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Revised manuscript for consideration for publication in Cell Biochem. Biophys.

Fluid Dynamics of Heart Development

Arvind Santhanakrishnan and Laura A. Miller

*

Dept. of Mathematics, The University of North Carolina at Chapel Hill

Chapel Hill, NC 27599-3250, U.S.A

Abstract

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 paper, 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.

_______________________________ *Author to whom all correspondence and reprint requests should be addressed. E-mail: lam9@email.unc.edu

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Introduction

Burggren (2004) 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 (Liao et al., 2002) and silent heart mutants are able to hatch and swim

(Sehnert, 2002).

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 (1918) 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

(Hove, 2004; Hove, 2006; Vennemann et al., 2006). Hove et al. (2003) 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

(2004) 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. (2003) on chick embryos

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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 (Wilkins-Haug et al., 2007; Selamet Tierney et al., 2007).

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 (Boldt et al., 2002). 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 paper 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

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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

Since vertebrate hearts are similar at the earliest stages of development, zebrafish (Thisse

and Zon, 2002; Glickman and Yelon, 2002), chicken (Hamburger and Hamilton, 1951), and mice

(Savolainen, 2009) embryos are commonly used as model organisms for the study of human

heart development. In all vertebrates, the heart first forms as a linear valveless tube. The tube

takes on a three-dimensional structure through a process known as cardiac looping. The heart

twists and bends with rightward looping to reorient from anterior-posterior polarity to left-right

polarity (Männer, 2000). Figure 1 shows a time sequence of cardiac looping in the mouse

embryonic heart. During looping, portions of this linear tube locally expand into the chambers of

the adult heart (ballooning), and cardiac cushions begin to form near the openings of the

chambers. The atria develop dorsally and expand laterally, while the ventricles expand ventrally.

Figure 2 shows a photograph from the embryonic mouse heart at gestation day 9. As the heart

tube elongates and begins to loop, the blood flows into the sinus venosus, then into the primitive

atria, the ventricles, and bulbous cordis before entering the visceral arch vessels.

When the heart tube begins to beat, specialized myocardial cells that are capable of

internal electrical activity drive the flow of blood through the tube using either peristalsis or

valveless suction pumping (see Pumping Mechanisms). In the avian embryo, the electrical

activity is initiated as early as the first week post fertilization in the form of a sinusoidal type

ECG (Moorman et al., 2004) corresponding to the morphology in the left panel in Figure 3A.

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The pumping of blood in the embryonic heart is generated by the wave of depolarization in the

myogenic cells that triggers contraction. The earliest electrical activity occurs in the pacemaker-

like cells present in the most venous (caudal) end of the heart tube (leftmost end in Figure 3).

The myocardial cells have varying degrees of intercellular coupling spatially, which allows

differential wave conduction velocities to occur across the tube. The presence of this polarity

between the caudal and cranial extremes ensures blood flows in a unidirectional manner. In

general, peristalsis requires slower conduction velocities (on the order of 1 cm/s in fish hearts)

and hence poor intercellular coupling. When the chambers begin to form, alternating contractions

of the atrium and ventricle force the blood to flow through the heart, as shown in Figure 3B. At

this stage, the ECG of the embryonic heart cycle closely resembles the adult (Moorman et al.,

2004). The cardiac cushions prevent backflow of blood at later development stages and

eventually become the valves in the adult heart. The chambers are observed to have a greater

degree of intercellular coupling and faster conduction velocities in comparison to the cardiac

cushion regions. Detailed reviews of chamber formation in relation to genetic factors both from a

developmental as well as evolutionary context can be found elsewhere (Moorman and

Christoffels, 2003a; Moorman et al., 2003b).

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

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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.

Figure 4 shows that the four-chambered structure of the heart is evident by day 10 (6

weeks for humans). The atrial chambers are connected to the ventricular chambers through the

atrio-ventricular canal. The blood flows through the atrial chambers and into the primitive left

ventricle through the atrio-ventricular canal. This canal is lined with cardiac cushions that later

remodel to form the mitral and tricuspid valves. The blood then flows through the primitive right

ventricle and out of the heart through the truncus arteriosus (highlighted in green).

By day 14 (week 8 in humans), the truncus arteriosus divides to form the pulmonary

trunk and aorta (see the right side of Figure 5). This is accomplished through the growth and

remodeling of the atrioventricular and outflow tract cushions. These cushions are formed when

the endothelial cells that line the heart migrate into the cardiac jelly and transform into

mesenchymal cells. In the atrioventricular (AV) canal, right, left, superior, and inferior cushions

grow to form four sides of the canal as seen in Figure 6. The AV canal is then separated into

right and left channels by the fusion of the superior and inferior cushions. The mitral valve which

separates left atrium and ventricle forms in the left AV canal. The tricuspid valve which

separates the right atrium and ventricle forms on the right AV canal. The conotruncal cushions in

the outflow tract also continue to grow and remodel at this stage to form the pulmonary trunk

and aorta. These cushions will eventually fuse to form the aorticopulmonary septum. This fusion

separates the flow so that the blood exits the left ventricle through the aorta and exits the right

ventricle through the pulmonary artery. At the base of the outflow tract, the cushions have a

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right-left orientation and become more dorsal and ventral to one another as ones moves along the

tract. This results in a spiraling of the aorticopulmonary septum.

