# Haemodynamic Analysis of Branched Endografts for Complex Aortic Arch Repair

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geometric Model

#### 2.2. Mathematical Model and Boundary Conditions

^{−3}and dynamic viscosity of 0.004 Pa.s for the properties of blood. Although laminar assumptions are often made when simulating blood flow in the aorta, the complex nature of the geometry here and the combination of peak Reynolds number (Re) and Womersley number (Wo) based on the inlet diameter (Peak Re = 3451 and Wo = 23) led to the decision to employ a model capable of capturing laminar to turbulence transition. To this end, the SST-Tran (shear stress transport—transitional) model was adopted as it has been successfully applied to low Re transitional flow in arterial stenoses [19] and the human aorta [19,20]. Numerical solutions were obtained using ANSYS CFX (v15.0, ANSYS Inc., Canonsburg, PA, USA).

#### 2.3. Haemodynamic Metrics

**τ**is the wall shear stress vector. TransWSS is defined as the “average over the cardiac cycle of WSS components perpendicular to the temporal mean WSS vector, with which endothelial cells are assumed to align” [27,28,29]. It can thus be used to quantify deviations in the direction of WSS vectors throughout the cycle, a phenomenon which demonstrates the multidirectionality and oscillations in flow and can be expressed as:

_{w}_{2}criterion for incompressible flows was evaluated and displayed as isosurfaces running through the fluid domain [30,31,32]. The λ

_{2}criterion can be expressed as:

_{2}is the second eigenvalue of the tensor

**S**, with

^{2}+ Ω^{2}**S**and

**Ω**being the symmetric and antisymmetric parts of the velocity gradient tensor, respectively.

**V**,

_{x}**v**,

_{y}**v**are the velocity components in the x, y, and z directions, respectively. Another measure of complex flow structures in the aorta is helicity, which can be quantified using a synthetic descriptor, namely, helical flow index (HFI) [33,34,35]. HFI can be calculated from local normalised helicity (LNH), which is defined as:

_{z}**s**) and time (t), with $\mathit{V}\left(\mathit{s},\text{}t\right)$ and $\omega \left(\mathit{s},\text{}t\right)$ being the velocity field and vorticity field, respectively, of the fluid domain. HFI is then computed from LNH using a set number of particles (N

_{p}) released at the model inlet and following the trajectory of each particle within a set time interval. Considering the path of an individual particle k, the helical flow index can be represented as:

_{k}is the number of points j (j = 1, … N

_{k}) in the kth trajectory in which LNH has been calculated. Combining this for all particles N

_{p}in the fluid domain, HFI is calculated as:

## 3. Results

#### 3.1. Flow Patterns and Pressure

#### 3.2. Wall Shear Stress

#### 3.3. Vortical and Helical Flow

_{2}criterion as shown in Figure 11. The λ

_{2}threshold was adjusted in order to properly isolate the relevant vortical flow through the vessel and the same has been used for all models for comparison. Figure 11 highlights the effect of geometry on the development of vortical flow, with Patient 1 and Patient 2 showing different vortical flow features along the aorta. Differences among the various cases for Patient 1 are small and are confined to the arch immediately downstream of the emerging arch branches. In addition, the helical flow index (HFI) was calculated for all cases, and the results are summarized in Table 2.

## 4. Discussion

#### 4.1. Patient Comparison

_{2}vortex identification criterion (Figure 11), Patient 1 has a greater degree of vortical structures in the aortic arch compared to Patient 2. This can be attributed to the narrowing of the lumen in the proximal arch, where the window for the emerging BSGs is located.

#### 4.2. Effect of Tunnel Branch Diameter

_{2}criterion as a means of generating isosurfaces in order to better identify the vortex cores developing in the region. The vortical structure of flow observed via the λ

