# Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Geometry

#### 2.2. Boundary Conditions

#### 2.3. Governing Equations in the Fluid and Solid Domains

^{−1}for the large arterial diameters; therefore, the blood could be modeled as a Newtonian fluid with a constant dynamic viscosity of 3.5 cP. For the shear strain rate beyond 50 s

^{−1}, the blood behaved as a homogeneous fluid with almost constant viscosity due to the high shear environment [5,45]. The mass density of the blood was used as 1056 kg/m

^{3}[8].

^{−5}for the solution convergence.

^{3}[9] and Poisson’s ratio of 0.45 [23] were employed for the healthy and calcified leaflets.

#### 2.4. FSI Coupling

#### 2.5. Finite-Time Lyapunov Exponent (FTLE) Analysis

^{−5}m in the horizontal and −0.01093 to 0.01133 m in the vertical directions to include both upstream and downstream in the domain. The element size for the computations was selected to be 0.7 × 10

^{−4}; therefore, a resolution of 388 × 318 elements was employed for the analysis.

## 3. Results

#### 3.1. Flow Streamlines and Pressure Contours

#### 3.2. FTLE Analysis Results

## 4. Discussion

## 5. Limitations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Gould, S.T.; Srigunapalan, S.; Simmons, C.A.; Anseth, K.S. Hemodynamic and Cellular Response Feedback in Calcific Aortic Valve Disease. Circ. Res.
**2013**, 113, 186–197. [Google Scholar] [CrossRef] [PubMed][Green Version] - Stewart, B.F.; Siscovick, D.; Lind, B.K.; Gardin, J.M.; Gottdiener, J.S.; Smith, V.E.; Kitzman, D.W.; Otto, C.M. Clinical Factors Associated with Calcific Aortic Valve Disease. J. Am. Coll. Cardiol.
**1997**, 29, 630–634. [Google Scholar] [CrossRef][Green Version] - Halevi, R.; Hamdan, A.; Marom, G.; Mega, M.; Raanani, E.; Haj-Ali, R. Progressive Aortic Valve Calcification: Three-Dimensional Visualization and Biomechanical Analysis. J. Biomech.
**2015**, 48, 489–497. [Google Scholar] [CrossRef] [PubMed] - De Hart, J.; Peters, G.W.M.; Schreurs, P.J.G.; Baaijens, F.P.T. A Two-Dimensional Fluid-Structure Interaction Model of the Aortic Value. J. Biomech.
**2000**, 33, 1079–1088. [Google Scholar] [CrossRef] - Amindari, A.; Saltik, L.; Kirkkopru, K.; Yacoub, M.; Yalcin, H.C. Assessment of Calcified Aortic Valve Leaflet Deformations and Blood Flow Dynamics Using Fluid-Structure Interaction Modeling. Inform. Med. Unlocked
**2017**, 9, 191–199. [Google Scholar] [CrossRef] - De Hart, J.; Peters, G.W.M.; Schreurs, P.J.G.; Baaijens, F.P.T. A Three-Dimensional Computational Analysis of Fluid—Structure Interaction in the Aortic Valve. J. Biomech.
**2003**, 36, 103–112. [Google Scholar] [CrossRef] - Spühler, J.H.; Jansson, J.; Jansson, N.; Hoffman, J. 3D Fluid–Structure Interaction Simulation of Aortic Valves Using a Unified Continuum ALE FEM Model. Front. Physiol.
**2018**, 9, 363. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mao, W.; Caballero, A.; McKay, R.; Primiano, C.; Sun, W. Fully-Coupled Fluid-Structure Interaction Simulation of the Aortic and Mitral Valves in a Realistic 3D Left Ventricle Model. PLoS ONE
**2017**, 12, e018472. [Google Scholar] [CrossRef][Green Version] - Halevi, R.; Hamdan, A.; Marom, G.; Lavon, K.; Ben-Zekry, S.; Raanani, E.; Bluestein, D.; Haj-Ali, R. Fluid–Structure Interaction Modeling of Calcific Aortic Valve Disease Using Patient-Specific Three-Dimensional Calcification Scans. Med. Biol. Eng. Comput.
**2016**, 54, 1683–1694. [Google Scholar] [CrossRef] - Haj-Ali, R.; Dasi, L.P.; Kim, H.S.; Choi, J.; Leo, H.W.; Yoganathan, A.P. Structural Simulations of Prosthetic Tri-Leaflet Aortic Heart Valves. J. Biomech.
