Caveolae in ventricular myocytes are required for stretch-dependent conduction slowing.

Mechanical stretch of cardiac muscle modulates action potential propagation velocity, causing potentially arrhythmogenic conduction slowing. The mechanisms by which stretch alters cardiac conduction remain unknown, but previous studies suggest that stretch can affect the conformation of caveolae in myocytes and other cell types. We tested the hypothesis that slowing of action potential conduction due to cardiac myocyte stretch is dependent on caveolae. Cardiac action potential propagation velocities, measured by optical mapping in isolated mouse hearts and in micropatterned mouse cardiomyocyte cultures, decreased reversibly with volume loading or stretch, respectively (by 19±5% and 26±4%). Stretch-dependent conduction slowing was not altered by stretch-activated channel blockade with gadolinium or by GsMTx-4 peptide, but was inhibited when caveolae were disrupted via genetic deletion of caveolin-3 (Cav3 KO) or membrane cholesterol depletion by methyl-ß-cyclodextrin. In wild-type mouse hearts, stretch coincided with recruitment of caveolae to the sarcolemma, as observed by electron microscopy. In myocytes from wild-type but not Cav3 KO mice, stretch significantly increased cell membrane capacitance (by 98±64%), electrical time constant (by 285±149%), and lipid recruitment to the bilayer (by 84±39%). Recruitment of caveolae to the sarcolemma during physiologic cardiomyocyte stretch slows ventricular action potential propagation by increasing cell membrane capacitance.

A novel computational model of mouse myocyte electrophysiology to assess the synergy between Na+ loading and CaMKII.

Ca(2+)-calmodulin-dependent protein kinase II (CaMKII) hyperactivity in heart failure causes intracellular Na(+) ([Na(+)]i) loading (at least in part by enhancing the late Na(+) current). This [Na(+)]i gain promotes intracellular Ca(2+) ([Ca(2+)]i) overload by altering the equilibrium of the Na(+)-Ca(2+) exchanger to impair forward-mode (Ca(2+) extrusion), and favour reverse-mode (Ca(2+) influx) exchange. In turn, this Ca(2+) overload would be expected to further activate CaMKII and thereby form a pathological positive feedback loop of ever-increasing CaMKII activity, [Na(+)]i, and [Ca(2+)]i. We developed an ionic model of the mouse ventricular myocyte to interrogate this potentially arrhythmogenic positive feedback in both control conditions and when CaMKIIδC is overexpressed as in genetically engineered mice. In control conditions, simulation of increased [Na(+)]i causes the expected increases in [Ca(2+)]i, CaMKII activity, and target phosphorylation, which degenerate into unstable Ca(2+) handling and electrophysiology at high [Na(+)]i gain. Notably, clamping CaMKII activity to basal levels ameliorates but does not completely offset this outcome, suggesting that the increase in [Ca(2+)]i per se plays an important role. The effect of this CaMKII-Na(+)-Ca(2+)-CaMKII feedback is more striking in CaMKIIδC overexpression, where high [Na(+)]i causes delayed afterdepolarizations, which can be prevented by imposing low [Na(+)]i, or clamping CaMKII phosphorylation of L-type Ca(2+) channels, ryanodine receptors and phospholamban to basal levels. In this setting, Na(+) loading fuels a vicious loop whereby increased CaMKII activation perturbs Ca(2+) and membrane potential homeostasis. High [Na(+)]i is also required to produce instability when CaMKII is further activated by increased Ca(2+) loading due to ß-adrenergic activation. Our results support recent experimental findings of a synergistic interaction between perturbed Na(+) fluxes and CaMKII, and suggest that pharmacological inhibition of intracellular Na(+) loading can contribute to normalizing Ca(2+) and membrane potential dynamics in heart failure.

A nonlinear model of passive muscle viscosity.

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.

Cardiac resynchronization: insight from experimental and computational models.