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.

Prior to birth, the flow of blood in the fetus is different from the infant, primarily due to

the fact that the lungs are not in use (see Figure 7). The blood flows to the fetus through the

umbilical vein to the ductus venosus and the liver. The ductus venosus joins the inferior vena

cava and the oxygenated blood from the placenta is mixed with the deoxygenated blood from the

body. The blood then moves to the right atrium, and most of it is shunted to the left atrium

through the foramen ovale. The rest of the blood moves through the right ventricle and into the

pulmonary trunk. Most of the blood in the pulmonary trunk moves through the ductus arteriosus

to the aortic arch and bypasses the lungs. From the descending aorta, the blood moves either to

the lower parts of the body or to the umbilical arteries.

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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

The Navier-Stokes equations are typically used to describe cardiac and vascular flows

(for details, see the Appendix). By non-dimensionalizing the Navier-Stokes equations, several

important dimensionless parameters can be obtained. One such parameter is the Reynolds

number (Re):

UL Re (1)

where ρ is the density of the blood, U is the characteristic velocity (such as the peak velocity), L

is the characteristic length scale (such as the diameter of a heart chamber), and µ is the viscosity.

The Reynolds number can be thought of as the ratio of inertial forces to viscous forces acting in

the fluid. An example of inertial dominated flows where Re >> 1 is the flow through the adult

aortic valve. An example of a viscous dominated flow where Re << 1 is the flow of red blood

cells through the capillaries. In the adult heart, inertia dominates over viscosity, and the

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Reynolds number is about 1000. When the embryonic heart first forms, viscous forces dominate

and the Reynolds number is about 0.02.

Another dimensionless number is the Womersley number (Wo) (Womersley, 1955)

which is used to quantify the unsteady effects of fluid flow. It is important in pulsatile systems

characteristic of cardiovascular flows and is given by the equation:

L = Wo (2)

where ω is the angular frequency of the pulse. Within the context of a blood vessel, when the

value of Wo is high, then the velocity is highest near the middle of the vessel and drops to zero at

the vessel well. The region of slow flow near the vessel wall is known as the boundary layer

where viscous effects are important. The flow at the center of the tube is inertial and pulsatile.

For low Wo, the velocity profile over the vessel cross-section is parabolic in nature, and the flow

is quasi-steady and viscous dominated. The transient effects can be ignored when Wo is

sufficiently small, and this is common in the case of microcirculation such as in capillaries and

arterioles where the effect of the heart pulsation does not change the character of the flow (Fung,

1996).

When the vertebrate embryonic heart first forms, its diameter is about 50 µm, the peak

flow velocity is approximately 1 mm/s, and the heart beats at a frequency of about 2.3 Hz. For

convenience, assume that the embryonic blood is about the density and viscosity of the adult

blood. In this case, the blood viscosity is set to 0.03 Poise, and the density is roughly 1.025

g/cm3. The Re of this flow is roughly 0.017, and the Wo is 0.11. Note that both values are less

than 1. Table 1 provides a summary of the dimensions and flow rates recorded for vertebrate

embryonic hearts at early stages of development. Table 2 shows the parameter values for flow

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through the aorta in mouse and chicken embryos. For comparison to other transport systems,

Figure 7 shows the distribution of the Reynolds number as a function of the characteristic flow

velocity imparted by various biological pumps.

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

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 (Fung, 1996). 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 (Rychter,

1955; McGrath, 2003). 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. (1970) 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

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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 (Moorman et al., 2004; Fishman and

Chien, 1997; Gilbert, 2000). 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

(Ichikawa and Hoshino, 1967), leeches (Wenning et al., 2004), and insects (Gerould, 1929;

McCann and Sanger, 1969).

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. (2007) 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.

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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. (2006) state

that the sensitivity of the flow rate to changes in heart rate is similar to what is observed during

impedance pumping. Both the zebra fish heart and the impedance pump exhibit resonant peaks in

the frequency-flow relationship.

Impedance pumping relies on differences in the resistance to the flow path between the

two possible flow directions emanating from the fixed actuation or active pumping location. This

method of pumping is fundamentally different from peristalsis where the actuating region travels

along the tube. A general consensus on the physical mechanism of impedance pumping has been

allusive. The earliest recorded demonstration of impedance pumping was conducted by Liebau

(1954). He showed that periodic compression of a rubber tube could drive the fluid against

gravity and act as a pump without any valves. Liebau (1955, 1956) extended this work to

demonstrate valveless pumping in a closed loop and showed that periodic compression of the

flexible tube at an off-center location generated net fluid motion in one direction. Figure 8

illustrates how this mechanism is thought to work qualitatively. One section of the tube is

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actively compressed or actuated. The deformations of the tube that occur away from this active

region are the result of the propagation of passive elastic waves and reflections. A mismatch in

impedance on either side of the actuation point is necessary to produce net flow and induce wave

reflection at the boundaries. Asymmetry can be introduced from the off-centered location of the

actuating mechanism. The location of the actuation point then determines the direction of fluid

motion. Wave reflection occurs at the boundaries of the elastic tube that are attached to stiffer

sections. A pressure differential is created on each side of the region of contraction which drives

the flow in one direction.

