Cell Biochemistry and Biophysics

, Volume 61, Issue 1, pp 33–45

Fluid Dynamics of Ventricular Filling in the Embryonic Heart

Authors

    • Department of MathematicsUniversity of North Carolina Chapel Hill
Original Paper

DOI: 10.1007/s12013-011-9157-9

Cite this article as:
Miller, L.A. Cell Biochem Biophys (2011) 61: 33. doi:10.1007/s12013-011-9157-9

Abstract

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

Keywords

BiofluidsHeart developmentShear stress

Introduction

The morphology, muscle mechanics, fluid dynamics, conduction properties, and molecular biology of the vertebrate embryonic heart have received much attention in recent years because of the importance of both fluid and elastic forces in shaping the heart [15] as well as the striking relationship between the heart’s evolution and development [6, 7]. It has been suggested that the early embryonic heart beat does not pump blood for the purpose of convective transport, but rather it aids in the shaping and maturation of the developing heart [1].Very few studies, however, have investigated the coupling between each of these components, although many researchers have noted the importance of such interactions [2, 4, 8].

One possible connection between cardiac fluid mechanics and cell biochemistry and biophysics is fluid shear sensing through mechanotransduction. More specifically, fluid shear stress is thought to trigger biochemical cascades within the endothelial cells of the developing heart that regulate chamber and valve morphogenesis [2, 9, 10]. Fluid shear is also known to increase transcription rates of several genes that are thought to also be involved in the regulation of cardiac morphogenesis, such as vascular endothelial growth factor [11, 12] and endothelial transforming growth factor [13]. One study has found that fluid shear might also increase the conduction velocities of action potentials in the myocardial layer of the developing heart [14]. Such changes in conduction properties and heart morphology will, in turn, alter the intracardiac fluid dynamics and shear stresses.

Transmural pressure is another hemodynamic signal that is thought to affect cardiac and vascular structure. In general, an increase in transmural pressure corresponds to an increase in circumferential stress in the cardiac and vessel walls. In the vascular system, larger pressures can lead to increased vessel wall thickness as an adaptation to reduce circumferential wall stress [1517]. When myocardial cells contract, circumferential stresses are generated in the heart, a force normal to the cardiac wall is applied to the fluid, and the intracardiac pressure increases. The normal force applied to the fluid per unit area is approximately equal to the pressure jump across the heart wall. The magnitude of the circumferential stress and transmural pressure depends on the intracardial fluid dynamics and the heart morphology. Few studies, however, have considered the role of transmural pressure or the forces acting normal to the cardiac wall in signaling cardiogenesis.

Recently, it has been suggested that either shear stress or myocardial activity could be a critical epigenetic factor in the formation of the endocardial cushions in the atrioventricular canal. Hove et al. [10] found that when flow was occluded at either the inflow or outflow tracts of the zebrafish heart, the valves and chambers did not develop correctly. As a result of the occlusion, the magnitude of the shear stress acting on the cardiac walls was reduced. However, pressure was presumably increased when the flow was occluded at the outflow tract and decreased when occluded at the inflow tract. They then argued that shear stress was the main epigenetic factor responsible for valve morphogenesis because the resulting heart in both cases had similar defects. In a later study, Bartman et al. [18] explored the role of myocardial function in the formation of the endocardial cushions. They administered various concentrations of an inhibitor of myofibril function that did not directly inhibit blood flow. 2,3-Butanedione monoxime (2,3-BDM) blocks myofibrillar ATPase and decreases myocardial force in a dose-dependent manner [19]. For concentrations of 2,3-BDM of 6 mM or higher, blood flow ceased in all embryos, yet 58% of them still formed endocardial rings 48 h post fertilization. Because endocardial cushions formed in the absence of flow (and hence shear), they concluded that myocardial function was the major factor in endocardial cushion formation. A detailed review of both studies can be found in Mironov et al. [20].

