# Modeling the Link between Left Ventricular Flow and Thromboembolic Risk Using Lagrangian Coherent Structures

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Experimental Studies

#### 2.2. Lagrangian Coherent Structures

#### 2.3. Implementation

#### 2.4. Selecting the Integration Time and Extracting LCS

#### 2.5. Lagrangian Particle Analysis

## 3. Results

#### 3.1. Dilated Cardiomyopathy (DCM) Model

#### 3.2. Effect of LVAD Support on LV Blood Transport

#### 3.3. Thrombus Development in the LVOT

#### 3.4. Regional Analysis of FTLE in the LVOT

## 4. Discussion

#### Limitations

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Experimental measurements of left ventricle (LV) flow for a model of dilated cardiomyopathy (DCM), a unique case of an LV with a thrombus in the left ventricular outflow tract (left ventricular outflow tract (LVOT) thrombus), and the addition of left ventricular assist device (LVAD) support were performed using a mock circulatory loop and particle image velocimetry. Regional analyses were calculated for two regions of interest (ROI) in the LVOT, identified as the distal (D) and proximal (P) ROI.

**Figure 2.**Reverse time integration of finite time Lyapunov exponent (FTLE) is performed to identify Lagrangian Coherent Structure (LCS) ridges. (

**a**) Integration time = 0.2 s; (

**b**) Integration time = 0.5 s; (

**c**) Integration time = 0.7 s; (

**d**) Integration time = 0.9 s; (

**e**) Integration time = 1.2 s.

**Figure 3.**Attracting LCS surfaces are extracted from the Reverse FTLE fields to illustrate the E and A waves through the mitral valve during diastolic filling. (

**a**) E wave; (

**b**) A wave.

**Figure 4.**LCS ridges extracted from the reverse FTLE illustrate the effect of the LVAD on LV filling dynamics.

**Figure 5.**Lagrangian particle analysis reflects the LCS structures identified from the FTLE analysis, illustrating how the altered filling dynamics affects transport through the LV.

**Figure 6.**Particle transport for the DCM model (

**left**) and the LVOT thrombus model (

**right**) after 1, 2 and 3 cardiac cycles.

**Figure 7.**Distribution of flow calculated for each condition. Direct flow is blood ejected during the first cardiac cycle, Delayed Ejection Flow is ejected during the second cycle, and the Retained Flow is blood with residence time longer than two cardiac cycles. The flow fractions are calculated from the particle analysis and scaled by the total aortic flow measured from the experimental studies. (

**a**) DCM LV Model; (

**b**) Small thrombus LV Model.

**Table 1.**Characteristics of the forward FTLE analysis for the entire LV, as well as for the regional analysis of flow structures in the LVOT adjacent to the location of the model thrombus.

Flow Condition | FTLE for Entire LV | Regional FTLE | |||
---|---|---|---|---|---|

Ave | Peak E | Peak A | Distal ROI | Proximal ROI | |

DCM | |||||

Pre-LVAD | 5.63 | 8.17 | 5.19 | 2.01 | 2.62 |

Parallel | 5.73 | 8.00 | 5.64 | 3.98 | 4.75 |

Series | 6.43 | 8.56 | 6.89 | 4.30 | 4.41 |

LVOT Thrombus | |||||

Pre-LVAD | 5.82 | 8.63 | 5.01 | 4.53 | 6.30 |

Parallel | 6.19 | 9.45 | 6.15 | 4.63 | 6.38 |

Series | 6.65 | 10.10 | 6.64 | 4.63 | 6.57 |

© 2016 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 license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

May-Newman, K.; Vu, V.; Herold, B.
Modeling the Link between Left Ventricular Flow and Thromboembolic Risk Using Lagrangian Coherent Structures. *Fluids* **2016**, *1*, 38.
https://doi.org/10.3390/fluids1040038

**AMA Style**

May-Newman K, Vu V, Herold B.
Modeling the Link between Left Ventricular Flow and Thromboembolic Risk Using Lagrangian Coherent Structures. *Fluids*. 2016; 1(4):38.
https://doi.org/10.3390/fluids1040038

**Chicago/Turabian Style**

May-Newman, Karen, Vi Vu, and Brian Herold.
2016. "Modeling the Link between Left Ventricular Flow and Thromboembolic Risk Using Lagrangian Coherent Structures" *Fluids* 1, no. 4: 38.
https://doi.org/10.3390/fluids1040038