# Vortex Dynamics in Trabeculated Embryonic Ventricles

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Embryonic Zebrafish Model Geometry

#### 2.2. Idealized Model Geometry

## 3. Results

#### 3.1. Steady Flow through an Embryonic Zebrafish Heart

#### 3.2. Pulsatile Flow through an Embryonic Zebrafish Heart

#### 3.3. Steady Flow through Idealized Trabeculated Chambers

#### 3.4. Pulsatile Flow through Idealized Trabeculated Chambers

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DPF | Days Post Fertilization |

HPF | Hours Post Fertilization |

WT | Wild Type embryo |

ZF | Zebrafish |

AV | Atrioventricular |

CFD | Computational Fluid Dynamics |

Re | Reynolds Number |

IB | Immersed Boundary Method |

WSS | Wall Shear Stress |

OSI | Oscillatory Shear Index |

## Appendix A. Immersed Boundary Method

**Table A1.**Inflow boundary conditions for both simulations, one pertaining to parabolic steady inflow and the other corresponding to a parabolic pulsatile inflow. The parameters used for the boundary conditions are f, the non-dimensional frequency, which is matched to the zebrafish heart at 96 hpf, and ${V}_{in}$, the maximum inflow velocity.

Case | Inflow BC |
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Steady Inflow | ${\mathbf{u}}_{\mathbf{in}}=\left(\begin{array}{c}\frac{{V}_{in}}{{d}_{AV}^{2}}tanh(2t)\left(\frac{1}{4{d}_{AV}^{2}}-{y}^{2}\right)\\ 0\end{array}\right)$ |

Pulsatile Inflow | ${\mathbf{u}}_{\mathbf{in}}=\left(\begin{array}{c}\frac{{V}_{in}}{{d}_{AV}^{2}}|sin(2\pi ft)|\left(\frac{1}{4{d}_{AV}^{2}}-{y}^{2}\right)\\ 0\end{array}\right)$ |

## Appendix B. Methods: Biologically Realistic Geometries

**Figure A1.**Extracting biologically realistic geometries of trabeculated geometries at different time-points from [21]. Geometries are given for a WT zebrafish at 80 hpf and 5 dpf (II).

## Appendix C. Results: Pulsatile Inflow for Biologically Realistic Geometries Results

**Figure A2.**Plots showing how the magnitude of velocity decays from the center of the atrioventricular (AV) canal to the ventricle wall for four different lines across the chamber using a 3 dpf WT embryo’s geometry.

**Figure A3.**Plots showing how the magnitude of velocity decays from the center of the atrioventricular (AV) canal to the ventricle wall for four different lines across the chamber using a 5 dpf WT (I) embryo’s geometry.

**Figure A4.**Plots showing how the magnitude of velocity decays from the center of the atrioventricular (AV) canal to the ventricle wall for five different lines across the chamber using the ErbB2-inhibited emrbyo’s geometry.

## Appendix D. Results: Computing WSS, TAWSS, and OSI

#### Appendix D.1. Results: Spatially-Averaged WSS for 3 dpf WT and 5 dpf WT (I)

**Figure A5.**Plots detailing what regions were considered for spatially-averaging the WSS on the trabeculae and intertrabecular regions.

## Appendix E. Results: Steady Inflow for Idealized Geometry

**Figure A6.**Measurements of the non-dimensional magnitude of velocity are provided for the $Re=10$ case quantified along a line from the intracardial center (labeled “0”) and extending to the ventricular lining (labeled “1”) for various intertrabecular regions and trabeculae heights. The velocity strictly decreases from the center until around the neighboring trabeculae heights, in which the velocity increases, in some cases, an order of magnitude, before dropping towards zero at the ventricle lining.

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**Figure 1.**Illustrations of significant stages of zebrafish heart development. At 1 dpf the heart begins pumping as a valveless heart tube, at $1.5$ dpf cardiac looping and chamber ballooning begins, at 2 dpf the heart is a growing two-chambered pumping system with endocardial cushions forming, and at 3 dpf ventricular trabeculation occurs. The blue arrows indicate blood flow while the black dotted arrows represent mechanical stresses re-configuring the heart.

**Figure 2.**The extracted geometries from trabeculated ventricles at different stages of development from Liu et al. [21]. There are two wild-type (WT) zebrafish cases shown (3 dpf and 5 dpf (I)) as well as the geometry taken from an ErbB2-inhibited zebrafish at 7 dpf.