It is difficult, however, to untangle the role of shear and myocardial activity in the formation of the endocardial cushions. Myocardial contractions generate circumferential stress in the heart chamber walls resulting in the contraction of the chamber. As a result of being incompressible and very viscous at this scale, the blood resists this motion and applies a force on the boundary that is normal to the chamber surface, and the heart wall applies an equal and opposite force on the fluid. Once the fluid is in motion, forces tangential to the heart’s surface are generated. If increases in shear stress increase cardiac conduction properties, then altering shear stress would also alter myocardial activity. This would then affect the velocity of the contraction wave, circumferential stresses, and transmural pressures. Finally, the embryonic blood is thought to be shear thinning and viscoelastic [21, 22], although the significance of the non-Newtonian effects has been debated, particularly for adult blood [23]. If non-Newtonian effects are significant in the embryonic heart, myocardial activity could potentially generate shear stresses in the absence of obvious fluid flow. A detailed discussion of non-Newtonian fluid dynamics and its implications for heart development is beyond the scope of this study, but there are a number of general resources available on the subject [24, 25].

In this study, a simple two-dimensional mathematical model was constructed that approximates the embryonic heart about 4.5 days post fertilization in the zebrafish. Prior to this stage of development, the heart undergoes a transition from a valveless tube that pumps blood via sinusoidal contractions to a valve and chamber pump [26]. During this transition, the heart tube loops to form a three-dimensional S-shape, and high rates of shear are generated in the chambers and atrioventricular canal [6, 10]. Finally, the electrocardiogram changes from a sinusoidal shape to an adult-like waveform as a result of the development of spatial variations in conduction velocities. This results in the distinct contraction of the atrium followed by the contraction of the ventricle. The simplified model of this stage in development was inspired by a schematic diagram that appears in Moorman and Christoffels [6]. The equations of fluid motion with an immersed elastic boundary representing the heart wall were solved using the immersed boundary method [27].

Since myocardial activity generates forces acting on the fluid normal to the chamber walls increasing the internal pressure, the aim of this study is to describe the spatial and temporal distribution of this force. Another goal of the study is to determine how normal force depends on the elastic properties of the heart and the size of the endocardial cushions. Finally, force distributions will be compared when the flow is driven by atrial contraction (modeling the moving boundary problem) and when the flow is simply pushed through a rigid tube with bulging chambers.

Methods

The immersed boundary method has been used successfully to model a variety of problems in biological fluid dynamics that typically involve the interactions between incompressible viscous fluids and deformable elastic boundaries. Some examples of problems that have been studied using the immersed boundary method include cardiac blood flow [28, 29], valveless pumping [30], and the formation of blood clots by platelet aggregation [31].

In this article, two-dimensional numerical simulations were constructed to be roughly similar in scale to the embryonic zebrafish heart 1.5–4.5 days post fertilization, as described in Hove et al. [10] (see Fig. 1). The simplified numerical model of the heart consisted of a straight channel with two staggered chambers shaped like hemiellipse bulging from each side of the channel. Endocardial cushions were placed on the dorsal and ventral sides of the inflow tract, the atrioventrical canal, and the outflow tract. The outer dimensions of the computational domain were set to be 1 × 1 mm. The diameter of the channel was set to 50 μm, and the radius of each chamber was set to 100 μm. The length of the atrioventricular canal separating the two chambers was 20 μm. The dynamic viscosity of the fluid was set to 0.0027 Pa·s, and the density was set to 1,025 kg/m3 which are equal to average values for adult blood. The blood flow was pumped through the model heart in two ways: (1) the flow was driven by applying an external force upstream of two rigid chambers (Fig. 2a), and (2) the flow was driven by the contraction of the atrium, causing the ventricle to be elastically stretched (Fig. 2b). In the first case, the maximum velocity upstream of the chambers was set to 1.05 mm s−1 [10]. In the second case, the atrium fully contracted over the period of 0.3 s.
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Fig. 1

The zebrafish embryonic heart at 4.5 days post fertilization from Hove et al. [10]. a The ventral view of a 4.5-days post fertilization zebrafish embryo with heart contained in the box. b High magnification of the 4.5-days post fertilization heart with the chamber boundaries outlined

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Fig. 2

The simplified mathematical models of the embryonic heart. a Sketch of immersed boundary simulations of flow forced through a channel with rigid walls. A parabolic velocity profile is prescribed upstream of the chamber by applying a force to the fluid that is proportional to the difference between the actual fluid velocity and the desired fluid velocity (d the diameter of the channel, c the depth of the chambers, and v the valve height). b Sketch of immersed boundary simulations of flow driven by the contraction of the atrium (a) to fill the ventricle (v). The dashed line represents the flexible region of the ventricle that is allowed to expand. The curved part of the atrium contracts with a prescribed motion. The solid line represents regions that are “nearly” rigid

Numerical Method

A “tethered boundary” version of the immersed boundary method was used to calculate the flow in the heart chambers. The particular numerical scheme used in this paper has been described in detail by Peskin and McQueen [29], and was tested against experimental data for the case of a flapping plate [32, 33] and flow through a physical model of the simplified heart [34].