_{2}isosurfaces tends to vary with the tunnel branch diameters (Figure 11), where isosurfaces near the wall tend to coincide with regions of high transWSS, which is due to changes in the direction of flow near the vessel wall. Furthermore, the helical flow index (HFI) computed here serves as an objective and reliably quantitative means for evaluating the helical nature of flow path. The values for the HFI for all cases considered here lie within the range (0.3–0.5) expected in a normal aorta [35].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Cheng, Z.; Tan, F.P.P.; Riga, C.V.; Bicknell, C.D.; Hamady, M.S.; Gibbs, R.G.J.; Wood, N.B.; Xu, X.Y. Analysis of flow patterns in a patient-specific aortic dissection model. J. Biomech. Eng.
**2010**, 132, 051007. [Google Scholar] [CrossRef] - Cheng, S. Aortic arch pathologies-incidence and natural history. Gefässchirurgie
**2016**, 21, 212–216. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nienaber, C.A.; Fattori, R.; Lund, G.; Dieckmann, C.; Wolf, W.; von Kodolitsch, Y.; Nicolas, V.; Pierangeli, A. Nonsurgical reconstruction of thoracic aortic dissection by stent–graft placement. N. Engl. J. Med.
**1999**, 340, 1539–1545. [Google Scholar] [CrossRef] [PubMed] - Bodell, B.D.; Taylor, A.C.; Patel, P.J. Thoracic endovascular aortic repair: Review of current devices and treatments options. Tech. Vasc. Interv. Radiol.
**2018**, 21, 137–145. [Google Scholar] [CrossRef] [PubMed] - Kuratani, T. Best surgical option for arch extension of type B dissection: The endovascular approach. Ann. Cardiothorac. Surg.
**2014**, 3, 292. [Google Scholar] - van Bakel, T.M.; Arthurs, C.J.; van Herwaarden, J.A.; Moll, F.L.; Eagle, K.A.; Patel, H.J.; Trimarchi, S.; Figueroa, C.A. A computational analysis of different endograft designs for Zone 0 aortic arch repair. Eur. J. Cardio-Thorac. Surg.
**2018**, 54, 389–396. [Google Scholar] [CrossRef] [Green Version] - van Bakel, T.M.; de Beaufort, H.W.; Trimarchi, S.; Marrocco-Trischitta, M.M.; Bismuth, J.; Moll, F.L.; Patel, H.J.; van Herwaarden, J.A. Status of branched endovascular aortic arch repair. Ann. Cardiothorac. Surg.
**2018**, 7, 406–413. [Google Scholar] [CrossRef] [Green Version] - Abraham, C.Z.; Rodriguez, V.M. Upcoming Technology for Aortic Arch Aneurysms. Endovasc. Today
**2015**, 46–52. Available online: https://evtoday.com/articles/2015-nov/upcoming-technology-for-aortic-arch-aneurysms (accessed on 10 August 2021). - Rylski, B.; Czerny, M. Relay
^{®}Branch: A Review of the Technology and Early Results. Suppl. Endovasc. Today Eur.**2002**, 6. Available online: https://vasculartube.com/2018/02/supplement2/relaybranch-a-review-of-the-technology-and-early-results/ (accessed on 10 August 2021). - Ferrer, C.; Cao, P. Endovascular arch replacement with a dual branched endoprosthesis. Ann. Cardiothorac. Surg.
**2018**, 7, 366–371. [Google Scholar] [CrossRef] [Green Version] - Heaton, D.H. The Next Generation of Aortic Endografts. Endovasc. Today
**2009**, 49–52. Available online: https://evtoday.com/articles/2009-jan/EVT0109_03-php (accessed on 21 August 2021). - Santos, I.C.; Rodrigues, A.; Figueiredo, L.; Rocha, L.A.; Tavares, J.M.R. Mechanical properties of stent–graft materials. J. Mater. Des. Appl.
**2012**, 226, 330–341. [Google Scholar] [CrossRef] [Green Version] - Prasad, A.; Xiao, N.; Gong, X.; Zarins, C.; Figueroa, C. A computational framework for investigating the positional stability of aortic endografts. Biomech. Model. Mechanobiol.
**2013**, 12, 869–887. [Google Scholar] [CrossRef] [PubMed] - Figueroa, C.A.; Zarins, C. Computational analysis of displacement forces acting on endografts used to treat aortic aneurysms. In Biomechanics and Mechanobiology of Aneurysms; Springer: Berlin/Heidelberg, Germany, 2011; Volume 7, pp. 221–246. [Google Scholar]
- Benard, N.; Coisne, D.; Donal, E.; Perrault, R. Experimental study of laminar blood flow through an artery treated by a stent implantation: Characterisation of intra- stent wall shear stress. J. Biomech.
