Yoram Lanir

Constitutive equations for fibrous connective tissues

A general multiaxial theory for the constitutive relations in fibrous connective tissues is developed on the basis of microstructural and thermodynamic considerations. It is compatible with existing general material theories. In elastic tissues, the theory considers the strain-energy function to be the sum of strain-energies of the tissues components. The stresses are derived from this strain-energy function. Viscoelastic constitutive relations are obtained in an analogous manner. Few examples are developed in detail. The results of the present strain-energy based theory are identical with those of the authors previous structural models (Lanir, 1979a, b) which are based on detailed equilibrium analysis. It turns out, however, that the analytical work involved in solving boundary value problems is considerably shorter if the present theory is used. The advantages of structural theories in avoiding ambiguity in material characterization and in offering an insight into the function, structure and mechanics of tissue components are discussed.

Time-dependent mechanical behavior of sheep digital tendons, including the effects of preconditioning.

The time-dependent mechanical properties of sheep digital extensor tendons were studied by sequences of stress-relaxation tests. The results exhibited irreversible preconditioning and reversible viscoelasticity. Preconditioning effects were manifested by stress decay during consecutive stretch cycles to the same strain level, accompanied by elongation of the tendons reference length. They intensified with increased strain level, and were reduced or became negligible as the strain decreased. The significance of intrinsic response mechanisms was studied via a structural model that includes viscoelasticity, preconditioning, and morphology of the tendons collagen fibers. Model/data comparisons showed good agreement and good predictive power, suggesting that preconditioning can be integrated into comprehensive material characterization of tendons.

A Transversely Isotropic Viscoelastic Constitutive Equation for Brainstem Undergoing Finite Deformation

The objective of this study was to define the constitutive response of brainstem undergoing finite shear deformation. Brainstem was characterized as a transversely isotropic viscoelastic material and the material model was formulated for numerical implementation. Model parameters were fit to shear data obtained in porcine brainstem specimens undergoing finite shear deformation in three directions: parallel, perpendicular, and cross sectional to axonal fiber orientation and determined using a combined approach of finite element analysis (FEA) and a genetic algorithm (GA) optimizing method. The average initial shear modulus of brainstem matrix of 4-week old pigs was 12.7 Pa, and therefore the brainstem offers little resistance to large shear deformations in the parallel or perpendicular directions, due to the dominant contribution of the matrix in these directions. The fiber reinforcement stiffness was 121.2 Pa, indicating that brainstem is anisotropic and that axonal fibers have an important role in the cross-sectional direction. The first two leading relative shear relaxation moduli were 0.8973 and 0.0741, respectively, with corresponding characteristic times of 0.0047 and 1.4538 s, respectively, implying rapid relaxation of shear stresses. The developed material model and parameter estimation technique are likely to find broad applications in neural and orthopaedic tissues.

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.

Why is the subendocardium more vulnerable to ischemia? A new paradigm

Myocardial ischemia is transmurally heterogeneous where the subendocardium is at higher risk. Stenosis induces reduced perfusion pressure, blood flow redistribution away from the subendocardium, and consequent subendocardial vulnerability. We propose that the flow redistribution stems from the higher compliance of the subendocardial vasculature. This new paradigm was tested using network flow simulation based on measured coronary anatomy, vessel flow and mechanics, and myocardium-vessel interactions. Flow redistribution was quantified by the relative change in the subendocardial-to-subepicardial perfusion ratio under a 60-mmHg perfusion pressure reduction. Myocardial contraction was found to induce the following: 1) more compressive loading and subsequent lower transvascular pressure in deeper vessels, 2) consequent higher compliance of the subendocardial vasculature, and 3) substantial flow redistribution, i.e., a 20% drop in the subendocardial-to-subepicardial flow ratio under the prescribed reduction in perfusion pressure. This flow redistribution was found to occur primarily because the vessel compliance is nonlinear (pressure dependent). The observed thinner subendocardial vessel walls were predicted to induce a higher compliance of the subendocardial vasculature and greater flow redistribution. Subendocardial perfusion was predicted to improve with a reduction of either heart rate or left ventricular pressure under low perfusion pressure. In conclusion, subendocardial vulnerability to a acute reduction in perfusion pressure stems primarily from differences in vascular compliance induced by transmural differences in both extravascular loading and vessel wall thickness. Subendocardial ischemia can be improved by a reduction of heart rate and left ventricular pressure.

