# Fluid–Structure Interaction Modeling of Ascending Thoracic Aortic Aneurysms in SimVascular

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Overview of Mathematical Models

#### 2.1.1. Fluid Domain

#### 2.1.2. Solid Domain

#### 2.1.3. Fluid–Structure Interaction

#### 2.2. Hyperelastic Constitutive Model

#### 2.3. Boundary Conditions

#### 2.4. Mesh Development Workflow in SimVascular

## 3. Results

#### 3.1. Sensitivity Analysis

#### 3.2. Hemodynamics and Structural Capabilities

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AD | Aortic Dissection |

ALE | Arbitrary Lagrangian–Eulerian |

AI | Artificial Intelligence |

ATAA | Ascending Thoracic Aortic Aneurysm |

CSM | Computational Solid Mechanics |

CFD | Computational Fluid Dynamics |

CT | Computed Tomography scan |

ESC | European Society of Cardiology |

FSI | Fluid–Structure Interaction |

LD | Luminal Domain |

MRI | Magnetic Resonance Imaging |

NAE | Normalized Amplitude Error |

NPAE | Normalized Phase Amplitude Error |

ROI | Region of Interest |

SD | Solid Domain |

WSS | Wall Shear Stress |

FEM | Finite Element Method |

FEA | Finite Element Analysis |

BAV | Bicuspid Aortic Valve |

## References

- Dieter, R.; Dieter, R.; Dieter, R., III. Diseases of the Aorta; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar] [CrossRef]
- Pasta, S.; Rinaudo, A.; Luca, A.; Pilato, M.; Scardulla, C.; Gleason, T.G.; Vorp, D.A. Difference in hemodynamic and wall stress of ascending thoracic aortic aneurysms with bicuspid and tricuspid aortic valve. J. Biomech.
**2013**, 46, 1729–1738. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Erbel, R.; Aboyans, V.; Boileau, C.; Bossone, E.; Bartolomeo, R.D.; Eggebrecht, H.; Evangelista, A.; Falk, V.; Frank, H.; Gaemperli, O.; et al. 2014 ESC guidelines on the diagnosis and treatment of aortic diseases. Eur. Heart J.
**2014**, 35, 2873–2926. [Google Scholar] [CrossRef] [Green Version] - Maiti, S.; Thunes, J.R.; Fortunato, R.N.; Gleason, T.G.; Vorp, D.A. Computational modeling of the strength of the ascending thoracic aortic media tissue under physiologic biaxial loading conditions. J. Biomech.
**2020**, 108, 109884. [Google Scholar] [CrossRef] - Farzaneh, S.; Trabelsi, O.; Avril, S. Inverse identification of local stiffness across ascending thoracic aortic aneurysms. Biomech. Model. Mechanobiol.
**2019**, 18, 137–153. [Google Scholar] [CrossRef] [PubMed] - Trimarchi, S.; Jonker, F.H.; Hutchison, S.; Isselbacher, E.M.; Pape, L.A.; Patel, H.J.; Froehlich, J.B.; Muhs, B.E.; Rampoldi, V.; Grassi, V.; et al. Descending aortic diameter of 5.5 cm or greater is not an accurate predictor of acute type B aortic dissection. J. Thorac. Cardiovasc. Surg.
**2011**, 142, e101–e107. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Youssefi, P.; Sharma, R.; Figueroa, C.A.; Jahangiri, M. Functional assessment of thoracic aortic aneurysms—The future of risk prediction? Br. Med. Bull.
**2017**, 121, 61–71. [Google Scholar] [CrossRef] [Green Version] - Gzik-Zroska, B.; Joszko, K.; Wolański, W.; Gzik, M. Development of New Testing Method of Mechanical Properties of Porcine Coronary Arteries. In Information Technologies in Medicine; Piętka, E., Badura, P., Kawa, J., Wieclawek, W., Eds.; Advances in Intelligent Systems and Computing; Springer International Publishing: Berlin/Heidelberg, Germany, 2016; Volume 472, pp. 289–297. [Google Scholar] [CrossRef]
- Youssefi, P.; Gomez, A.; He, T.; Anderson, L.; Bunce, N.; Sharma, R.; Figueroa, C.A.; Jahangiri, M. Patient-specific computational fluid dynamics—Assessment of aortic hemodynamics in a spectrum of aortic valve pathologies. J. Thorac. Cardiovasc. Surg.