Cardiac resynchronization therapy (CRT) is a promising therapy for heart failure patients with a conduction disturbance, such as left bundle branch block. The aim of CRT is to resynchronize contraction between and within ventricles. However, about 30% of patients do not respond to this therapy. Therefore, a better understanding is needed for the relation between electrical and mechanical activation. In this paper, we focus on to what extent animal experiments and mathematical models can help in order to understand the pathophysiology of asynchrony to further improve CRT.

Age-associated cardiac dysfunction in Drosophila melanogaster.

The fruit fly, Drosophila melanogaster, has served as a valuable model/organism for the study of aging and was the first organism possessing a circulatory system to have its genome completely sequenced. However, little is known about the function of the heartlike organ of flies during the aging process. We have developed methods for studying cardiac function in vivo in adult flies. Using 2 different cardiovascular stress methods (elevated ambient temperature and external electrical pacing), we found that maximal heart rate is significantly and reproducibly reduced with aging in Drosophila, analogous to observations in elderly humans. We also describe for the first time several other aspects of the cardiac physiology of young adult and aging Drosophila, including an age-associated increase in rhythm disturbances. These observations suggest that the study of declining cardiac function in aging flies may serve as a genetically tractable model for genome-wide mutational screening for genes that participate in or protect against cardiac aging and disease.

Myocardial mechanics and collagen structure in the osteogenesis imperfecta murine (oim).

Because the amount and structure of type I collagen are thought to affect the mechanics of ventricular myocardium, we investigated myocardial collagen structure and passive mechanical function in the osteogenesis imperfecta murine (oim) model of pro-alpha2(I) collagen deficiency, previously shown to have less collagen and impaired biomechanics in tendon and bone. Compared with wild-type littermates, homozygous oim hearts exhibited 35% lower collagen area fraction (P:<0.05), 38% lower collagen fiber number density (P:<0.05), and 42% smaller collagen fiber diameter (P:<0.05). Compared with wild-type, oim left ventricular (LV) collagen concentration was 45% lower (P:<0.0001) and nonreducible pyridinoline cross-link concentration was 22% higher (P:<0.03). Mean LV volume during passive inflation from 0 to 30 mm Hg in isolated hearts was 1.4-fold larger for oim than wild-type (P:=NS). Uniaxial stress-strain relations in resting right ventricular papillary muscles exhibited 60% greater strains (P:<0.01), 90% higher compliance (P:=0.05), and 64% higher nonlinearity (P:<0.05) in oim. Mean opening angle, after relief of residual stresses in resting LV myocardium, was 121+/-9 degrees in oim compared with 45+/-4 degrees in wild-type (P:<0.0001). Mean myofiber angle in oim was 23+/-8 degrees greater than wild-type (P:<0.02). Decreased myocardial collagen diameter and amount in oim is associated with significantly decreased fiber and chamber stiffness despite modestly increased collagen cross-linking. Altered myofiber angles and residual stress may be beneficial adaptations to these mechanical alterations to maintain uniformity of transmural fiber strain. In addition to supporting and organizing myocytes, myocardial collagen contributes directly to ventricular stiffness at high and low loads and can influence stress-free state and myofiber architecture.

Transmural distribution of three-dimensional systolic strains in stunned myocardium.

