Residual strain in the ventricle of the stage 16-24 chick embryo

Ventricular anisotropic deformation and contractile function of the developing heart of zebrafish in vivo

BACKGROUND

The developing zebrafish ventricle generates higher intraventricular pressure (IVP) with increasing stroke volume and cardiac output at different developmental stages to meet the metabolic demands of the rapidly growing embryo (Salehin et al. Ann Biomed Eng, 2021;49(9): 2080-2093). To understand the changing role of the developing embryonic heart, we studied its biomechanical characteristics during in vivo cardiac cycles. By combining changes in wall strains and IVP measurements, we assessed ventricular wall stiffness during diastolic filling and the ensuing systolic IVP-generation capacity during 3-, 4-, and 5-day post fertilization (dpf). We further examined the anisotropy of wall deformation, in different regions within the ventricle, throughout a complete cardiac cycle.

RESULTS

We found the ventricular walls grow increasingly stiff during diastolic filling with a corresponding increase in IVP-generation capacity from 3- to 4- and 5-dpf groups. In addition, we found the corresponding increasing level of anisotropic wall deformation through cardiac cycles that favor the latitudinal direction, with the most pronounced found in the equatorial region of the ventricle.

CONCLUSIONS

From 3- to 4- and 5-dpf groups, the ventricular wall myocardium undergoes increasing level of anisotropic deformation. This, in combination with the increasing wall stiffness and IVP-generation capacity, allows the developing heart to effectively pump blood to meet the rapidly growing embryo's needs.

Fluid mechanics of the left atrial ligation chick embryonic model of hypoplastic left heart syndrome

Left atrial ligation (LAL) of the chick embryonic heart at HH21 is a model of the hypoplastic left heart syndrome (HLHS) disease, demonstrating morphological and hemodynamic features similar to human HLHS cases. Since it relies on mechanical intervention without genetic or pharmacological manipulations, it is a good model for understanding the biomechanics origins of such HLHS malformations. To date, however, the fluid mechanical environment of this model is poorly understood. In the current study, we performed 4D ultrasound imaging of LAL and normal chick embryonic hearts and 4D cardiac flow simulations to help shed light on the mechanical environment that may lead to the HLHS morphology. Results showed that the HH25 LAL atrial function was compromised, and velocities in the ventricle were reduced. The HH25 LAL ventricles developed a more triangular shape with a sharper apex, and in some cases, the atrioventricular junction shifted medially. These changes led to more sluggish flow near the ventricular free wall and apex, where more fluid particles moved in an oscillatory manner with the motion of the ventricular wall, while slowly being washed out, resulting in lower wall shear stresses and higher oscillatory indices. Consequent to these flow conditions, at HH28, even before septation is complete, the left ventricle was found to be hypoplastic while the right ventricle was found to be larger in compensation. Our results suggest that the low and oscillatory flow near the left side of the heart may play a role in causing the HLHS morphology in the LAL model.

Residual Stress Estimates from Multi-cut Opening Angles of the Left Ventricle

PURPOSE

Residual stress tensor has an essential influence on the mechanical behaviour of soft tissues and can be particularly useful in evaluating growth and remodelling of the heart and arteries. It is currently unclear if one single radial cut using the opening angle method can accurately estimate the residual stress. In many previous models, it has been assumed that a single radial cut can release the residual stress in a ring of the artery or left ventricle. However, experiments by Omens et al. (Biomech Model Mechanobiol 1:267-277, 2003) on mouse hearts, have shown that this is not the case. The aim of this paper is to answer this question using a multiple-cut mathematical model.

METHODS

In this work, we have developed models of multiple cuts to estimate the residual stress in the left ventricle and compared with the one-cut model. Both two and four-cut models are considered. Given that the collagen fibres are normally coiled in the absence of loading, we use the isotropic part of the Holzapfel-Ogden strain energy function to model the unloaded myocardium.

RESULTS

The estimated residual hoop stress from our multiple-cut model is around 8 to 9 times greater than that of a single-cut model. Although in principle infinite cuts are required to release the residual stress, we find four cuts seem to be sufficient as the model agrees well with experimental measurements of the myocardial thickness. Indeed, even the two-cut model already gives a reasonable estimate of the maximum residual hoop stress. We show that the results are not significantly different using homogeneous or heterogeneous material models. Finally, we explain that the multiple cuts approach also applies to arteries.

CONCLUSION

We conclude that both radial and circumferential cuts are required to release the residual stress in the left ventricle; using multiple radial cuts alone is not sufficient. A multiple-cut model gives a marked increase of residual stress in a left ventricle ring compared to that of the commonly used single-cut model.

Validating the Paradigm That Biomechanical Forces Regulate Embryonic Cardiovascular Morphogenesis and Are Fundamental in the Etiology of Congenital Heart Disease

The goal of this review is to provide a broad overview of the biomechanical maturation and regulation of vertebrate cardiovascular (CV) morphogenesis and the evidence for mechanistic relationships between function and form relevant to the origins of congenital heart disease (CHD). The embryonic heart has been investigated for over a century, initially focusing on the chick embryo due to the opportunity to isolate and investigate myocardial electromechanical maturation, the ability to directly instrument and measure normal cardiac function, intervene to alter ventricular loading conditions, and then investigate changes in functional and structural maturation to deduce mechanism. The paradigm of "Develop and validate quantitative techniques, describe normal, perturb the system, describe abnormal, then deduce mechanisms" was taught to many young investigators by Dr. Edward B. Clark and then validated by a rapidly expanding number of teams dedicated to investigate CV morphogenesis, structure-function relationships, and pathogenic mechanisms of CHD. Pioneering studies using the chick embryo model rapidly expanded into a broad range of model systems, particularly the mouse and zebrafish, to investigate the interdependent genetic and biomechanical regulation of CV morphogenesis. Several central morphogenic themes have emerged. First, CV morphogenesis is inherently dependent upon the biomechanical forces that influence cell and tissue growth and remodeling. Second, embryonic CV systems dynamically adapt to changes in biomechanical loading conditions similar to mature systems. Third, biomechanical loading conditions dynamically impact and are regulated by genetic morphogenic systems. Fourth, advanced imaging techniques coupled with computational modeling provide novel insights to validate regulatory mechanisms. Finally, insights regarding the genetic and biomechanical regulation of CV morphogenesis and adaptation are relevant to current regenerative strategies for patients with CHD.