**2017**, 153, 8–20.e3. [Google Scholar] [CrossRef] [Green Version] - Mourato, A.; Brito, M.; Xavier, J.; Gil, L.; Tomás, A. On the RANS modelling of the patient-specific thoracic aortic aneurysm. In Advances and Current Trends in Biomechanics; CRC Press: Boca Raton, FL, USA, 2021; pp. 98–102. [Google Scholar]
- Shang, E.K.; Nathan, D.P.; Sprinkle, S.R.; Vigmostad, S.C.; Fairman, R.M.; Bavaria, J.E.; Gorman, R.C.; Gorman, J.H., III; Chandran, K.B.; Jackson, B.M. Peak wall stress predicts expansion rate in descending thoracic aortic aneurysms. Ann. Thorac. Surg.
**2013**, 95, 593–598. [Google Scholar] [CrossRef] [Green Version] - Gültekin, O.; Hager, S.P.; Dal, H.; Holzapfel, G.A. Computational modeling of progressive damage and rupture in fibrous biological tissues: Application to aortic dissection. Biomech. Model. Mechanobiol.
**2019**, 18, 1607–1628. [Google Scholar] [CrossRef] [Green Version] - Mousavi, S.J.; Jayendiran, R.; Farzaneh, S.; Campisi, S.; Viallon, M.; Croisille, P.; Avril, S. Coupling hemodynamics with mechanobiology in patient-specific computational models of ascending thoracic aortic aneurysms. Comput. Methods Programs Biomed.
**2021**, 205, 106107. [Google Scholar] [CrossRef] - Taghizadeh, H.; Tafazzoli-Shadpour, M.; Shadmehr, M.B. Analysis of arterial wall remodeling in hypertension based on lamellar modeling. J. Am. Soc. Hypertens.
**2015**, 9, 735–744. [Google Scholar] [CrossRef] [PubMed] - Liu, M.; Liang, L.; Sun, W. Estimation of in vivo mechanical properties of the aortic wall: A multi-resolution direct search approach. J. Mech. Behav. Biomed. Mater.
**2018**, 77, 649–659. [Google Scholar] [CrossRef] [PubMed] - Thunes, J.R.; Phillippi, J.A.; Gleason, T.G.; Vorp, D.A.; Maiti, S. Structural modeling reveals microstructure-strength relationship for human ascending thoracic aorta. J. Biomech.
**2018**, 71, 84–93. [Google Scholar] [CrossRef] [PubMed] - Capellini, K.; Vignali, E.; Costa, E.; Gasparotti, E.; Biancolini, M.E.; Landini, L.; Positano, V.; Celi, S. Computational fluid dynamic study for aTAA hemodynamics: An integrated image-based and radial basis functions mesh morphing approach. J. Biomech. Eng.
**2018**, 140, 111007. [Google Scholar] [CrossRef] - Thunes, J.R.; Pal, S.; Fortunato, R.N.; Phillippi, J.A.; Gleason, T.G.; Vorp, D.A.; Maiti, S. A structural finite element model for lamellar unit of aortic media indicates heterogeneous stress field after collagen recruitment. J. Biomech.
**2016**, 49, 1562–1569. [Google Scholar] [CrossRef] [Green Version] - Ban, E.; Cavinato, C.; Humphrey, J.D. Differential propensity of dissection along the aorta. Biomech. Model. Mechanobiol.
**2021**, 20, 895–907. [Google Scholar] [CrossRef] - Wang, R.; Yu, X.; Gkousioudi, A.; Zhang, Y. Effect of Glycation on Interlamellar Bonding of Arterial Elastin. Exp. Mech.
**2021**, 61, 81–94. [Google Scholar] [CrossRef] - Mousavi, S.J.; Farzaneh, S.; Avril, S. Patient-specific predictions of aneurysm growth and remodeling in the ascending thoracic aorta using the homogenized constrained mixture model. Biomech. Model. Mechanobiol.
**2019**, 18, 1895–1913. [Google Scholar] [CrossRef] [Green Version] - Hackstein, U.; Krickl, S.; Bernhard, S. Estimation of ARMA-model parameters to describe pathological conditions in cardiovascular system models. Inform. Med. Unlocked
**2020**, 18, 100310. [Google Scholar] [CrossRef] - Adam, C.; Fabre, D.; Mougin, J.; Zins, M.; Azarine, A.; Ardon, R.; d’Assignies, G.; Haulon, S. Pre-surgical and Post-surgical Aortic Aneurysm Maximum Diameter Measurement: Full Automation by Artificial Intelligence. Eur. J. Vasc. Endovasc. Surg.