Recent experiments have used physical models to further investigate impedance pumping

(Hickerson et al., 2005; Hickerson & Gharib, 2006; Bringley et al, 2008). These studies have

highlighted the complexity of the underlying mechanism of impedance pumping. Experiments

using an open loop system developed by Hickerson et al. (2005, 2006) 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 travelling waves

on the surface and reflections at the ends of the tube. Closed loop experiments were also

conducted by Bringley et al. (2008) 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 of that observed by Hickerson et al. (2005). They

developed a simple mathematical model to explain the frequency flow relationship that did not

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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 (1978) 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 & Peskin (2002) 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 (2003) 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. (2006) 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 travelling 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 (Bringley et al., 2006; Jung and Peskin, 2001), there has not been a careful

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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

Experimental studies have shown that the morphology of the developing heart is

important to the dynamics of fluid flow within the chambers and through the atrio-ventricular

canal. Using particle image velocimetry, Hove et al. (2003) observed the formation of vortices

within the atrium, ventricle, and bulbus 4.5 d.p.f. in wildtype zebra fish hearts (see Figure 9). In a

similar study, Foroughar et al. (2006) showed that chamber vortices were not present in the zebra

fish tubular heart 36 h.p.f. Both studies used the red blood cells as passive fluid markers to

reconstruct the flow fields. The absence of vortices in Stage 15 chick tubular hearts was also

reported by Vennemann et al. (2006). They performed in vitro flow field measurements using

liposome particles as tracers that were artificially introduced into the flow. They synchronized

their measurements to different portions of the cardiac cycle and did not report vortex formation

(see Figure 10). If and when chamber vortices form, flow reversals can occur near the cardiac

walls, changing the magnitude and direction of shear stress. Cardiac endothelial can sense

changes in both the direction and the magnitude of flow, and this signal is thought to feed into

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the biochemical pathways that activate genes required for morphological development (Jones,

2005).

A couple of recent numerical investigations have described the blood flow through

sophisticated three-dimensional models of the vertebrate embryonic heart. DeGroff et al. (2003)

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., (2007) 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. (2009) 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. 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.

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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

Recent work by Hove et al. (2003) suggests that shear stress plays an important signaling

role in heart looping, bulbus formation, and valvulogenesis in the zebrafish Danio rerio. Normal

morphology of the zebrafish heart 37 h.p.f. and 4.5 d.p.f. is shown in Figure 12. Blood flow

through the heart was occluded in vivo by inserting 50 m spheres at different locations around

the heart tube (see Figure 13). The heart developed normally in the control case, but valve and

chamber morphogenesis was disrupted when the beads occluded the flow at either the inflow or

outflow tracts. They argue that the intracardial pressures were higher when the outflow tract was

blocked and lower when the inflow tract was blocked. Since shear stress is reduced in both cases,

they propose that shear stress is an essential signal for the formation and development of the

heart valves. They go on to suggest that these flow driven forces provide a biomechanical

stimulus to the endothelial surface layer, which then feeds into the biochemical regulatory

networks that initiate morphogenesis.

Bartman et al. (2004) 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 (Herrmann et al., 1992) 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

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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 (Bruneau et al.,

2001; Biben et al., 2000; Camenisch et al., 2000). 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.

It should be noted that myocardial activity directly affects intracardial fluid dynamics by

elevating the transmural pressure. The relationship between myocardial activity and pressure can

be understood by considering the hoop stress acting on a cylindrical tube. As the myocardial

cells contract, stresses around the heart wall are generated that must be balanced by the internal

pressure of the fluid within the tube. The relationship between the hoop stress and the transmural

pressure is given by the equation:

h = pR

t (3)

where h is the hoop stress, p is the pressure, R is the radius of the tube, and t is the thickness of

the tube (see Figure 14). In the idealized case, the longitudinal stress, p can be related to the

pressure as:

p = pR

2t (4)

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

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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. (2004), shear stress can be generated in some

non-Newtonian fluids without fluid motion. These fluids have nonzero yield stress (the stress at

which the fluid begins to deform). Although the rheology of the embryonic blood is not well

known, previous work has shown that the adult blood does have nonzero yield stress (Picart et

al., 1998; Cokelet et al., 1963; Merrill et al., 1965).