The equations of motion for the fluid are given by the Navier–Stokes equations in Eulerian form:
$$ \rho \left( {{\frac{\partial {\mathbf{u}}}{\partial t}} + \mathbf{u}\left( {{\mathbf{x}},t} \right) \cdot \nabla {\mathbf{u}}\left( {{\mathbf{x}},t} \right)} \right) = - \nabla {\text{p}}\left( {{\mathbf{x}},t} \right) + \mu \Updelta {\mathbf{u}}\left( {x,t} \right) + {\mathbf{f}}\left( {{\mathbf{x}},t} \right) $$
(1)
$$ \nabla \cdot {\mathbf{u}}\left( {{\mathbf{x}},t} \right) = 0 $$
(2)
where u(x, t) is the fluid velocity, p(x, t) is the pressure, \( {\mathbf{f}}\left( {{\mathbf{x}},t} \right) \) is the force per unit volume applied to the fluid by the immersed boundary, ρ is the density of the fluid, and μ is the dynamic viscosity of the fluid. The independent variables are the time t and the position x. Note that bold letters represent vector quantities. Equation 2 is the condition that the fluid is incompressible. We use three-dimensional terminology in this description even though our model is two-dimensional. This is consistent if we regard the model as a cross-section of a three-dimensional apparatus that looks the same in every cross-section parallel to the x, y-plane.

In the rigid heart model simulations, parabolic flow within the channel was prescribed upstream of the chambers by applying an external force, fext, to the fluid proportional to the difference between the desired fluid velocity and the actual fluid velocity [34]. This force was applied to a 10-µm strip of the fluid that was 100 µm upstream of the atrium. The difference between the actual and desired velocity was controlled by a “stiffness” parameter, kext, such that the difference between the two was less than 0.1%.

In the simulations where the flow was driven by the contraction of the atrium, this force was set to zero. The interaction and forcing equations between the fluid and the boundary were then given by the following:
$$ {\mathbf{f}}\left( {{\mathbf{x}},t} \right)\,=\,\int \mathop{\mathbf{F}}\left( {r,t} \right)\delta \left( {{\mathbf{x}} -{\mathbf{X}}\left( {r,t} \right)} \right)dr +{\mathbf{f}}_{\text{ext}} \left( {{\mathbf{x}},t} \right) $$
(3)
$$ {\frac{{\partial {\mathbf{X}}\left( {r,t} \right)}}{\partial t}} = {\mathbf{U}}\left( {{\mathbf{X}}\left( {r,t} \right)} \right) = \mathop \int {\mathbf{u}}\left( {{\mathbf{x}},t} \right)\delta \left( {{\mathbf{x}} - {\mathbf{X}}\left( {r,t} \right)} \right)d{\mathbf{x}} $$
(4)
where F(r, t) is the force per unit area applied by the boundary to the fluid as a function of Lagrangian position and time, fext is the external force applied to drive the flow, δ(x) is a two-dimensional delta function, X(r, t) gives the Cartesian coordinates at time t of the material point labeled by the Lagrangian parameter r. Equation 3 applies force from the boundary to the fluid grid and adds an external force term, and Eq. 4 evaluates the local fluid velocity at the boundary. The boundary is then moved at the local fluid velocity, and this enforces the no-slip condition. Each of these equations involves a two-dimensional Dirac delta function δ, which acts in each case as the kernel of an integral transformation. These equations convert Lagrangian variables to Eulerian variables and vice versa.
To construct the boundary, the channel and chamber walls were made nearly rigid by tethering each boundary point to a target point that does not move and does not interact with the fluid. The target and actual immersed boundary points were then attached with virtual springs, and these spring attachments applied a force to the actual boundary that was proportional to the distance between corresponding points of the two boundaries. This force was then used to calculate the force applied to the fluid. The force equations were given by the following:
$$ {\mathbf{F}}_{\text{targ}} \left( {r,t} \right) = k_{\text{targ}} \left( {{\mathbf{Y}}\left( {r,t} \right) - X\left( {r,t} \right)} \right) $$
(5)
$$ {\mathbf{F}}_{\text{str}} \left( {r,t} \right) = k_{\text{str}} {\frac{\partial }{\partial r}}\left\{ {\left( {\left| {{\frac{{\partial {\mathbf{X}}}}{\partial r}} - 1} \right|} \right){\frac{{\partial {\mathbf{X}}/\partial r}}{{\left| {\partial {\mathbf{X}}/\partial r} \right|}}}} \right\} $$
(6)
$$ {\mathbf{f}}_{\text{ext}} \left( {x,t} \right) = - k_{\text{ext}} \left( {{\mathbf{u}}\left( {{\mathbf{x}},t} \right) - {\mathbf{u}}_{\text{targ}} \left( {{\mathbf{x}},t} \right)} \right) $$
(7)
$$ {\mathbf{u}}_{\text{targ}} \left( {{\mathbf{x}},t} \right) = \left[ {\begin{array}{*{20}c} {U_{ \max } \left\{ {1 - \left( {{\frac{0.5\,l - x}{d/2}}} \right)^{2} } \right\}} \\ 0 \\ \end{array} } \right] $$
(8)
$$ {\mathbf{f}}\left( {{\mathbf{x}},t} \right) = {\mathbf{f}}_{\text{ext}} + {\mathbf{f}}_{\text{targ}} + {\mathbf{f}}_{\text{str}}. $$
(9)