**2021**, 62, 869–877. [Google Scholar] [CrossRef] - Cao, L.; Shi, R.; Ge, Y.; Xing, L.; Zuo, P.; Jia, Y.; Liu, J.; He, Y.; Wang, X.; Luan, S.; et al. Fully automatic segmentation of type B aortic dissection from CTA images enabled by deep learning. Eur. J. Radiol.
**2019**, 121, 108713. [Google Scholar] [CrossRef] - Sazonov, I.; Xie, X.; Nithiarasu, P. An improved method of computing geometrical potential force (GPF) employed in the segmentation of 3D and 4D medical images. Comput. Methods Biomech. Biomed. Eng. Imaging Vis.
**2017**, 5, 287–296. [Google Scholar] [CrossRef] - Rueckel, J.; Reidler, P.; Fink, N.; Sperl, J.; Geyer, T.; Fabritius, M.; Ricke, J.; Ingrisch, M.; Sabel, B. Artificial intelligence assistance improves reporting efficiency of thoracic aortic aneurysm CT follow-up. Eur. J. Radiol.
**2021**, 134, 109424. [Google Scholar] [CrossRef] [PubMed] - He, X.; Avril, S.; Lu, J. Prediction of local strength of ascending thoracic aortic aneurysms. J. Mech. Behav. Biomed. Mater.
**2021**, 115, 104284. [Google Scholar] [CrossRef] - Lindquist Liljeqvist, M.; Bogdanovic, M.; Siika, A.; Gasser, T.C.; Hultgren, R.; Roy, J. Geometric and biomechanical modeling aided by machine learning improves the prediction of growth and rupture of small abdominal aortic aneurysms. Sci. Rep.
**2021**, 11, 18040. [Google Scholar] [CrossRef] - Liu, M.; Liang, L.; Sun, W. Estimation of in vivo constitutive parameters of the aortic wall using a machine learning approach. Comput. Methods Appl. Mech. Eng.
**2019**, 347, 201–217. [Google Scholar] [CrossRef] [PubMed] - Holzapfel, G.A.; Linka, K.; Sherifova, S.; Cyron, C.J. Predictive constitutive modelling of arteries by deep learning. J. R. Soc. Interface
**2021**, 18, 20210411. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; Yang, W.; Lan, I.S.; Marsden, A.L. Fluid–structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation. Mech. Res. Commun.
**2020**, 107, 103556. [Google Scholar] [CrossRef] - Liang, L.; Liu, M.; Martin, C.; Sun, W. A machine learning approach as a surrogate of finite element analysis–based inverse method to estimate the zero-pressure geometry of human thoracic aorta. Int. J. Numer. Methods Biomed. Eng.
**2018**, 34, e3103. [Google Scholar] [CrossRef] [PubMed] - Liang, L.; Liu, M.; Martin, C.; Sun, W. A deep learning approach to estimate stress distribution: A fast and accurate surrogate of finite-element analysis. J. R. Soc. Interface
**2018**, 15, 20170844. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Lu, Q.; Feng, J.; Yu, P.; Zhang, S.; Teng, Z.; Gillard, J.H.; Song, R.; Jing, Z. A pilot study exploring the mechanisms involved in the longitudinal propagation of acute aortic dissection through computational fluid dynamic analysis. Cardiology
**2014**, 128, 220–225. [Google Scholar] [CrossRef] [PubMed] - Long Ko, J.K.; Liu, R.W.; Ma, D.; Shi, L.; Ho Yu, S.C.; Wang, D. Pulsatile hemodynamics in patient-specific thoracic aortic dissection models constructed from computed tomography angiography. J. X-ray Sci. Technol.
**2017**, 25, 233–245. [Google Scholar] [CrossRef] - Kaspera, W.; Ćmiel Smorzyk, K.; Wolański, W.; Kawlewska, E.; Hebda, A.; Gzik, M.; Ładziński, P. Morphological and Hemodynamic Risk Factors for Middle Cerebral Artery Aneurysm: A Case-Control Study of 190 Patients. Sci. Rep.