BACKGROUND: Regional function in stunned myocardium is usually thought to be more depressed in the endocardium than the epicardium. This has been attributed to the greater loss of blood flow at the endocardium during ischemia. METHODS AND RESULTS: We measured transmural distributions of 3D systolic strains relative to local myofiber axes in open-chest anesthetized dogs before 15 minutes of left anterior descending coronary artery occlusion and during 2 hours of reperfusion. During ischemia, regional myocardial blood flow was reduced 84% at the endocardium and 32% at the epicardium (P<0.005, n=7), but changes in end-systolic fiber length from baseline were transmurally uniform. Relative to baseline, radial segments in stunned tissue were significantly thinner at the endocardium than the epicardium at end systole (24+/-5% versus 16+/-3%; P<0.05, n=8), consistent with previous reports. Unlike radial and cross-fiber segments, however, the increase of end-systolic fiber lengths in stunned myocardium had no significant transmural gradient (23+/-8% epicardium versus 21+/-4% endocardium). We also observed significant 3D diastolic dysfunction in the ischemic-reperfused region transmurally. CONCLUSIONS: Myocardial ischemia/reperfusion in the dog results in a significant transmural gradient of dysfunction between epicardial and endocardial layers in radial and cross-fiber segments, but not for fiber segments, despite a gradient in blood flow reduction during ischemia. Perhaps systolic fiber dysfunction rather than the degree of perfusion deficit during the preceding ischemic period may be the main determinant of myocardial dysfunction during reperfusion.

Cell proliferation rates in an artificial tissue-engineered environment.

Worldwide, and particularly in Europe, Japan and the USA, cardiovascular disease is a major killer. It can be treated using tissue or organ transplant surgery, but donor organs may be scarce. Tissue engineering is the integration of engineering principles and biology to produce satisfactory synthetic replacement body parts, using viable cells in a suitable matrix, for regenerative medicine. The aim of this study was to measure and compare cell proliferation kinetics after different time intervals of myofibroblasts in a synthetic matrix, thus to be able to deduce the period that a transplanted-cell population can be expected to survive in a tissue-engineered environment. Porcine aortic wall cells were grown in a porous sponge scaffold, that later could be fashioned into aortic or heart valve substitutes. Freshly acquired cells were seeded on identical sponges and were grown under normal culture conditions for a period of 4 weeks. Seeding concentration was a million cells per sponge. Cells progressively populated the sponges, both covering the surface and infiltrating the depth of the matrix, via sponge pores. Samples were taken at 1 week and at 4 weeks, and the rate of cell proliferation was determined by the metaphase arrest technique. Specimens were also taken for light and electron microscopy to determine whether these transplanted cells were capable of synthesizing their own extracellular matrix.

Three-dimensional analysis of regional cardiac function: a model of rabbit ventricular anatomy.

The three-dimensional geometry and anisotropic properties of the heart give rise to nonhomogeneous distributions of stress, strain, electrical activation and repolarization. In this article we review the ventricular geometry and myofiber architecture of the heart, and the experimental and modeling studies of three-dimensional cardiac mechanics and electrophysiology. The development of a three-dimensional finite element model of the rabbit ventricular geometry and fiber architecture is described in detail. Finally, we review the experimental results, from the level of the cell to the intact organ, which motivate the development of coupled three-dimensional models of cardiac electromechanics and mechanoelectric feedback.

Homogenization modeling for the mechanics of perfused myocardium.

Altered coronary perfusion can change the apparent diastolic stiffness of ventricular myocardium--the 'garden hose' effect. Our recent findings showed that myocardial strains are reduced during ventricular filling, primarily along the directions transverse to the coronary microvessels. In this article, we review hypotheses and theoretical models regarding the role that regional wall stress plays in the mechanical interaction between myocardium and coronary circulation. Various mechanisms have been used to explain the effects of the tissue stress on coronary flow, as well as the effect of coronary dynamics on myocardial mechanics. Many models of coronary pressure-flow relations using lumped parameter circuit analogs. Poroelasticity and swelling theories have been used to model the mechanics of perfused muscle. Here, we describe a new mathematical model of the mechanics of perfused myocardium derived using homogenization theory. In this model, perfused myocardium is treated as a nonlinear anisotropic elastic solid embedded with cylindrical vessels of known distensibility. The solid compartment is incompressible but the vascular compartment may change volume according to a simple relation between vessel diameter and perfusion pressure. The work done by the perfusion pressure in changing vascular volume contributes to the macroscopic strain energy and hence affects the stress and stiffness of the composite. Conversely, the stress in the tissue affects microvessel diameter and volume, since tractions transverse to the vessel axis oppose the internal blood pressure. Finite element simulations of passive filling show good agreement of model with experimental results.

