# Computational Methods for Fluid-Structure Interaction Simulation of Heart Valves in Patient-Specific Left Heart Anatomies

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## Abstract

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## 1. Problem Formulation

## 2. Prosthetic Heart Valves

#### 2.1. Mechanical Valves

**M**${}_{0}$ is the moment coefficient:

#### 2.2. Bioprosthetic Valves

## 3. Fluid–Structure Interaction of Valves and Blood Flows

Algorithm 1: Fixed-point iteration algorithm to solve the Equation (25) |

$l=0$ ${\varphi}^{0}={\varphi}_{n-1}$ while
$|{\varphi}^{l}-{\varphi}^{l-1}|>tolerance$
do$l=l+1$ $\tilde{{\varphi}^{l+1}}=\mathbb{S}\circ \mathfrak{F}\left({\varphi}^{l}\right)$ ${e}^{l}=\tilde{{\varphi}^{l+1}}-{\varphi}^{l}$ ${\varphi}^{l+1}={\lambda}^{l}\tilde{{\varphi}^{l+1}}+\left(1-{\lambda}^{l}\right){\varphi}^{l}$ end while${\varphi}_{n}={\varphi}^{l+1}$ |

## 4. Patient-Specific Anatomy and the Dynamics of ${\Gamma}_{LV}$

## 5. Continuum Approaches for Solving the Flow Dynamics in Ω_{f}

#### 5.1. Governing Equations for the Fluid Domain ${\Omega}_{f}$

#### 5.2. Finite Difference Methods

#### 5.3. Immersed Finite Element Methods

## 6. Particle Methods

## 7. Current Challenges and Future Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALE | Arbitrary Lagrangian–Eulerian |

BMHV | Bi-leaflet Mechanical Heart Valve |

BGK | Bhatnagar–Gross–Krook |

BDI | Blood Damage Index |

CT | Computed Tomography |

DNS | Direct Numerical Simulation |

ELB | Entropic Lattice Boltzmann |

EBF | External Boundary Force |

FSI | Fluid–Structure Interaction |

IBM | Immersed Boundary Method |

LV | Left Ventricle |

LVOT | Left Ventricle Outflow Tract |

LCA | Left Coronary Artery |

NCA | Non-Cusp Coronary Artery |

LBM | Lattice Boltzmann Method |

MRI | Magnetic Resonance Imaging |

PSM | Patient-Specific Modeling |

RCA | Right Coronary Artery |

SIB | Sharp-Interface Immersed Boundary |

SBB | Standard Bounce Back |

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**Figure 1.**(

**A**) The anatomy of the left heart including the left atrium, left ventricle, aortic valve, mitral valve, and the sinuses. (

**B**) The sinuses and the valve leaflets are shown in colors to emphasize their anatomical location: (i) left coronary artery (LCA); (ii) right coronary artery (RCA); and (iii) non-cusp coronary artery (NCA). The mitral leaflets are shown in red. The right ventricle is shown in the shadow to highlight the orientation of the left heart. The ventricular structures are visualized using the public computed tomography data of a human subject, which can be found at: http://www.gimias.org/index.php?option=com_content&view=article&id=26&Itemid=18 (accessed on 15 August 2021).

**Figure 2.**The configuration of the computational domain. The fluid domain (${\Omega}_{f}$) of the left heart is divided into three main regions: (i) the left atrium (LA); (ii) the left ventricle (LV); and (iii) the aorta. The motion of the left ventricle is tracked on the boundary ${\Gamma}_{LV}$. The valve motions (${\Omega}_{s}$) are computed from the fluid–structure interaction algorithms by exchanging the kinematics and loading conditions on (${\Gamma}_{fsi}$). Blood flow comes from the lung via the inlets at ${\Gamma}_{inlet}$ and exits at the outlet ${\Gamma}_{outlet}$. The motion of the mechanical valve is tracked with the opening angle $\varphi $.