**2008**, 41, 1510–1519. [Google Scholar] [CrossRef] - Dumont, K.; Vierendeels, J.; Kaminsky, R.; Van Nooten, G.; Verdonck, P.; Bluestein, D. Comparison of the Hemodynamic and Thrombogenic Performance of Two Bileaflet Mechanical Heart Valves Using a CFD/FSI Model. J. Biomech. Eng.
**2007**, 129, 558–565. [Google Scholar] [CrossRef][Green Version] - Bavo, A.M.; Rocatello, G.; Iannaccone, F.; Degroote, J.; Vierendeels, J.; Segers, P. Fluid-Structure Interaction Simulation of Prosthetic Aortic Valves: Comparison between Immersed Boundary and Arbitrary Lagrangian-Eulerian Techniques for the Mesh Representation. PLoS ONE
**2016**, 11, e0154517. [Google Scholar] [CrossRef][Green Version] - Borazjani, I. Fluid–Structure Interaction, Immersed Boundary-Finite Element Method Simulations of Bio-Prosthetic Heart Valves. Comput. Methods Appl. Mech. Eng.
**2013**, 257, 103–116. [Google Scholar] [CrossRef] - Chandran, K.B.; Vigmostad, S.C. Patient-Specific Bicuspid Valve Dynamics: Overview of Methods and Challenges. J. Biomech.
**2013**, 46, 208–216. [Google Scholar] [CrossRef] [PubMed][Green Version] - Conti, C.A.; Votta, E.; Della Corte, A.; Del Viscovo, L.; Bancone, C.; Cotrufo, M.; Redaelli, A. Dynamic Finite Element Analysis of the Aortic Root from MRI-Derived Parameters. Med. Eng. Phys.
**2010**, 32, 212–221. [Google Scholar] [CrossRef] - Cao, K.; BukaČ, M.; Sucosky, P. Three-Dimensional Macro-Scale Assessment of Regional and Temporal Wall Shear Stress Characteristics on Aortic Valve Leaflets. Comput. Methods Biomech. Biomed. Eng.
**2016**, 19, 603–613. [Google Scholar] [CrossRef] - Cao, K.; Sucosky, P. Computational Comparison of Regional Stress and Deformation Characteristics in Tricuspid and Bicuspid Aortic Valve Leaflets. Int. J. Numer. Method. Biomed. Eng.
**2017**, 33, e02798. [Google Scholar] [CrossRef] - Gilmanov, A.; Sotiropoulos, F. Comparative Hemodynamics in an Aorta with Bicuspid and Trileaflet Valves. Theor. Comput. Fluid Dyn.
**2016**, 30, 67–85. [Google Scholar] [CrossRef] - Weinberg, E.J.; Kaazempur Mofrad, M.R. A Multiscale Computational Comparison of the Bicuspid and Tricuspid Aortic Valves in Relation to Calcific Aortic Stenosis. J. Biomech.
**2008**, 41, 3482–3487. [Google Scholar] [CrossRef] [PubMed] - Chandra, S.; Rajamannan, N.M.; Sucosky, P. Computational Assessment of Bicuspid Aortic Valve Wall-Shear Stress: Implications for Calcific Aortic Valve Disease. Biomech. Model. Mechanobiol.
**2012**, 11, 1085–1096. [Google Scholar] [CrossRef] - Kuan, M.Y.S.; Espino, D.M. Systolic Fluid–Structure Interaction Model of the Congenitally Bicuspid Aortic Valve: Assessment of Modelling Requirements. Comput. Methods Biomech. Biomed. Eng.
**2015**, 18, 1305–1320. [Google Scholar] [CrossRef] [PubMed] - Marom, G.; Peleg, M.; Halevi, R.; Rosenfeld, M.; Raanani, E.; Hamdan, A.; Haj-Ali, R. Fluid-Structure Interaction Model of Aortic Valve with Porcine-Specific Collagen Fiber Alignment in the Cusps. J. Biomech. Eng.
**2013**, 135, 101001–101006. [Google Scholar] [CrossRef][Green Version] - Grande, K.J.; Cochran, R.P.; Reinhall, P.G.; Kunzelma, K.S. Stress Variations in the Human Aortic Root and Valve: The Role of Anatomic Asymmetry. Ann. Biomed. Eng.
**1998**, 26, 534–545. [Google Scholar] [CrossRef] - Votta, E.; Le, T.B.; Stevanella, M.; Fusini, L.; Caiani, E.G.; Redaelli, A.; Sotiropoulos, F. Toward Patient-Specific Simulations of Cardiac Valves: State-of-the-Art and Future Directions. J. Biomech.