Theoretical studies also support the idea that both shear stress and pressure are important

to the development of the cardiac valves. Biechler et al. (2010) 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 (submitted) 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 (Reckova et al., 2003). In experiments where

20

shear stress was reduced in vivo, Hove et al. (2003) 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. (1988) 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. Griffth et al. (2010) 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

(Henriquez, 1993) 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 (Peskin, 2002). 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

21

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) (Topper and Gimbrone, 1999). Shear stress levels as low as 0.2 dyn/cm2

can be sensed by cultured vascular endothelial cells ex vivo through mechanotransduction

(Olesen et al., 1988). 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 (Dewey et al., 1991; Davies, 1995). A few recent studies have shown that

laminar flow can alter gene expression in the embryonic heart (Groenendijk et al., 2005, 2007).

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. (2008)

were able to demonstrate that these patterns of gene expression overlapped regions of high or

low shear generated by the numerical simulations.

The exact mechanism of shear sensing in endothelial cells is not yet clearly understood

(Weinbaum et al., 2003). One proposed mechanism of shear stress sensing through the

endothelial glycocalyx (Reitsma, 2007; Jones, 2004). The endothelial glycocalyx projects from

the luminal side of endothelial cells and consists of glycoproteins, proteoglycans, and

polysaccharides (see Figure 15). This polysaccharide-rich layer was termed the glycocalyx,

meaning sweet husk, by Bennet (1963) and was visualized by Luft (1966). The proteoglycans

function as the backbone of the layer and consist of a core protein with attached

glycosaminoglycan chains. Glycoproteins are also anchored in the cell membrane and include

endothelial cell adhesion molecules that play an important role in cell signaling (Reitsma, 2007).

22

In mechanotransduction, the basic idea is that the layer deforms as blood flows over it, and the

mechanical forces are transmitted through the cytoskeleton to sites where the transduction of

mechanical force to biochemical response may occur. For the case of cytoskeletal rearrangement,

flow studies performed on endothelial cells with and without the ESL revealed that without

specific glycocalyx components, mechanotransduction and subsequent cytoskeletal

rearrangement did not occur (Thi, et al., 2004; Yao et al., 2007).

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 (Brinkman,

1947; Damino et al., 2004; Vincent et al., 2008). For example, Weinbaum et al. (2003) 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. (2008) 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

23

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 (Marshall and Nonaka, 2006;

Singla and Reiter, 2006). It has recently been discovered that primary cilia are present in both

endothelial and endocardial cells (Iomini et al., 2004; Nauli et al, 2008; Van der Heiden et al.,

2008) and during embryonic development (Van der Heiden et al., 2006). 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 (McGrath, 2003;

Yost, 2003) and the adult kidney (Nauli et al., 2003; Praetorius, 2001). 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 (Dekker et al., 2002;

Groenendijk et al., 2004).

Summary

Numerous studies that use flow manipulation in the embryonic heart indicate that fluid

shear stress and pressure act as epigenetic signals for cardiogenesis. Hoggers et al. (1997, 1995)

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. (2004) found that the dynamics of

ventricular filling changed after using the venous clip in a chick embryo. The clipped embryos

24

exhibited reduced passive filling in favor of atrial contraction to fill the ventricle at stage 24.

Hove et al. (2003) 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. (2005, 2007) provide an excellent example of how gene

expression could be connected to regions of high and low shear stress.

Challenges and Future Directions

25

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. (2003) 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. (2006) 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 (2006) 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.

(2010) used scanning micro particle image velocimetry 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

26

seeding material (Santiago et al., 1998; Tretheway and Meinhart, 2004). 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 (Peskin and McQueen, 1996; Mittal, 2005). More recently,

electrophysiological models have been coupled to models of muscle mechanics (Lee et al.,

2010). 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

27

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 (Bove et al, 2003). 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 (Cortez, 2000). 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 (Crowl and Fogelson, 2009) or treat the blood as a

homogenized non-Newtonian fluid (Shelley et al., 2008) may be appropriate.

Acknowledgements

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

28

heart used in this review. This work was funded by Miller's Burroughs Wellcome Fund Career

Award at the Scientific Interface.

Appendix

Navier-Stokes equations

In this section, the governing equations of incompressible, constant viscosity fluid flow is

presented. Consider a fluid of constant density (mass per unit volume) and dynamic viscosity

(indicative of the effects of frictional forces/mixing in a fluid), flowing through a system with

velocity components u, v, and w in x, y, and z coordinates, respectively. Let p represent the local

fluid pressure. Equation A.1 represents the conservation of mass, which requires that the mass

flux (mass per unit time) of fluid entering a system must be equal to the mass flux of fluid

leaving the system,

0 = + + z

w

y

v

x

u

(A.1)

Equations A.2-A.4 represent the conservation of linear momentum in each coordinate direction,

and this principle requires that the inertial force must be equal and opposite to the sum of the

pressure force, viscous force, and body force (due to gravity in most cases). The inertial forces

on the left hand side of the Equations A.2–A.4 arise due to the fluid flow and include both

unsteady (or transient) and convective (flow velocity and velocity gradient dependent)

contributions.

xBfz

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

u,2

2

2

2

2

2

1

=

(A.2)

29

yBfz

v

y

v

x

v

y

p

z

vw

y

vv

x

vu

t

v,2

2

2

2

2

2

1

=

(A.3)

zBfz

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

w,2

2

2

2

2

2

+ 1

=

(A.4)

Note that is the kinematic viscosity of the fluid, which is the ratio of the coefficient of dynamic

viscosity to the density of the fluid ( = /), while fB indicates the body force acting on the fluid

flow. The collective set of governing conservation equations (1)–(4) are also known as the

Navier–Stokes equations. The nonlinear mathematical nature of these partial differential

equations renders it difficult to solve, and only a few exact analytical solutions for specific

problems are known. Further details on these equations may be obtained in the book by

Schlichting and Gersten (2000), for example.