Trabecular Architecture Determines Impulse Propagation Through the Early Embryonic Mouse Heart

Most embryonic ventricular cardiomyocytes are quite uniform, in contrast to the adult heart, where the specialized ventricular conduction system is molecularly and functionally distinct from the working myocardium. We thus hypothesized that the preferential conduction pathway within the embryonic ventricle could be dictated by trabecular geometry. Mouse embryonic hearts of the Nkx2.5:eGFP strain between ED9.5 and ED14.5 were cleared and imaged whole mount by confocal microscopy, and reconstructed in 3D at 3.4 μm isotropic voxel size. The local orientation of the trabeculae, responsible for the anisotropic spreading of the signal, was characterized using spatially homogenized tensors (3 × 3 matrices) calculated from the trabecular skeleton. Activation maps were simulated assuming constant speed of spreading along the trabeculae. The results were compared with experimentally obtained epicardial activation maps generated by optical mapping with a voltage-sensitive dye. Simulated impulse propagation starting from the top of interventricular septum revealed the first epicardial breakthrough at the interventricular grove, similar to experimentally obtained activation maps. Likewise, ectopic activation from the left ventricular base perpendicular to dominant trabecular orientation resulted in isotropic and slower impulse spreading on the ventricular surface in both simulated and experimental conditions. We conclude that in the embryonic pre-septation heart, the geometry of the A-V connections and trabecular network is sufficient to explain impulse propagation and ventricular activation patterns.

Contribution of left ventricular residual stress by myocytes and collagen: existence of inter-constituent mechanical interaction

We quantify the contribution of myocytes, collagen fibers and their interactions to the residual stress field found in the left ventricle (LV) using both experimental and theoretical methods. Ring tissue samples extracted from normal rat, male and female, LV were treated with collagenase and decellularization to isolate myocytes and collagen fibers, respectively. Opening angle tests were then performed on these samples as well as intact tissue samples containing both constituents that served as control. Our results show that the collagen fibers are the main contributor to the residual stress fields found in the LV. Specifically, opening angle measured in collagen-only samples (106.45[Formula: see text] ± 23.02[Formula: see text]) and myocytes-only samples (21.00[Formula: see text] ± 4.37[Formula: see text]) was significantly higher and lower than that of the control (57.88[Formula: see text] ± 12.29[Formula: see text]), respectively. A constrained mixture (CM) modeling framework was then used to infer these experimental results. We show that the framework cannot reproduce the opening angle found in the intact tissue with measurements made on the collagen-only and myocytes-only samples. Given that the CM framework assumes that each constituent contributes to the overall mechanics simply by their mere presence, this result suggests the existence of some myocyte-collagen mechanical interaction that cannot be ignored in the LV. We then propose an extended CM formulation that takes into account of the inter-constituent mechanical interaction in which constituents are deformed additionally when they are physically combined into a mixture. We show that the intact tissue opening angle can be recovered in this framework.

Transmural gradients of myocardial structure and mechanics: Implications for fiber stress and strain in pressure overload

Although a truly complete understanding of whole heart activation, contraction, and deformation is well beyond our current reach, a significant amount of effort has been devoted to discovering and understanding the mechanisms by which myocardial structure determines cardiac function to better treat patients with cardiac disease. Several experimental studies have shown that transmural fiber strain is relatively uniform in both diastole and systole, in contrast to predictions from traditional mechanical theory. Similarly, mathematical models have largely predicted uniform fiber stress across the wall. The development of this uniform pattern of fiber stress and strain during filling and ejection is due to heterogeneous transmural distributions of several myocardial structures. This review summarizes these transmural gradients, their contributions to fiber mechanics, and the potential functional effects of their remodeling during pressure overload hypertrophy.

Blood flow through the embryonic heart outflow tract during cardiac looping in HH13-HH18 chicken embryos

Blood flow is inherently linked to embryonic cardiac development, as haemodynamic forces exerted by flow stimulate mechanotransduction mechanisms that modulate cardiac growth and remodelling. This study evaluated blood flow in the embryonic heart outflow tract (OFT) during normal development at each stage between HH13 and HH18 in chicken embryos, in order to characterize changes in haemodynamic conditions during critical cardiac looping transformations. Two-dimensional optical coherence tomography was used to simultaneously acquire both structural and Doppler flow images, in order to extract blood flow velocity and structural information and estimate haemodynamic measures. From HH13 to HH18, peak blood flow rate increased by 2.4-fold and stroke volume increased by 2.1-fold. Wall shear rate (WSR) and lumen diameter data suggest that changes in blood flow during HH13-HH18 may induce a shear-mediated vasodilation response in the OFT. Embryo-specific four-dimensional computational fluid dynamics modelling at HH13 and HH18 complemented experimental observations and indicated heterogeneous WSR distributions over the OFT. Characterizing changes in haemodynamics during cardiac looping will help us better understand the way normal blood flow impacts proper cardiac development.

A general framework for application of prestrain to computational models of biological materials

It is often important to include prestress in computational models of biological tissues. The prestress can represent residual stresses (stresses that exist after the tissue is excised from the body) or in situ stresses (stresses that exist in vivo, in the absence of loading). A prestressed reference configuration may also be needed when modeling the reference geometry of biological tissues in vivo. This research developed a general framework for representing prestress in finite element models of biological materials. It is assumed that the material is elastic, allowing the prestress to be represented via a prestrain. For prestrain fields that are not compatible with the reference geometry, the computational framework provides an iterative algorithm for updating the prestrain until equilibrium is satisfied. The iterative framework allows for enforcement of two different constraints: elimination of distortion in order to address the incompatibility issue, and enforcing a specified in situ fiber strain field while allowing for distortion. The framework was implemented as a plugin in FEBio (www.febio.org), making it easy to maintain the software and to extend the framework if needed. Several examples illustrate the application and effectiveness of the approach, including the application of in situ strains to ligaments in the Open Knee model (simtk.org/home/openknee). A novel method for recovering the stress-free configuration from the prestrain deformation gradient is also presented. This general purpose theoretical and computational framework for applying prestrain will allow analysts to overcome the challenges in modeling this important aspect of biological tissue mechanics.