**2013**, 46, 217–228. [Google Scholar] [CrossRef] [PubMed][Green Version] - Katayama, S.; Umetani, N.; Sugiura, S.; Hisada, T. The Sinus of Valsalva Relieves Abnormal Stress on Aortic Valve Leaflets by Facilitating Smooth Closure. J. Thorac. Cardiovasc. Surg.
**2008**, 136, 1528–1535.e1. [Google Scholar] [CrossRef][Green Version] - Balachandran, K.; Sucosky, P.; Yoganathan, A.P. Hemodynamics and Mechanobiology of Aortic Valve Inflammation and Calcification. Int. J. Inflamm.
**2011**, 2011, 263870. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hutcheson, J.D.; Goettsch, C.; Rogers, M.A.; Aikawa, E. Revisiting Cardiovascular Calcification: A Multifaceted Disease Requiring a Multidisciplinary Approach. Semin. Cell Dev. Biol.
**2015**, 46, 68–77. [Google Scholar] [CrossRef] [PubMed][Green Version] - Seaman, C.; McNally, A.; Biddle, S.; Jankowski, L.; Sucosky, P. Generation of Simulated Calcific Lesions in Valve Leaflets for Flow Studies. J. Heart Valve Dis.
**2015**, 24, 115–125. [Google Scholar] [PubMed] - Arjunon, S.; Rathan, S.; Jo, H.; Yoganathan, A.P. Aortic Valve: Mechanical Environment and Mechanobiology. Ann. Biomed. Eng.
**2013**, 41, 1331–1346. [Google Scholar] [CrossRef] [PubMed][Green Version] - Olcay, A.B.; Amindari, A.; Kirkkopru, K.; Yalcin, H.C. Characterization of Disturbed Hemodynamics Due to Stenosed Aortic Jets with a Lagrangian Coherent Structures Technique. J. Appl. Fluid Mech.
**2018**, 11, 375–384. [Google Scholar] [CrossRef] - Shadden, S.C.; Astorino, M.; Gerbeau, J.-F. Computational Analysis of an Aortic Valve Jet with Lagrangian Coherent Structures. Chaos Interdiscip. J. Nonlinear Sci.
**2010**, 20, 017512. [Google Scholar] [CrossRef][Green Version] - 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. [Google Scholar] [CrossRef] - Green, M.A.; Rowley, C.W.; Smits, A.J. Using Hyperbolic Lagrangian Coherent Structures to Investigate Vortices in Bioinspired Fluid Flows. Chaos
**2010**, 20, 017510. [Google Scholar] [CrossRef][Green Version] - Shadden, S.C.; Arzani, A. Lagrangian Postprocessing of Computational Hemodynamics. Ann. Biomed. Eng.
**2015**, 43, 41–58. [Google Scholar] [CrossRef] - Olcay, A.B.; Krueger, P.S. Measurement of Ambient Fluid Entrainment during Laminar Vortex Ring Formation. Exp. Fluids
**2008**, 44, 235–247. [Google Scholar] [CrossRef] - Olcay, A.B.; Pottebaum, T.S.; Krueger, P.S. Sensitivity of Lagrangian Coherent Structure Identification to Flow Field Resolution and Random Errors. Chaos
**2010**, 20, 017506. [Google Scholar] [CrossRef] [PubMed] - Olcay, A.B.; Krueger, P.S. Momentum Evolution of Ejected and Entrained Fluid during Laminar Vortex Ring Formation. Theor. Comput. Fluid Dyn.
**2010**, 24, 465–482. [Google Scholar] [CrossRef] - Olcay, A.B. Investigation of a Wake Formation for Flow over a Cylinder Using Lagrangian Coherent Structures. Prog. Comput. Fluid Dyn.
**2016**, 16, 126–130. [Google Scholar] [CrossRef] - Mutlu, O.; Olcay, A.B.; Bilgin, C.; Hakyemez, B. Evaluating the Effectiveness of 2 Different Flow Diverter Stents Based on the Stagnation Region Formation in an Aneurysm Sac Using Lagrangian Coherent Structure. World Neurosurg.
**2019**, 127, e727–e737. [Google Scholar] [CrossRef] [PubMed] - Mutlu, O.; Olcay, A.B.; Bilgin, C.; Hakyemez, B. Evaluating the Effect of the Number of Wire of Flow Diverter Stents on the Nonstagnated Region Formation in an Aneurysm Sac Using Lagrangian Coherent Structure and Hyperbolic Time Analysis. World Neurosurg.
**2020**, 133, e666–e682. [Google Scholar] [CrossRef] [PubMed] - Mutlu, O.; Olcay, A.B.; Bilgin, C.; Hakyemez, B. Understanding the Effect of Effective Metal Surface Area of Flow Diverter Stent’s on the Patient-Specific Intracranial Aneurysm Numerical Model Using Lagrangian Coherent Structures. J. Clin. Neurosci.
