Fluid Dynamics of Heart Development
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- Santhanakrishnan, A. & Miller, L.A. Cell Biochem Biophys (2011) 61: 1. doi:10.1007/s12013-011-9158-8
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.
KeywordsHeart developmentHemodynamicsShear stressMathematical modelingFluid dynamics
Burggren  suggests that the embryonic heart beat is not required for the purpose of nutrition but rather aids in the growth, shaping, and morphogenesis of the heart itself. This proposition is based upon previous experimental work in fish, amphibian and bird embryos. When cardiac output was disrupted either through mutation or surgical intervention, these organisms continued to develop normally for some time using the diffusion of oxygen, nutrients, metabolic wastes, and hormones. In the specific case of zebrafish embryos, for example, this idea is supported by the fact that mutant embryos lacking erythrocytes display no vascular defects and can be raised to adulthood  and silent heart mutants are able to hatch and swim .
It has been proposed that the purpose of the embryonic heartbeat is to produce forces that play a role in the formation of the heart and the underlying vascular network . This idea began with Chapman  nearly a hundred years ago who surgically removed the heart of chicken embryos and documented the resultant malformation of the circulatory system. Recent advances in quantitative flow visualization techniques at spatial scales on the order of several micrometers have made in vivo exploration of the fluid dynamics of the vertebrate embryonic heart possible [42, 43, 106]. Hove et al.  experimentally showed that shear stress imparted on the cardiac walls by the blood flow is important to proper morphological development of the zebrafish heart (Danio rerio). They also noted that proper formation of the heart valves was particularly sensitive to changes in flow. Gruber and Epstein  found that congenital heart abnormalities such as the hypoplastic left heart syndrome (HLHS) in which the left ventricle is either small or absent may be triggered by improper blood flow to the developing ventricle. A recent study by Reckova et al.  on chick embryos showed that the maturation of the conduction cells responsible for ventricular contraction depended on the forces imparted by the blood flow.
Accurate descriptions of the normal hemodynamics during each stage of development as well as an understanding of how fluid forces shape the heart could be important for both the diagnosis and correction of congenital heart disease. In utero surgical interventions for severe aortic stenosis have already shown great promise in improving ventricular function and possibly preventing the development of HLHS [91, 111]. Selamet Tierney et al. showed that in utero aortic valvuloplasty improves left ventricular systolic function for mid-gestation fetuses that show severe aortic stenosis. In utero echocardiography, which can be used to detect abnormalities in bloodflow (such as backflow), has been used to diagnose general structural heart diseases from 16 weeks onward . Such methods have been used for the early detection of univentricular heart (UVH), ventricular septal defect (VSD), as well as HLHS. There is hope that early detection and in utero surgical intervention could improve outcomes for other congenital heart diseases.
While there is increasing evidence that points to blood flow driven forces as being an important and essential factor influencing both proper cardiovascular development, most of the physical details of the fluid dynamics through the embryonic heart, especially at the level of shear sensing components in the endothelium, remain unclear. The objective of this article is to review the field of hemodynamics pertinent to early cardiovascular development. This review begins with a description of the morphology of the developing heart. In the following two sections, the pumping mechanisms and intracardial fluid dynamics in the early stages of development are presented. A description of the electrophysiology of the embryonic heart is then discussed. A brief review of the fundamental fluid dynamical theory is provided, followed by a discussion of some recent investigations that provide in vivo information on the flow patterns in vertebrate embryos. Finally, the possible epigenetic triggers of cardiac cushion formation and mechanisms of pumping in the embryonic heart are presented.
Morphological Overview of the Developing Heart
At later stages of development, differences in fish, avian, and mammalian hearts are apparent since the final design of each heart is fundamentally different. Fish hearts are two-chambered with a single atrium and ventricle. The blood flows through the sinus venosus to the atrium, is then pumped into the ventricle, and finally exists the heart through the conus arteriosus. Valves develop at the sinoatrial, atrioventricular, and ventriculoconal junctions to prevent backflow into the preceding compartment. The atrium is positioned dorsally and the ventricle is positioned ventrally, creating the S-shape of the adult fish heart. This S-shape is a common feature of all vertebrate hearts. The adult avian and mammalian hearts have four chambers and four valves configured in a parallel-arrangement. For the remainder of this section, the discussion will be focused on the development of the mouse embryonic heart as a model for the human heart.