**Figure 3.**The complexity of left ventricular flow [18]. The motion of the heart is monitored using magnetic resonance imaging in (

**A**). The numerical simulation using the prescribed kinematics (${\Gamma}_{LV}\left(t\right)$) is shown in (

**B**). The mitral valve is not included in the simulation. Instead, the 4D-Flow MRI data is used to prescribe the incoming jet from the left atrium. The mitral vortex ring (MVR) forms during early diastole (E-wave) as the jet accelerates. The MVR is visualized using Q-criterion [73].

**Figure 4.**Fluid–structure interaction of a tri-leaflet valve in an axisymmetric aorta [91] with the peak systolic Reynolds number of 2580, which is based on the peak systolic velocity U = 0.78 m/s and the diameter of the valve D = 25.4 mm. The figures respectively show the time instantsat (

**A**) ${t}_{A}=200$ milliseconds, (

**B**) ${t}_{B}=260$ milliseconds and (

**C**) ${t}_{C}=380$ milliseconds, respectively. The top row shows the evolution of the leading vortex ring, which is visualized by the contour of out-of-plane vorticity. The bottom row shows the three-dimensional structures, which are visualized by the iso-surface of Q-criterion [73].

**Figure 6.**Flow patterns at the aortic sinus under healthy (tricuspid) and diseased (bicuspid) aortic valves. The flow is visualized by streamlines colorized by velocity magnitude. The black arrow indicates the impingement location of the aortic jet. Reprinted by permission from Springer Theoretical and Computational Fluid Dynamics [88].

**Figure 7.**Illustration of (

**a**) three-dimensional discretization scheme, namely D3Q19 and (

**b**) representative process of streaming and collision in two-dimensional discretization [61].

**Figure 8.**Visualization of blood damage index analysis. (

**a**) Modeled platelet with surface mesh of 292 triangular elements, 3 $\mathsf{\mu}$m major axis diameter, 1.3 $\mathsf{\mu}$m minor axis diameter. (

**b**) Platelets in flow through a BMHV colored by instantaneous surface shear stress magnitude. Perpendicular viewpoints of platelet pathline while (

**c**) traversing near leaflet and (

**d**) platelet caught in recirculation near sinus expansion wall, (

**e**,

**f**) corresponding damage accumulation overtime, respectively. Reprinted by permission from the Journal of Biomechanical Engineering [136].

**Figure 9.**Pulsatile flow visualization of three-dimensional BMHV simulations. (

**a**–

**e**) Vorticity field and (

**f**–

**j**) Q-criterion for (

**a**,

**f**) opening phase, (

**b**,

**g**) acceleration phase, (

**c**,

**h**) peak flow, (

**d**,

**i**) closing phase, and (

**e**,

**j**) leakage phase. Reprinted by permission from the Journal of Fluid Mechanics, Cambridge University Press [137].

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**MDPI and ACS Style**

Le, T.B.; Usta, M.; Aidun, C.; Yoganathan, A.; Sotiropoulos, F.
Computational Methods for Fluid-Structure Interaction Simulation of Heart Valves in Patient-Specific Left Heart Anatomies. *Fluids* **2022**, *7*, 94.
https://doi.org/10.3390/fluids7030094

**AMA Style**

Le TB, Usta M, Aidun C, Yoganathan A, Sotiropoulos F.
Computational Methods for Fluid-Structure Interaction Simulation of Heart Valves in Patient-Specific Left Heart Anatomies. *Fluids*. 2022; 7(3):94.
https://doi.org/10.3390/fluids7030094

**Chicago/Turabian Style**

Le, Trung Bao, Mustafa Usta, Cyrus Aidun, Ajit Yoganathan, and Fotis Sotiropoulos.
2022. "Computational Methods for Fluid-Structure Interaction Simulation of Heart Valves in Patient-Specific Left Heart Anatomies" *Fluids* 7, no. 3: 94.
https://doi.org/10.3390/fluids7030094