Fluid dynamic scaling

To obtain a physical perspective, it is a useful exercise to non-dimensionalize the terms in the

above governing conservation equations using equivalent scaling characteristics as shown below,

x = x

L, y =

y

L, z =

z

L (A.5)

u = u

U, v =

v

U, w =

w

U (A.6)

t = t, p = p

U 2 (A.7)

where L, U, and 1/ are characteristic flow length, velocity, and time scales respectively, As an

example, in the case of blood flow through the arteries of the adult human circulatory system, the

diameter of the vessel, the velocity along the centerline of the artery (after some distance devoid

of any entrance effects), and the pumping rate of the heart are typically chosen to be the

30

characteristic length, velocity, and time scales respectively. The application of these terms to

equations 1–4 results in the following set of equations after neglecting the body force

contribution (as its importance in cardiovascular flows is insignificant),

0 = z

w

y

v

x

u

(A.8)

2

2

2

2

2

2

= z

u

y

u

x

uUL

x

p

z

uw

y

uv

x

uu

t

u

U

L

(A.9)

2

2

2

2

2

2

= z

v

y

v

x

vUL

y

p

z

vw

y

vv

x

vu

t

v

U

L

(A.10)

2

2

2

2

2

2

= z

w

y

w

x

wUL

z

p

z

ww

y

wv

x

wu

t

w

U

L

(A.11)

In the context of vertebrate embryonic heart development, it is useful to examine the limit

of low Reynolds and Womersley numbers in the vector form of the Navier–Stokes equations

A.1–A.4 as given below:

g 1

= 2

upuu

t

u

(A.12)

As the Re is sufficiently small, the inertial terms in the momentum equations as well as the

gravitational force can be ignored, and for small values of Wo the transient term can be

neglected, resulting in the Stokes equations below:

up2 = , 0 u

(A.13)

In this limit of very low Re and Wo, the flow is entirely driven through a balance between the

pressure gradient and viscous diffusion.

Shear stress on blood vessels

31

To analyze the flow within a blood vessel, it is useful to start with a simplified model of internal

flow through a pipe. Consider the steady, incompressible, two dimensional (radial r and axial x,

see definitions in Figure A.1), incompressible, internal flow of a fluid of density and uniform

dynamic viscosity through a cylinder of radius R. The flow velocity is assumed to have no

swirling component (u = 0), and the flow is considered to be axisymmetric about the central axis

of the pipe, such that 𝜕 𝜕𝜃 = 0 .

0 = 1

rr ru

rrx

u

(A.14)

r

u

rr

u

x

u

x

p

r

uu

x

uu xxxx

r

x

x

1

1 =

2

2

2

2

(A.15)

22

2

2

2

1

1

= r

u

r

u

rr

u

x

u

r

p

r

uu

x

uu rrrrr

rr

x

(A.16)

This problem can be solved analytically by considering a few simplifying assumptions.

Typically, in the case of microcirculation, the flow reaches a fully developed state at a short

distance from its entrance (small multiple of the vessel diameter) such that there is no variation

in the flow velocity along the primary axial direction thereafter ( / x = 0). This reduces the

above equation set A.14–A.16 to the following:

0 = 1

rrurr

(A.17)

r

u

rr

u

x

p

r

uu xxx

r

1

1 =

2

2

(A.18)

22

2

1

1

= r

u

r

u

rr

u

r

p

r

uu rrrr

r

(A.19)

32

The above equations of mass and momentum are subject to the following conditions at specific

boundaries in the problem domain:

ur r = 0 = 0 (A.20)

ur r = R = 0 (A.21)

ux r = R = 0 (A.22)

0 = 0 = r

x

r

u

(A.23)

The boundary conditions A.20 and A.23 are obtained by considering symmetry about the

centerline. The boundary condition A.21 ensures that there is no normal flow through the vessel

wall, and condition A.22 means that the layer of fluid that is in contact with the vessel wall

remains at rest (“no slip” of fluid on the solid surface). From the continuity equation A.17, rru

must be constant. Applying the boundary conditions A.20 and A.21, it can be seen that there is

no radial flow throughout the vessel, i.e ur = 0 everywhere. This simplifies the r-momentum

equation A.19 to the form given below:

0 = 1

r

p

(A.24)

As the flow is incompressible, this means that the dynamic pressure is invariant in the radial

direction, and is only a function of the axial location, i.e. p = p(x). The axial momentum

conservation equation A.18 now becomes

r

u

rr

u

dx

dp xx 1 =

12

2

(A.25)

which can be written as,

r

ur

rdx

dpr x =

(A.26)