Modelling the mechanical properties of cardiac muscle.

A model of passive and active cardiac muscle mechanics is presented, suitable for use in continuum mechanics models of the whole heart. The model is based on an extensive review of experimental data from a variety of preparations (intact trabeculae, skinned fibres and myofibrils) and species (mainly rat and ferret) at temperatures from 20 to 27 degrees C. Experimental tests include isometric tension development, isotonic loading, quick-release/restretch, length step and sinusoidal perturbations. We show that all of these experiments can be interpreted with a four state variable model which includes (i) the passive elasticity of myocardial tissue, (ii) the rapid binding of Ca2+ to troponin C and its slower tension-dependent release, (iii) the kinetics of tropomyosin movement and availability of crossbridge binding sites and the length dependence of this process and (iv) the kinetics of crossbridge tension development under perturbations of myofilament length.

Structural basis of regional dysfunction in acutely ischemic myocardium.

OBJECTIVE: Impaired systolic function in the normally perfused myocardium adjacent to an ischemic region - the functional border zone - is thought to result from mechanical interactions across the perfusion boundary. We investigated how segment orientation and vessel involved affect regional strains in the functional border zone and whether altered stresses associated with a step transition in contractility can explain the functional border zone. METHODS AND RESULTS: Regional epicardial strain distributions were obtained from measured displacements of radiopaque markers in open-chest anesthetized canines, and related to local myofiber angles and blood flows. The functional border zone for fiber strain was significantly narrower than that for cross-fiber strain and significantly wider for left anterior descending (LAD) than left circumflex (LCx) coronary occlusion (1.23 vs. 0.45 cm). A detailed three-dimensional computational model with a one-to-one relation between perfusion and myofilament activation and no transitional zone of intermediate contractility showed close agreement with these observations and significantly elevated stresses in the border zone. Differences between LAD and LCx occlusions in the model were due to differences in left ventricular systolic pressure and not to differences in perfusion boundary or muscle fiber orientation. The border zone was narrower for fiber strain than cross-fiber strain because systolic stiffness is greatest along the muscle fiber direction. CONCLUSION: Abnormal regional mechanics in the acute ischemic border arise from increased wall stresses without a transitional zone of intermediate contractility. Perfusion is more tightly coupled to fiber than cross-fiber strain, and a wider functional border zone of fiber strain during LAD than LCx occlusion is primarily due to higher regional wall stresses rather than anatomic variations.

Flow-function relations during graded coronary occlusions in the dog: effects of transmural location and segment orientation.

OBJECTIVE: The sensitive relationship between regional myocardial perfusion and local systolic deformation during acute myocardial ischemia is not independent of the transmural location or segment orientation. The aim of this study was to determine the effects of fiber orientation and transmural location on the relationships between regional myocardial flow and three-dimensional systolic wall strain during graded coronary artery occlusions. METHODS: Transmural distributions of three-dimensional strain (by biplane radiography of implanted radiopaque markers) and myocardial blood flows (using fluorescent microspheres) were measured in the ischemic region during graded left anterior descending (LAD) coronary artery occlusions in 12 anesthetized dogs. RESULTS: Occlusion of the coronary artery did not significantly alter mean heart rate or end-systolic pressure. As flow decreased during graded occlusions, ischemia significantly changed systolic circumferential, longitudinal, radial, fiber and cross-fiber strains (p < 0.004). There was a significant effect of transmural position on circumferential, cross-fiber and radial strains, but not on fiber or longitudinal strains. Ischemia significantly altered all normal strains: circumferential, longitudinal, fiber, cross-fiber and radial. There was a strong interaction effect between transmural location and blood flow for circumferential, cross-fiber and radial strains, but not fiber or longitudinal strains. CONCLUSION: During non-transmural ischemia, there is evidence of strong transmural tethering in the cross-fiber direction, whereas the fiber-strain flow relation is independent of transmural position. Thus, whether the relationship between local myocardial bloodflow and systolic strain during acute ischemia is dependent on transmural location, depends on segment orientation.