**2003**, 36, 991–998. [Google Scholar] [CrossRef] - Kandail, H.; Hamady, M.; Xu, X.Y. Patient-specific analysis of displacement forces acting on fenestrated stent grafts for endovascular aneurysm repair. J. Biomech.
**2014**, 47, 3546–3554. [Google Scholar] [CrossRef] [PubMed] - Ong, C.; Xiong, F.; Kabinejadian, F.; Praveen Kumar, G.; Cui, F.; Chen, G.; Ho, P.; Leo, H. Hemodynamic analysis of a novel stent graft design with slit perforations in thoracic aortic aneurysm. J. Biomech.
**2019**, 85, 210–217. [Google Scholar] [CrossRef] [PubMed] - Raptis, A.; Xenos, M.; Kouvelos, G.; Giannoukas, A.; Matsagkas, M. Haemodynamic performance of AFX and Nellix endografts: A computational fluid dynamics study. Interact. Cardiovasc. Thorac. Surg.
**2018**, 26, 826–833. [Google Scholar] [CrossRef] [PubMed] - Tan, F.P.P.; Soloperto, G.; Bashford, S.; Wood, N.B.; Thom, S.; Hughes, A.; Xu, X.Y. Analysis of flow disturbance in a stenosed carotid artery bifurcation using two-equation transitional and turbulence models. J. Biomech. Eng.
**2008**, 130, 061008. [Google Scholar] [CrossRef] - Kousera, C.A.; Wood, N.B.; Seed, W.A.; Torii, R.; O’Regan, D.; Xu, X.Y. A numerical study of aortic flow stability and comparison with in vivo flow measurements. J. Biomech. Eng.
**2013**, 135, 011003. [Google Scholar] [CrossRef] - Tan, F.P.P.; Borghi, A.; Mohiaddin, R.H.; Wood, N.B.; Thom, S.; Xu, X.Y. Analysis of flow patterns in a patient-specific thoracic aortic aneurysm model. Comput. Struct.
**2009**, 87, 680–690. [Google Scholar] [CrossRef] - Khanafer, K.; Berguer, R. Fluid–structure interaction analysis of turbulent pulsatile flow within a layered aortic wall as related to aortic dissection. J. Biomech.
**2009**, 42, 2642–2648. [Google Scholar] [CrossRef] - Pirola, S.; Cheng, Z.; Jarral, O.A.; O’Regan, D.P.; Pepper, J.R.; Athanasiou, T.; Xu, X.Y. On the choice of outlet boundary conditions for patient-specific analysis of aortic flow using computational fluid dynamics. J. Biomech.
**2017**, 60, 15–21. [Google Scholar] [CrossRef] - Vignon-Clementel, I.E.; Figueroa, C.A.; Jansen, K.E.; Taylor, C.A. Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput. Methods Biomech. Biomed. Eng.
**2010**, 13, 625–640. [Google Scholar] [CrossRef] [Green Version] - Mohamied, Y.; Rowland, E.; Bailey, E.; Sherwin, S.; Schwartz, M.; Weinberg, P. Change of direction in the biomechanics of atherosclerosis. Ann. Biomed. Eng.
**2015**, 43, 16–25. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kazakidi, A.; Plata, A.M.; Sherwin, S.J.; Weinberg, P.D. Effect of reverse flow on the pattern of wall shear stress near arterial branches. J. R. Soc. Interface
**2011**, 8, 1594–1603. [Google Scholar] [CrossRef] [PubMed] - Mohamied, Y.; Sherwin, S.J.; Weinberg, P.D. Understanding the fluid mechanics behind transverse wall shear stress. J. Biomech.
**2017**, 50, 102–109. [Google Scholar] [CrossRef] [Green Version] - Peiffer, V.; Sherwin, S.J.; Weinberg, P.D. Computation in the rabbit aorta of a new metric—the transverse wall shear stress—to quantify the multidirectional character of disturbed blood flow. J. Biomech.
**2013**, 46, 2651–2658. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Andersson, M.; Lantz, J.; Ebbers, T.; Karlsson, M. Multidirectional WSS disturbances in stenotic turbulent flows: A pre- and post-intervention study in an aortic coarctation. J. Biomech.
**2016**, 51, 8–16. [Google Scholar] [CrossRef] [PubMed] - Dong, Y.; Yan, Y.; Liu, C. New visualization method for vortex structure in turbulence by lambda2 and vortex filaments. Appl. Math. Model.
**2016**, 40, 500–509. [Google Scholar] [CrossRef] - Jeong, J.; Hussain, F. On the identification of a vortex. J. Fluid Mech.
**1995**, 285, 69–94. [Google Scholar] [CrossRef] - Chong, M.Y.; Gu, B.; Chan, B.T.; Ong, Z.C.; Xu, X.Y.; Lim, E. Effect of intimal flap motion on flow in acute type B aortic dissection by using fluid-structure interaction. Int. J. Numer. Meth. Biomed. Engng.
