Fluid Dynamics of Ventricular Filling in the Embryonic Heart

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.

Appendix

The system of differentio-integral equations given by Eqs. 1–9 was solved on a rectangular grid with periodic boundary conditions in both directions. The Navier–Stokes equations were discretized on a fixed Eulerian grid, and the immersed boundaries were discretized on a Lagrangian array of points.

Consider the discretization of the equations that describe the force applied to the fluid as a result of the distance between the actual boundary and the tether points (Eq. 5). Let Δt be the duration of the time step, let Δs be the length of a boundary spatial step, let n be the time step index, and let s define the location of a boundary point in the Lagrangian framework (s = mΔs, m is an integer). Also, let ${\mathbf{X}}^{n} \left( s \right) = {\mathbf{X}}\left( {s,n\Updelta t} \right),\,{\mathbf{Y}}^{n} \left( s \right) = {\mathbf{Y}}\left( {s,n\Updelta t} \right),\,{\mathbf{u}}^{n} \left( {\mathbf{x}} \right) = {\mathbf{u}}\left( {{\mathbf{x}},n\Updelta t} \right),\,$ and ${\text{p}}^{n} \left( {\mathbf{x}} \right) = {\text{p}}\left( {{\mathbf{x}},n\Updelta t} \right)$. For any function ϕ(s), let (D _s ϕ)(s) be defined by the following equation:

$$ \left( {D_{s} \phi } \right)\left( s \right) = {\frac{{\phi \left( {s + {\frac{{{{\Updelta}}s}}{2}}} \right) - \phi \left( {s - {\frac{{{{\Updelta}}s}}{2}}} \right)}}{{{{\Updelta}}s}}}. $$

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Using these definitions, the force applied to the fluid by the target points, F _targ, defined by Eq. 5, is discretized as follows:

$$ {\mathbf{F}}_{\text{targ}}^{n} = k_{\text{targ}} \left( {{\mathbf{Y}}^{n} \left( s \right) - {\mathbf{X}}^{n} \left( s \right)} \right). $$

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The stretch force, F _str, defined by Eq. 6, is discretized as follows:

$$ {\mathbf{F}}_{\text{str}}^{n} \left( s \right) = k_{\text{str}} D_{s} \left\{ {\left( {D_{s} {\mathbf{X}}^{n} \left( s \right) - 1} \right){\frac{{D_{s} {\mathbf{X}}^{n} \left( s \right)}}{{\left| {D_{s} {\mathbf{X}}^{n} \left( s \right)} \right|}}}} \right\}. $$

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The external force applied to the fluid in order to drive the flow, f _ext, defined in Eq. 7 and the desired flow profile upstream, u _targ(x, t), defined by Eq. 8 can be discretized as follows:

$$ {\mathbf{f}}_{\text{ext}}^{n} = \left\{ {\begin{array}{*{20}c} { - k_{\text{ext}} \left( {{\mathbf{u}}^{n} - {\mathbf{u}}_{\text{targ}}^{n} } \right), \quad {\frac{l - d}{2}} \le jh \le {\frac{l + d}{2}},10 \le i \le 15} \\ {0, \quad {\text{otherwise}}} \\ \end{array} } \right. $$

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$$ {\mathbf{u}}_{\text{targ}}^{n} \left( {i,j} \right) = \left[ {\begin{array}{*{20}c} {U_{ \max } \left( {1 - \left( {{\frac{0.5 - jh}{d/2}}} \right)^{2} } \right)} \\ 0 \\ \end{array} } \right] $$

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where x = (ih, jh), i and j are integers, and h is the mesh width.

Finally, the total force applied to the fluid by the boundary is given by the following equation:

$$ {\mathbf{f}}^{n} = {\mathbf{f}}_{\text{str}}^{n} + {\mathbf{f}}_{\text{targ}}^{n} + {\mathbf{f}}_{\text{ext}}^{n}. $$

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Since the Lagrangian array of boundary points does not necessarily coincide with the fixed Eulerian lattice used for the computation of the fluid velocities, a smoothed approximation to the Dirac delta function (Eqs. 3 and 4) is used to handle the fluid-boundary interaction. This approximate delta function applies a force from the boundary to the fluid and interpolates the fluid velocity. The approximate two-dimensional Dirac delta function, δ _h , used in these calculations is given by the following equations:

$$ \begin{gathered} \delta_{h} \left( x \right) = h^{ - 2} \phi\left( {\frac{x}{h}} \right)\phi \left( {\frac{y}{h}} \right)\hfill \\ \phi \left( r \right) = \left\{ {\begin{array}{ll}{\frac{1}{4}\left( {1 + \cos \left( {{\frac{\pi r}{2}}} \right)}\right), \quad \left| r \right| \le 2} \\ {0, \quad{\text{otherwise}}}\\ \end{array} } \right. \hfill \\ \end{gathered} $$

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where h is the size of the spatial step in the Eulerian grid and x = (x, y). Peskin and McQueen [29] have previously described this choice of the delta function. Other approximate delta functions have also been used successfully [27].