**2020**, 133, e666–e682. [Google Scholar] [CrossRef] - Chester, A.H.; El-Hamamsy, I.; Butcher, J.T.; Latif, N.; Bertazzo, S.; Yacoub, M.H. The Living Aortic Valve: From Molecules to Function. Glob. Cardiol. Sci. Pract.
**2014**, 2014, 11. [Google Scholar] [CrossRef][Green Version] - Girfoglio, M.; Quaini, A.; Rozza, G. A Finite Volume Approximation of the Navier-Stokes Equations with Nonlinear Filtering Stabilization. Comput. Fluids
**2019**, 187, 27–45. [Google Scholar] [CrossRef][Green Version] - Frolov, S.V.; Sindeev, S.V.; Lischouk, V.A.; Gazizova, D.S.; Liepsch, D.; Balasso, A. A Lumped Parameter Model of Cardiovascular System with Pulsating Heart for Diagnostic Studies. J. Mech. Med. Biol.
**2017**, 17, 1750056. [Google Scholar] [CrossRef] - Young, D.F. Fluid Mechanics of Arterial Stenoses. J. Biomech. Eng.
**1979**, 101, 157–175. [Google Scholar] [CrossRef] - Meslem, A.; Bode, F.; Croitoru, C.; Nastase, I. Comparison of Turbulence Models in Simulating Jet Flow from a Cross-Shaped Orifice. Eur. J. Mech. B Fluids
**2014**, 44, 100–120. [Google Scholar] [CrossRef] - Benra, F.K.; Dohmen, H.J.; Pei, J.; Schuster, S.; Wan, B. A Comparison of One-Way and Two-Way Coupling Methods for Numerical Analysis of Fluid-Structure Interactions. J. Appl. Math.
**2011**, 2011, 853560. [Google Scholar] [CrossRef] - Salman, H.E.; Ramazanli, B.; Yavuz, M.M.; Yalcin, H.C. Biomechanical Investigation of Disturbed Hemodynamics-Induced Tissue Degeneration in Abdominal Aortic Aneurysms Using Computational and Experimental Techniques. Front. Bioeng. Biotechnol.
**2019**, 7, 111. [Google Scholar] [CrossRef] - Bathe, K.J.; Zhang, H.; Ji, S. Finite Element Analysis of Fluid Flows Fully Coupled with Structural Interactions. Comput. Struct.
**1999**, 72, 1–16. [Google Scholar] [CrossRef][Green Version] - Haller, G. Distinguished Material Surfaces and Coherent Structures in Three-Dimensional Fluid Flows. Phys. D Nonlinear Phenom.
**2001**, 149, 248–277. [Google Scholar] [CrossRef][Green Version] - Shadden, S.C.; Lekien, F.; Marsden, J.E. Definition and Properties of Lagrangian Coherent Structures from Finite-Time Lyapunov Exponents in Two-Dimensional Aperiodic Flows. Phys. D Nonlinear Phenom.
**2005**, 212, 271–304. [Google Scholar] [CrossRef] - Lekien, F.; Shadden, S.C.; Marsden, J.E. Lagrangian Coherent Structures in N-Dimensional Systems. J. Math. Phys.
**2007**, 48, 065404. [Google Scholar] [CrossRef] - Shadden, S.C.; Taylor, C.A. Characterization of Coherent Structures in the Cardiovascular System. Ann. Biomed. Eng.
**2008**, 36, 1152–1162. [Google Scholar] [CrossRef] [PubMed] - Weinberg, E.J.; Mofrad, M.R.K. Three-Dimensional, Multiscale Simulations of the Human Aortic Valve. Cardiovasc. Eng.
**2007**, 7, 140–155. [Google Scholar] [CrossRef] - Weska, R.F.; Aimoli, C.G.; Nogueira, G.M.; Leirner, A.A.; Maizato, M.J.S.; Higa, O.Z.; Polakievicz, B.; Pitombo, R.N.M.; Beppu, M.M. Natural and Prosthetic Heart Valve Calcification: Morphology and Chemical Composition Characterization. Artif. Organs
**2010**, 34, 311–318. [Google Scholar] [CrossRef] [PubMed] - Nandy, S.; Tarbell, J.M. Flush Mounted Hot Film Anemometer Measurement of Wall Shear Stress Distal to a Tri-Leaflet Valve for Newtonian and Non-Newtonian Blood Analog Fluids. Biorheology
**1987**, 24, 483–500. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) 3D aortic valve model. The grey surface is the cross-sectional plane in the longitudinal direction. The arrows show the direction of flow at the inlet and outlet. (