Equation 5 describes the force applied to the fluid by the boundary in Lagrangian coordinates. Ftarg is the force per unit area as a result of the target boundary, ktarg is a stiffness coefficient, and Y(r, t) is the prescribed position of the target boundary. Equation 6 describes the force applied to the fluid as a result of the resistance to stretching by boundary given as Fstr, where kstr is the corresponding stiffness coefficient. Equation 7 describes the force applied to the fluid in Eulerian coordinates as a result of the external force necessary to drive the channel flow. Equation 8 describes utarg, the desired parabolic flow profile upstream of the chambers, where d is the channel diameter, l is the domain length, and Umax is the desired maximum velocity in the channel. The total force applied to the fluid is then given as the sum of each of the forces in Eq. 9. Note that ftarg and fstr represent the target and stretching forces after they are converted to the Eulerian coordinate system.

The stiffness coefficient of the virtual springs attaching tether points to boundary points was chosen to reduce the deformation of the boundary to acceptable levels. Spatial and time step sizes were chosen to provide stability and convergence. The computational domain for the fluid was set to 600 × 600 spatial steps for all simulations. The boundary was constructed such that the distance between each boundary point was approximately equal to dx/2, where dx is the spatial step.

In the case of ventricular filling, the atrium was made to contract by moving the position of the target points to which the actual boundary points are tethered. The elastic region of the ventricle was not tethered and was free to expand. The position of the atrial target points as a function of time is given by the following:
$$ {\mathbf{s}}_{\text{targ}} \left( {r,t} \right) = \left[ {\begin{array}{*{20}c} {a_{1} - c\cos \left( {{\frac{\pi r}{{2c\tilde{r}}}}} \right)} \\ {a_{2} - c\left( {1 - {\frac{t}{{t_{\text{beat}} }}}} \right)\sin \left( {{\frac{\pi r}{{2c\tilde{r}}}}} \right)} \\ \end{array} } \right] $$
(10)
where the center of the half-circle that forms the atrium is located at the point (a1, a2), c is the chamber radius, \( \tilde{r} \) gives the number of boundary points along the moving part of the atrium, and tbeat is the time it takes the atrium to contract.

Relating Shear Stress and Pressure to the Force Acting on the Wall

As stated previously, the normal force per unit area applied to the fluid is approximately equal to the pressure jump across the heart wall, and the tangential force per unit area is approximately equal to the tangential component of the intracardiac shear stress. This can be stated mathematically as follows:
$$ {\mathbf{F}} \cdot {\mathbf{n}} = \left[ p \right]\left| {\partial {\mathbf{X}}/\partial s} \right| $$
(11)
$$ {\mathbf{F}} \cdot \tau = - \mu \tau \cdot \left[ {{\frac{{\partial {\mathbf{u}}}}{\partial n}}} \right]\left| {\partial {\mathbf{X}}/\partial s} \right| $$
(12)
Equation 11 gives the normal component of the force per unit area acting on the fluid, where n is the unit vector normal to the boundary and [p] is the jump in the pressure across the boundary. Equation 12 gives the tangential component of the force per unit area acting on the fluid, where τ is the unit vector tangential to the boundary, \( \partial /\partial n \) is the normal derivative, and \( \left[ {\partial {\mathbf{u}}/\partial n} \right] \) is the jump in the normal derivative of the velocity across the boundary. For small strains, \( \left| {\partial {\mathbf{X}}/\partial s} \right| \approx 1 \). A derivation of Eqs. 11 and 12 can be found in Peskin and Printz [35].