The flow between the right and left sides of the heart is separated by the formation of septa in the atrial and ventricular chambers. The formation of the interventricular septum divides the two ventricles. The upper posterior portion of this septum is called the membranous ventricular septum since it is thin and membranous. The rest of the septum is thick and muscular. The left and right atria are divided by the septum primum and the septum secundum. The septum primum grows down the chambers into the atrial cavity to eventually fuse with the endocardial cushions. During its growth, the gap between the septum primum and the cushions is known as the ostium primum. Perforations also appear in the superior part of the septum primum creating an opening known as the ostium secundum which eventually forms part of the fossa ovalis. The septum secundum grows downward from the upper wall of the atrium to the right of the primary septum. The septum secundum remains incomplete, leaving an opening called the foramen ovale.
At birth, the initial inflation of the lungs reduces the resistance to blood flow through the lungs. The ductus arteriosus closes so that blood flow is increased to the lungs and is no longer shunted into the aorta. The increased venous return from the lungs raises pressure in the left atrium. This pressure differential closes the foramen ovale. At this point the heart is separated into two pumps. Blood flows from the vena cava to the right atrium, through the tricuspid valve, and into the right ventricle. The blood is then pumped through the pulmonary valve and into the pulmonary artery to the lungs. The blood then moves from the lungs to the left atrium through the pulmonary vein. From the left atrium, the blood is pumped to the left ventricle through the mitral valve. The left ventricle finally pumps the blood through the aortic valve into the aorta.
Review of Fluid Dynamics
Reynolds and Womersley numbers for embryonic hearts at several stages of development given in hours post fertilization (h.p.f.), days post fertilization (d.p.f.), days post conception (d.p.c.), and Hamburger and Hamilton  stages (HH)
Flow rate (mm/s)
Zebrafish, 26 h.p.f.
Zebrafish, 4.5 d.p.f.
Mouse, 8.5 d.p.c.
Mouse, 9.5 d.p.c.
Mouse, 10.5 d.p.c.
Reynolds and Womersley numbers for the aorta at several stages of development given in days post conception (dpc) and Hamburger and Hamilton  stages (HH)
Flow rate (mm/s)
Chicken, stage 18
Chicken, stage 24
Mouse, 11.5 d.p.c.
Mouse, 12.5 d.p.c.
Mouse, 13.5 d.p.c.
Mouse, 14.5 d.p.c.
Most studies of embryonic flows are built upon the fundamental assumption that blood behaves a Newtonian fluid and can be described by the Navier–Stokes equation. In Newtonian flows, the shear stress is linearly related to shear strain, with the coefficient of dynamic viscosity μ being the constant of proportionality. This approximation works well for simple fluids such as air and water. Blood does exhibit non-Newtonian rheological behavior, but this effect may be negligible under many circumstances . Some mathematical models for non-Newtonian fluids (including blood) are given in the Appendix. Non-Newtonian effects, when present, are primarily due to the presence of erythrocytes. In the adult, red blood cells have a biconcave shape with fairly close packing amounting to roughly 45% of the volume of the blood. In the embryonic circulation, red blood cells are spherical shaped and make up low percentages (estimated 10–15%) of the volume of the blood in the early stages of development [63, 83]. The hematocrit (percent blood cell volume) will alter the effective viscosity of the blood as well as its apparent viscosity. In addition, the effective viscosity will lower with increasing shear rates. Based on in vitro experiments on human erythrocytes, Chien et al.  observed that this shear-thinning behavior was due to cell–cell interaction and cell–protein interaction, with the former being the more important factor. The diameter of the vessel affects the shear-thinning nature of blood, and this is pronounced especially in the case of microcirculation (such as in capillaries or the embryonic heart) where the cells interact with the tube walls thereby altering the shear rate and the viscosity. In addition to the above factors, cell aggregation and hardening increase the dynamic viscosity; while increases in cell deformability decrease the effective blood viscosity. The importance of non-Newtonian effects in the embryonic circulation, however, has not been carefully examined.