**b**) 2D view of the cross-sectional plane. Only the top half of the model is shown due to the symmetry. The dashed line represents the symmetry line. (

**c**) Boundary conditions shown on the cross-sectional plane. (

**d**) Transient inlet velocity profile of the blood flow. (

**e**) Mesh of the fluid domain. (

**f**) Mesh of the solid domain.

**Figure 2.**Velocity streamlines on the 3D aortic valve model. The configuration of the maximum valve opening for the healthy aortic valve is given at the bottom of the figure.

**Figure 3.**Pressure on the selected fluid particles for healthy, mildly calcified, and severely calcified aortic valves. The results are shown on the top half of the cross-sectional plane given in Figure 1. The flow was from right to left. The values presented are pressures relative to the valve outlet. The pressure was set to zero at the outlet boundary.

**Figure 4.**Backward finite-time Lyapunov exponent (FTLE) field plots of healthy (

**first column**), mildly calcified (

**second column**), and severely calcified (

**third column**) aortic valves.

**Figure 5.**Forward finite-time Lyapunov exponent (FTLE) field plots of healthy (

**first column**), mildly calcified (

**second column**), and severely calcified (

**third column**) aortic valves.

Healthy Aortic Valve | Mildly Calcified Aortic Valve | Severely Calcified Aortic Valve | |
---|---|---|---|

400th step (added for the first time) | 104,291 | 104,291 | 104,291 |

339th step (Total with readded particles) | 261,418 | 398,350 | 434,337 |

330th step (remaining particles) | 153,900 | 285,259 | 325,046 |

329th step (total with readded particles) | 254,945 | 382,472 | 421,736 |

320th step (remaining particles) | 155,146 | 268,018 | 305,521 |

319th step (total with re-added particles) | 255,743 | 367,599 | 403,394 |

310th step (remaining particles) | 165,389 | 276,347 | 309,397 |

309th step (total with readded particles) | 263,070 | 376,494 | 407977 |

300th step (remaining particles) | 185,322 | 295,500 | 323,787 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mutlu, O.; Salman, H.E.; Yalcin, H.C.; Olcay, A.B. Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis. *Fluids* **2021**, *6*, 203.
https://doi.org/10.3390/fluids6060203

**AMA Style**

Mutlu O, Salman HE, Yalcin HC, Olcay AB. Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis. *Fluids*. 2021; 6(6):203.
https://doi.org/10.3390/fluids6060203

**Chicago/Turabian Style**

Mutlu, Onur, Huseyin Enes Salman, Huseyin Cagatay Yalcin, and Ali Bahadir Olcay. 2021. "Fluid Flow Characteristics of Healthy and Calcified Aortic Valves Using Three-Dimensional Lagrangian Coherent Structures Analysis" *Fluids* 6, no. 6: 203.
https://doi.org/10.3390/fluids6060203