Results

For flows externally driven through rigid chambers, three chamber depths and two valve heights are considered. For flows driven by atrial contraction, four valve heights and four ventricular wall stiffnesses are considered. In all simulations, kstr = ktarg, unless otherwise noted. The streamline plots, spatial and temporal distributions of normal forces acting on the boundary, and contour plots of the spatial distributions of shear stresses are reported for representative cases.

Flows Externally Driven Through Rigid Chambers

The simple case of flow driven externally through a rigid channel with bulging chambers was considered first. The depth of the chambers was set at 25, 50, and 100 µm, and the height of the endocardial cushions was set to 0 and 10 μm. The variations in the pattern of flow as the endocardial cushions and chambers grow may be seen by studying Fig. 3. Figure 3 contains graphs of the streamlines of the fluid flow through the chambers at steady state. The streamlines are curves that have the same direction as the instantaneous fluid velocity u(x, t) at each point. They were drawn by making a contour map of the stream function because the stream function is constant along streamlines. The stream function Ψ(x, t) in 2-D is defined by the following equations:
$$ u\left( {{\mathbf{x}},t} \right) = {\frac{{\partial {{\Uppsi}}\left( {{\mathbf{x}},t} \right)}}{\partial y}}, v\left( {{\mathbf{x}},t} \right) = - {\frac{{\partial {{\Uppsi}}\left( {{\mathbf{x}},t} \right)}}{\partial x}} $$
(13)
where u(x, t) and v(x, t) are components of the fluid velocity \( {\mathbf{u}}\left( {{\mathbf{x}},t} \right) = \left[ {u\left( {{\mathbf{x}},t} \right), v({\mathbf{x}},t)} \right] \). Because the density of the streamlines is proportional to the speed of the flow, one can see that as the chamber depth increase, the flow speed within the chambers decreases (see Fig. 3a–c). Similarly, as the height of the cushions increases, the velocity of the fluid between the cushions increases (see Fig. 3c and d).
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Fig. 3

Streamline plots of flow driven through rigid chamber walls from left to right. The density of streamlines is proportional to the velocity of the fluid. a Fluid through the heart tube as the chambers begin to form with a chamber depth of 25 µ. b Flow through chambers that are 50 µ in depth. c Flow through chambers that are 100 µ in depth. Note that the flow velocities within the chambers decrease as the chamber depth increases. d Flow through 100 µ chambers and endocardial cushions that are 10 µ high

Normal force and shear stress distributions for a simulation of blood flow through a heart tube with a chamber depth of 100 μm and an endocardial cushion height of 10 μm are given in Figs. 4 and 5, respectively. The normal force per unit area reaches a maximum absolute value of about 0.02 dynes/cm2. The normal force acting on the walls, and hence the internal cardiac pressure, drops almost in steps moving from chamber to chamber downstream. This normal force per unit area is rather small in comparison to shear stress, as seen in Fig. 5. The fluid shear stress reaches values of nearly 2.0 dynes/cm2 between the endocardial cushions. Shear stress drops significantly within the cardiac chambers.
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Fig. 4

The spatial distribution of the force per unit area normal to the heart walls in a rigid tube. The chamber depth is equal to 100 μm, and the cushion height is equal to 10 μm. The magnitude of the force decreases almost in discrete steps as one moves downstream from section to section

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Fig. 5

Absolute value of fluid shear stress within the rigid heart tube. The bar on the right gives the scale of the stress in dynes/cm2. The chamber depth is equal to 100 μm, and the cushion height is equal to 10 μm. The shear stress is greatest between the endocardial cushions