33

Integrating both sides of the above equation in terms of r, we obtain

Ar

ur

dx

dpr x = 2

2

(A.27)

where A is the constant of integration, the value of which is determined by applying A.23 to the

above equation. The resultant equation can be integrated once again in terms of r to solve for the

axial velocity profile ux(r),

r2

4

dp

dx = ux B (A.28)

The constant of integration is determined by using boundary condition A.22. The solution for the

axial velocity is thus given by

ux r = R2

4

dp

dx

1

r2

R2

= umax 1

r2

R2

(A.29)

The internal flow through a cylindrical vessel under the previously stated assumptions has a

parabolic velocity profile with the peak located along the centerline, the magnitude of which

depends on the pressure gradient at the particular axial location of interest, the dynamic viscosity

of the fluid, and the vessel radius. The pressure gradient is referred to be adverse when dp/dx > 0

resulting in a decelerating flow, and is favorable when dp/dx < 0 and the flow accelerates.

The viscous fluid flow exerts a tangential shear stress, which can be determined as the

gradient of the axial velocity as given below:

dx

dpr

R

ru

r

u x

rxxr

2 =

2 = = =

2

max

(A.30)

Of special importance in developmental physiology is the shear stress imposed by blood flow on

the walls of blood vessels, which is given by,

w = xr r = R =

2umax

R =

R

2

dp

dx (A.31)

34

The mean flow velocity through the vessel can be calculated by integrating the axial velocity

profile over the cross section,

2

1

= max2

0 02

uddrrru

Ru

R

xx

(A.32)

The shear stress can be redefined in terms of the volumetric flow rate Q based on mean flow

velocity as

w = 4u x

R =

4Q

R3 =

32Q

D3 (A.33)

Blood rheology

One way to model the non-Newtonian properties of the blood is to consider it as a

generalized Newtonian fluid, where the shear stress is a function of the shear rate at the

particular time, and the fluid dynamics do not depend upon the history of deformation. This

approach has been used previously to model the blood as a Cross fluid (Broboana et al., 2007)

and as a power law fluid (Hron et al., 2000; Yoganathan et al., 2007). In both cases, the

constitutive equations are the same as the traditional incompressible Navier–Stokes equations

with the exception that the viscosity is no longer constant and depends upon the shear stress

and/or the shear rate. For a power law fluid, the shear stress and effective viscosities are given by

the equations:

n

y

uK

= (A.34)

1

=

n

effy

uK (A.35)

35

where K is the flow consistency index, 𝜕𝑢 𝜕𝑦 is the velocity gradient perpendicular to the plane

of shear, n is the flow behavior index, and eff is the effective viscosity. Note that for the

Newtonian case n = 1. The disadvantage of this model is that it is only appropriate for shear rates

over the range for which it was fitted. Notice that the effective viscosity goes to infinity as the

shear rate approaches zero for shear thinning fluids (n < 1). A more reasonable generalized

Newtonian model might be the Cross model. In this case, the effective viscosity is a function of

the shear rate and is given by the equation:

neff 1

*

0

0

1

=

(A.36)

where 0, * and n are experimentally determined coefficients. The shear rate, , is set to the

gradient of the velocity of the fluid. In this model, the fluid behaves as a Newtonian fluid at low

shear rates (0 << *) and as a power law fluid at high shear rates (0 >> *).

For modeling purposes, eff is usually fit in the biologically relevant range of the non-

Newtonian characteristics of the blood. These models are typically used for intermediate sized

blood vessels (diameter > 22 m) where blood is treated as a homogenous fluid. If the cell

diameter is comparable to the vessel diameter (or on the same order of magnitude), this

continuum approximation is not appropriate. A Newtonian fluid approximation for blood

viscosity is acceptable typically for larger vessels where the diameter of the vessel (typically >

0.5 mm) is well above the diameter of the red blood cells (roughly 8 m). Skalak and Özkaya

(1989) present a detailed review of blood rheology, and may be referred to for further

information.

36

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Species Flow Rate

(mm/s)

Diameter

(mm)

Frequency

(Hz) Wo Re Reference

Zebrafish, 26 hpf 1 0.05 2.3 0.111 0.017 Forouhar et al., 2006

Zebrafish, 4.5 dpf 10 0.1 2 0.207 0.342 Hove et al., 2003

Chicken, HH15 26 0.2 2 0.414 1.777

Vennemann et al.,

2006

Mouse, 8.5 dpc 3 0.075 2.8 0.184 0.077 Jones et al., 2004

Mouse, 9.5 dpc 4 0.125 2.1 0.265 0.171 Jones et al., 2004

Mouse, 10.5 dpc 4 0.15 2.4 0.340 0.205 Jones et al., 2004

Table 1: Reynolds and Womersley numbers for embryonic hearts at several stages of

development given in hours post fertilization (hpf), days post fertilization (dpf), days post

conception (dpc), and Hamburger & Hamilton (1951) stages (HH). Peak flow rates and

maximum diameters of the heart were used in the calculation. It was assumed that the dynamic

viscosity of the blood was 0.003 N s/m2, and the density of the blood was 1025 kg/m

3. Note that

the calculation of the dimensionless numbers is sensitive to the choice of the characteristic

length, velocity, viscosity, and density. These calculations may be different than those reported

in the references.

