A macroscopic description of stress relaxation in a muscle subject to stepwise deformation

Stress relaxation regimes arising in a muscle subject to stepwise deformation are described on the basis of a recent phenomenological model of fully activated muscle tissue which is presented in the form of a second-order constitutive equation. It is shown that this model reproduces the qualitative form of relaxation curves observed experimentally. Relations between rheological parameters which correspond to different types of stress relaxation are found for the case where the jump duration is much smaller than the relaxation times of the sample. As illustrated by the simplest model of a slow length jump (linear deformation), the qualitative form of the stress relaxation depends on the jump duration in this case. This effect can lead to rough errors in determination of rheological and molecular parameters of muscle tissue in mechanical experiments in which the relation between jump duration and relaxation times is not controlled.     

Kinetics of stress relaxation properties of oat coleoptile cell wall after geotropic stimulation

This study describes the stress relaxation of the cell wall of oat (Avena sativa) coleoptiles after different periods of geotropic stimulation. The upper and lower tissues (with respect to gravity) of geotrophically stimulated coleoptiles exhibit different wall properties. The lower tissues are less resistant to deformation than the upper. The ratio of stress to strain is significatly less in the lower than in the upper tissue. Similarly, the relaxation time and the minimum relaxation time, derived from the Maxwell model which describes the physical characteristics of polymers, are also shorter in the lower tissue. However, the maximum relaxation time shows no difference between the upper and lower tissues of a geotropically stimulated coleoptile. The differences between the tissues begin at about 8 minutes after the commencement of stimulation, similar to the time for the initiation of dictyosome redistribution, and precede the onset of geotropism. The above responses of the cell wall of the lower tissue are similar to those induced by indoleacetic acid. The parameters of wall properties of the coleoptiles of both the control and the geostimulated fluctuate rhythmically with time. The periodic changes in wall properties of the coleoptile are compared to other cyclic physiological phenomena.     

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.     

Stress relaxation in fine fibrin films: Comparison of films prepared with thrombin and ancrod

Measurements of stress relaxation in uniaxial extension have been made on fibrin film prepared from fine bovine fibrin clots (i.e., clots in which there is minimal lateral aggregation of protofibrils), both ligated and unligated, and polymerized with both thrombin and ancrod, plasticized with either aqueous buffer or glycerol. The stress 100 s after imposition of strain was approximately proportional to In λ, where λ is the stretch ratio. Ligated thrombin films showed comparatively little relaxation over a period of one day and almost complete recovery after release of stress. In unligated thrombin films, there was substantial relaxation in two stages, as previously observed for coarse films, and substantial irrecoverable deformation. The extent of relaxation and the proportion of strain that was irrecoverable increased with the magnitude of the strain. In ancrod films (unligated), there was much more relaxation (stress decaying by as much as a factor of 10) and much more irrecoverable deformation (about 70% of the initial deformation); these results did not depend on the magnitude of the strain. When an ancrod film was released after relaxation and submitted to a second stretch, the extent of the second relaxation was much less. These observations are discussed in relation to the structure of fine films and possible mechanisms for relaxation and irrecoverable deformation.     

Late systolic onset of regional LV relaxation demonstrated in three-dimensional space by MRI tissue tagging

Left ventricular (LV) relaxation entails myocardial deformation that induces LV filling. Yet, the precise mechanisms of the earliest changes in tissue properties that characterize myocardial relaxation remain incompletely understood. Ten healthy volunteers (seven males), 25-43 yr, underwent tagged and cine MRI with high temporal resolution (25-35 ms). Normal strains including radial (E(rr)), circumferential (E(cc)), and longitudinal (E(ll)) strains, shear strains including E(cl) (circumferential-longitudinal), E(cr) (circumferential-radial), and E(rl) (radial-longitudinal), and principal strains (E(1), E(2), and E(3)) were calculated using a displacement field-fitting method. Temporal changes in angular strains indicative of shear and torsion release and normal strains were studied during late systole and early relaxation. The onset of individual relaxation strains was heterogeneous relative to LV filling. Shear strains (E(cr), E(rl), and E(cl)) and radial thinning were first to develop. Times of onset of E(cr), E(rl), E(cl), and E(rr) occurred 108, 93, 67, and 73 ms before aortic valve closure, respectively. E(ll), E(cc), and LV volume change commenced significantly later after the onset of diastolic shear strains and radial thinning. The onset of E(cc), E(ll), and LV volume change was noted 38 ms before aortic valve closure (P < or = 0.05 relative to the onset of shear strains and E(rr)). Myocardial relaxation is characterized by a three-dimensional unfolding deformation that includes release of torsion, shear, and radial thinning beginning before aortic valve closure. This unfolding pattern precedes longitudinal and circumferential elongation and may facilitate early diastolic filling.     