Pumping Mechanisms of the Heart Tube
Peristalsis and impedance pumping have both been proposed as mechanisms through which the embryonic heart tube pumps blood. Both of these mechanisms do not require the presence of valves for providing a net output flow. Historically, the examination of contraction kinematics and electrocardiograms lead researchers to assume that the blood is pumped by peristaltic contractions when the heart tube first forms [24, 28, 68]. Peristalsis in biological systems can be described as a wave of axially symmetric contractions that propagate down a muscular tube to drive the fluid within. Peristalsis is commonly observed in the esophagus and gastrointestinal tract, and a variety of mechanical devices also use peristalsis to move fluids. In addition to the embryonic heart, peristalsis has been described as the pumping mechanism of the tubular hearts of ascidians , leeches , and insects [27, 62].
In vivo measurements have shown that the flow within the embryonic heart tube becomes pulsatile early in development before the valves form. Pulsatile flow is not characteristic of typical peristalsis. Taber et al.  explored the peristaltic pumping mechanism in the heart tube using a computational fluid model. Results from this model showed that the formation of the endocardial cushions induces a transition from peristaltic to pulsatile flow. The flow velocities and pressures generated from their model show good agreement with published experimental data.
Recent in vivo work based on particle image velocimetry (PIV) suggests that the heart might pump using valveless suction pumping (e.g., impedance pumping) rather than peristalsis. Forouhar tested the peristaltic hypothesis against three features of this pumping mechanism: (a) the wave traveling down the heart tube should be unidirectional, (b) the magnitude of the flow velocities should be bounded by the velocity of the traveling wave (assuming constant diameter and spatially uniform flow), and (c) the volumetric flow rate should increase linearly with heart rate. They found that bidirectional waves propagated from the region of the pacemaker cells, the maximum velocity of the blood exceeded the wall wave speed, and the relationship between heart rate and volumetric flow rate was nonlinear. From these results, they suggest that the pumping mechanism of the embryonic heart tube is not peristalsis. Forouhar et al.  state that the sensitivity of the flow rate to changes in heart rate is similar to what is observed during impedance pumping. Both the zebrafish heart and the impedance pump exhibit resonant peaks in the frequency-flow relationship.
Recent experiments have used physical models to further investigate impedance pumping [7, 37, 38]. These studies have highlighted the complexity of the underlying mechanism of impedance pumping. Experiments using an open loop system developed by Hickerson et al.  indicate that the flow rate is sensitive to both the actuation frequency and the duty cycle (fraction of pumping cycle during which the tube is actuated). Their experiments were performed for Womerseley numbers range of 10–30, and the results indicated the maximum non-dimensional flow rate was slightly better than peristaltic flow. Visualization of the tube clearly showed the presence of traveling waves on the surface and reflections at the ends of the tube. Closed-loop experiments were also conducted by Bringley et al.  for a system consisting of an elastic section attached to an inelastic section. These experiments also revealed that the flow rate was a function of the actuation frequency. Although a change in flow direction was observed with increasing frequency, the flow direction was opposite to that observed by Hickerson et al. . They developed a simple mathematical model to explain the frequency flow relationship that did not account for any wave phenomena and concluded that the net flow generated is a function of the nonlinear term in the momentum equation.
There have been a number of theoretical attempts to explain impedance pumping using both numerical and analytical techniques. Thomann  computed the flow in the closed-loop Liebau setup for inviscid flows (μ = 0) and accounted for wave reflection at the rigid tube. His results indicated that net flow was generated by the higher pressure on one side of the chamber due to wave reflection. He also predicted flow reversals with changes in pumping frequency. Two-dimensional numerical simulations using the immersed boundary method were performed by Jung and Peskin  for the Liebau phenomena in a closed loop. The range of the Womersely numbers for their computations was 3–27. Similar to other studies, the magnitude and direction of net flow were found to be functions of the actuation frequency. The simulations also showed a traveling wave along the elastic section of the tube. A simplified one-dimensional numerical model was developed by Ottesen  for the closed-loop system with viscosity. The numerical results were compared with experiments and the magnitude and direction of the net flow again depended upon the frequency of actuation and elasticity of the tubes. Manopoulos et al.  investigated the mechanism of impedance pumping in a closed loop using a qausi one-dimensional unsteady model derived from the integration of the continuity and momentum equations over the tube cross-sectional area. The periodic compression of the soft part of the tube generated unidirectional flow under certain conditions. They attributed this net flow to the pressure difference created across the tube due to the phase difference in the traveling waves.
