# The Hemodynamic Effect of Modified Blalock–Taussig Shunt Morphologies: A Computational Analysis Based on Reduced Order Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. The Mesh Morphing Technique

#### 2.2. Reduced Order Modeling

**x**(x,y,z) is considered as the aggregation of the product of the response surface factor and the mode data for all of the modes r:

## 3. Materials and Methods

#### 3.1. Patient-Specific Data Pre-Processing

^{®}SpaceClaim

^{®}software. Exploiting the advanced features of this software, a semi-automated procedure was implemented to create the CAD model of MBTS. In order to set up this parametric-based procedure, the characteristic geometrical quantities of the MBTS were assigned. These tools allowed us to create the geometry of the MBTS with a high level of flexibility. The obtained aortic CAD model with the different components highlighted is shown in Figure 3.

#### 3.2. Mesh Morphing Set-Up

^{®}Fluent

^{®}in order to obtain an initial computational mesh to be morphed. Source points organized into four sets were selected on the implant’s district, as depicted in Figure 4a,b. The morphing action was constrained inside the borders of the blue cylinder indicated in Figure 4a. The four different sets of source points were generated and placed at:

- the inferior boundary of the shunt’s geometry (ISP)
- the central segment of the shunt, correspondingto the maximum cross-sectional diameter (DSP)
- the cylindrical periphery of the shunt’s geometry (CSP)
- the superior boundary of the shunt’s geometry (SSP)

- dl-1-vol—Rigid motion of ISP along the ±x direction: sliding of the shunt’s root along the length of the pulmonary artery (Figure 5a).
- dl-2-vol—Rigid motion of SSP along the ±x direction: sliding of the shunt’s top segment along the length of the right subclavian artery (Figure 5b).
- dr-1-vol—Rigid motion of ISP along the ±y direction: sliding of the shunt’s root along the width of the pulmonary artery (Figure 5c).
- dr-2-vol—Rigid motion of SSP along the ±y direction: sliding of the shunt’s top segment along the width of the right subclavian artery (Figure 5d).
- mid-dl-vol2—Rigid motion of DSP along ±x direction: inflation or deflation towards the ±x axis (Figure 5e).
- mid-dl-vol—Rigid motion of CSP along ±x direction: inflation or deflation towards the ±x axis (Figure 5f).
- mid-dr-vol2—Rigid motion of DSP along ±y direction: inflation or deflation towards the ±y axis (Figure 5g).
- mid-dr-vol—Rigid motion of CSP along ±y direction: inflation or deflation towards the ±y axis (Figure 5h).
- rl-1-vol—Rotation of ISP with respect to the y axis of LCRS: the shunt’s root, which is located on pulmonary boundary, is revealed or hidden on the zx plane (Figure 5i).
- rl-2-vol—Rotation of SSP with respect to the y axis of LCRS: the shunt’s upper segment which is located on the right subclavian aortic boundary, is revealed or hidden on the zx plane (Figure 5j).
- rr-1-vol—Rotation of ISP with respect to the x axis of LCRS: the shunt’s root is revealed or hidden on the yx plane (Figure 5k).
- rr-2-vol—Rotation of SSP with respect to the x axis of LCRS: the shunt’s upper segment is revealed or hidden on the yx plane (Figure 5l).