Using this choice of δ _h , Eqs. 3and 4 can be discretized as follows:

$$ {\mathbf{f}}^{n} \left( {\mathbf{x}} \right) = \mathop \sum \limits_{s} {\mathbf{F}}^{n} \left( s \right)\delta_{h} \left( {{\mathbf{x}} - {\mathbf{X}}^{n} \left( s \right)} \right)\Updelta s + {\mathbf{f}}_{\text{ext}}^{n} \left( {\mathbf{x}} \right) $$

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$$ \begin{gathered} {\mathbf{U}}^{n + 1} \left( s \right) = \mathop \sum \limits_{{\mathbf{x}}} {\mathbf{u}}^{n + 1} \left( {\mathbf{x}} \right)\delta_{h} \left( {{\mathbf{x}} - {\mathbf{X}}^{n} \left( s \right)} \right)h^{2} \hfill \\ {\frac{{{\mathbf{X}}^{n + 1} \left( s \right) - {\mathbf{X}}^{n} \left( s \right)}}{\Updelta t}} = {\mathbf{U}}^{n + 1} \hfill \\ \end{gathered} $$

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where ${{\Upsigma}}_{\text{s}}$ is the sum over all of the discrete collection of points of the form s = mΔs, and ${{\Upsigma}}_{{\mathbf{x}}}$ is the sum over of the discrete points of the form x = (ih, jh), where i, j, and m are integers and h is the mesh width.

The final step is to discretize the incompressible Navier–Stokes equations (Eqs. 1 and 2). The method used here is implicitly defined as follows:

$$ \rho \left( {{\frac{{{\mathbf{u}}^{n + 1} - {\mathbf{u}}^{n} }}{\Updelta t}} + S_{h} \left( {{\mathbf{u}}^{n} } \right){\mathbf{u}}^{n} } \right) + D^{0} {\text{p}}^{n + 1} = \mu \mathop \sum \limits_{\alpha = 1}^{2} D_{\alpha }^{ + } D_{\alpha }^{ - } {\mathbf{u}}^{n + 1} + {\mathbf{F}}^{n} $$

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$$ D^{0} \cdot {\mathbf{u}}^{n + 1} $$

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where ρ is the density of the fluid and μ is the viscosity of the fluid. D ⁰ is the central difference approximation to $\nabla.$ It is defined as follows:

$$ D^{0} = \left( {D_{1}^{0} ,D_{2}^{0} } \right) $$

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$$ \left( {D_{\alpha }^{0} \phi } \right)\left( {\mathbf{x}} \right) = {\frac{{\phi \left( {{\mathbf{x}} + h{\mathbf{e}}_{\alpha } } \right) - \phi \left( {{\mathbf{x}} - h{\mathbf{e}}_{\alpha } } \right)}}{2h}} $$

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where {e₁, e₂} is the standard basis of ${\mathbf{R}}^{2}.$ The operators $D_{\alpha }^{ \pm }$ are forward and backward difference approximations of $\partial /\partial x_{\alpha }.$ They are defined as follows:

$$ \left( {D_{\alpha }^{ + } \phi } \right)\left( {\mathbf{x}} \right) = {\frac{{\phi \left( {{\mathbf{x}} + h{\mathbf{e}}_{\alpha } } \right) - \phi \left( {\mathbf{x}} \right)}}{h}} $$

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$$ \left( {D_{\alpha }^{ - } \phi } \right)\left( {\mathbf{x}} \right) = {\frac{{\phi \left( {\mathbf{x}} \right) - \phi \left( {{\mathbf{x}} - h{\mathbf{e}}_{\alpha } } \right)}}{h}}. $$

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Thus, the viscous term given as $\mathop \sum \limits_{\alpha = 1}^{2} D_{\alpha }^{ + } D_{\alpha }^{ - }$ is a difference approximation to the Laplace operator.

Finally, S _h (u) is a skew-symmetric difference operator which serves as a difference approximation of the nonlinear term ${\mathbf{u}} \cdot \nabla {\mathbf{u}}$. This skew-symmetric difference operator is defined as follows:

$$ S_{h} \left( {\mathbf{u}} \right)\phi = \frac{1}{2}{\mathbf{u}} \cdot {\mathbf{D}}_{\text{h}}^{0} \phi + \frac{1}{2}{\mathbf{D}}_{\text{h}}^{0} \cdot \left( {{\mathbf{u}}\phi } \right). $$

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Since the equations are linear in the unknowns u ⁿ⁺¹ and p ⁿ⁺¹, the Fast Fourier Transform (FFT) algorithm was used to solve for u ⁿ⁺¹ and p ⁿ⁺¹ from u ⁿ, p ⁿ, and ${\mathbf{f}}^{n}$ [29, 37].