**2020**, 10, 2016. [Google Scholar] [CrossRef] - Pasta, S.; Agnese, V.; Gallo, A.; Cosentino, F.; Di Giuseppe, M.; Gentile, G.; Raffa, G.M.; Maalouf, J.F.; Michelena, H.I.; Bellavia, D.; et al. Shear Stress and Aortic Strain Associations with Biomarkers of Ascending Thoracic Aortic Aneurysm. Ann. Thorac. Surg.
**2020**, 110, 1595–1604. [Google Scholar] [CrossRef] [PubMed] - Condemi, F.; Campisi, S.; Viallon, M.; Troalen, T.; Xuexin, G.; Barker, A.; Markl, M.; Croisille, P.; Trabelsi, O.; Cavinato, C.; et al. Fluid-and biomechanical analysis of ascending thoracic aorta aneurysm with concomitant aortic insufficiency. Ann. Biomed. Eng.
**2017**, 45, 2921–2932. [Google Scholar] [CrossRef] [PubMed] - Simão, M.; Ferreira, J.M.; Tomas, A.C.; Fragata, J.; Ramos, H.M. Aorta ascending aneurysm analysis using CFD models towards possible anomalies. Fluids
**2017**, 2, 31. [Google Scholar] [CrossRef] [Green Version] - Tomaszewski, M.; Sybilski, K.; Małachowski, J.; Wolański, W.; Buszman, P.P. Numerical and experimental analysis of balloon angioplasty impact on flow hemodynamics improvement. Acta Bioeng. Biomech.
**2020**, 22, 169–183. [Google Scholar] [CrossRef] - Yeh, H.H.; Rabkin, S.W.; Grecov, D. Hemodynamic assessments of the ascending thoracic aortic aneurysm using fluid–structure interaction approach. Med. Biol. Eng. Comput.
**2018**, 56, 435–451. [Google Scholar] [CrossRef] - Chen, H.; Peelukhana, S.; Berwick, Z.; Kratzberg, J.; Krieger, J.; Roeder, B.; Chambers, S.; Kassab, G. Editor’s Choice–Fluid–Structure Interaction Simulations of Aortic Dissection with Bench Validation. Eur. J. Vasc. Endovasc. Surg.
**2016**, 52, 589–595. [Google Scholar] [CrossRef] [Green Version] - Mendez, V.; Di Giuseppe, M.; Pasta, S.; Giuseppe, M.D.; Pasta, S. Comparison of hemodynamic and structural indices of ascending thoracic aortic aneurysm as predicted by 2-way FSI, CFD rigid wall simulation and patient-specific displacement-based FEA. Comput. Biol. Med.
**2018**, 100, 221–229. [Google Scholar] [CrossRef] - Alimohammadi, M.; Sherwood, J.M.; Karimpour, M.; Agu, O.; Balabani, S.; Díaz-Zuccarini, V. Aortic dissection simulation models for clinical support: Fluid–structure interaction vs. rigid wall models. Biomed. Eng. Online
**2015**, 14, 34. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zouggari, L.; Bou-said, B.; Massi, F.; Culla, A.; Millon, A. The Role of Biomechanics in the Assessment of Carotid Atherosclerosis Severity: A Numerical Approach. World J. Vasc. Surg.
**2018**, 1, 1007. [Google Scholar] - Ateshian, G.A.; Shim, J.J.; Maas, S.A.; Weiss, J.A. Finite Element Framework for Computational Fluid Dynamics in FEBio. J. Biomech. Eng.
**2018**, 140, 021001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lopes, D.; Agujetas, R.; Puga, H.; Teixeira, J.; Lima, R.; Alejo, J.; Ferrera, C. Analysis of finite element and finite volume methods for fluid–structure interaction simulation of blood flow in a real stenosed artery. Int. J. Mech. Sci.
**2021**, 207, 106650. [Google Scholar] [CrossRef] - Wolański, W.; Gzik-Zroska, B.; Joszko, K.; Gzik, M.; Sołtan, D. Numerical Analysis of Blood Flow Through Artery with Elastic Wall of a Vessel. In Innovations in Biomedical Engineering; Gzik, M., Tkacz, E., Paszenda, Z., Piętka, E., Eds.; Advances in Intelligent Systems and Computing; Springer International Publishing: Berlin/Heidelberg, Germany, 2017; Volume 526, pp. 193–200. [Google Scholar] [CrossRef]
- Donea, J.; Huerta, A.; Ponthot, J. Chapter 14: Arbitrary Lagrangian–Eulerian Methods. Encycl. Comput.