4D subject-specific inverse modeling of the chick embryonic heart outflow tract hemodynamics

Blood flow plays a critical role in regulating embryonic cardiac growth and development, with altered flow leading to congenital heart disease. Progress in the field, however, is hindered by a lack of quantification of hemodynamic conditions in the developing heart. In this study, we present a methodology to quantify blood flow dynamics in the embryonic heart using subject-specific computational fluid dynamics (CFD) models. While the methodology is general, we focused on a model of the chick embryonic heart outflow tract (OFT), which distally connects the heart to the arterial system, and is the region of origin of many congenital cardiac defects. Using structural and Doppler velocity data collected from optical coherence tomography, we generated 4D ([Formula: see text]) embryo-specific CFD models of the heart OFT. To replicate the blood flow dynamics over time during the cardiac cycle, we developed an iterative inverse-method optimization algorithm, which determines the CFD model boundary conditions such that differences between computed velocities and measured velocities at one point within the OFT lumen are minimized. Results from our developed CFD model agree with previously measured hemodynamics in the OFT. Further, computed velocities and measured velocities differ by [Formula: see text]15 % at locations that were not used in the optimization, validating the model. The presented methodology can be used in quantifications of embryonic cardiac hemodynamics under normal and altered blood flow conditions, enabling an in-depth quantitative study of how blood flow influences cardiac development.

Noninvasive in vivo determination of residual strains and stresses

Vascular growth and remodeling during embryonic development are associated with blood flow and pressure induced stress distribution, in which residual strains and stresses play a central role. Residual strains are typically measured by performing in vitro tests on the excised vascular tissue. In this paper, we investigated the possibility of estimating residual strains and stresses using physiological pressure-radius data obtained through in vivo noninvasive measurement techniques, such as optical coherence tomography or ultrasound modalities. This analytical approach first tested with in vitro results using experimental data sets for three different arteries such as rabbit carotid artery, rabbit thoracic artery, and human carotid artery based on Fung's pseudostrain energy function and Delfino's exponential strain energy function (SEF). We also examined residual strains and stresses in the human swine iliac artery using the in vivo experimental ultrasound data sets corresponding to the systolic-to-diastolic region only. This allowed computation of the in vivo residual stress information for loading and unloading states separately. Residual strain parameters as well as the material parameters were successfully computed with high accuracy, where the relative errors are introduced in the range of 0-7.5%. Corresponding residual stress distributions demonstrated global errors all in acceptable ranges. A slight discrepancy was observed in the computed reduced axial force. Results of computations performed based on in vivo experimental data obtained from loading and unloading states of the artery exhibited alterations in material properties and residual strain parameters as well. Emerging noninvasive measurement techniques combined with the present analytical approach can be used to estimate residual strains and stresses in vascular tissues as a precursor for growth estimates. This approach is also validated with a finite element model of a general two-layered artery, where the material remodeling states and residual strain generation are investigated.

Stress and strain adaptation in load-dependent remodeling of the embryonic left ventricle

Altered pressure in the developing left ventricle (LV) results in altered morphology and tissue material properties. Mechanical stress and strain may play a role in the regulating process. This study showed that confocal microscopy, three-dimensional reconstruction, and finite element analysis can provide a detailed model of stress and strain in the trabeculated embryonic heart. The method was used to test the hypothesis that end-diastolic strains are normalized after altered loading of the LV during the stages of trabecular compaction and chamber formation. Stage-29 chick LVs subjected to pressure overload and underload at stage 21 were reconstructed with full trabecular morphology from confocal images and analyzed with finite element techniques. Measured material properties and intraventricular pressures were specified in the models. The results show volume-weighted end-diastolic von Mises stress and strain averaging 50-82 % higher in the trabecular tissue than in the compact wall. The volume-weighted-average stresses for the entire LV were 115, 64, and 147 Pa in control, underloaded, and overloaded models, while strains were 11, 7, and 4 %; thus, neither was normalized in a volume-weighted sense. Localized epicardial strains at mid-longitudinal level were similar among the three groups and to strains measured from high-resolution ultrasound images. Sensitivity analysis showed changes in material properties are more significant than changes in geometry in the overloaded strain adaptation, although resulting stress was similar in both types of adaptation. These results emphasize the importance of appropriate metrics and the role of trabecular tissue in evaluating the evolution of stress and strain in relation to pressure-induced adaptation.

Biomechanics of the chick embryonic heart outflow tract at HH18 using 4D optical coherence tomography imaging and computational modeling

During developmental stages, biomechanical stimuli on cardiac cells modulate genetic programs, and deviations from normal stimuli can lead to cardiac defects. Therefore, it is important to characterize normal cardiac biomechanical stimuli during early developmental stages. Using the chicken embryo model of cardiac development, we focused on characterizing biomechanical stimuli on the Hamburger–Hamilton (HH) 18 chick cardiac outflow tract (OFT), the distal portion of the heart from which a large portion of defects observed in humans originate. To characterize biomechanical stimuli in the OFT, we used a combination of in vivo optical coherence tomography (OCT) imaging, physiological measurements and computational fluid dynamics (CFD) modeling. We found that, at HH18, the proximal portion of the OFT wall undergoes larger circumferential strains than its distal portion, while the distal portion of the OFT wall undergoes larger wall stresses. Maximal wall shear stresses were generally found on the surface of endocardial cushions, which are protrusions of extracellular matrix onto the OFT lumen that later during development give rise to cardiac septa and valves. The non-uniform spatial and temporal distributions of stresses and strains in the OFT walls provide biomechanical cues to cardiac cells that likely aid in the extensive differential growth and remodeling patterns observed during normal development.

Hemodynamic patterning of the avian atrioventricular valve

In this study, we develop an innovative approach to rigorously quantify the evolving hemodynamic environment of the atrioventricular (AV) canal of avian embryos. Ultrasound generated velocity profiles were imported into Micro-Computed Tomography generated anatomically precise cardiac geometries between Hamburger-Hamilton (HH) stages 17 and 30. Computational fluid dynamic simulations were then conducted and iterated until results mimicked in vivo observations. Blood flow in tubular hearts (HH17) was laminar with parallel streamlines, but strong vortices developed simultaneous with expansion of the cushions and septal walls. For all investigated stages, highest wall shear stresses (WSS) are localized to AV canal valve-forming regions. Peak WSS increased from 19.34 dynes/cm(2) at HH17 to 287.18 dynes/cm(2) at HH30, but spatiotemporally averaged WSS became 3.62 dynes/cm(2) for HH17 to 9.11 dynes/cm(2) for HH30. Hemodynamic changes often preceded and correlated with morphological changes. These results establish a quantitative baseline supporting future hemodynamic analyses and interpretations.

Differential growth and residual stress in cylindrical elastic structures

Cylindrical forms are among one of Nature's fundamental building blocks. They serve many different purposes, from sustaining body weight to carrying flows. Their mechanical properties are generated through the often complex arrangements of the walls. In particular, in many structures that have elastic responses, such as stems and arteries, the walls are in a state of tension generated by differential growth. Here, the effect of differential growth and residual stress on the overall mechanical response of the cylindrical structure is studied within the framework of morpho-elasticity.