#### 3.3. CFD Set-Up

^{®}Fluent

^{®}- Release 21R1 was used to solve the governing flow equations. A steady state flow regime was adopted and the k-$\omega $ SST model was employed to integrate the turbulence [40,41]. An unstructured mesh consisting of 1.79 million polyhedral elements was built. In order to capture the laminar ongoing phenomena close to the boundaries, ten inflation layers with a growth rate of 1.05 and a total thickness of 2.5 mm were introduced. The blood was considered as a Newtonian fluid with a density and viscosity equal to 1060 kg/m ${}^{3}$ and 3.5 $\xb7{10}^{-3}$ Pa s, respectively. The systolic peak flow data were used as the inlet boundary condition by applying a velocity condition equal to 0.24 m/s at the aortic valve section. For the aortic branches, the Right/Left Common Carotid Arteries (RCCA/LCCA), Left Subclavian Artery (LSA), and descending aorta (DA) were handled with a constant pressure condition of 55 mmHg. The Right Subclavian Artery (RSA) was treated with a 48 mmHg condition. For the pulmonary branch, the right and the left pulmonary artery (RPA/LPA) boundary conditions were set to a constant pressure condition of 8 mmHg and 7 mmHg, respectively. These pressure values correspond to the physiological pressure conditions of an infant. A wall condition was imposed on the pulmonary valve boundary to simulate the pulmonary atresia condition.

#### 3.4. ROM Set-Up

^{®}DesignXplorer

^{®}-Release 22R1. In order to ensure the optimal spatial allocation of the investigated scenarios, an Optimal Space-Filling algorithm was employed. Starting from the baseline configuration, a total of 150 runs were performed by combining the shape modifiers described in Section 3.2. The associated snapshots were imported in ROM Builder (ANSYS

^{®}Electronics Desktop ${}^{\mathrm{TM}}$ - Release 22R1) and used to feed the SVD algorithm. The number of modes chosen for the Digital Twin was 27 for the pressure, 18 for the velocity, and 22 for the wall shear stress (WSS) evaluation. The decision regarding the number of selected modes was carried out with the aim of reaching the best compromise between the accuracy and responsiveness of ROM. For evaluation of the accuracy of the generated ROM, the original full CFD and the reduced computational models were compared. The error between the two models was calculated according to Equation (9)

## 4. Results

#### 4.1. Mesh Morphing Verification

#### 4.2. ROM Verification

#### 4.3. ROM Consumption

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CAE | Computer-Aided Engineering |

CFD | Computational Fluid Dynamics |

CPU | Central Processing Unit |

CSP | Cylindrical Source Points |

CT | Computed Tomography |

DA | Descending Aorta |

DOE | Design of Experiments |

DSP | Diameter Source Points |

GARS | Genetic Aggregation Response Surface |

ISP | Inferior Source Points |

LCRS | Local Coordinate Reference System |

LCRS | Local Coordinate Reference System |

MBTS | Modified Blalock–Taussig Shunt |

MDT | Medical Digital Twin |

RBFs | Radial Basis Functions |

RCCA/LCCA | Right/Left Common Carotid Arteries |

ROM | Reduced Order Modeling |

RPA/LPA | Right/Left Pulmonary Artery |

RS | Response Surface |

RSA/LSA | Right/Left Subclavian Artery |

SSP | Superior Source Points |

SVD | Singular Value Decomposition |

WSS | Wall Shear Stresses |

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**Figure 1.**ROM workflow with the description of its main sections and the estimation of each step’s time frame.

**Figure 4.**Location of the source points on the shunt’s geometry: SSP, DSP, and ISP sets with the cylindrical morphing domain (