离体心舰顺应性综合评定的实验研究

采用精度高、稳定性好的标准等长收缩灌流装置,从数学和弹性力学角度计算、分析正常及缺氧状态下大鼠乳头肌的应力-应变、应力松弛和蠕变三种生物物理特性的变化。以前者反映顺应性的静态变化,尝试以后二者反映顺应性的动态变化。结果表明,缺氧状态下,不仅心肌的静态顺应性降低(应力—应变曲线明显左移),其动态顺应性也明显降低(应力松驰及蠕变的程度和速度明显降低)。结果提示,本实验方法可用于不同状态下离体心肌顺应性的综合评定。     

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.     

A constitutive model for the mechanical behaviour of soft connective tissues

A model of the mechanical behaviour of soft connective tissue has been developed by considering the role of the collagen and glycosaminoglycan (GAG) components within the tissue in order to examine the mechanism by which a variation in the GAG components may exert a control over the mechanical properties of the tissue. It is proposed that the strain energy stored within the collagen fibrils of the loaded tissue can be transferred into a potential field created by the charged GAG components and their electrostatic interaction with the collagen fibrils. A fundamental mechanical unit is described to simulate this energy transfer and a combination of such units is used to represent the tissue. The computer implementation of the proposed tissue model shows it to reproduce many features which have been recognised in the rate dependent mechanical behaviour of soft tissues. These include the characteristic non-linearity of the force-deformation behaviour and the approximate invariance of the stress relaxation behaviour with deformation. The model is also consistent with earlier constitutive representations of tissue behaviour.     

Structural finite deformation model of the left ventricle during diastole and systole

A model of left ventricular function is developed based on morphological characteristics of the myocardial tissue. The passive response of the three-dimensional collagen network and the active contribution of the muscle fibers are integrated to yield the overall response of the left ventricle which is considered to be a thick wall cylinder. The deformation field and the distributions of stress and pressure are determined at each point in the cardiac cycle by numerically solving three equations of equilibrium. Simulated results in terms of the ventricular deformation during ejection and isovolumic cycles are shown to be in good qualitative agreement with experimental data. It is shown that the collagen network in the heart has considerable effect on the pressure-volume loops. The particular pattern of spatial orientation of the collagen determines the ventricular recoil properties in early diastole. The material properties (myocardial stiffness and contractility) are shown to affect both the pressure-volume loop and the deformation pattern of the ventricle. The results indicate that microstructural consideration offer a realistic representation of the left ventricle mechanics.     

Fast Simulation of Mechanical Heterogeneity in the Electrically Asynchronous Heart Using the MultiPatch Module

Cardiac electrical asynchrony occurs as a result of cardiac pacing or conduction disorders such as left bundle-branch block (LBBB). Electrically asynchronous activation causes myocardial contraction heterogeneity that can be detrimental for cardiac function. Computational models provide a tool for understanding pathological consequences of dyssynchronous contraction. Simulations of mechanical dyssynchrony within the heart are typically performed using the finite element method, whose computational intensity may present an obstacle to clinical deployment of patient-specific models. We present an alternative based on the CircAdapt lumped-parameter model of the heart and circulatory system, called the MultiPatch module. Cardiac walls are subdivided into an arbitrary number of patches of homogeneous tissue. Tissue properties and activation time can differ between patches. All patches within a wall share a common wall tension and curvature. Consequently, spatial location within the wall is not required to calculate deformation in a patch. We test the hypothesis that activation time is more important than tissue location for determining mechanical deformation in asynchronous hearts. We perform simulations representing an experimental study of myocardial deformation induced by ventricular pacing, and a patient with LBBB and heart failure using endocardial recordings of electrical activation, wall volumes, and end-diastolic volumes. Direct comparison between simulated and experimental strain patterns shows both qualitative and quantitative agreement between model fibre strain and experimental circumferential strain in terms of shortening and rebound stretch during ejection. Local myofibre strain in the patient simulation shows qualitative agreement with circumferential strain patterns observed in the patient using tagged MRI. We conclude that the MultiPatch module produces realistic regional deformation patterns in the asynchronous heart and that activation time is more important than tissue location within a wall for determining myocardial deformation. The CircAdapt model is therefore capable of fast and realistic simulations of dyssynchronous myocardial deformation embedded within the closed-loop cardiovascular system.     