Regional heterogeneity of function in nonischemic dilated cardiomyopathy.

OBJECTIVE: To quantify regional three-dimensional (3D) motion and myocardial strain using magnetic resonance (MR) tissue tagging in patients with non-ischemic dilated cardiomyopathy (DCM). METHODS: MR grid tagged images were obtained in multiple short- and long-axis planes in thirteen DCM patients. Regional 3D displacements and strains were calculated with the aid of a finite element model. Five of the patients were also imaged after LV volume reduction by partial left ventriculectomy (PLV), combined with mitral and tricuspid valve repair. RESULTS: DCM patients showed consistent, marked regional heterogeneity. Systolic lengthening occurred in the septum in both circumferential (%S(C) -5+/-7%) and longitudinal (%S(L) -2+/-5%) shortening components (negative values indicating lengthening). In contrast, the lateral wall showed relatively normal systolic shortening (%S(C) 12+/-6% and %S(L) 6+/-5%, P<0.001 lateral vs. septal walls). A geometric estimate of regional stress was correlated with shortening on a regional basis, but could not account for the differences in shortening between regions. In the five patients imaged post-PLV, septal function recovered (%S(C) 9+/-5%,%S(L) 6+/-5%, P<0.02 pre vs. post) with normalization of wall stress, whereas lateral wall shortening was reduced (%S(C) 7+/-6%,%S(L) 3+/-3%, P<0.02 pre vs. post) around the site of surgical resection. CONCLUSIONS: A consistent pattern of regional heterogeneity of myocardial strain was seen in all patients. Reduced function may be related to increased wall stress, since recovery of septal function is possible after PLV. However, simple geometric stress determinants are not sufficient to explain the functional heterogeneity observed.

Heart valve and arterial tissue engineering.

In the industrialized world, cardiovascular disease alone is responsible for almost half of all deaths. Many of the conditions can be treated successfully with surgery, often using transplantation techniques; however, autologous vessels or human-donated organs are in short supply. Tissue engineering aims to create specific, matching grafts by growing cells on appropriate matrices, but there are many steps between the research laboratory and the operating theatre. Neo-tissues must be effective, durable, non-thrombogenic and non-immunogenic. Scaffolds should be bio-compatible, porous (to allow cell/cell communication) and amenable to surgery. In the early days of cardiovascular tissue engineering, autologous or allogenic cells were grown on inert matrices, but patency and thrombogenicity of grafts were disappointing. The current ethos is toward appropriate cell types grown in (most often) a polymeric matrix that degrades at a rate compatible with the cells' production of their own extracellular matrical proteins, thus gradually replacing the graft with a living counterpart. The geometry is crucial. Computer models have been made of valves, and these are used as three-dimensional patterns for mass-production of implant scaffolds. Vessel walls have integral connective tissue architecture, and application of physiological level mechanical forces conditions bio-engineered components to align in precise orientation. This article reviews the concepts involved and successes achieved to date.

Anterior-posterior strain variation in normally hydrated and swollen rabbit cornea.