**2004**, 1, 1–25. [Google Scholar] - Karimi, S.; Dabagh, M.; Vasava, P.; Dadvar, M.; Dabir, B.; Jalali, P. Effect of rheological models on the hemodynamics within human aorta: CFD study on CT image-based geometry. J. Non–Newton. Fluid Mech.
**2014**, 207, 42–52. [Google Scholar] [CrossRef] - Di Martino, E.S.; Guadagni, G.; Fumero, A.; Ballerini, G.; Spirito, R.; Biglioli, P.; Redaelli, A.; Martino, E.S.D.; Guadagni, G.; Fumero, A.; et al. Fluid–structure interaction within realistic three-dimensional models of the aneurysmatic aorta as a guidance to assess the risk of rupture of the aneurysm. Med. Eng. Phys.
**2001**, 23, 647–655. [Google Scholar] [CrossRef] - Wan Ab Naim, W.N.; Ganesan, P.B.; Sun, Z.; Chee, K.H.; Hashim, S.A.; Lim, E. A perspective review on numerical simulations of hemodynamics in aortic dissection. Sci. World J.
**2014**, 2014, 652520. [Google Scholar] [CrossRef] - Canchi, T.; Kumar, S.D.; Ng, E.Y.K.; Narayanan, S. A Review of Computational Methods to Predict the Risk of Rupture of Abdominal Aortic Aneurysms. BioMed Res. Int.
**2015**, 2015, 861627. [Google Scholar] [CrossRef] [Green Version] - Azadani, A.N.; Chitsaz, S.; Matthews, P.B.; Jaussaud, N.; Leung, J.; Tsinman, T.; Ge, L.; Tseng, E.E. Comparison of mechanical properties of human ascending aorta and aortic sinuses. Ann. Thorac. Surg.
**2012**, 93, 87–94. [Google Scholar] [CrossRef] - Deplano, V.; Boufi, M.; Gariboldi, V.; Loundou, A.D.; D’Journo, X.B.; Cautela, J.; Djemli, A.; Alimi, Y.S. Mechanical characterisation of human ascending aorta dissection. J. Biomech.
**2019**, 94, 138–146. [Google Scholar] [CrossRef] [PubMed] - Kim, B.; Lee, S.B.; Lee, J.; Cho, S.; Park, H.; Yeom, S.; Park, S.H. A comparison among Neo-Hookean model, Mooney-Rivlin model, and Ogden model for chloroprene rubber. Int. J. Precis. Eng. Manuf.
**2012**, 13, 759–764. [Google Scholar] [CrossRef] - Bäumler, K.; Vedula, V.; Sailer, A.M.; Seo, J.; Chiu, P.; Mistelbauer, G.; Chan, F.P.; Fischbein, M.P.; Marsden, A.L.; Fleischmann, D. Fluid–structure interaction simulations of patient-specific aortic dissection. Biomech. Model. Mechanobiol.
**2020**, 19, 1607–1628. [Google Scholar] [CrossRef] [PubMed] - Vignon-Clementel, I.E.; Alberto Figueroa, C.; Jansen, K.E.; Taylor, C.A. Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 3776–3796. [Google Scholar] [CrossRef] - Hsu, M.C.; Bazilevs, Y. Blood vessel tissue prestress modeling for vascular fluid–structure interaction simulation. Finite Elem. Anal. Des.
**2011**, 47, 593–599. [Google Scholar] [CrossRef] - Fonken, J.H.; Maas, E.J.; Nievergeld, A.H.; van Sambeek, M.R.; van de Vosse, F.N.; Lopata, R.G. Ultrasound-Based Fluid–Structure Interaction Modeling of Abdominal Aortic Aneurysms Incorporating Pre-stress. Front. Physiol.
**2021**, 12, 717593. [Google Scholar] [CrossRef] - Puyvelde, J.V.; Verbeken, E.; Verbrugghe, P.; Herijgers, P.; Meuris, B. Aortic wall thickness in patients with ascending aortic aneurysm versus acute aortic dissection. Eur. J. Cardio-Thorac. Surg.
**2015**, 49, 756–762. [Google Scholar] [CrossRef] [Green Version] - Agnese, V.; Pasta, S.; Michelena, H.I.; Minà, C.; Romano, G.M.; Carerj, S.; Zito, C.; Maalouf, J.F.; Foley, T.A.; Raffa, G.; et al. Patterns of ascending aortic dilatation and predictors of surgical replacement of the aorta: A comparison of bicuspid and tricuspid aortic valve patients over eight years of follow-up. J. Mol. Cell. Cardiol.
**2019**, 135, 31–39. [Google Scholar] [CrossRef] - DeCampli, W.M. Ascending aortopathy with bicuspid aortic valve: More, but not enough, evidence for the hemodynamic theory. J. Thorac. Cardiovasc. Surg.
**2017**, 153, 6–7. [Google Scholar] [CrossRef] [Green Version]