The orthotropic viscoelastic behavior of aortic elastin

In this paper, we studied the viscoelastic behaviors of isolated aortic elastin using combined modeling and experimental approaches. Biaxial stress relaxation and creep experiments were performed to study the time-dependent behavior of elastin. Experimental results reveal that stress relaxation preconditioning is necessary in order to obtain repeatable stress relaxation responses. Elastin exhibits less stress relaxation than intact or decellularized aorta. The rate of stress relaxation of intact and decellularized aorta is linearly dependent on the initial stress levels. The rate of stress relaxation for elastin increases linearly at stress levels below about 60 kPa; however, the rate changes very slightly at higher initial stress levels. Experimental results also show that creep response is negligible for elastin, and the intact or decellularized aorta. A quasi-linear viscoelasticity model was incorporated into a statistical mechanics based eight-chain microstructural model at the fiber level to simulate the orthotropic viscoelastic behavior of elastin. A user material subroutine was developed for finite element analysis. Results demonstrate that this model is suitable to capture both the orthotropic hyperelasticity and viscoelasticity of elastin.     

An elasto-visco-plastic model of cell aggregates

Concentrated cell suspensions exhibit different mechanical behavior depending on the mechanical stress or deformation they undergo. They have a mixed rheological nature: cells behave elastically or viscoelastically, they can adhere to each other whereas the carrying fluid is usually Newtonian. We report here on a new elasto-visco-plastic model which is able to describe the mechanical properties of a concentrated cell suspension or aggregate. It is based on the idea that the rearrangement of adhesion bonds during the deformation of the aggregate is related to the existence of a yield stress in the macroscopic constitutive equation. We compare the predictions of this new model with five experimental tests: steady shear rate, oscillatory shearing tests, stress relaxation, elastic recovery after steady prescribed deformation, and uniaxial compression tests. All of the predictions of the model are shown to agree with these experiments.     

Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage

The dominant mechanism giving rise to the viscoelastic response of articular cartilage during compression is the nonlinear diffusive interaction of the fluid and solid phases of the tissue as they flow relative to one another. The present study is concerned with the role of this interaction under uniaxial stress relaxation in compression. The model is a biphasic mixture of fluid and solid which incorporates the strain-dependent permeability found earlier from permeation experiments. When a ramp-displacement is imposed on the articular surface, simple, but accurate, asymptotic approximations are derived for the deformation and stress fields in the tissue for slow and moderately fast rates of compression. They are shown to agree very well with experiment and they provide a simple means for determining the material parameters. Moreover, they lead to important insights into the role of the flow-dependent viscoelastic nature of articular cartilage and other hydrated biological tissues.     

A three-dimensional constitutive model for the stress relaxation of articular ligaments

A new nonlinear constitutive model for the three-dimensional stress relaxation of articular ligaments is proposed. The model accounts for finite strains, anisotropy, and strain-dependent stress relaxation behavior exhibited by these ligaments. The model parameters are identified using published uniaxial stress–stretch and stress relaxation data on human medial collateral ligaments (MCLs) subjected to tensile tests in the fiber and transverse to the fiber directions (Quapp and Weiss in J Biomech Eng Trans ASME 120:757–763, 1998; Bonifasi-Lista et al. in J Orthop Res 23(1):67–76, 2005). The constitutive equation is then used to predict the nonlinear elastic and stress relaxation response of ligaments subjected to shear deformations in the fiber direction and transverse to the fiber direction, and an equibiaxial extension. A direct comparison with stress relaxation data collected by subjecting human MCLs to shear deformation in the fiber direction is presented in order to demonstrate the predictive capabilities of the model.     

Thyroid hormone prevention of disorders of cardiac contractile function during stress

The depression of cardiac contractility has been observed in rats during the immobilized stress in state of relative physiological rest and maximal load. In the animals pretreated with thyroid after stress the indexes of intensity and rate of myocardial contraction and relaxation didn't differ from the control, and during the maximal load the myocardium was characterized by the less expressed decrease of the structure functioning intensity and the higher rate of relaxation. The data obtained show that the physiological doses of thyroid hormones prevent the myocardium from contractile disorders during stress.     