PURPOSE: To investigate the variation in anterior and posterior straining under intraocular pressure changes for the central cornea of normally hydrated and swollen rabbit eyes. METHODS: A new method of measuring regional corneal strains, by imaging a specific tissue location at various intraocular pressures, was developed. Sixteen freshly enucleated, New Zealand White rabbit eyes were investigated either in their normal hydration state or after swelling of the deepithelialized cornea. The eyes were mounted on a specially designed eye fixture, and laser-scanning confocal microscopic images of a selected region in the anterior stroma or endothelium were taken at intraocular pressures of 5, 12.5, 20, 35, and 65 mm Hg. The positions of individual keratocytes or endothelial cells were used to calculate the nonhomogeneous two-dimensional strain field over the image. Corneal thickness was measured at the lowest and highest intraocular pressures (5 mm Hg and 65 mm Hg). RESULTS: All pressure strain curves were highly nonlinear for intraocular pressures between 5 mm Hg and 65 mm Hg; the maximal posterior strains (normally hydrated, 2.1 +/- 0.1%; swollen, 4.8 +/- 0.8%) were larger than the maximal anterior strains (normally hydrated, 1.8 +/- 0.1%; swollen, 1.5 +/- 0.2%). Swelling significantly decreased the anterior strain response but increased the posterior one. The corneal thickness decreased 7.4 +/- 0.4% for the normally hydrated and 6.3 +/- 0.5% for the swollen corneas for an intraocular pressure step from 5 mm Hg to 65 mm Hg. CONCLUSIONS: Bending was found to play a significant role in central corneal deformation of swollen eyes but not in the normal hydration state. Microscopic strain measurements of the cornea, using a laser-scanning confocal microscope, are a valuable tool for the assessment of regional nonhomogeneous strains in various depths and locations of the cornea.

Oesophageal morphometry and residual strain in a mouse model of osteogenesis imperfecta.

Recently, it was demonstrated in the oesophagus that the zero-stress state is not a closed cylinder but an open circular cylindrical sector. The closed cylinder with no external loads applied is called the no-load state and residual strain is the difference in strain between the no-load state and the zero-stress state. To understand the physiology and pathology of the oesophagus, it is necessary to know the zero-stress state and the stress-strain relationships of the tissues in the oesophagus, and the changes of these states and relationships due to biological remodelling of the tissues under stress. The aim of this study was to investigate the morphological and biomechanical remodelling at the no-load and zero-stress states in mutant osteogenesis imperfecta murine (oim) mice with collagen deficiency. The oesophagi of seven oim and seven normal wild-type mice were excised, cleaned, and sectioned into rings in an organ bath containing calcium-free Krebs solution with dextran and EGTA. The rings were photographed in the no-load state and cut radially to obtain the zero-stress state. Equilibrium was awaited for 30 min and the specimens were photographed again. Circumferences, submucosa and muscle layer thicknesses and areas, and the opening angle were measured from the digitized images. The oesophagi in oim mice had smaller layer thicknesses and areas compared to the wild types. The largest reduction in layer thickness in oim mice was found in the submucosa (approximately 36%). Oim mice had significantly larger opening angles (120.2 +/- 4.5 degrees ) than wild-type mice (93.0 +/- 11.2 degrees ). The residual strain was compressive at the mucosal surface and tensile at the serosal surface in both oim and wild types. In the oim mice, the residual strains at the serosal and mucosal surfaces and the mucosa-submucosal-muscle layer interface were higher than in the wild types (P < 0.05). The gradient of residual strain per unit thickness was higher in oim mice than in wild-type mice, and was highest in submucosa (P < 0.05). The only morphometric measure that was similar in oim and wild-type mice was the inner circumference in the no-load state. In conclusion, our data show significant differences in the residual strain distribution and morphometry between oim mice and wild-type mice. The data suggest that the residual stress in oesophagus is caused by the tension in the muscle layer rather than the stiffness of the submucosa in compression and that the remodelling process in the oim oesophagus is due mainly to morphometric and biomechanical alterations in the submucosa.

Non-homogeneous analysis of three-dimensional transmural finite deformation in canine ventricular myocardium.

