Flow within models of the vertebrate embryonic heart

Human Cardiac Development

Many aspects of heart development are topographically complex and require three-dimensional (3D) reconstruction to understand the pertinent morphology. We have recently completed a comprehensive primer of human cardiac development that is based on firsthand segmentation of structures of interest in histological sections. We visualized the hearts of 12 human embryos between their first appearance at 3.5 weeks and the end of the embryonic period at 8 weeks. The models were presented as calibrated, interactive, 3D portable document format (PDF) files. We used them to describe the appearance and the subsequent remodeling of around 70 different structures incrementally for each of the reconstructed stages. In this chapter, we begin our account by describing the formation of the single heart tube, which occurs at the end of the fourth week subsequent to conception. We describe its looping in the fifth week, the formation of the cardiac compartments in the sixth week, and, finally, the septation of these compartments into the physically separated left- and right-sided circulations in the seventh and eighth weeks. The phases are successive, albeit partially overlapping. Thus, the basic cardiac layout is established between 26 and 32 days after fertilization and is described as Carnegie stages (CSs) 9 through 14, with development in the outlet component trailing that in the inlet parts. Septation at the venous pole is completed at CS17, equivalent to almost 6 weeks of development. During Carnegie stages 17 and 18, in the seventh week, the outflow tract and arterial pole undergo major remodeling, including incorporation of the proximal portion of the outflow tract into the ventricles and transfer of the spiraling course of the subaortic and subpulmonary channels to the intrapericardial arterial trunks. Remodeling of the interventricular foramen, with its eventual closure, is complete at CS20, which occurs at the end of the seventh week. We provide quantitative correlations between the age of human and mouse embryos as well as the Carnegie stages of development. We have also set our descriptions in the context of variations in the timing of developmental features.

Capturing functional relations in fluid-structure interaction via machine learning

While fluid-structure interaction (FSI) problems are ubiquitous in various applications from cell biology to aerodynamics, they involve huge computational overhead. In this paper, we adopt a machine learning (ML)-based strategy to bypass the detailed FSI analysis that requires cumbersome simulations in solving the Navier-Stokes equations. To mimic the effect of fluid on an immersed beam, we have introduced dissipation into the beam model with time-varying forces acting on it. The forces in a discretized set-up have been decoupled via an appropriate linear algebraic operation, which generates the ground truth force/moment data for the ML analysis. The adopted ML technique, symbolic regression, generates computationally tractable functional forms to represent the force/moment with respect to space and time. These estimates are fed into the dissipative beam model to generate the immersed beam's deflections over time, which are in conformity with the detailed FSI solutions. Numerical results demonstrate that the ML-estimated continuous force and moment functions are able to accurately predict the beam deflections under different discretizations.

A pictorial account of the human embryonic heart between 3.5 and 8 weeks of development

Heart development is topographically complex and requires visualization to understand its progression. No comprehensive 3-dimensional primer of human cardiac development is currently available. We prepared detailed reconstructions of 12 hearts between 3.5 and 8 weeks post fertilization, using Amira® 3D-reconstruction and Cinema4D®-remodeling software. The models were visualized as calibrated interactive 3D-PDFs. We describe the developmental appearance and subsequent remodeling of 70 different structures incrementally, using sequential segmental analysis. Pictorial timelines of structures highlight age-dependent events, while graphs visualize growth and spiraling of the wall of the heart tube. The basic cardiac layout is established between 3.5 and 4.5 weeks. Septation at the venous pole is completed at 6 weeks. Between 5.5 and 6.5 weeks, as the outflow tract becomes incorporated in the ventricles, the spiraling course of its subaortic and subpulmonary channels is transferred to the intrapericardial arterial trunks. The remodeling of the interventricular foramen is complete at 7 weeks.