**Figure 4.**Standard mesh shape with our new approach using a two-layer solid domain (red) and lumen domain (blue) containing concentric mesh. “Point A”, “Point B” and “Slice 1” location for results analysis.

**Figure 8.**Velocity variation of each numerical model for the time frame 245 $\mathrm{m}\mathrm{s}$ in parallel with the computational time required for the “SD mesh size variation” analysis.

**Figure 12.**Velocity variation of each numerical model for the time frame 245 $\mathrm{m}\mathrm{s}$ in parallel with the computational time required for the “LD mesh size variation” analysis.

**Figure 13.**Velocity magnitude, displacement magnitude and wall shear stress distribution analysis at 150 and 300 $\mathrm{m}\mathrm{s}$.

Fluid density | $\rho $ | 1.060 $\mathrm{g}\xb7{\mathrm{cm}}^{-3}$ |

Fluid viscosity | $\mu $ | 0.04 $\mathrm{P}$ |

Solid density | ${\rho}_{s}$ | 1.120 $\mathrm{g}\xb7{\mathrm{cm}}^{-3}$ |

Young’s modulus | E | 10 $\mathrm{Mdyn}\xb7{\mathrm{cm}}^{-2}$ |

Poisson ratio | $\nu $ | $0.49$ |

${\mathit{R}}_{\mathit{p}}$ (dyn$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$s$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$cm${}^{-5}$) | C (cm${}^{5}$$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$dyn${}^{-1}$) | ${\mathit{R}}_{\mathit{d}}$ (dyn$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$s$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$cm${}^{-5}$) | |
---|---|---|---|