Contribution of the extracellular matrix to the viscoelastic behavior of the urinary bladder wall

We previously reported that when the stress relaxation response of urinary bladder wall (UBW) tissue was analyzed using a single continuous reduced relaxation function (RRF), we observed non-uniformly distributed, time-dependent residuals (Ann Biomed Eng 32(10):1409-1419, 2004). We concluded that the single relaxation spectrum was inadequate and that a new viscoelastic model for bladder wall was necessary. In the present study, we report a new approach composed of independent RRFs for smooth muscle and the extracellular matrix components (ECM), connected through a stress-dependent recruitment function. In order to determine the RRF for the ECM component, biaxial stress relaxation experiments were first performed on decellularized extracellular matrix network of the bladder obtained from normal and spinal cord injured rats. While it was assumed that smooth muscle followed a single spectrum RRF, modeling the UBW ECM required a dual-Gaussian spectrum. Experimental results revealed that the ECM stress relaxation response was insensitive to the initial stress level. Thus, the average ECM RRF parameters were determined by fitting the average stress relaxation data. The resulting stress relaxation behavior of whole bladder tissue was modeled by combining the ECM RRF with the RRF for the smooth muscle component using an exponential recruitment function representing the recruitment of collagen fibers at higher stress levels. In summary, the present study demonstrated, for the first time, that stress relaxation response of bladder tissue can be better modeled when divided into the contributions of the extracellular matrix and smooth muscle components. This modeling approach is suitable for prediction of mechanical behaviors of the urinary bladder and other organs that exhibit rapid tissue remodeling (i.e., smooth muscle hypertrophy and altered ECM synthesis) under various pathological conditions.     

A quasi-linear, viscoelastic, structural model of the plantar soft tissue with frequency-sensitive damping properties

Little is known about the structural properties of plantar soft-tissue areas other than the heel; nor is it known whether the structural properties vary depending on location. Furthermore, although the quasi-linear viscoelastic (QLV) theory has been used to model many soft-tissue types, it has not been employed to model the plantar soft tissue. The structural properties of the plantar soft tissue were quantified via stress relaxation experiments at seven regions (subcalcaneal, five submetatarsal, and subhallucal) across eight cadaveric feet. The cadaveric feet were 36.9 +/- 17.4 (mean +/- S.D.) years of age, all free from vascular diseases and orthopedics disorders. All tests were performed at a constant environmental temperature of 35 degrees C. Stress relaxation experiments were performed; different loads were employed for different areas based on normative gait data. A modification of the relaxation spectrum employed within the QLV theory allowed for the inclusion of frequency-sensitive relaxation properties in addition to nonlinear elastic behavior. The tissue demonstrated frequency-dependent damping properties that made the QLV theory ill suited to model the relaxation. There was a significant difference between the elastic structural properties (A) of the subcalcaneal tissue and all other areas (p = 0.004), and a trend (p = 0.067) for the fifth submetatarsal to have less viscous damping (c1) than the subhallucal, or first, second, or third submetatarsal areas. Thus, the data demonstrate that the structural properties of the foot can vary across regions, but careful consideration must be given to the applied loads and the manner in which the loads were applied.     

The interdependent contributions of gravitational and structural features to perfusion distribution in a multiscale model of the pulmonary circulation

Recent experimental and imaging studies suggest that the influence of gravity on the measured distribution of blood flow in the lung is largely through deformation of the parenchymal tissue. To study the contribution of hydrostatic effects to regional perfusion in the presence of tissue deformation, we have developed an anatomically structured computational model of the pulmonary circulation (arteries, capillaries, veins), coupled to a continuum model of tissue deformation, and including scale-appropriate fluid dynamics for blood flow in each vessel type. The model demonstrates that both structural and the multiple effects of gravity on the pulmonary circulation make a distinct contribution to the distribution of blood. It shows that postural differences in perfusion gradients can be explained by the combined effect of tissue deformation and extra-acinar blood vessel resistance to flow in the dependent tissue. However, gravitational perfusion gradients persist when the effect of tissue deformation is eliminated, highlighting the importance of the hydrostatic effects of gravity on blood distribution in the pulmonary circulation. Coupling of large- and small-scale models reveals variation in microcirculatory driving pressures within isogravitational planes due to extra-acinar vessel resistance. Variation in driving pressures is due to heterogeneous large-vessel resistance as a consequence of geometric asymmetry in the vascular trees and is amplified by the complex balance of pressures, distension, and flow at the microcirculatory level.