Patterns of muscular strain in the embryonic heart wall

The hypothesis that inner layers of contracting muscular tubes undergo greater strain than concentric outer layers was tested by numerical modeling and by confocal microscopy of strain within the wall of the early chick heart. We modeled the looped heart as a thin muscular shell surrounding an inner layer of sponge-like trabeculae by two methods: calculation within a two-dimensional three-variable lumped model and simulated expansion of a three-dimensional, four-layer mesh of finite elements. Analysis of both models, and correlative microscopy of chamber dimensions, sarcomere spacing, and membrane leaks, indicate a gradient of strain decreasing across the wall from highest strain along inner layers. Prediction of wall thickening during expansion was confirmed by ultrasonography of beating hearts. Degree of stretch determined by radial position may thus contribute to observed patterns of regional myocardial conditioning and slowed proliferation, as well as to the morphogenesis of ventricular trabeculae and conduction fascicles. Developmental Dynamics 238:1535-1546, 2009. (c) 2009 Wiley-Liss, Inc.

Aortic arch morphogenesis and flow modeling in the chick embryo

Morphogenesis of the "immature symmetric embryonic aortic arches" into the "mature and asymmetric aortic arches" involves a delicate sequence of cell and tissue migration, proliferation, and remodeling within an active biomechanical environment. Both patient-derived and experimental animal model data support a significant role for biomechanical forces during arch development. The objective of the present study is to quantify changes in geometry, blood flow, and shear stress patterns (WSS) during a period of normal arch morphogenesis. Composite three-dimensional (3D) models of the chick embryo aortic arches were generated at the Hamburger-Hamilton (HH) developmental stages HH18 and HH24 using fluorescent dye injection, micro-CT, Doppler velocity recordings, and pulsatile subject-specific computational fluid dynamics (CFD). India ink and fluorescent dyes were injected into the embryonic ventricle or atrium to visualize right or left aortic arch morphologies and flows. 3D morphology of the developing great vessels was obtained from polymeric casting followed by micro-CT scan. Inlet aortic arch flow and cerebral-to-lower body flow split was obtained from 20 MHz pulsed Doppler velocity measurements and literature data. Statistically significant variations of the individual arch diameters along the developmental timeline are reported and correlated with WSS calculations from CFD. CFD simulations quantified pulsatile blood flow distribution from the outflow tract through the aortic arches at stages HH18 and HH24. Flow perfusion to all three arch pairs are correlated with the in vivo observations of common pharyngeal arch defect progression. The complex spatial WSS and velocity distributions in the early embryonic aortic arches shifted between stages HH18 and HH24, consistent with increased flow velocities and altered anatomy. The highest values for WSS were noted at sites of narrowest arch diameters. Altered flow and WSS within individual arches could be simulated using altered distributions of inlet flow streams. Thus, inlet flow stream distributions, 3D aortic sac and aortic arch geometries, and local vascular biologic responses to spatial variations in WSS are all likely to be important in the regulation of arch morphogenesis.

Engineered cardiac organoid chambers: toward a functional biological model ventricle

A growing area in the field of tissue engineering is the development of tissue equivalents as model systems for in vitro experimentation and high-throughput screening applications. Although a variety of strategies have been developed to enhance the structure and function of engineered cardiac tissues, an inherent limitation with traditional myocardial patches is that they do not permit evaluation of the fundamental relationships between pressure and volume that characterize global contractile function of the heart. Therefore, in the following study we introduce fully biological, living engineered cardiac organoids, or simplified heart chambers, that beat spontaneously, develop pressure, eject fluid, contain residual stress, exhibit a functional Frank-Starling mechanism, and generate positive stroke work. We also demonstrate regional variations in pump function following local cryoinjury, yielding a novel engineered tissue model of myocardial infarction. With the unique ability to directly evaluate relevant pressure-volume characteristics and regulate wall stress, this organoid chamber culture system provides a flexible platform for developing a controllable biomimetic cardiac niche environment that can be adapted for a variety of high-throughput and long-term investigations of cardiac pump function.

Arterial adaptations to chronic changes in haemodynamic function: coupling vasomotor tone to structural remodelling

Healthy mature arteries are usually extremely quiescent tissues with cell proliferation rates much below 1%/day and with extracellular matrix constituents exhibiting half-lives of years to decades. However, chronic physiological or pathological changes in haemodynamic function elicit arterial remodelling processes that may involve substantial tissue synthesis, degradation or turnover. Although these remodelling processes accommodate changing demands placed upon the cardiovascular system by physiological adaptations, they can compromise further perfusion in the context of arterial occlusive disease and they entrench hypertension and may exacerbate its progression. Recent findings indicate that some of the most important such remodelling responses involve the integrated effects of persistently altered vascular tone that feed into restructuring responses, with common signalling pathways frequently interacting in the control of both phases of the response. Current efforts to define these signals and their targets may provide new directions for therapeutic interventions to treat important vascular disorders.

Computational modeling of morphogenesis regulated by mechanical feedback

Mechanical forces cause changes in form during embryogenesis and likely play a role in regulating these changes. This paper explores the idea that changes in homeostatic tissue stress (target stress), possibly modulated by genes, drive some morphogenetic processes. Computational models are presented to illustrate how regional variations in target stress can cause a range of complex behaviors involving the bending of epithelia. These models include growth and cytoskeletal contraction regulated by stress-based mechanical feedback. All simulations were carried out using the commercial finite element code ABAQUS, with growth and contraction included by modifying the zero-stress state in the material constitutive relations. Results presented for bending of bilayered beams and invagination of cylindrical and spherical shells provide insight into some of the mechanical aspects that must be considered in studying morphogenetic mechanisms.

Quantifying cardiovascular flow dynamics during early development

The relationship between developing biologic tissues and their dynamic fluid environments is intimate and complex. Increasing evidence supports the notion that these embryonic flow-structure interactions influence whether development will proceed normally or become pathogenic. Genetic, pharmacological, or surgical manipulations that alter the flow environment can thus profoundly influence morphologic and functional cardiovascular phenotypes. Functionally deficient phenotypes are particularly poorly described as there are few imaging tools with sufficient spatial and temporal resolution to quantify most intra-vital flows. The ability to visualize biofluids flow in vivo would be of great utility in functionally phenotyping model animal systems and for the elucidation of the mechanisms that underlie flow-related mechano-sensation and transduction in living organisms. This review summarizes the major methodological advances that have evolved for the quantitative characterization of intra-vital fluid dynamics with an emphasis on assessing cardiovascular flows in vertebrate model organisms.