Thoracic aorta | 39 | $4.82\times {10}^{-4}$ | 1016 |

Brachiocephalic trunk | 139 | $8.74\times {10}^{-5}$ | 3637 |

Left common carotid artery | 520 | $7.70\times {10}^{-5}$ | 13,498 |

Left subclavian artery | 420 | $9.34\times {10}^{-5}$ | 10,969 |

**Table 3.**Variation of the number of elements in both domains in the “SD mesh size variation” analysis.

Lumen Domain | Solid Domain | |||
---|---|---|---|---|

Nomenclature | Elem. Size (mm) | Elem. Number | Elem. Size (mm) | Elem. Number |

E1 | 1.5 | 1,132,012 | 1.4 | 93,960 |

E2 | 1.5 | 1,132,012 | 1.3 | 106,208 |

E3 | 1.5 | 1,132,012 | 1.2 | 128,791 |

E4 | 1.5 | 1,132,012 | 1.1 | 154,963 |

E5 | 1.5 | 1,132,012 | 1.0 | 184,184 |

**Table 4.**Normalized amplitude error, ${A}_{\chi}$, and normalized phase amplitude error, ${\phi}_{\chi}$, for all previous measurements of “SD mesh size variation” using “E5” as reference.

Error | E1 | E2 | E3 | E4 | E5 | |
---|---|---|---|---|---|---|

Velocity | ${A}_{\chi}$ | 0.972 | 0.984 | 1.006 | 1.001 | 1 |

${\phi}_{\chi}$ | 0.028 | 0.016 | 0.006 | 0.001 | 0 | |

Area Variation | ${A}_{\chi}$ | 0.879 | 0.942 | 1.060 | 1.009 | 1 |

${\phi}_{\chi}$ | 0.122 | 0.058 | 0.060 | 0.009 | 0 | |

WSS | ${A}_{\chi}$ | 0.971 | 1.010 | 1.018 | 0.997 | 1 |

${\phi}_{\chi}$ | 0.029 | 0.010 | 0.018 | 0.003 | 0 |

**E1**…

**E5**represent the five mesh densities of the models in ascending order.

**Table 5.**Variation of the number of elements in both domains in the “LD mesh size variation” analysis.

Lumen Domain | Solid Domain | |||
---|---|---|---|---|

Nomenclature | Elem. Size (mm) | Elem. Number | Elem. Size (mm) | Elem. Number |

F1 | 1.50 | 1,132,012 | 1.1 | 154,963 |

F2 | 1.35 | 1,531,120 | 1.1 | 264,022 |

F3 | 1.30 | 1,700,764 | 1.1 | 280,207 |

**Table 6.**Normalized amplitude error, ${A}_{\chi}$, and normalized phase amplitude error, ${\phi}_{\chi}$, for all previous measurements of “LD mesh size variation” using “F3” as the reference.

Error | F1 | F2 | F3 | |
---|---|---|---|---|

Velocity | ${A}_{\chi}$ | 1.118 | 1.034 | 1 |

${\phi}_{\chi}$ | 0.118 | 0.034 | 0 | |

Area Variation | ${A}_{\chi}$ | 1.186 | 1.045 | 1 |

${\phi}_{\chi}$ | 0.186 | 0.045 | 0 | |

WSS | ${A}_{\chi}$ | 1.355 | 0.998 | 1 |

${\phi}_{\chi}$ | 0.355 | 0.001 | 0 |

**F1**…

**F3**represent the three mesh densities of the models, in ascending order.

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

© 2022 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**

Valente, R.; Mourato, A.; Brito, M.; Xavier, J.; Tomás, A.; Avril, S.
Fluid–Structure Interaction Modeling of Ascending Thoracic Aortic Aneurysms in SimVascular. *Biomechanics* **2022**, *2*, 189-204.
https://doi.org/10.3390/biomechanics2020016

**AMA Style**

Valente R, Mourato A, Brito M, Xavier J, Tomás A, Avril S.
Fluid–Structure Interaction Modeling of Ascending Thoracic Aortic Aneurysms in SimVascular. *Biomechanics*. 2022; 2(2):189-204.
https://doi.org/10.3390/biomechanics2020016

**Chicago/Turabian Style**

Valente, Rodrigo, André Mourato, Moisés Brito, José Xavier, António Tomás, and Stéphane Avril.
2022. "Fluid–Structure Interaction Modeling of Ascending Thoracic Aortic Aneurysms in SimVascular" *Biomechanics* 2, no. 2: 189-204.
https://doi.org/10.3390/biomechanics2020016