**2020**, 36, e3399. [Google Scholar] [CrossRef] - Tasso, P.; Lodi Rizzini, M.; Raptis, A.; Matsagkas, M.; de Nisco, G.; Gallo, D.; Xenos, M.; Morbiducci, U. In-stent graft helical flow intensity reduces the risk of migration after endovascular aortic repair. J. Biomech.
**2019**, 94, 170–179. [Google Scholar] [CrossRef] - Morbiducci, U.; Ponzini, R.; Rizzo, G.; Cadioli, M.; Esposito, A.; de Cobelli, F.; del Maschio, A.; Montevecchi, F.; Redaelli, A. In vivo quantification of helical blood flow in human aorta by time-resolved three-dimensional cine phase contrast magnetic resonance imaging. Ann. Biomed. Eng.
**2009**, 37, 516–531. [Google Scholar] [CrossRef] [PubMed] - Morbiducci, U.; Ponzini, R.; Rizzo, G.; Cadioli, M.; Esposito, A.; Montevecchi, F.; Redaelli, A. Mechanistic insight into the physiological relevance of helical blood flow in the human aorta: An in vivo study. Biomech. Model. Mechanobiol.
**2011**, 10, 339–355. [Google Scholar] [CrossRef] - Grigioni, M.; Daniele, C.; Morbiducci, U.; del Gaudio, C.; D’Avenio, G.; Balducci, A.; Barbaro, V. A mathematical description of blood spiral flow in vessels: Application to a numerical study of flow in arterial bending. J. Biomech.
**2005**, 38, 1375–1386. [Google Scholar] [CrossRef] - Qiao, Y.; Fan, J.; Ding, Y.; Zhu, T.; Luo, K. A primary computational fluid dynamics study of pre- and post-TEVAR with intentional left subclavian artery coverage in a type B aortic dissection. J. Biomech. Eng.
**2019**, 141, 111002. [Google Scholar] [CrossRef] - Zhu, G.; Yuan, Q.; Yang, J.; Yeo, J.H. The role of the circle of Willis in internal carotid artery stenosis and anatomical variations: A computational study based on a patient-specific three-dimensional model. Biomed. Eng. Online
**2015**, 14, 107. [Google Scholar] [CrossRef] [Green Version] - DeVault, K.; Gremaud, P.A.; Novak, V.; Olufsen, M.S.; Vernières, G.; Zhao, P. Blood flow in the circle of willis: Modeling and calibration. Multiscale Modeling Simul.
**2008**, 7, 888–909. [Google Scholar] [CrossRef] [Green Version] - Qiao, Y.; Mao, L.; Ding, Y.; Fan, J.; Luo, K.; Zhu, T. Effects of in situ fenestration stent-graft of left subclavian artery on the hemodynamics after thoracic endovascular aortic repair. Vascular
**2019**, 27, 369–377. [Google Scholar] [CrossRef] [PubMed] - Qiao, Y.; Mao, L.; Ding, Y.; Fan, J.; Zhu, T.; Luo, K. Hemodynamic consequences of TEVAR with in situ double fenestrations of left carotid artery and left subclavian artery. Med. Eng. Phys.
**2020**, 76, 32–39. [Google Scholar] [CrossRef] [PubMed] - Kandail, H.; Hamady, M.; Xu, X.Y. Comparison of blood flow in branched and fenestrated stent-grafts for endovascular repair of abdominal aortic aneurysms. J. Endovasc. Ther.
**2015**, 22, 578–590. [Google Scholar] [CrossRef] [Green Version] - Qiao, Y.; Mao, L.; Zhu, T.; Fan, J.; Luo, K. Biomechanical implications of the fenestration structure after thoracic endovascular aortic repair. J. Biomech.
**2020**, 99, 109478. [Google Scholar] [CrossRef] - Liu, X.; Pu, F.; Fan, Y.; Deng, X.; Li, D.; Li, S. A numerical study on the flow of blood and the transport of LDL in the human aorta: The physiological significance of the helical flow in the aortic arch. Am. J. Physiol. Heart Circ. Physiol.
**2009**, 297, 163–170. [Google Scholar] [CrossRef] [PubMed] - Mazzi, V.; Morbiducci, U.; Calò, K.; De Nisco, G.; Lodi Rizzini, M.; Torta, E.; Caridi, G.C.A.; Chiastra, C.; Gallo, D. Wall shear stress topological skeleton analysis in cardiovascular flows: Methods and applications. Mathematics
**2021**, 9, 720. [Google Scholar] [CrossRef] - Liu, X.; Fan, Y.; Deng, X. Effect of spiral flow on the transport of oxygen in the Aorta: A numerical study. Ann. Biomed. Eng.
**2010**, 38, 917–926. [Google Scholar] [CrossRef] [PubMed] - Zhu, Y.; Zhan, W.; Hamady, M.; Xu, X.Y. A pilot study of aortic hemodynamics before and after thoracic endovascular repair with a double-branched endograft. Med. Nov. Technol. Devices
**2019**, 4, 100027. [Google Scholar] [CrossRef]