On the effects of residual stress in microindentation tests of soft tissue structures

Microindentation methods are commonly used to determine material properties of soft tissues at the cell or even sub-cellular level. In determining properties from force-displacement (FD) data, it is often assumed that the tissue is initially a stress-free, homogeneous, linear elastic half-space. Residual stress, however, can strongly influence such results. In this paper, we present a new microindentation method for determining both elastic properties and residual stress in soft tissues that, to a first approximation, can be regarded as a pre-stressed layer embedded in or adhered to an underlying relatively soft, elastic foundation. The effects of residual stress are shown using two linear elastic models that approximate specific biological structures. The first model is an axially loaded beam on a relatively soft, elastic foundation (i.e., stress-fiber embedded in cytoplasm), while the second is a radially loaded plate on a foundation (e.g., cell membrane or epithelium). To illustrate our method, we use a nonlinear finite element (FE) model and experimental FD and surface contour data to find elastic properties and residual stress in the early embryonic chick heart, which, in the region near the indenter tip, is approximated as an isotropic circular plate under tension on a foundation. It is shown that the deformation of the surface in a microindentation test can be used along with FD data to estimate material properties, as well as residual stress, in soft tissue structures that can be regarded as a plate under tension on an elastic foundation. This method may not be as useful, however, for structures that behave as a beam on a foundation.

Defects in cardiac function precede morphological abnormalities in fish embryos exposed to polycyclic aromatic hydrocarbons

Fish embryos exposed to complex mixtures of polycyclic aromatic hydrocarbons (PAHs) from petrogenic sources show a characteristic suite of abnormalities, including cardiac dysfunction, edema, spinal curvature, and reduction in the size of the jaw and other craniofacial structures. To elucidate the toxic mechanisms underlying these different defects, we exposed zebrafish (Danio rerio) embryos to seven non-alkylated PAHs, including five two- to four-ring compounds that are abundant in crude oil and two compounds less abundant in oil but informative for structure-activity relationships. We also analyzed two PAH mixtures that approximate the composition of crude oil at different stages of weathering. Exposure to the three-ring PAHs dibenzothiophene and phenanthrene alone was sufficient to induce the characteristic suite of defects, as was genetic ablation of cardiac function using a cardiac troponin T antisense morpholino oligonucleotide. The primary etiology of defects induced by dibenzothiophene or phenanthrene appears to be direct effects on cardiac conduction, which have secondary consequences for late stages of cardiac morphogenesis, kidney development, neural tube structure, and formation of the craniofacial skeleton. The relative toxicity of the different mixtures was directly proportional to the amount of phenanthrene, or the dibenzothiophene-phenanthrene total in the mixture. Pyrene, a four-ring PAH, induced a different syndrome of anemia, peripheral vascular defects, and neuronal cell death, similar to the effects previously described for potent aryl hydrocarbon receptor ligands. Therefore, different PAH compounds have distinct and specific effects on fish at early life history stages.

A new method for predicting the opening angle for soft tissues

The purpose of this paper is to present a simple new method for calculating the opening angle produced by a given residual stress field in a soft biological tissue. The method uses minimization of potential energy, and is therefore named the MPE method. The accuracy of the MPE method is evaluated by comparing the opening angle it predicts to results from a finite element model of the opening angle experiment. We show that the MPE method provides good predictions of the opening angle, and that it is significantly more accurate than two other methods previously used in the literature.

A model of the structural and functional development of the normal human fetal left ventricle based on a global growth law

The purpose of this research is to study the growth of the normal human left ventricle (LV) during the fetal period from 14 to 40 weeks of gestation. A new constitutive law for the active myocardium describing the mechanical properties of the active muscle during the whole cardiac cycle has been proposed. The LV model is a thick-walled, incompressible, hyperelastic cylinder, with families of helicoidal fibers running on cylindrical surfaces [1]. Based on the works of Lin and Taber [2] done on the embryonic chick heart, we use for the human fetal heart a growth law in which the growth rate depends on the wall stresses. The parameters of the growth law are adapted to agree with sizes and volumes inferred from two dimensional ultrasound measurements performed on 18 human fetuses.Then calculations are performed to extrapolate the cardiac performance during normal growth of the fetal LV. The results presented support the idea that a growth law in which the growth rate depends linearly on the mean wall stresses averaged through the space and during whole cardiac cycle, is adapted to the normal human fetal LV development.

Biomechanics of cardiovascular development

It long has been known that mechanical forces play a role in the development of the cardiovascular system, but only recently have biomechanical engineers begun to explore this field. This paper reviews some of this work. First, an overview of the relevant biology is discussed. Next, a mechanical theory is presented that can be used to model developmental processes. The theory includes the effects of finite volumetric growth and active contractile forces. Finally, applications of this and other theories to problems of cardiovascular development are discussed, and some future directions are suggested. The intent is to stimulate further interest among engineers in this important area of research.

Stress-modulated growth, residual stress, and vascular heterogeneity

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.

Cardiac morphology and blood pressure in the adult zebrafish

Zebrafish has become a popular model for the study of cardiovascular development. We performed morphologic analysis on 3 months postfertilization zebrafish hearts (n > or = 20) with scanning electron microscopy, hematoxylin and eosin staining and Masson's trichrome staining, and morphometric analysis on cell organelles with transmission electron photomicrographs. We measured atrial, ventricular, ventral, and dorsal aortic blood pressures (n > or = 5) with a servonull system. The atrioventricular orifice was positioned on the dorsomedial side of the anterior ventricle, surmounted by the single-chambered atrium. The atrioventricular valve was free of tension apparati but supported by papillary bands to prevent retrograde flow. The ventricle was spanned with fine trabeculae perpendicular to the compact layer and perforated with a subepicardial network of coronary arteries, which originated from the efferent branchial arteries by means of the main coronary vessel. Ventricular myocytes were larger than those in the atrium (P < 0.05) with abundant mitochondria close to the sarcolemmal. Sarcoplasmic reticulum was sparse in zebrafish ventricle. Bulbus arteriosus was located anterior to the ventricle, and functioned as an elastic reservoir to absorb the rapid rise of pressure during ventricular contraction. The dense matrix of collagen interspersed across the entire bulbus arteriosus exemplified the characteristics of vasculature smooth muscle. There were pressure gradients from atrium to ventricle, and from ventral to dorsal aorta, indicating that the valves and the branchial arteries, respectively, were points of resistance to blood flow. These data serve as a framework for structure-function investigations of the zebrafish cardiovascular system.