A new method has been developed for analyzing transmural distributions of finite deformation in canine ventricular myocardium without the need to assume that the strain in a finite volume of the wall is homogeneous. The three-dimensional nodal geometric parameters of bilinear-cubic or bilinear-quadratic finite elements are fitted by least squares to the measured coordinates of 12-18 radiopaque markers implanted in the left ventricular free wall. For six dog hearts, root-mean-squared errors in the fitted in-plane coordinates ranged from 0.079-0.556 mm in the end-diastolic reference state and 0.142-0.622 mm at end-systole. The corresponding error ranges in the radial coordinate were 0.042-0.264 mm at end-diastole and 0.106-0.279 mm at end-systole. Smoothly continuous transmural profiles of wall strain computed as the element deformed during the cardiac cycle from end-diastole to end-systole showed good agreement with the discrete results of conventional homogeneous analysis. Using the kinematics of a thick-walled incompressible cylinder, overall absolute errors due to the non-homogeneity of myocardial deformation were found to be reduced in the new analysis by 30-35% for typical experimental parameters. Overall relative errors were also reduced (from 23 to 20%). Since measurement errors in the reconstructed marker coordinates were spatially smoothed by the fitting procedure, noise in the computed deformations was also substantially attenuated, and transmural gradients of three-dimensional strain components could be obtained with improved accuracy. Hence physiological factors affected by transmural stress and strain distributions, such as myocardial blood flow, ischemia and hypertrophy, may be better understood.

Stress-dependent finite growth in soft elastic tissues.

Growth and remodeling in tissues may be modulated by mechanical factors such as stress. For example, in cardiac hypertrophy, alterations in wall stress arising from changes in mechanical loading lead to cardiac growth and remodeling. A general continuum formulation for finite volumetric growth in soft elastic tissues is therefore proposed. The shape change of an unloaded tissue during growth is described by a mapping analogous to the deformation gradient tensor. This mapping is decomposed into a transformation of the local zero-stress reference state and an accompanying elastic deformation that ensures the compatibility of the total growth deformation. Residual stress arises from this elastic deformation. Hence, a complete kinematic formulation for growth in general requires a knowledge of the constitutive law for stress in the tissue. Since growth may in turn be affected by stress in the tissue, a general form for the stress-dependent growth law is proposed as a relation between the symmetric growth-rate tensor and the stress tensor. With a thick-walled hollow cylinder of incompressible, isotropic hyperelastic material as an example, the mechanics of left ventricular hypertrophy are investigated. The results show that transmurally uniform pure circumferential growth, which may be similar to eccentric ventricular hypertrophy, changes the state of residual stress in the heart wall. A model of axially loaded bone is used to test a simple stress-dependent growth law in which growth rate depends on the difference between the stress due to loading and a predetermined growth equilibrium stress.

Measurement of strain and analysis of stress in resting rat left ventricular myocardium.

A technique has been developed for measuring two-dimensional strains in the left ventricle of the isolated arrested rat heart subjected to passive ventricular loading. The pressure-volume relationship was found in eight hearts during inflation of a left ventricular balloon. With the zero-pressure state as reference, in-plane strain components were determined using a triangle of ultrasonic dimension transducers (0.6-0.8 mm diameter) placed 3-6 mm apart in the midwall of the left ventricle. Mean circumferential (fiber) strain was larger than longitudinal (cross-fiber) strain (0.108 +/- 0.045, 0.055 +/- 0.045, respectively, at 11 mmHg), and shear strain (-0.048 +/- 0.029) was negative, consistent with left-handed torsion. The in-plane angle of greatest stretch was uniform with inflation (range = -26.5 degrees to -34.5 degrees). The equatorial region of the left ventricle was modeled with finite element analysis of a transversely isotropic thick-walled cylindrical shell subjected to internal loading and axial forces. The material parameters of an exponential strain energy function were optimized so that the least-squares difference between the predicted and the measured midwall strains was minimized. Material properties, stress and strain in the rat heart were compared to values predicted for the dog. In both species the tissue was stiffer in the fiber direction than in the cross-fiber direction. The ratio of fiber to cross-fiber stiffness was lower in the rat (2.50) than in the dog (5.24) at low loads and approximately equal at higher loads (1.63 and 1.39, respectively). The computational and experimental analyses showed that the larger shear strain and more nonuniform in-plane extension in the rat may be an indication of significantly different anisotropic material properties in these two species, and implies differences in the collagen ultrastructure.