Further work is needed to support the proposition that the vertebrate embryonic heart acts as an impedance pump. Although this is a viable mechanism of pumping fluid over a wide range of Reynolds numbers [7, 49], there has not been a careful study that matches duty cycle and Reynolds number or Womersley number to the embryonic heart case. Previous work has also assumed that the blood is Newtonian and that the heart has a simple cylindrical geometry. Finally, the mechanics of the pumping mechanism should be integrated with biologically realistic methods of actuation and muscle mechanics. Integrated models and simulations of this stage of development that combine fluid dynamics, muscle mechanics, and electrophysiology could provide key insights into the early development of the heart and the cardiac conduction system.
Vortex Formation and Scale
A couple of recent numerical investigations have described the blood flow through sophisticated three-dimensional models of the vertebrate embryonic heart. DeGroff et al.  used a sequence of two-dimensional cross-sectional images to reconstruct the three-dimensional surface of human heart embryos at stages 10 and 11. In their paper, the heart walls did not move, and steady and pulsatile flows were obtained using finite volume CFD. Their study also showed that streaming was present in the heart tube (particles released on one side of the lumen did not cross over or mix with particles released from the opposite side), and no coherent vortex structures were observed. Liu et al.  quantified the hemodynamic forces on a three-dimensional model of a chick embryonic heart using a finite element model. They focused on pulsatile flow through the outflow tract during stage HH21 (after about 3.5 days of incubation) and included flexibility in the walls of the tract. They did not include cardiac cushions in their simulations. Maximum velocities were observed in regions of constrictions and vortices were observed during the ejection phase near the inner curvature of the outflow tract, corresponding to a maximum Reynolds number of 6.9.
Santhanakrishnan et al.  used simple physical and mathematical models to show that the conditions required for vortex formation are significantly affected by flow Reynolds number and are highly sensitive to the chamber and cushion dimensions. In general, chamber vortices were observed for Reynolds numbers on the order of 10 and higher. The transition to vortical flow was particularly sensitive to changes in chamber depth and cushion height for Reynolds numbers in this range (see Fig. 11). It is likely that this transition also depends upon the unsteady or pulsatile behavior of the flow, although sensitivity to such unsteady effects was not explored. Since the large scale structure of the blood flow is critically sensitive to small changes in scale and morphology, detailed studies of intracardial flow carefully matched to each developmental stage are needed to understand the complex relationship between structure and flow.
Shear Stress, Pressure, and Myocardial Activity
Bartman et al.  argue that myocardial function, not shear stress, is required for the formation of the endocardial cushions. They used various concentrations of 2,3-butanedione monoxime (2,3-BDM) to block myofibrillar ATPase  and reduce the myocardial force generated in a dose-dependent manner in zebrafish embryos. They found that as the embryos were treated with increasing amounts of 2,3-BDM at 36 h.p.f., the blood flow abruptly stopped. The percentage of embryos that formed endocardial rings at 48 h.p.f. decreased continuously. They concluded that since 58% of embryos treated with 6 mM or more of 2,3-BDM formed endocardial rings in the absence of blood flow, myocardial activity must be the required signal in endocardial cushion formation. They found similar results using the anesthetic tricaine. Studies in mice also suggest a strong relationship between myocardial activity and heart morphogenesis. The mutation of a single gene can disrupt both [3, 10, 12]. The authors concede that further studies are needed to unravel the roles of myocardial activity and endothelial shear stress on cardiac cushion and valve formation since the two processes are fundamentally coupled.
Since myocardial activity, pressure, and shear stress are mechanically coupled, it is difficult to untangle which signal(s) may be responsible for valvulogenesis. Given that fluids flow from regions of high to low pressure, the generation of hoop stress in the heart tube moves the blood within it, and this movement produces shear stress. To further complicate the interpretation of the results from Bartman et al. , shear stress can be generated in some non-Newtonian fluids without significant fluid motion. Although the rheology of the embryonic blood is not well known, previous work suggests a nonlinear relationship between stress and strain for adult blood [16, 64, 78].