**Figure 3.**(

**a**) a microscopy image of an embryonic zebrafish’s trabeculated ventricle at 4 dpf from Liu et al. [21]. This snapshot was taken immediately before the systolic phase. The protrusions into the ventricle are trabeculae. The image was taken from Tg(cmlc2:dsRed)s879; Tg(flk1:mcherry)s843 embryos expressing fluorescent proteins that label the myocardium and endocardium, respectively [21]; (

**b**) simplified diagram showing the basic idea behind our idealized geometry. Blood flows from the atrio-ventricular canal into the ventricle and then proceeds into the bulbus arteriosus; (

**c**) further idealization of the computational model which is now flattened. The geometric parameters are as follows: ${a}_{V}$ and ${b}_{V}$ are the semi-major and semi-minor axis of the elliptical chamber, ${h}_{T}$ and ${r}_{T}$ are the height and radii of the trabeculae, and ${w}_{AV}$ and ${w}_{SV}$ are the widths of the atrioventricular (AV) canal and sinus venosus, respectively.

**Figure 4.**Schematic diagram showing the parameters for the biologically realistic simulations. Note that ${a}_{V}$ is the characteristic length for selecting the correct $Re$.

**Figure 5.**Streamlines showing the direction of steady flow through realistic trabeculated ventricular geometries for the biologically relevant $Re$, $Re=1$. The orange lines detail closed streamlines, illustrating vortical flow patterns, while blue laminar, non-vortical flow patterns.

**Figure 6.**Simulation results for a 3 and 5 dpf WT (I) and 7 dpf ErbB2 inhibited zebrafish for $Re=1$. The magnitude of velocity (cm/s) is given by the colormap, and the arrows show the direction of flow once steady state is reached (left). The panels on the right show the magnitude of velocity along lines drawn from the center of the AV canal to the ventricular lining between the trabeculae. Note that these lines are shown in the middle panels.

**Figure 7.**Temporal snapshots of the streamlines at different points during a pulsation cycle, described by a percentage of the pulsation period, T, within the realistic trabeculated ventricles of embryonic zebrafish from different stages of development. These simulations were run for biologically relevant $Re$, $Re=1$.

**Figure 8.**Temporal snapshots of the flow at different points during a pulsation cycle, described by a percentage of the pulsation period, T, for the 5 dpf (I) WT ventricle at $Re=1$. The magnitude of velocity (cm/s) is given by the colormap and the direction of flow is given by the arrows.

**Figure 9.**Temporal snapshots of the flow at different points during a pulsation cycle, described by a percentage of the pulsation period, T, for an ErbB2-inhibited based geometry at 7 dpf at $Re=1$. The magnitude of velocity (cm/s) is shown by the colormap along with the velocity vectors.

**Figure 10.**Plots showing how the log magnitude of velocity decays from the center of the AV canal to the ventricle wall for five different lines across the chamber using a 3 dpf WT embryo’s geometry at $Re=1$. The velocity decays until it reaches a trabeculae height away from the ventricle wall. Within pronounced trabecular valleys, the velocity increases before decaying further when measured closer to the ventricular lining. Each sub-figure corresponds to a different time during the pulsation cycle, described by a percentage of the pulsation period, T.

**Figure 11.**Plots showing how the log magnitude of velocity decays from the center of the AV canal to the ventricle wall for four different lines across the chamber using a 5 dpf WT (I) embryo’s geometry at $Re=1$. The velocity decays until it reaches a trabeculae height away from the ventricle wall. Within pronounced trabecular valleys, the velocity increases before decaying further when measured closer to the ventricular lining. Each sub-figure corresponds to a different time during the pulsation cycle, described by a percentage of the pulsation period, T.

**Figure 12.**Plots showing how the log magnitude of velocity decays from the center of the AV canal to the ventricle wall for five different lines across the chamber using a 7 dpf ErbB2-inhibited embryo’s geometry at $Re=1$. The velocity decays until it reaches a trabeculae height away from the ventricle wall. Within pronounced trabecular valleys, the velocity increases before decaying further when measured closer to the ventricular lining. Each sub-figure corresponds to a different time during the pulsation cycle, described by a percentage of the pulsation period, T.

**Figure 13.**A plot showing the spatially-averaged magnitude of wall shear-stress (WSS) over the ventricle during the first 4 pulsation cardiac cycles for the 3 dpf WT, 5 dpf WT, and 7 dpf ErbB2-inhibited zebrafish (ZF) for $Re=1$. The highest spatially-averaged WSS was in the 3 dpf WT ZF with the least in the 7 dpf ErbB2-inhibited ZF geometry.

**Figure 14.**Plot giving the spatially-averaged magnitude of WSS over the trabeculae, the intertrabecular regions, and the entire ventricle during the first 4 pulsation cardiac cycles for the 3 dpf WT (

**a**) and 5 dpf WT (I) (

**b**) ZF for $Re=1$. In both cases, intertrabecular regions experience less WSS than the trabeculae themselves.

**Figure 15.**Plot giving the oscillatory shear index (OSI) over the ventricle during the 4th pulsation cardiac cycle for $Re=1$. High OSI occurs in intertrabecular regions and on the trabeculae themselves.

**Figure 16.**Streamline analysis performed for the case of steady flow into the trabeculated ventricle of a zebrafish at 4 dpf for varying $Re$ and trabeculae heights.

