# Using Discrete Multiphysics Modelling to Assess the Effect of Calcification on Hemodynamic and Mechanical Deformation of Aortic Valve

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## Abstract

**:**

## 1. Introduction

## 2. Discrete Multiphysics

#### 2.1. Smooth Particle Hydrodynamics

_{i}and density ρ

_{i}to obtain

#### 2.2. Lattice Spring Model (LSM)

_{0}is the initial distance between the two particles, r is the distance at time t, and k is a Hookean constant

## 3. The Model

_{0}= 400 m s

^{−2}, ω = 2πf is the angular frequency, n = 13, and ϕ = π/10 as discussed in Steven et al., 2003 [40]. The value of G

_{0}is determined to achieve full opening of the valve that gives an average flow rate around 600 mL s

^{−1}for valves at normal condition, which is consistent with the literature, e.g., [40,43,44].

_{0}, in such a way that the valve opens fully and the flow rate for a healthy and non-calcified valve is 600 mL s

^{−1}as discussed above.

_{0}in Equation (6) were chosen in such a way to make sure the flow in the valve and its opening are consistent with available observations (e.g., [40,43,44,46,47]). Convergence of the results on the number of particles used to discretise the system was carried out and the numbers reported in Table 1 represent the best compromise between accuracy and computational times. Since we refer to the human body, the available observations of real valves show very scattered results because of individual variability. Therefore, we cannot provide a systematic validation like in the case of experiments carried out in a laboratory under controlled conditions. This is even more true for calcified valves, where, on top of physiological variations, we have additional variations due to the severity and the course of the disease. Given all of the above, the best we can do is to make sure that the results are consistent with real data. In our study, we achieved this by making sure that the flow in the valve and the opening of the leaflets are both within the range of real observations.

## 4. Results and Discussion

#### 4.1. Stages of Calcification

_{H}is the stiffness of the healthy valve. Figure 3 shows the valve during maximal opening for four different degrees of calcification.

_{CR}= 3, after which it decreases sharply. In Figure 5, γ

_{CR}corresponds to a flow rate of 200 mL s

^{−1}. Our calculations are consistent with the medical literature where a flow rate ≥250 mL s

^{−1}is considered acceptable, whereas <200 mL s

^{−1}is associated with an increase of mortality rate in cases of patients with aortic stenosis [44,49]. The maximum orifice diameter and the average stress on the valve (see next section) can also be used to monitor the progression of the valve stenosis. Table 2 summarizes the parameters as the condition worsens from normal to severe.

#### 4.2. Stress Distribution on the Membrane

## 5. Conclusions

^{−1}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Mesh generation using a pre-processing solver (

**a**) and the generated particle distribution (

**b**).

- Creation or importation of the CAD geometry
- Writing of the data file
- Generation of the bond and coefficient files
- Implementation of the input file

**Figure A2.**Tricuspid valve CAD part (

**a**), tricuspid valve meshed CAD part (

**b**), nodes generation using the CAD software (

**c**), and external data file (

**d**).

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**Table 1.**Model parameters used in the simulation; for the meaning of SPH parameters such as α or h, refer to Liu and Liu, 2003 [21].

Parameters | Values |
---|---|

Number of SPH wall particle | 56,660 |

Number of SPH fluid particle | 342,358 |

Number of SPH leaflets | 19,725 |

Mass of each particle (fluid) | 6.7 × 10^{−8} kg |

Mass of each particle (Solid) | 14 × 10^{−8} kg |

Smoothing length h | 1.0 × 10^{−3} m |

Length L | 6 × 10^{−2} m |

Diameter D | 2 × 10^{−2} m |

Particle spacing l | 0.4 × 10^{−3} m |

Fluid Density ρ | 1060 kg m^{−3} |

Frequency f | 1.167 s^{−1} (70 beats min^{−1}) |

Pseudo-gravity G_{0} | 400 m s^{−2} |

Viscosity μ | 0.003 Pa∙s |

Elastic constant k | 10−14,500 N m^{−1} |

Sound speed c_{0} | 16 m s^{−1} |

Time step Δt | 1 × 10^{−6} s |

Calcification | Maximum Orifice Diameter [cm] | Mean Flow × 10^{−4} [m^{3}s^{−1}] | Average Stress [kPa] |
---|---|---|---|

Normal (γ = 0.0) | 1.81 | 5.72 | 10.60 |

Mild (γ = 2.0) | 1.41 | 3.21 | 91.68 |

Moderate (γ = 2.7) | 1.29 | 2.17 | 181.61 |

Severe (γ = 3.1) | 1.17 | 1.58 | 324.27 |

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**MDPI and ACS Style**

Mohammed, A.M.; Ariane, M.; Alexiadis, A.
Using Discrete Multiphysics Modelling to Assess the Effect of Calcification on Hemodynamic and Mechanical Deformation of Aortic Valve. *ChemEngineering* **2020**, *4*, 48.
https://doi.org/10.3390/chemengineering4030048

**AMA Style**

Mohammed AM, Ariane M, Alexiadis A.
Using Discrete Multiphysics Modelling to Assess the Effect of Calcification on Hemodynamic and Mechanical Deformation of Aortic Valve. *ChemEngineering*. 2020; 4(3):48.
https://doi.org/10.3390/chemengineering4030048

**Chicago/Turabian Style**

Mohammed, Adamu Musa, Mostapha Ariane, and Alessio Alexiadis.
2020. "Using Discrete Multiphysics Modelling to Assess the Effect of Calcification on Hemodynamic and Mechanical Deformation of Aortic Valve" *ChemEngineering* 4, no. 3: 48.
https://doi.org/10.3390/chemengineering4030048