Theoretical studies also support the idea that both shear stress and pressure are important to the development of the cardiac valves. Biechler et al.  used a two-dimensional mathematical model of flow through a rigid channel to show that shear stress and pressure over the simple atrioventricular cushions are about the same order of magnitude. Their simulations were performed for Reynolds numbers in the range of 1–10, corresponding to an HH-stage 25 chick heart. Miller  also found that pressure and shear are of the same order of magnitude in a simplified two-dimensional beating heart model for Reynolds numbers on the order of 0.1. In this case, shear stress is maximized on the luminal side of the cushions, and pressure is maximized on the chamber walls during contraction.
Electrophysiology and Relationship to Fluid Dynamics
The electrophysiology of the embryonic heart is clearly significant to its internal fluid dynamics since electrical activity triggers the contraction of the myocardial cells that drive the blood flow. On the other hand, fluid shear may impact the electrophysiology of the developing heart. Increasing shear stress is known to increase the conduction velocities of action potentials in the myocardial layer of the developing heart . In experiments where shear stress was reduced in vivo, Hove et al.  found that the timing of muscle contraction and presumably the conduction velocities of the heart tube were reduced relative to the control case. Tucker et al.  found that the heart beat is involved in the proper formation of the pacemaker and other cardiac conduction tissue in early chicken embryos. Such changes in conduction properties, in turn, alter the intracardiac fluid dynamics and shear stresses.
Although work that has attempted to integrate the electrophysiology of the heart with its pumping kinematics and fluid dynamics is limited, recent improvements in numerical methods and scientific computing are starting to make such studies possible. Griffith and Peskin  are using an immersed boundary formulation of the bidomain equations to study cardiac electrophysiology of a beating adult heart with moving boundaries. The bidomain equations  describe the dynamics of intracellular and extracellular voltage and current in cardiac tissue. Although other electrophysiology models could be used, the bidomain equations take into account the strong difference in electrical anisotropy between the intracellular and extracellular spaces. Their method is analogous to Peskin’s traditional immersed boundary method . Lagrangian curvilinear coordinates are used for the intracellular space, which is confined to the myocardium, and Cartesian coordinates are used for the extracellular space, which extends beyond the myocardium, into the electrically conducting blood and extracardiac tissue. The local membrane potential is then used to trigger the contraction of the myocardium. Modifications of this method for the embryonic case could be used to understand how the pumping mechanism and fluid dynamics of the heart tube can be integrated with its electrophysiology.
Overview of Shear Sensing
It has long been noted that the endothelial cells lining the blood vessels respond to three kinds of biomechanical stimuli which include the flow shear, fluid hydrostatic pressure, and cyclic strain (stretch) . Shear stress levels as low as 0.2 dyn/cm2 can be sensed by cultured vascular endothelial cells ex vivo through mechanotransduction . It is also known that exposure to flow causes endothelial cell actin microfilaments to change from banded to parallel fiber patterns which affects the stiffness of endothelial cells [19, 22]. A few recent studies have shown that laminar flow can alter gene expression in the embryonic heart [31, 32]. During the looping process in the chicken embryonic heart, they found that enodothelin-1 (ET1) was expressed in regions of low shear such as the chambers where the heart widens. Krüppel-like factor-2 (KLF2) and endothelial nitric oxide synthase (NOS3) were expressed in regions of high shear (the AV canal and outflow tract). Using computational fluid dynamics, Hierck et al.  were able to demonstrate that these patterns of gene expression overlapped regions of high or low shear generated by the numerical simulations.