**Figure 17.**Measurements of the non-dimensional magnitude of velocity are provided for the $Re=1$ case quantified along a line from the intracardial center (labeled “0”) and extending to the ventricular lining (labeled “1”) for various intertrabecular regions and trabeculae heights. The velocity magnitude strictly decreases from the center until around the neighboring trabeculae heights. The velocity magnitude then increases towards the center of the intratrabecular region, in some cases an order of magnitude, before dropping towards zero at the ventricle lining.

**Figure 18.**Measurements of the non-dimensional magnitude of velocity are provided for the $Re=100$ case quantified along a line from the intracardial center (labeled “0”) and extending to the ventricular lining (labeled “1”) for various intertrabecular regions and trabeculae heights. Each subfigure (

**A**–

**E**), corresponds to a different line between the intracardial center and ventricular lining in which the magnitude of velocity is measured. The velocity magnitude decreases from the center and then increases before decreasing again as one approaches a distance from the wall that is equal to the neighboring trabeculae heights. As one moves between the trabeculae, the velocity magnitude again increases, in some cases an order of magnitude, before dropping towards zero at the ventricle lining.

**Figure 19.**Streamline analysis taken at different time points within a single pulsation cycle (described by a percentage of a single pulsation period, T) for the case of pulsatile flow into the trabeculated ventricle of a zebrafish at 4 dpf for $Re=0.1$ and varying trabeculae heights.

**Figure 20.**Streamline analysis taken at different time points within a single pulsation cycle (described by a percentage of a single pulsation period, T) for the case of pulsatile flow into the trabeculated ventricle of a zebrafish at 4 dpf for $Re=1.0$ and varying trabeculae heights.

**Figure 21.**Snapshots during a pulsation cycle in the $Re=1$ case of the horizontal velocity measured from the intracardial center to the intertrabecular region directly above for multiple trabeculae heights. Each sub-figure corresponds to a different time point described as a percentage of the pulsation cycle, T.

**Figure 22.**Snapshots during a pulsation cycle in the $Re=1$ case of the horizontal velocity measured from the intracardial center (labeled “0”) to the intertrabecular region (labeled “1”) directly above for multiple trabeculae heights. The trabeculae cause a drop in velocity as one nears the ventricular wall. Each sub-figure corresponds to a different time point described as percentge of the pulsation cycle, T.

**Figure 23.**Streamline analysis taken at different time points within a single pulsation cycle (described by a percentage of a single pulsation period, T) for the case of pulsatile flow into the idealized trabeculated ventricle of a zebrafish at 4 dpf for $Re=10.0$ and varying trabeculae heights.

**Figure 24.**Snapshots during a pulsation cycle in the $Re=10$ case of the horizontal velocity measured from the intracardial center to the intertrabecular region directly above for multiple trabeculae heights. Each sub-figure corresponds to a different time point described as a percentage of the pulsation cycle, T.

**Figure 25.**Streamline analysis performed for the case of pulsatile flow into the trabeculated ventricle of a zebrafish at 4 dpf for $Re=100$ and varying trabeculae heights.

**Figure 26.**Snapshots during a pulsation cycle in the $Re=100$ case of the horizontal velocity measured from the intracardial center to the intertrabecular region directly above for multiple trabeculae heights.

**Table 1.**Table of geometric parameters used in the idealized numerical model. The height of trabeculae, ${h}_{T}$, were varied for numerical experiments.

Parameter | Value |
---|---|

${a}_{V}$ | 1.0 |

${b}_{V}$ | 0.8 |

${w}_{AV}$ | 0.8 |

${w}_{SV}$ | 0.8 |

${r}_{T}$ | 0.10 |

$\frac{{h}_{T}}{{b}_{V}}$ | {0, 0.02, 0.04, …, 0.16} |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Battista, N.A.; Douglas, D.R.; Lane, A.N.; Samsa, L.A.; Liu, J.; Miller, L.A. Vortex Dynamics in Trabeculated Embryonic Ventricles. *J. Cardiovasc. Dev. Dis.* **2019**, *6*, 6.
https://doi.org/10.3390/jcdd6010006

**AMA Style**

Battista NA, Douglas DR, Lane AN, Samsa LA, Liu J, Miller LA. Vortex Dynamics in Trabeculated Embryonic Ventricles. *Journal of Cardiovascular Development and Disease*. 2019; 6(1):6.
https://doi.org/10.3390/jcdd6010006

**Chicago/Turabian Style**

Battista, Nicholas A., Dylan R. Douglas, Andrea N. Lane, Leigh Ann Samsa, Jiandong Liu, and Laura A. Miller. 2019. "Vortex Dynamics in Trabeculated Embryonic Ventricles" *Journal of Cardiovascular Development and Disease* 6, no. 1: 6.
https://doi.org/10.3390/jcdd6010006