Fibre Orientation in Human Fetal Heart and Ventricular Mechanics : A Small Perturbation Analysis

The study of the topological organisation of myocardial cells is a basic requirement for understanding the mechanical design of the normal and pathological heart. Anatomical observations show that cardiac muscle tissue has a highly specialized architecture. We have made new quantitative measurements of fibre orientation through the heart wall by means of polarized light analysis on some thick sections of human fetal heart embedded in a resin and polymerized. A small perturbation method to find an equilibrium solution in a cylindrical left ventricular (LV) geometry with fibres running on toroidal shells of revolution is used to investigate the mechanical behaviour of three human fetal hearts (FH) of 14, 20 and 33 weeks of gestational age. The results of fibre strains and stresses presented for end-systolic state show significant differences when compared to results of the cylindrical geometry with regular helicoidal fibres running on cylindrical surfaces. Moreover, the toroidal shells of revolution explain shear stresses and strains in the transverse plane which also exist in the adult heart.

Theoretical model for myocardial trabeculation

During the morphogenetic process of myocardial trabeculation, most of the cardiac jelly of the initially smooth-walled heart is replaced by sponge-like muscle. The mechanisms that drive and regulate this important process are poorly understood. Using a theoretical model, we examined the possible role that cytoskeletal contraction plays during the initial stages of trabeculation. The myocardium is modeled as a thin viscoelastic membrane consisting of contractile (stress) fibers embedded in an isotropic incompressible matrix, with the interaction of myocardial cells and cardiac jelly fibers providing long-range mechanical effects. The stress fibers are assumed to behave like smooth muscle and to normally operate on the descending limb of their stress-stretch curve. Mechanical instability due to the effectively negative stiffness then leads to the creation of pattern. As a first approximation, computations were carried out for a flat rectangular membrane with stress fibers aligned along a single direction. The computed deformation patterns depend strongly on the magnitude and anisotropy of the long-range effects. Given plausible assumptions about the mechanical properties of the embryonic heart, the model predicts trabecular patterns similar to those observed in the embryo, including the development of circumferential ridges and relatively thin regions ("holes") in the trabecular sheets.

Maturation of end-systolic stress-strain relations in chick embryonic myocardium

The embryonic myocardium increases functional performance geometrically during cardiac morphogenesis. We investigated developmental changes in the in vivo end-systolic stress-strain relations of embryonic chick myocardium in stage 17, 21, and 24 white Leghorn chick embryos (n = 10 for each stage). End-systolic stress-strain relations were linear in all developmental stages. End-systolic strain decreased from 0.50 +/- 0.02 to 0.31 +/- 0.01 (mean +/- SE, P < 0.05), while average end-systolic wall stress was similar at 3.29 +/- 0.34 to 4.19 +/- 0.43 mmHg (P = 0.14) from stage 17 to 24. Normalized end-systolic myocardial stiffness, a load-independent index of ventricular contractility, increased from 2.98 +/- 0.19 to 6.03 +/- 0.39 mmHg from stage 17 to 24 (P < 0.05). Zero-stress midwall volume increased from 0.024 +/- 0.002 to 0.124 +/- 0.004 microl from stage 17 to 24 (P < 0.05). These results suggest that the embryonic ventricle increases normalized ventricular "contractility" while maintaining average end-systolic wall stress over a relatively narrow range during cardiovascular morphogenesis.

End-systolic myocardial stiffness is a load-independent index of contractility in stage 24 chick embryonic heart

Cardiac morphogenesis and function are interrelated during cardiovascular development. We evaluated the effects of acute alteration of loading condition to chick embryonic ventricular contractility using end-systolic myocardial stiffness based on the incremental elastic modulus concept. End-systolic stress-strain relations including geometric factor and end-systolic myocardial stiffness were determined from the simultaneous measurement of ventricular pressure and chamber dimension in the following four groups of stage 24 White Leghorn chick embryos: volume infusion (n = 9), conotruncal occlusion (n = 9), calcium suffusion (n = 10), and verapamil suffusion (n = 8). The end-systolic stress-strain relationship was linear in each embryo. There was no correlation between end-systolic myocardial stiffness and end-systolic stress. End-systolic myocardial stiffness increased with calcium suffusion (P < 0.05 vs. volume infusion). The geometric factor increased after verapamil suffusion (P < 0.05). End-systolic myocardial stiffness normalized by geometric factor was not changed by alteration of preload or afterload, increased after calcium suffusion, and decreased after verapamil administration (P < 0.05). These results suggest that normalized end-systolic myocardial stiffness is a load-independent index of ventricular contractility in the developing embryonic chick ventricle.

Mechanical aspects of cardiac development

Heart development depends on a dynamic interaction between genetic and epigenetic factors. This paper discusses some of the biomechanical processes that help shape the heart in the embryo. First, an overview is given of some of the critical events that occur during cardiac development. Next, mechanics and modeling strategies are discussed for the morphogenetic processes of cardiac tube formation, cardiac looping, myocardial trabeculation, septation, valve formation, and muscle-fiber alignment. Finally, some considerations for future work in this area are listed.

A model for aortic growth based on fluid shear and fiber stresses

Stress-modulated growth in the aorta is studied using a theoretical model. The model is a thick-walled tube composed of two pseudoelastic, orthotropic layers representing the intima/media and the adventitia. Both layers are assumed to follow a growth law in which the time rates of change of the growth stretch ratios depend linearly on the local smooth muscle fiber stress and on the shear stress due to blood flow on the endothelium. Using finite elasticity theory modified to include volumetric growth, we computed temporal changes in stress, geometry, and opening angle (residual strain) during development and following the onset of sudden hypertension. For appropriate values of the coefficients in the growth law, the model yields results in reasonable agreement with published data for global and local growth of the rat aorta.

Uniform stress state in bone structure with residual stress

Residual stress and strain in living tissues have been investigated from the viewpoint of mechanical optimality maintained by adaptive remodeling. In this study, the residual stresses in the cortical-cancellous bone complex of bovine coccygeal vertebrae were examined. Biaxial strain gages were bonded onto the cortical surface, so that the gage axes were aligned in the cephalocaudal and circumferential directions. Strains induced by removal of the end-plate and the cancellous bone were recorded sequentially. The results revealed the existence of compressive residual stress in the cortical bone and tensile residual stress in the cancellous bone in both the cephalocaudal and the circumferential direction. The observed strains were examined on the basis of the uniform stress hypothesis using a three-bar model for the cephalocaudal direction and a three-layered cylinder model for the circumferential direction. In this model study, the magnitude of effective stresses, which is defined as the macroscopic stress divided by the area fraction of bone material, was found not to differ significantly between cephalocaudal and circumferential directions, although they were evaluated using independent models. These results demonstrate that the uniform stress state of the cortical-cancellous bone structure is consistent with results obtained in the cutting experiment when the existence of residual stress is taken into account.