Effects of Strain Level and Proteoglycan Depletion on Preconditioning and Viscoelastic Responses of Rat Dorsal Skin

AbstractThe mechanical response of rat dorsal skin was experimentally studied under cyclic uniaxial ramp stretches to various strain levels. Special emphasis was paid to the effects of the preconditioning protocol on the stress–strain relationship, and to the effects of ramp strain level and proteoglycan (PG) depletion, on viscoelasticity and preconditioning responses. The results show that preconditioning significantly reduced both the slope of the low strain stress–strain relationship, and the stress levels at consecutive stretch cycles. Following a short rest there was a significant partial recovery. Stress decay due to preconditioning was significant at all strain levels, and increased with strain. Stress relaxation was significant at all strain levels, but varied little with strain. Recovery following a 10 min rest was minor at all strain levels and varied little with strain. PG-depleted samples manifested similar response patterns. These results are consistent with the following notion: (1) skin consists of three mechanical components: elastin and proteoglycan which dominate the low strain response and are effected by preconditioning and (PG) depletion, and collagen which dominates the high strain response and is unaffected by preconditioning and PG depletion; (2) that the viscoelasticity of elastin and PG vs that of collagen are similar, so that rat dorsal skin can be regarded quasilinear viscoelastic.

Large-Scale 3-D Geometric Reconstruction of the Porcine Coronary Arterial Vasculature Based on Detailed Anatomical Data

The temporal and spatial distribution of coronary blood flow, pressure, and volume are determined by the branching pattern and three-dimensional (3-D) geometry of the coronary vasculature, and by the mechanics of heart wall and vascular tone. Consequently, a realistic simulation of coronary blood flow requires, as a first step, an accurate representation of the coronary vasculature in a 3-D model of the beating heart. In the present study, a large-scale stochastic reconstruction of the asymmetric coronary arterial trees (right coronary artery, RCA; left anterior descending, LAD; and left circumflex, LCx) of the porcine heart has been carried out to set the stage for future hemodynamic analysis. The model spans the entire coronary arterial tree down to the capillary vessels. The 3-D tree structure was reconstructed initially in rectangular slab geometry by means of global geometrical optimization using parallel simulated annealing (SA) algorithm. The SA optimization was subject to constraints prescribed by previously measured morphometric features of the coronary arterial trees. Subsequently, the reconstructed trees were mapped onto a prolate spheroid geometry of the heart. The transformed geometry was determined through least squares minimization of the related changes in both segments lengths and their angular characteristics. Vessel diameters were assigned based on a novel representation of diameter asymmetry along bifurcations. The reconstructed RCA, LAD and LCx arterial trees show qualitative resemblance to native coronary networks, and their morphological statistics are consistent with the measured data. The present model constitutes the first most extensive reconstruction of the entire coronary arterial system which will serve as a geometric foundation for future studies of flow in an anatomically accurate 3-D coronary vascular model.

Optimal design of biaxial tests for structural material characterization of flat tissues

A rational methodology is developed for optimal design of biaxial stretch tests intended for estimating material parameters of flat tissues. It is applied to a structural model with a variety of constitutive equations and test protocols, and for a wide range of parameter levels. The results show nearly identical optimal designs under all circumstances. Optimality is obtained with two uniaxial stretch tests at mutually normal directions inclined by 22.5 deg to the axes of material symmetry. Protocols which include additional equibiaxial tests provide superior estimation with lower variance of estimates. Tests performed at angles 0, 45, and 90 deg to the axes of material symmetry provide unreliable estimates. The optimal sampling is variable and depends on the protocols and model parameters. In conclusion, the results indicate that biaxial tests can be improved over presently common procedures and show that this conclusion applies for a variety of circumstances.

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.

Mechanisms of myocardium-coronary vessel interaction

The mechanisms by which the contracting myocardium exerts extravascular forces (intramyocardial pressure, IMP) on coronary blood vessels and by which it affects the coronary flow remain incompletely understood. Several myocardium-vessel interaction (MVI) mechanisms have been proposed, but none can account for all the major flow features. In the present study, we hypothesized that only a specific combination of MVI mechanisms can account for all observed coronary flow features. Three basic interaction mechanisms (time-varying elasticity, myocardial shortening-induced intracellular pressure, and ventricular cavity-induced extracellular pressure) and their combinations were analyzed based on physical principles (conservation of mass and force equilibrium) in a realistic data-based vascular network. Mechanical properties of both vessel wall and myocardium were coupled through stress analysis to simulate the response of vessels to internal blood pressure and external (myocardial) mechanical loading. Predictions of transmural dynamic vascular pressure, diameter, and flow velocity were determined under each MVI mechanism and compared with reported data. The results show that none of the three basic mechanisms alone can account for the measured data. Only the combined effect of the cavity-induced extracellular pressure and the shortening-induced intramyocyte pressure provides good agreement with the majority of measurements. These findings have important implications for elucidating the physical basis of IMP and for understanding coronary phasic flow and coronary artery and microcirculatory disease.