Contraction of Atrium to Fill Ventricle

Streamline plots of blood flow driven through the atrioventricular canal and into the ventricle by the contraction of the atrium during one contraction cycle are shown for a representative case in Fig. 6. In this case, the chamber depth was set to 100 μm and the valve height was set to 0 μm. The motion of the atrial wall is driven by the motion of the target points to which the boundary is tethered. The streamlines are shown at four times during the contraction, t = T/4, T/2, 3T/4, and T, where T = tbeat. The dorsal surface of the ventricle is elastic and expands under the force exerted by the fluid. Streamlines appear external to the heart tube because the region outside of the heart tube is also a fluid. As the chambers expand and contract, the external fluid must also move with the boundary as it is either pushed out of the way or made to fill in the space that was once occupied by the chamber. Note the striking difference between the direction of flows that are externally driven through rigid chambers (Fig. 3) and flows driven by atrial contraction (Fig. 6).
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Fig. 6

Streamline plots of flow driven through the atrioventricular canal and into the ventricle by the contraction of the atrium during one contraction cycle at at = tbeat/4, bt = 2, ct = 3tbeat/4, and dt = tbeat. The density of streamlines is proportional to the velocity of the fluid. The motion of the atrial wall is driven by the motion of the target points to which the boundary is tethered. The dorsal surface of the ventricle is elastic and expands under the force exerted by the fluid. Note the difference in the direction of flow for externally driven flows through rigid chambers (Fig. 3) to the fluid motion driven by atrial contraction

The spatial distribution of the normal force per unit area during atrial contraction and ventricular filling is shown in Fig. 7. In this particular case, the chamber depth is equal to 100 μm, and the cushion height is equal to 15 μm. The left graph of Fig. 7 shows the normal force on the wall opposite the atrium and containing the ventricle. The right graph of Fig. 7 shows the wall on the same side as the atrium. Large forces are applied to the fluid as the atrium contracts that are over an order of magnitude higher than those calculated in the case of flow through a rigid heart model. The magnitude of this force increases over time because inertia in this system is negligible and the elastic ventricular wall resists stretching.
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Fig. 7

The spatial distribution of the force per unit area normal to the heart walls during ventricular filling. The chamber depth is equal to 100 μm, and the cushion height is equal to 15 μm. Force is plotted at times, t = T/4, T/2, 3T/4, and T, where T = tbeat. The left graph shows the normal force on the wall opposite the atrium. The right graph shows the wall on the same side as the atrium. Large forces are applied to the fluid as the atrium contracts. The magnitude of this force increases over time because inertia in this system is negligible, and the elastic ventricular wall resists stretching

The distribution of the shear stress over time is depicted in the plots shown in Fig. 8. The chamber depth is set to 100 μm, and the cushion height is set to 10 μm. The plots are shown at four times during the contraction, t = T/4, T/2, 3T/4, and T. The peak magnitude of the shear stress within the atrioventricular canal is 13.6 dynes/cm2. The shear through the canal is saturated in the plot so that chamber shear stresses can be viewed. The maximum shear stress was also calculated for three other cushion heights, v = 5, 10, 15, and 20 μm. The results are plotted in Fig. 9. When the endocardial cushions occupy more than 50% of the atrioventricular canal, peak shear stress increases dramatically. These values are consistent with those reported by Hove et al. [10.] who measured shear stresses ranging from 2.5 dynes/cm2 in the 37-h post fertilization heart to 76 dynes/cm2 in the 4.5 days post fertilization heart.
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Fig. 8

Magnitude of fluid shear stress over time during atrial contraction in dynes/cm2. The chamber depth is 100 μm and the cushion height is 10 μm. The plots are shown at four times during the contraction, t = T/4, T/2, 3T/4, and T. The peak magnitude of the shear stress within the atrioventricular canal is 13.6 dynes/cm2. The shear through the canal is saturated in the plot so that chamber shear stresses can be viewed

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Fig. 9

Magnitude of fluid shear stress at 20% of the contraction time in dynes/cm2. The chamber depth is set to 100 μm, and the cushion heights are set to a 0 μm, b 5 μm, c 10 μm, d 15 μm, and e 20 μm. These cushion heights correspond to 0, 20, 40, 60, and 80% blockage of the atrioventricular canal. When the endocardial cushions occupy more than 50% of the atrioventricular canal, peak shear stress increases dramatically

The forces acting on the chamber walls also increase significantly as the cushion height increases because of the resistance to flow through the small atrioventricular canal. Spatial distributions of the force per unit area normal to the heart walls for several endocardial cushion heights are shown in Fig. 10. The chamber depth is equal to 100 μm, and the cushion heights were set to 0, 5, 10, and 15 μm. The maximum normal force acting on the heart wall increases more than fivefold as the cushion height increases from 0 to 15 μm, corresponding to 0–60% blockage of the atrioventricular canal. The maximum force acting normal to the boundary in the center of the atrium with a cushion height of 20 μm (not shown) reached 3.5 dynes/cm2, further illustrating the sharp increase in the force required to contract the atrium for large cushion heights.
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Fig. 10