Species Flow Rate

(mm/s)

Diameter

(mm)

Frequency

(Hz) Wo Re Reference

Chicken, Stage 18 170 0.083 2 0.172 4.82 Wang et al., 2009

Chicken, Stage 24 250 0.14 2 0.290 11.96 Wang et al., 2009

Mouse, 11.5 dpc 127 0.33 3.78 0.940 14.32 Phoon et al., 2002

Mouse, 12.5 dpc 158 0.36 4.07 1.064 19.43 Phoon et al., 2002

Mouse, 13.5 dpc 173 0.35 4.40 1.076 20.69 Phoon et al., 2002

Mouse, 14.5 dpc 226 0.34 4.35 1.039 26.25 Phoon et al., 2002

Table 2: Reynolds and Womersley numbers for the aorta at several stages of development given

in days post conception (dpc) and Hamburger & Hamilton (1951) stages (HH). Peak flow rates

and the average diameter of the aorta were used in the calculations. It was assumed that the

dynamic viscosity of the blood was 0.003 N s/m2, and the density of the blood was 1025 kg/m

3.

Note that the calculations are sensitive to the choice of the characteristic length, velocity,

viscosity, and density.

55

FIGURES

Figure 1: Time sequence of heart looping in the mouse embryonic heart. The frontal view of the

heart is shown for 8 to 10 days post conception (d. p. c.). The tube takes on a three-dimensional

structure through a process known as cardiac looping. The heart twists and bends with right-ward

looping to reorient from anterior/posterior polarity to left/right polarity. The expanded ventricle

is clearly seen in image E. SEM images courtesy of Dr. Kathleen K. Sulik.

56

Figure 2: Frontal view of the embryonic mouse heart during day 9 (approximate human age is 25

days) and cartoon schematic. As the heart tube elongates and begins to loop, the blood flows into

the sinus venosus (green), then into the primitive atria(blue), the ventricles (purple) and bulbous

cordis (yellow) before entering the visceral arch vessels. SEM images courtesy of Dr. Kathleen

K. Sulik. Diagram redrawn from Sadler (1995).

57

Figure 3: Flow in the vertebrate embryonic heart, drawn after Moorman et al. (2004). The left

panel shows the initial form of the heart tube, with the flow being driven by peristaltic

contractions dictated by the electrophysiological activity in the myocardium. Chamber formation

is initiated at the next developmental stage as shown in the right panel, where a and v indicate

the locations of the atria and ventricle respectively, and alternating contractions of these regions

force the blood to flow within the tube. The gray regions in the right panel denote the cardiac

cushions that later develop into the valves required for maintaining unidirectional flow. At each

stage of development, the fluid flow and cardiac activity are presented with increasing time,

looking from top to bottom. Arrows inside the heart tube are used to denote the blood flow

direction. External arrows show the direction of chamber contraction.

58

Figure 4: Schematic diagrams and SEM images of the mouse embryonic heart 10 (A, C) and 14

(B, D) days post conception. At 10 days, the blood flows through the atrial chambers and into the

primitive left ventricle through the atrio-ventricular canal. The blood then flows through the

primitive right ventricle and out of the heart through the truncus arteriosus (highlighted in green).

At 14 days, the fusion of the endocardial cushions that line the outflow tract results in separation

of the blood flow. The blood moves from the left ventricle to the aorta and moves from the right

ventricle to the pulmonary artery. SEM images courtesy of Dr. Kathleen K. Sulik. Diagrams

redrawn from Sadler (1995).

59

Figure 5: Schematic diagrams and SEM images of the mouse embryonic heart 14 days post

conception. (A, B) show the superior and inferior cushions in the atrioventricular canal (yellow)

fuse. This separates the canal into right and left channels. (C, D) show the conotruncal cushions

(purple) which spiral from a left-right to a dorsal-ventral orientation as one moves through the

outflow tract. These cushions fuse to form the aorticopulmonary septum. This divides the

outflow tract into the aortic and pulmonary trunks. SEM images courtesy of Dr. Kathleen K.

Sulik. Diagrams redrawn from Sadler (1995).

60

Figure 6: Schematic diagram of the fetal circulation prior to birth redrawn from Sadler (1995).

(A) and the circulation in the infant after birth (B). In the fetal circulation, blood bypasses the

liver via the ductus venosus. The circulation of blood through the collapsed lungs is reduced by

the foramen ovale (connecting the left and right atria) and ductus arteriorsus (connecting the

pulmonary trunk to the descending aorta). At birth, the lungs are filled and the resistance to

blood flow through the lungs is drastically reduced. The ductus arteriosus closes, and blood in

the pulmonary trunk is no longer shunted to the aorta. The pressure in the left atrium increases,

and this closes the foramen ovale.