Swelling and curling behaviors of articular cartilage

A new experimental method was developed to quantify parameters of swelling-induced shape change in articular cartilage. Full-thickness strips of cartilage were studied in free-swelling tests and the swelling-induced stretch, curvature, and areal change were measured. In general, swelling-induced stretch and curvature were found to increase in cartilage with decreasing ion concentration, reflecting an increasing tendency to swell and "curl" at higher swelling pressures. An exception was observed at the articular surface, which was inextensible for all ionic conditions. The swelling-induced residual strain at physiological ionic conditions was estimated from the swelling-induced stretch and found to be tensile and from 3-15 percent. Parameters of swelling were found to vary with sample orientation, reflecting a role for matrix anisotropy in controlling the swelling-induced residual strains. In addition, the surface zone was found to be a structurally important element, which greatly limits swelling of the entire cartilage layer. The findings of this study provide the first quantitative measures of swelling-induced residual strain in cartilage ex situ, and may be readily adapted to studies of cartilage swelling in situ.

Passive stress-strain measurements in the stage-16 and stage-18 embryonic chick heart

The first stress-strain measurements on embryonic cardiovascular tissue are described here, obtained from cyclic uniaxial loading of the primitive ventricle. An excised ventricular segment from Hamburger/Hamilton stage-16 or stage-18 chicks (2-1/2 and 3 days of a 21-day incubation period) was mounted longitudinally between two small wires in oxygenated Krebs-Henseleit cardioplegia solution. One wire was attached to an ultrasensitive force transducer and the other to a Huxley micromanipulator controlled by remote motor drive. A real-time video tracking system calculated three myocardial surface strains based on the positions of three surface markers while the heart was deformed in a triangular wave pattern. Force transducer output was filtered, digitally sampled, and stored with strains and time. Results were plotted as strain (longitudinal, circumferential, shear, and principal) versus time, stress versus time, and stress versus longitudinal strain. The stress-strain curves were nonlinear, even at low strain levels. The hysteresis loops were large; mean hysteresis energy as a proportion of total cycle stored strain energy was 36 percent (stage 16) and 41 percent (stage 18). We created a finite element model of the ventricle and fit the model behavior to the experimental behavior to determine parameters for a stage-18 pseudoelastic strain-energy function of exponential form. The calculated exponential parameter is significantly lower than that found in corresponding uniaxial studies of mature myocardium, possibly indicating the lower fiber content of the immature tissue. The results of this study are the first step in characterizing material properties for comparisons with later developmental stages and with impaired and altered myocardium. The long-term goal is to aid in identifying the biomechanical factors regulating growth and morphogenesis.

Optimization of cardiac fiber orientation for homogeneous fiber strain at beginning of ejection

Mathematical models of left ventricular (LV) wall mechanics show that fiber stress depends heavily on the choice of muscle fiber orientation in the wall. This finding brought us to the hypothesis that fiber orientation may be such that mechanical load in the wall is homogeneous. Aim of this study was to use the hypothesis to compute a distribution of fiber orientation within the wall. In a finite element model of LV wall mechanics, fiber stresses and strains were calculated at beginning of ejection (BE). Local fiber orientation was quantified by helix (HA) and transverse (TA) fiber angles using a coordinate system with local r-, c-, and l-directions perpendicular to the wall, along the circumference and along the meridian, respectively. The angle between the c-direction and the projection of the fiber direction on the cl-plane (HA) varied linearly with transmural position in the wall. The angle between the c-direction and the projection of the fiber direction on the cr-plane (TA) was zero at the epicardial and endocardial surfaces. Midwall TA increased with distance from the equator. Fiber orientation was optimized so that fiber strains at BE were as homogeneous as possible. By optimization with TA = 0 degree, HA was found to vary from 81.0 degrees at the endocardium to -35.8 degrees at the epicardium. Inclusion of TA in the optimization changed these angles to respectively 90.1 degrees and -48.2 degrees while maximum TA was 15.3 degrees. Then the standard deviation of fiber strain (epsilon f) at BE decreased from +/- 12.5% of mean epsilon f to +/- 9.5%. The root mean square (RMS) difference between computed HA and experimental data reported in literature was 15.0 degrees compared to an RMS difference of 11.6 degrees for a linear regression line through the latter data.

Three-dimensional residual strain in midanterior canine left ventricle

All previous studies of residual strain in the ventricular wall have been based on one- or two-dimensional measurements. Transmural distributions of three-dimensional (3-D) residual strains were measured by biplane radiography of columns of lead beads implanted in the midanterior free wall of the canine left ventricle (LV). 3-D bead coordinates were reconstructed with the isolated arrested LV in the zero-pressure state and again after local residual stress had been relieved by excising a transmural block of tissue. Nonhomogeneous 3-D residual strains were computed by finite element analysis. Mean +/- SD (n = 8) circumferential residual strain indicated that the intact unloaded myocardium was prestretched at the epicardium (0.07 +/- 0.06) and compressed in the subendocardium (-0.04 +/- 0.05). Small but significant longitudinal shortening and torsional shear residual strains were also measured. Residual fiber strain was tensile at the epicardium (0.05 +/- 0.06) and compressive in the subendocardium (-0.01 +/- 0.04), with residual extension and shortening, respectively, along structural axes parallel and perpendicular to the laminar myocardial sheets. Relatively small residual shear strains with respect to the myofiber sheets suggest that prestretching in the plane of the myocardial laminae may be a primary mechanism of residual stress in the LV.

Pathogenetic mechanisms of congenital cardiovascular malformations revisited

Rapid advances in cardiovascular science have expanded our knowledge of the mechanisms of heart development. Epidemiologists have defined the prevalence of congenital cardiovascular malformations, developmental biologists have delineated cascades of cell lineage, and molecular geneticists have identified mutations and loci associated with familial heart and vascular defects. We are well on the way to a molecular understanding of congenital cardiovascular malformations. Thus, it seems appropriate to review the pathogenetic classification of congenital cardiovascular malformations in light of this new clinical and scientific evidence. This schema serves as a template for the scientist to organize clinical information relevant to the pathogenesis of cardiac defects and as a tool for the clinician in approaching the difficult task of counseling parents of children with congenital cardiovascular malformations.