In order to understand glycocalyx-mediated mechanotransduction, one must understand the profile of the blood flow above and through this layer . For example, the flow profile will determine the amount of shear stress that is felt on the luminal surface of the layer and at the cell membrane. The amount of flow through the ESL will also contribute to the movement of molecules into and out of the layer if the convection of particles is on the same order or greater than the rate of diffusion through the layer. Since spatially resolved measurements of flow above and within the ESL are limited, a number of researchers have used mathematical models to determine flow rates and shear stresses within this layer. One of the more popular models uses the Brinkman equation where the ESL is treated as a homogenized porous layer [8, 18, 107]. For example, Weinbaum et al.  modeled the glycocalyx as a Brinkman layer and calculated the value of the hydraulic conductivity using estimates of the volume fraction of core proteins and by assuming that the layer has a quasi-periodic structure. To obtain the flow profile for the entire vessel, they matched the flow through this layer to Stokes flow above the layer. They found that the majority of shear stress was imposed on the tip of the core proteins and relatively little was imposed at the membrane. Leiderman et al.  modeled the endothelial surface layer as clumps of a Brinkman medium immersed in a Newtonian fluid. They varied the width and spacing of each clump, the hydraulic permeability, and the height of the ESL. They found that spatial inhomogeneities altered the magnitude and location of maximum shear stress within the layer.
Another mechanism for shear sensing is the primary cilium [61, 93]. It has recently been discovered that primary cilia are present in both endothelial and endocardial cells [47, 71, 105] and during embryonic development . Van der Heiden suggests that primary cilia act as a shear sensor in the embryonic heart. This role has also been attributed to primary cilia on the epithelial cells in Hensen’s node in the embryo [63, 115] and the adult kidney [70, 80]. In these cases, the primary cilia transduce mechanical signals into an intracellular Ca2+ response. In the embryonic heart, Van der Heiden et al. found that the primary cilia dissociate under high shear conditions (such as the AV canal and outflow tract) and are more prevalent in regions of low shear (such as the chambers). They also found that the distribution of primary cilia distribution coincided with the expression of and KLF-2 which is considered a high shear stress marker [21, 31].
Numerous studies that use flow manipulation in the embryonic heart indicate that fluid shear stress and pressure act as epigenetic signals for cardiogenesis. Hogers et al. [40, 41] ligated the right lateral vitelline vein in stage 17 chick embryos and found subaortic ventricular septal defects, semilunar valve anomalies, atrioventricular anomalies, and pharyngeal arch artery malformations at later stages of development. Ursem et al.  found that the dynamics of ventricular filling changed after using the venous clip in a chick embryo. The clipped embryos exhibited reduced passive filling in favor of atrial contraction to fill the ventricle at stage 24. Hove et al.  describe the presence of high-shear vortical flow at two important stages of heart development and suggest that shear stress plays a fundamental role in chamber and valve morphogenesis in the zebrafish embryo.
To fully understand the role of fluid dynamics in heart development, connections need to be made between the intracardiac flows, myocardial activity, molecular regulatory networks, and cardiac electrophysiology since all of these functions are coupled. The timing of the myocardial contractions is controlled by the electrical activity of the heart, deformations of the muscular heart wall influence the electrophysiology of the heart via stretch activated transmembrane ion channels, and the contraction of the myocardial cells move the intracardial blood. An integrated embryonic model of the heart could also be used to address a number of issues related to electro-mechanical coupling in the developing heart. For example, simulations and experiments with physical models can help to clarify the nature of the pumping mechanism employed when the heart tube first forms. Numerical simulations and physical models could also be used to determine more precisely the developmental stage(s) at which fluid dynamic transitions occur. If proteins responsible for heart morphogenesis are translated or activated at the same developmental stages at which fluid transitions occur, then this could support the idea that fluid dynamic transitions signal morphogenesis. More broadly, flow studies can be used to determine which genes involved in heart development may be up- or down-regulated by shear. For example, studies by Groenendijk et al. [31, 32] provide an excellent example of how gene expression could be connected to regions of high and low shear stress.
Challenges and Future Directions
Measuring spatially and temporally resolved flow-fields in vivo is challenging, particularly near the endocardial wall and through the AV-canal and outflow tract. Some of the major obstacles include measuring flows on the submicron level and obtaining visual access. Hove et al.  measured the flow field within the heart tube of the zebrafish embryo in vivo using PIV, and the erythrocytes in the blood were tracked as the fluid parcel markers. They obtained excellent information on the flow rates and the larger scale fluid dynamics of the blood flow through the chambers. They were not, however, able to resolve flow profiles in the AV canal or near the chamber walls. The heart tube diameter was approximately 50 microns, which is only an order of magnitude greater than the size of the red blood cells. In order to resolve fluid motion at these fine scales, the seeding particle size has to be sufficiently small in comparison to the flow domain.