Building Valveless Impedance Pumps From Biological Components: Progress and Challenges

Valveless pumping based on Liebau mechanism entails asymmetrical positioning of the compression site relative to the attachment sites of the pump's elastic segment to the rest of the circuit. Liebau pumping is believed to play a key role during heart development and be involved in several other physiological processes. Until now studies of Liebau pump have been limited to numerical analyses, in silico modeling, experiments using non-biological elements, and a few indirect in vivo measurements. This review aims to stimulate experimental efforts to build Liebau pumps using biologically compatible materials in order to encourage further exploration of the fundamental mechanisms behind valveless pumping and its role in organ physiology. The covered topics include the biological occurrence of Liebau pumps, the main differences between them and the peristaltic flow, and the potential uses and body sites that can benefit from implantable valveless pumps based on Liebau principle. We then provide an overview of currently available tools to build such pumps and touch upon limitations imposed by the use of biological components. We also talk about the many variables that can impact Liebau pump performance, including the concept of resonant frequencies, the shape of the flowrate-frequency relationship, the flow velocity profiles, and the Womersley numbers. Lastly, the choices of materials to build valveless impedance pumps and possible modifications to increase their flow output are briefly discussed.

A semi-automated finite difference mesh creation method for use with immersed boundary software IB2d and IBAMR

Numerous fluid-structure interaction problems in biology have been investigated using the immersed boundary method. The advantage of this method is that complex geometries, e.g., internal or external morphology, can easily be handled without the need to generate matching grids for both the fluid and the structure. Consequently, the difficulty of modeling the structure lies often in discretizing the boundary of the complex geometry (morphology). Both commercial and open source mesh generators for finite element methods have long been established; however, the traditional immersed boundary method is based on a finite difference discretization of the structure. Here we present a software library for obtaining finite difference discretizations of boundaries for direct use in the 2D immersed boundary method. This library provides tools for extracting such boundaries as discrete mesh points from digital images. We give several examples of how the method can be applied that include passing flow through the veins of insect wings, within lymphatic capillaries, and around starfish using open-source immersed boundary software.

Vortex Dynamics in Trabeculated Embryonic Ventricles

Proper heart morphogenesis requires a delicate balance between hemodynamic forces, myocardial activity, morphogen gradients, and epigenetic signaling, all of which are coupled with genetic regulatory networks. Recently both in vivo and in silico studies have tried to better understand hemodynamics at varying stages of veretebrate cardiogenesis. In particular, the intracardial hemodynamics during the onset of trabeculation is notably complex—the inertial and viscous fluid forces are approximately equal at this stage and small perturbations in morphology, scale, and steadiness of the flow can lead to significant changes in bulk flow structures, shear stress distributions, and chemical morphogen gradients. The immersed boundary method was used to numerically simulate fluid flow through simplified two-dimensional and stationary trabeculated ventricles of 72, 80, and 120 h post fertilization wild type zebrafish embryos and ErbB2-inhibited embryos at seven days post fertilization. A 2D idealized trabeculated ventricular model was also used to map the bifurcations in flow structure that occur as a result of the unsteadiness of flow, trabeculae height, and fluid scale (Re). Vortex formation occurred in intertrabecular regions for biologically relevant parameter spaces, wherein flow velocities increased. This indicates that trabecular morphology may alter intracardial flow patterns and hence ventricular shear stresses and morphogen gradients. A potential implication of this work is that the onset of vortical (disturbed) flows can upregulate Notch1 expression in endothelial cells in vivo and hence impacts chamber morphogenesis, valvulogenesis, and the formation of the trabeculae themselves. Our results also highlight the sensitivity of cardiac flow patterns to changes in morphology and blood rheology, motivating efforts to obtain spatially and temporally resolved chamber geometries and kinematics as well as the careful measurement of the embryonic blood rheology. The results also suggest that there may be significant changes in shear signalling due to morphological and mechanical variation across individuals and species.