The effect of age on residual strain in the rat aorta

Residual stress is observed in many parts of the cardiovascular system and is thought to reduce transmural stress gradients due to intravascular pressure. Its development is closely associated with normal growth and pathological remodeling, although there appear to be few previous reports of the relationship between aging and residual stress. We have estimated residual strain (an indicator of the magnitude of residual stress) at ten sites along the aorta of rats aged 2.5 to 56 weeks by measuring the degree to which rings of vessel spring open when cut (opening angle). At all ages the opening angle decreased along the aorta, reaching a minimum near the renal arteries and increasing toward the aorto-iliac bifurcation, a result that confirms previous studies. During growth, although the unloaded circumference of the aorta increased steadily, the wall thickness and medial surface area fell to a minimum at the age of 6 weeks before continuing a steady increase. Similarly, the opening angle decreased between the ages of 2.5 and 6 weeks, thereafter increasing with age. In the abdominal aorta, a strong correlation between opening angle and wall thickness relative to midwall radius (h/R) was seen; whereas in the thoracic segment, in which no increase in h/R with age occurred, no such relationship was found. These observations are in keeping with a recently proposed hypothesis that residual stress will change in response to growth-related changes in vessel geometry driven by a tendency to minimize the nonuniform stress distribution inevitably found in pressurized thick-walled cylinders.

Cervical flexion: its contribution to normal and abnormal cardiac morphogenesis

Although His in 1881 and Patten in 1922 suggested that cervical flexion could play an important role in normal cardiac morphogenesis, until recently this hypothesis has been largely neglected. The purpose of this report is to present data indicating that prevention of cervical flexion leads to double outlet right ventricle (DORV), which is unrelated to any affect on neural crest cell migration. At stage 11-12, suture material was inserted into the neural tube to prevent cervical flexion. Six out of 22 experimental embryos survived until stage 38 and 5 out of 6 had DORV. Neither abnormalities of the aortico-pulmonary septum nor interruption of the aortic arch were observed at dissection. Hemodynamic studies performed at stage 18 revealed distinctive characteristics that are inconsistent with hemodynamic studies previously reported following neural crest ablation. With respect to immunohistochemical studies using neural crest associated antigen HNK-1 antibody, normal migration of neural crest cells was noted in the outflow tract in the experimental embryos at stage 22. These hemodynamic and immunohistochemical studies suggest that insertion of suture material into the neural tube at stage 11-12 does not jeopardize neural crest migration. We propose that reduction in cervical flexion increases the distance between the future aorta and the left ventricle, which prevents the transition of intracardiac flow pattern from a serial circulation to a parallel one, leaving persistence of a "double outlet" from the right ventricle.

A model for stress-induced growth in the developing heart

Mechanical loads affect growth and morphogenesis in the developing heart. Using a theoretical model, we studied stress-modulated growth in the embryonic chick ventricle during stages 21-29 (4-6 days of a 21-day incubation period). The model is a thick-walled, compressible, pseudoelastic cylinder, with finite volumetric growth included by letting the rate of change of the local zero-stress configuration depend linearly on the Cauchy stresses. After investigating the fundamental behavior of the model, we used it to study global and local growth in the primitive ventricle due to normal and abnormal cavity pressures. With end-diastolic pressure taken as the growth-modulating stimulus, correlating theoretical and available experimental results yielded the coefficients of the growth law, which was assumed to be independent of time and loading conditions. For both normal and elevated pressures, the predicted changes in radius and wall volume during development were similar to experimental measurements. In addition, the residual stress generated by differential growth agreed with experimental data. These results suggest that wall stress may be a biomechanical factor that regulates growth in the embryonic heart.

Mechanics of cardiac looping

During the early stages of embryonic development, the heart is a smooth-walled, muscle-wrapped tube that bends and rotates in a vital, but poorly understood, morphogenetic process called looping. Since looping involves biomechanical forces, this paper examines two mechanically based hypotheses for the bending component of cardiac looping. The first hypothesis is that an initial tension in or near the dorsal mesocardium (DM), a longitudinal structure along the outside of the ventricle, drives the deformation. To relieve the bending stresses in the tube, the myocytes change shape passively, and then they deform actively to continue the process to completion of a full loop. In the second hypothesis, contraction of circumferentially arranged actin macrofilaments produces circumferential compression and longitudinal expansion (due to incompressibility) of the myocytes. The DM locally constrains the longitudinal deformation, forcing the tube to bend. The feasibility of these hypotheses was evaluated using theoretical models and published experimental results. The models, which consist of beams composed of two layers representing the DM and the ventricular myocardium, show that the hypotheses are consistent with most of the known data, but further studies are necessary. In this regard, the models provide a conceptual framework for designing experiments to investigate the mechanics of looping.

A model approach to the adaptation of cardiac structure by mechanical feedback in the environment of the cell

The uniformity of the mechanical load of the cardiac fibers in the wall is maintained by continuous remodeling. In this proposed model the myocyte changes direction in optimizing systolic sarcomere shortening. Early systolic stretch and contractility increases the mass of contractile proteins. Cyclic strain of the myocardial tissue diminishes passive stiffness, resulting in the control of ventricular end-diastolic volume. Utilizing these rules of remodeling in our mathematical model yields that the natural helical pathways of the myocardial fibers in the wall are formed automatically.

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.

Mechanical effects of looping in the embryonic chick heart

During early embryonic development, the heart bends into a curved tube in a vital morphogenetic process called looping. Since looping involves poorly understood biomechanical forces that are difficult to measure, this paper presents a theoretical model for the tubular chick heart, whose development is similar to that of the human heart. Representing the basic morphology of the looped ventricle, the model is a thick-walled, isotropic, pressurized curved tube composed of three layers representing the myocardium, cardiac jelly, and endocardium. The model is analyzed with nonlinear elasticity theory, modified to include residual strain and muscle activation, and material properties are determined by correlating theoretical and experimental pressure-volume relations. The results show that longitudinal curvature significantly influences the biomechanical behavior of the embryonic heart. As the curvature increases, the compliance of the tube increases, especially at end systole. Stress concentrations, which develop in the endocardium during diastole and in the myocardium during systole, also increase with the curvature. The largest wall stress during the cardiac cycle occurs near the beginning of systolic ejection in the myocardial layer at the inner curvature of the tube. Relative to end diastole, the model predicts epicardial strains that are nearly equal in the circumferential and meridional directions, in agreement with experimental measurements. These results provide insight into the interrelation between biomechanical forces and morphogenesis during cardiac looping.