**a**) and CSP set with the LCRS system (

**b**).

**Figure 5.**Morphing effect of the twelve RBF shape modifiers: dl-1-vol (

**a**); dl-2-vol (

**b**); dr-1-vol (

**c**); dr-2-vol (

**d**); mid-dl-vol2 (

**e**); mid-dl-vol (

**f**); mid-dr-vol2 (

**g**); mid-dr-vol (

**h**); rl-1-vol (

**i**); rl-2-vol (

**j**); rr-1-vol (

**k**); rr-2-vol (

**l**).

**Figure 6.**Velocity map comparison of the shunt’s longitudinal cross-section between the CFD (

**a**–

**d**) and the ROM (

**e**–

**h**) flow field (

**e**–

**h**) for four different validation scenarios: No. 135 (

**a**–

**e**), No. 1 (

**b**–

**f**), No. 34 (

**c**–

**g**), No. 146 (

**d**–

**h**).

**Figure 7.**Pressuredistribution comparison between the CFD (

**a**–

**d**) and the ROM (

**e**–

**h**) flow field (

**e**–

**h**) for four different validation scenarios using two perspectives: No. 76 (

**a**–

**e**), No. 87 (

**b**–

**f**), No. 99 (

**c**–

**g**) and No. 101 (

**d**–

**h**).

**Figure 8.**Wall Shear Stress distribution comparison between the CFD (

**a**–

**d**) and the ROM (

**e**–

**h**) flow field (

**e**–

**h**) for four different validation scenarios using two perspectives: No.65 (

**a**–

**e**), No. 52 (

**b**–

**f**), No. 63 (

**c**–

**g**), No. 87 (

**d**–

**h**).

**Figure 9.**Maximumabsolute error of the velocity in scenario No. 34 (

**a**), pressure in scenario No. 101 (

**b**), and wall-shear stress in scenario No. 87 (

**c**) ROM.

**Figure 10.**Visualization of the Digital Twin GUI with the interactive environment of the ROM consumption.

**Figure 11.**Timeframe comparison between the CFD and the ROM for exploration of possible MBTS configurations.

**Table 1.**Four worst-case scenarios presenting the greatest mesh quality degradation. The amplification shape factors close to the range boundaries are shown in bold font.

Shape factor | Scenario 23 | Scenario 43 | Scenario 123 | Scenario 142 |
---|---|---|---|---|

dl-1-vol | −0.26 | −0.49 | −0.36 | −0.39 |

dl-2-vol | −0.03 | −0.27 | 0.48 | −0.46 |

dr-1-vol | −0.17 | 3.30 | 0.70 | −2.23 |

dr-2-vol | −3.30 | 0.90 | 4.23 | −0.70 |

mid-dl-vol2 | −0.04 | 0.30 | −0.08 | 0.34 |

mid-dl-vol | −0.02 | 0.18 | −0.18 | 0.14 |

mid-dr-vol2 | 0.48 | −0.36 | -0.08 | 0.34 |

mid-dr-vol | −0.25 | 0.24 | −0.34 | 0.05 |

rl-1-vol | −4.63 | 4.43 | −1.83 | 3.30 |

rl-2-vol | −2.83 | 0.57 | 3.90 | 2.43 |

rr-1-vol | −0.57 | 2.50 | −1.37 | −3.10 |

rr-2-vol | 4.77 | −0.37 | 3.17 | −2.37 |

Cell Squish | 0.982 | 0.994 | 0.906 | 0.973 |

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**MDPI and ACS Style**

Kardampiki, E.; Vignali, E.; Haxhiademi, D.; Federici, D.; Ferrante, E.; Porziani, S.; Chiappa, A.; Groth, C.; Cioffi, M.; Biancolini, M.E.;
et al. The Hemodynamic Effect of Modified Blalock–Taussig Shunt Morphologies: A Computational Analysis Based on Reduced Order Modeling. *Electronics* **2022**, *11*, 1930.
https://doi.org/10.3390/electronics11131930

**AMA Style**

Kardampiki E, Vignali E, Haxhiademi D, Federici D, Ferrante E, Porziani S, Chiappa A, Groth C, Cioffi M, Biancolini ME,
et al. The Hemodynamic Effect of Modified Blalock–Taussig Shunt Morphologies: A Computational Analysis Based on Reduced Order Modeling. *Electronics*. 2022; 11(13):1930.
https://doi.org/10.3390/electronics11131930

**Chicago/Turabian Style**

Kardampiki, Eirini, Emanuele Vignali, Dorela Haxhiademi, Duccio Federici, Edoardo Ferrante, Stefano Porziani, Andrea Chiappa, Corrado Groth, Margherita Cioffi, Marco Evangelos Biancolini,
and et al. 2022. "The Hemodynamic Effect of Modified Blalock–Taussig Shunt Morphologies: A Computational Analysis Based on Reduced Order Modeling" *Electronics* 11, no. 13: 1930.
https://doi.org/10.3390/electronics11131930