Fluid dynamics in heart development: effects of hematocrit and trabeculation

Recent in vivo experiments have illustrated the importance of understanding the haemodynamics of heart morphogenesis. In particular, ventricular trabeculation is governed by a delicate interaction between haemodynamic forces, myocardial activity, and morphogen gradients, all of which are coupled to genetic regulatory networks. The underlying haemodynamics at the stage of development in which the trabeculae form is particularly complex, given the balance between inertial and viscous forces. Small perturbations in the geometry, scale, and steadiness of the flow can lead to changes in the overall flow structures and chemical morphogen gradients, including the local direction of flow, the transport of morphogens, and the formation of vortices. The immersed boundary method was used to solve the two-dimensional fluid-structure interaction problem of fluid flow moving through a two chambered heart of a zebrafish (Danio rerio), with a trabeculated ventricle, at 96 hours post fertilization (hpf). Trabeculae heights and hematocrit were varied, and simulations were conducted for two orders of magnitude of Womersley number, extending beyond the biologically relevant range (0.2-12.0). Both intracardial and intertrabecular vortices formed in the ventricle for biologically relevant parameter values. The bifurcation from smooth streaming flow to vortical flow depends upon the trabeculae geometry, hematocrit, and Womersley number, $Wo$. This work shows the importance of hematocrit and geometry in determining the bulk flow patterns in the heart at this stage of development.

Kinking and Torsion Can Significantly Improve the Efficiency of Valveless Pumping in Periodically Compressed Tubular Conduits. Implications for Understanding of the Form-Function Relationship of Embryonic Heart Tubes

Valveless pumping phenomena (peristalsis, Liebau-effect) can generate unidirectional fluid flow in periodically compressed tubular conduits. Early embryonic hearts are tubular conduits acting as valveless pumps. It is unclear whether such hearts work as peristaltic or Liebau-effect pumps. During the initial phase of its pumping activity, the originally straight embryonic heart is subjected to deforming forces that produce bending, twisting, kinking, and coiling. This deformation process is called cardiac looping. Its function is traditionally seen as generating a configuration needed for establishment of correct alignments of pulmonary and systemic flow pathways in the mature heart of lung-breathing vertebrates. This idea conflicts with the fact that cardiac looping occurs in all vertebrates, including gill-breathing fishes. We speculate that looping morphogenesis may improve the efficiency of valveless pumping. To test the physical plausibility of this hypothesis, we analyzed the pumping performance of a Liebau-effect pump in straight and looped (kinked) configurations. Compared to the straight configuration, the looped configuration significantly improved the pumping performance of our pump. This shows that looping can improve the efficiency of valveless pumping driven by the Liebau-effect. Further studies are needed to clarify whether this finding may have implications for understanding of the form-function relationship of embryonic hearts.

IB2d: a Python and MATLAB implementation of the immersed boundary method

The development of fluid-structure interaction (FSI) software involves trade-offs between ease of use, generality, performance, and cost. Typically there are large learning curves when using low-level software to model the interaction of an elastic structure immersed in a uniform density fluid. Many existing codes are not publicly available, and the commercial software that exists usually requires expensive licenses and may not be as robust or allow the necessary flexibility that in house codes can provide. We present an open source immersed boundary software package, IB2d, with full implementations in both MATLAB and Python, that is capable of running a vast range of biomechanics models and is accessible to scientists who have experience in high-level programming environments. IB2d contains multiple options for constructing material properties of the fiber structure, as well as the advection-diffusion of a chemical gradient, muscle mechanics models, and artificial forcing to drive boundaries with a preferred motion.

The role of the pericardium in the valveless, tubular heart of the tunicate Ciona savignyi

Tunicates, small invertebrates within the phylum Chordata, possess a robust tubular heart which pumps blood through their open circulatory systems without the use of valves. This heart consists of two major components: the tubular myocardium, a flexible layer of myocardial cells that actively contracts to drive fluid down the length of the tube; and the pericardium, a stiff, outer layer of cells that surrounds the myocardium and creates a fluid-filled space between the myocardium and the pericardium. We investigated the role of the pericardium through in vivo manipulations on tunicate hearts and computational simulations of the myocardium and pericardium using the immersed boundary method. Experimental manipulations reveal that damage to the pericardium results in aneurysm-like bulging of the myocardium and major reductions in the net blood flow and percentage closure of the heart's lumen during contraction. In addition, varying the pericardium-to-myocardium (PM) diameter ratio by increasing damage severity was positively correlated with peak dye flow in the heart. Computational simulations mirror the results of varying the PM ratio experimentally. Reducing the stiffness of the myocardium in the simulations reduced mean blood flow only for simulations without a pericardium. These results indicate that the pericardium has the ability to functionally increase the stiffness of the myocardium and limit myocardial aneurysms. The pericardium's function is likely to enhance flow through the highly resistive circulatory system by acting as a support structure in the absence of connective tissue within the myocardium.