61

Figure 7: Graph showing the Reynolds number (Re) verses the diameter for a variety of internal

flows in biology. The Re is calculated using the kinematic viscosity of the fluid and the average

flow rate and diameter of the vessel or chamber.

62

Figure 8: Diagram of the proposed valveless suction pumping mechanism for the vertebrate

embryonic heart redrawn from Forouhar et al. (2006). (A) The inflow tract (ift) and outflow tract

(oft) are stiff compared to the elastic heart tube walls. (B) The initial contraction occurs near the

ift. (C) Bidirectional waves travel away from the site of the initial contraction. (D) The upstream

wave hits the stiff ift and reflects downstream. (E) The reflected wave travels downstream with

the second wave producing net flow in that direction. (F) The wave is reflected at the oft and the

contraction cycle repeats.

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Figure 9: Confocal sections of BODIPY-ceramide stained zebra fish embryos 4.5 d.p.f. taken

from Hove et al. (2003). (A) Atrial systole and ventricular filling. (B) Atrial diastole and

ventricular systole. (C, D) Overlay of digital particle image velocimetry (DPIV) velocity field.

The magnitude and direction of flow is denoted by the arrows. Red denotes the greatest velocity

magnitudes and blue denotes the lowest. (E, F) Contour map of the vorticity field. Vortices

appear behind the AV constriction during ventricular filling, and a vortex pair forms in the

bulbus during ventricular systole.

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Figure 10: Velocity distributions in the Stage 15 chicken embryonic heart obtained through

particle image velocimetry (from Venn, 2006), along with scanning electron micrographs of the

heart (from Männer, 2000). Flow through the developing ventricle is shown on the left and

through the atrium on the right. The relative magnitude and direction of flow is given by the

length and direction of the arrows, and the lumen boundary is denoted with a dashed line.

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Figure 11: The formation of a vortex in a simple physical model of a cardiac chamber at

Reynolds number 24. Cushions are placed upstream of the chamber. (A) The flow over the

cushion with a height of 0.12 of the channel diameter does not separate and now flow reversals

are observed. (B) The fluid flows over the cushion with a height of 0.36 of the channel depth,

and a vortex forms in the chamber. This results in a change in the magnitude and direction of

shear at the chamber wall.

A B

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Figure 12: Diagram of the zebrafish embryonic heart about 37 d.p.f. (A) and about 4.5 d.p.f (B)

redrawn from Hove et al. (2003). The labels on the heart tube represent the regions that will

become the primitive atrium (PA), the primitive ventricle (PV), and the bulbus arteriosus (BA).

The later stage shows the the embryonic heart after looping and chamber formation with atrium

(A), ventricle (V), bulbus (B), atrio-ventricular valve (avv) and ventriculo-bulbar valve (vbv).

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Figure 13: Diagram of the flow perturbation experiments performed by Hove et al. (2003). (A) A

50 m bead was inserted close to the sinus venosus 37 h.p.f. without blocking the flow. (B) To

block the inflow, beads were inserted in front of the sinus. (C) To block outflow, beads were

inserted behind the outflow tract. For the control case (A), valve and chamber formation occur

normally. For cases (B) and (C), valve and chamber formation did not occur, and the primitive

peristaltic-like contractions persist. This implies that the adult waveform as would be seen in an

ECG does not develop.

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Figure 14: Simplified diagram of a cross section of a cylindrical heart tube. When myocardial

cells contract, hoop stresses, , are generated around the heart wall and are balanced by internal

pressures, p. The pressure jump across the heart wall is equivalent to the normal force acting on

the wall. R gives the radius of the heart tube.

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Figure 15: Simplified diagram of the endothelial surface layer. Proteoglycans with

glycosaminoglycan side chains (GAG chains) are anchored in the endothelial cell membrane.

Glycoproteins are also anchored in the cell membrane and have short branched carbohydrate side

chains. Plasma and endothelium-derived soluble components, including hyaluronic acid and

proteoglycans, make up the top of the layer.

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Figure A.1: Internal flow of a fluid through a circular cylinder of radius R (also known as Hagen

– Poiseuille flow). Under the simplifications of two-dimensional, time-invariant, axisymmetric

(no fluid rotation, u = 0), incompressible flow of a fluid with uniform density and viscosity, the

axial velocity ux remains unchanged along the longitudinal direction x after a certain length

(typically 2–4 multiples of the radius) from the entrance, and this condition is also known as a

fully developed flow. However, the radial variation of the axial velocity ux(r) is parabolic about

the centerline axis as shown. The shear stress imposed by the flow in the radial direction varies

linearly from the centerline, and the maximum value w occurs at the walls. Note that indicates

the thickness of the boundary layer, which is the region near the solid boundaries where the

deceleration of the flow on account of fluid viscosity is non-negligible. The coordinate system

used for the analysis of this problem (r–radial, x–axial, –rotational) is also shown.