**Figure 1.**(

**Top**) Multislice CT images used to generate anatomically accurate 3D reconstruction of the aortic arch. (

**Bottom**) Reconstructed aorta model for Patient 1 with a transparent view indicating inner tunnel branches within the main endograft. The original inner tunnel branches are 12 mm in diameter and have been artificially modified to create models with branches of 10 mm and 8 mm diameter, respectively, as illustrated by the red outlines in a cross-sectional view of the ascending aorta, with the arrows indicating the three different branch diameters being investigated here.

**Figure 2.**(

**Left**) Representation of reconstructed 3D geometry showing prescribed boundary conditions, with inflow waveform imposed at the model inlet and 3−EWM prescribed at the three model outlets. (

**Right**) Pulsatile velocity waveform prescribed at the inlet.

**Figure 3.**Comparison of instantaneous velocity streamlines obtained for Patient 1 and Patient 2 at three characteristic time points in the cardiac cycle.

**Figure 4.**Comparison of instantaneous velocity streamlines obtained for cases of different diameters of inner tunnel branches for the same patient geometry, at three characteristic time points in the cardiac cycle. Patient 1 (

**top**row), model 1A (

**middle**row) and model 1B (

**bottom**row) have inner tunnel diameters of 12 mm, 10 mm and 8 mm, respectively.

**Figure 5.**Comparison between Patient 1 and model 1B of instantaneous velocity magnitudes at different cross-sectional planes at peak systole and mid-systolic deceleration phases of the cardiac cycle.

**Figure 6.**Pressure distribution throughout the vessel at peak systole for Patient 1 (left) with the pressure difference measured between planes S1 and S2 (right).

**Figure 9.**TAWSS distribution in the repaired aorta of Patient 1 with different tunnel branch diameters (1–12 mm, 1A–10 mm, 1B–8 mm).

**Figure 10.**TransWSS distribution in the repaired aorta of Patient 1 with different tunnel branch diameters (1–12 mm, 1A–10 mm, 1B–8 mm).

**Figure 11.**Isosurfaces used to depict vortical structures isolated using the λ

_{2}criterion computed for each model at mid-systolic deceleration. (1A–10 mm, 1B–8 mm).

Model | Tunnel Branch Diameter (mm) | Δ Pressure (mmHg) |
---|---|---|

Patient 1 | 12 | 10.45 |

1A | 10 | 9.95 |

1B | 8 | 9.53 |

**Table 2.**Geometric dimensions and helical flow index (HFI) for different models, where Patient 1 and Patient 2 have different overall geometries, while Patient 1, models 1A and 1B have the same aorta geometry but different tunnel branch diameters. C.S. stands for cross-sectional.

Model | Tunnel Branch Diameter (mm) | Lumen C.S. Area at Tunnel Branch Mouth (mm^{2}) | % of Lumen C.S. Area Taken up by Tunnel Branches | HFI |
---|---|---|---|---|

Patient 1 | 12 | 706 | 32.03 | 0.391 |

Model 1A | 10 | 706 | 22.25 | 0.380 |

Model 1B | 8 | 706 | 14.24 | 0.397 |

Patient 2 | 12 | 1219 | 18.55 | 0.476 |

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**MDPI and ACS Style**

Sengupta, S.; Hamady, M.; Xu, X.-Y.
Haemodynamic Analysis of Branched Endografts for Complex Aortic Arch Repair. *Bioengineering* **2022**, *9*, 45.
https://doi.org/10.3390/bioengineering9020045

**AMA Style**

Sengupta S, Hamady M, Xu X-Y.
Haemodynamic Analysis of Branched Endografts for Complex Aortic Arch Repair. *Bioengineering*. 2022; 9(2):45.
https://doi.org/10.3390/bioengineering9020045

**Chicago/Turabian Style**

Sengupta, Sampad, Mohamad Hamady, and Xiao-Yun Xu.
2022. "Haemodynamic Analysis of Branched Endografts for Complex Aortic Arch Repair" *Bioengineering* 9, no. 2: 45.
https://doi.org/10.3390/bioengineering9020045