Vennemann et al.  used liposomes of the order of several hundred nanometers as tracers to examine the blood flow within the heart of a chick embryo using PIV. They obtained more reliable estimates of flow field characteristics near the wall, but their spatial resolution was limited on account of the low seeding intensity. Kim and Lee  proposed an X-ray based PIV method for measuring blood flow without using any tracers in the fluid. However, this technique was found to only work in the limit of blood vessels larger than 1 cm. Poelma et al.  used scanning micro PIV to obtain in vivo measurements of the three-dimensional distribution of wall shear stress in the outflow tract of an embryonic chicken heart. They were able to obtain three-dimensional shear stress and velocity fields with a spatial resolution of 15-20 μm. However, their estimates of velocity near the wall and wall shear stress had errors on the order of 20% due to the sparse distribution of particles in this region. One method to improve the signal to noise ratio in would be to use fluorescent nano-particles as the seeding material [86, 101]. Such particles are roughly of the order of few hundred nanometers in diameter, and are typically coated with a fluorescent material that absorbs and emits light at specific wavelengths. By closely matching the wavelength of absorption and emission of the particle with the illumination source, extraneous reflections that affect the contrast of the PIV image can be avoided.
One of the main challenges in mathematically modeling the embryonic heart is to balance the complexity of the model so that it is biological relevant and simple enough so that the problem is still tractable. Computational advances in solving fluid-structure interaction problems have allowed researchers to numerically simulate virtual hearts that move fluid through the contraction of muscles [66, 76]. More recently, electrophysiological models have been coupled to models of muscle mechanics . These muscle models are then used to drive the blood flow in numerical simulations of the heart. The computational task associated with the use of such detailed models is significant. To represent wavefront propagation of the action potential, spatial resolution must be on the order tenths of microns, whereas the complete organ is on the order of hundreds of microns. In three dimensions, hundreds of millions of nodes are required in a three-dimensional computation. Events in the cell membrane happen on the millisecond scale, with the entire heartbeat lasting about half a second, and stability considerations require time steps on the order of microseconds, therefore requiring millions of time steps. The fluid flow within the heart must be resolved at a scale of tenths of microns near the chamber walls, again requiring hundreds of millions of nodes for three-dimensional calculations of the fluid flow. Simplifications of the mathematical models must be made in order to reasonably study the fluid mechanics of heart development. The challenge is to model the heart in such a way that the minimum amount of complexity is included to capture the fundamental features of the system.
It is possible that some of the advances in computation developed for pediatric cardiovascular research could be applied to the embryonic heart, particularly at the later stages of development. For example, computational fluid dynamics has been used to improve the treatment of HLHS after the Norwood procedure . They compared hydraulic performance between the hemi-Fontan and bidirectional Glenn procedures, with the hopes of improving the design these surgical operations. Perhaps similar studies could be used to understand flow patterns in fetuses with HLHS and other congenital heart diseases with the hopes of early detection and treatment. For earlier stages of development, there are fundamental differences in scale that may require alternative mathematical models and numerical methods. For example, the Reynolds number of the embryonic heart tube is on the order of 0.01 so the hemodynamics could be modeled using the Stokes equations rather than the full Navier–Stokes equations. The Stokes equations allow for a number of alternative numerical methods such as the Method of Regularized Stokeslets . Also due to differences in scale, the effects of the presence of red blood cells on the flow may also be non-negligible. In this case, methods that include the flexible red blood cells  or treat the blood as a homogenized non-Newtonian fluid  may be appropriate.
We would like to thank the University of Utah Mathematical Biology Group and the UNC Fluids and Integrative & Mathematical Physiology Groups for their suggestions and insight. We would also like to thank Dr. Kathy K. Sulik for her excellent SEM images of the mouse embryonic heart used in this review. This work was funded by Miller’s Burroughs Wellcome Fund Career Award at the Scientific Interface.