium

Principal axis formulations are regularly used in isotropic elasticity, but they are not often used in dealing with anisotropic problems. In this paper, based on a principal axis technique, we develop a physical invariant orthotropic constitutive equation for incompressible solids, where it contains only a one variable (general) function. The corresponding strain energy function depends on six invariants that have immediate physical interpretation. These invariants are useful in facilitating an experiment to obtain a specific constitutive equation for a particular type of materials. The explicit appearance of the classical ground-state constants in the constitutive equation simplifies the calculation for their admissible values. A specific constitutive model is proposed for passive myocardium, and the model fits reasonably well with existing simple shear and biaxial experimental data. It is also able to predict a set of data from a simple shear experiment.     

A continuous cable method for determining the transient potential in passive dendritic trees of known geometry

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.     

Improvements to Hoang et al.'s method for measuring passive length–tension properties of human gastrocnemius muscle in vivo

While the passive mechanical properties of a musculo-articular complex can be determined using the relationship between the articular angle and the passive torque developed in resistance to motion, the properties of different structures of the musculo-articular complex cannot be easily assessed. Recently, an elegant method has been proposed to estimate the passive length–tension properties of gastrocnemius muscle–tendon unit (Hoang et al., 2005). In the present paper, two improvements of this method are proposed to decrease the number of parameters required to assess the passive length–tension relationship from 9 to 2. Furthermore, these two parameters have physical meaning as they represent a passive muscle–tendon stiffness index (α) and the muscle–tendon slack length (l₀). α and l₀ are relevant clinical parameters to study the chronic effects of aging, training protocols or neuromuscular pathologies on the passive mechanical properties of the muscle–tendon unit. Eight healthy subjects performed passive loading/unloading cycles at 5°/s with knee angle at 6 knee angles to assess the torque–angle relationships and to apply the modified method. Numerical optimization was used to minimize the squared error between the experimental and the modeled relationships. The experiment was performed twice to assess the reliability of α and l₀ across days. The results showed that the reliability of the two parameters was good (α: ICC=0.82, SEM=6.1 m^?1, CV=6.3% and l₀: ICC=0.83, SEM=0.29 cm, CV=0.9%). Using a sensitivity analysis, it was shown that the numerical solution was unique. Overall, the findings may provide increased interest in the method proposed by Hoang et al. (2005).     

Active and passive stresses in the myocardium

A mathematical approach that can be used to calculate the passive stress in the ventricular wall is presented. The active fiber stress (force/unit area) generated by the muscular fibers in the ventricular wall is expressed by means of body force (force/unit volume of the myocardium). It is shown that the total intramyocardial passive stress induced in the passive medium of the myocardium can be expressed as the sum of a passive stress induced by the left ventricular pressure and a passive stress induced by the active fiber stress. Applications to experimental data published in the literature are given. New results are presented that show the relation among those two components of the intramyocardial passive stress. New relations between the intramyocardial passive stress, the slope (elastance) of the pressure-volume relation, and the residual volume are also derived. The results obtained give a better understanding of some aspects of the mechanics of cardiac contraction and can provide a more detailed interpretation of clinical conditions.     

Passive material properties of intact ventricular myocardium determined from a cylindrical model

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.     

In vivo passive mechanical properties of the human gastrocnemius muscle belly

The purpose of the present study was to determine the in vivo passive mechanical properties, including the length below the slack length, of the gastrocnemius muscle (GAS) belly in humans. Transverse ultrasound images of the medial head of the GAS were taken in 11 subjects during passive knee extension from 80 degrees to 5 degrees with a constant ankle joint angle of 10 degrees (0 degrees is the neutral ankle position: positive values for dorsiflexion). The change in passive ankle joint moment (Mp), which is produced only by the GAS length change, was also measured during passive knee extension. The onset of Mp during passive knee extension was found to be 43+/-8 degrees (mean+/-SD) when the baseline of the Mp was set at the average Mp in the range of 55-60 degrees where the Mp was almost constant (SD<0.03 Nm). At this onset, the muscle fascicle length of the GAS (Lf) was 46+/-7 mm (slack length; Lfs). Lf at 80 degrees was 6+/-4 mm (13+/-6%) less than the Lfs, and Lf at 5 degrees was 12+/-5 mm (27+/-11%) greater than the Lfs. The passive force-resisting compression of the GAS did not produce a dorsiflexion moment in the joint angle range adopted. The passive ankle joint moment increased linearly with Lf (coefficient of determination (R2)=0.85-0.96), and the slopes of the relationships between Lf and Mp, and between the relative Lf to Lfs and Mp were 0.093+/-0.038 Nm/mm and 0.043+/-0.021 Nm/%Lfs. The findings of the present study can be implemented in musculoskeletal modeling, which would provide a more accurate evaluation of the passive mechanical properties of muscle during movement.     

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.