Fluid dynamics of heart development

The morphology, muscle mechanics, fluid dynamics, conduction properties, and molecular biology of the developing embryonic heart have received much attention in recent years due to the importance of both fluid and elastic forces in shaping the heart as well as the striking relationship between the heart's evolution and development. Although few studies have directly addressed the connection between fluid dynamics and heart development, a number of studies suggest that fluids may play a key role in morphogenic signaling. For example, fluid shear stress may trigger biochemical cascades within the endothelial cells of the developing heart that regulate chamber and valve morphogenesis. Myocardial activity generates forces on the intracardiac blood, creating pressure gradients across the cardiac wall. These pressures may also serve as epigenetic signals. In this article, the fluid dynamics of the early stages of heart development is reviewed. The relevant work in cardiac morphology, muscle mechanics, regulatory networks, and electrophysiology is also reviewed in the context of intracardial fluid dynamics.

Fluid dynamics of ventricular filling in the embryonic heart

The vertebrate embryonic heart first forms as a valveless tube that pumps blood using waves of contraction. As the heart develops, the atrium and ventricle bulge out from the heart tube, and valves begin to form through the expansion of the endocardial cushions. As a result of changes in geometry, conduction velocities, and material properties of the heart wall, the fluid dynamics and resulting spatial patterns of shear stress and transmural pressure change dramatically. Recent work suggests that these transitions are significant because fluid forces acting on the cardiac walls, as well as the activity of myocardial cells that drive the flow, are necessary for correct chamber and valve morphogenesis. In this article, computational fluid dynamics was used to explore how spatial distributions of the normal forces acting on the heart wall change as the endocardial cushions grow and as the cardiac wall increases in stiffness. The immersed boundary method was used to simulate the fluid-moving boundary problem of the cardiac wall driving the motion of the blood in a simplified model of a two-dimensional heart. The normal forces acting on the heart walls increased during the period of one atrial contraction because inertial forces are negligible and the ventricular walls must be stretched during filling. Furthermore, the force required to fill the ventricle increased as the stiffness of the ventricular wall was increased. Increased endocardial cushion height also drastically increased the force necessary to contract the ventricle. Finally, flow in the moving boundary model was compared to flow through immobile rigid chambers, and the forces acting normal to the walls were substantially different.

Quantifying blood flow and wall shear stresses in the outflow tract of chick embryonic hearts

Wall shear stresses (WSS) exerted by blood flow on cardiac tissues modulate growth and development of the heart. To study the role of hemodynamic conditions on cardiac morphogenesis, here, we present a methodology that combines imaging and finite element modeling to quantify the in vivo blood flow dynamics and WSS in the cardiac outflow tract (OFT) of early chicken embryos (day 3 out of 21-day incubation period). We found a distinct blood flow field and heterogeneous distribution of WSS in the chicken embryonic heart OFT, which have physiological implications for OFT morphogenesis.

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.

Mathematical modeling of flow-generated forces in an in vitro system of cardiac valve development

Heart valve defects are the most common cardiac defects. Therefore, defining the mechanisms of cardiac valve development is critical to our understanding and treatment of these disorders. At early stages of embryonic cardiac development, the heart begins as a simple tube that then becomes constricted into separate atrial and ventricular regions by the formation of small, mound-like structures, called atrioventricular (AV) cushions. As valve development continues, these mounds fuse and then elongate into valve leaflets. A longstanding hypothesis proposes that blood flow-generated shear stress and pressure are critical in shaping the cushions into leaflets. Here we show results from a two-dimensional mathematical model that simulates the forces created by blood flow present in a developing chick heart and in our in vitro, tubular model system. The model was then used to predict flow patterns and the resulting forces in the in vitro system. The model indicated that forces associated with shear stress and pressure have comparable orders of magnitude and collectively produce a rotational profile around the cushion in the direction of flow and leaflet growth. Further, it was concluded that the replication of these forces on a cushion implanted in our tubular in vitro system is possible. Overall, the two-dimensional, mathematical model provides insight into the forces that occur during early cardiac valve elongation.