The spatial distribution of the force per unit area normal to the heart walls for several values of endocardial cushion height, v. The chamber depth is equal to 100 μm, and the cushion heights were set to 0, 5, 10, and 15 μm. The left graph shows the normal force on the wall opposite the atrium. The right graph shows the wall on the same side as the atrium. The maximum normal force acting on the heart wall increases more than fivefold as the cushion height increases from 0 to 15 μm. A height of μm means that the maximum blockage of the atrioventricular canal is 60% (30 μm/50 μm)

Spatial distributions of the normal force per unit area for several values of ventricular wall stiffness are shown in Fig. 11. The chamber depth is equal to 100 μm, and the cushion height was set to 15 μm. The left graph of Fig. 11 shows the normal force on the wall opposite the atrium. The right graph of Fig. 11 shows the wall on the same side as the atrium. Wall stiffness, kstr was set to 0.2 ktarg, 0.5 ktarg, ktarg, and 2 ktarg. As the ventricular wall stiffness increases, the force required to fill the chamber also increases. The relative change in normal force is greatest for the ventricular walls. As the wall stiffens, the resistance to chamber filling is increased. This leads to an increase in transmural pressure in the ventricle that is coupled to an increase in the normal force as the ventricular wall applies larger forces on the fluid to resist filling.
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Fig. 11

Spatial distribution of the normal force per unit area for several values of ventricular wall stiffness. The chamber depth is equal to 100 μm and the cushion height was set to 15 μm. The left graph shows the normal force on the wall opposite the atrium. The right graph shows the wall on the same side as the atrium. Wall stiffness, kstr, was set to 0.2 ktarg, 0.5 ktarg, ktarg, and 2 ktarg. As the ventricular wall stiffness increases, the force required to fill the chamber also increases

Discussion

The two-dimensional simulations in this article suggest that fluid forces acting normal to chamber walls and transmural pressures in simple models of the vertebrate embryonic heart are only accurately captured when the moving boundary problem is considered. Flow patterns through the heart are markedly different when blood flow is driven by chamber contraction rather than when flow is externally driven through rigid chambers. This result highlights the importance of developing embryonic heart models that consider moving chamber walls and the resulting fluid dynamics of the heart.

The development of the endocardial cushions significantly alters the distribution and magnitude of fluid shear stress in the heart and normal forces acting on the chamber walls. Shear stress peaks in the atrioventricular canal. Peak shear is also highly sensitive to the height of the endocardial cushions, reaching values of up to 96 dynes/cm2 when 80% of the atrioventricular canal cross-section was occupied by cushion. Force per unit area acting normal to the chamber walls also increases significantly as the cushion height increases because of the resistance to flow through the small atrioventricular canal. Peak values ranged from about 0.1 dynes/cm2 when the cushion height equaled 0 μm to 3.5 dynes/cm2 when the cushion height equaled 20 μm.

This work illustrates the importance of considering the moving boundary problem, but much more work is needed before a complete picture of the mechanics of heart development is developed. A complete three-dimensional model will need to couple the electrophysiology, elasticity, and fluid dynamics to molecular regulatory networks responsible for chamber and valve morphogenesis. Such a model could serve as a powerful tool for understanding mechanisms of mechanotransduction if the activation of genes responsible for heart morphogenesis is tied either temporally or spatially to changes in fluid shear and transmural pressure as the heart forms. Such a model will also need to describe changes in the cardiac electrophysiology during development, and waves of action potentials in the myocardial layer will need to be connected to myocardial activity. Recent advances in computational methods for these coupled problems now make the study of these problems possible [36], creating exciting new opportunities for using mathematical models to understand such complex systems.

Acknowledgments

The author would like to thank Arvind Santhanakrisnan, Anil Shenoy, and Charles Peskin for meaningful discussions concerning the biofluid mechanics of heart development. The author would also like to thank the Applied Mathematics Fluids Lab Group at the University of North Carolina Chapel Hill for their input and intuition. This work was funded by BWF CASI Award ID # 1005782.01.

Copyright information

© Springer Science+Business Media, LLC 2011