# Wall Shear Stress Analysis and Optimization in Tissue Engineering TPMS Scaffolds

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Scaffold Design

^{®}[25] was used to apply a three-step Laplacian smoothing process followed by a three-step isotropic explicit remeshing process to obtain the scaffold with smooth surfaces (Figure 2b).

#### 2.2. Surface Topology CFD Analysis

^{®}Ansys

^{®}(Ansys Inc., Canonsburg, PA, USA), which has already proven to be a useful tool for analyzing the fluidic properties of scaffolds [26,27]. In terms of the parameters for the Fluent solver, the fluid chosen for the simulations was water, in accordance with previous studies [22,28], with a density of 1000 kg/m

^{3}and dynamic viscosity of 0.001 Pa.s. The fluid was also assumed to pass through the scaffold in a steady state laminar flow.

^{2}; ΔP is the pressure drop expressed in Pa; L is the length of the scaffold expressed in m; v is the inlet velocity of the fluid and µ is the dynamic viscosity of the water which is 0.001 Pa*s.

#### 2.3. Structural Optimization Process

_{max}); and the upper (U

_{j}) and lower (L

_{j}) boundaries of each parameter. In this work, a cooling schedule variable of 0.9 was chosen, alongside a maximum of 100 iterations. The objective function was defined as the normalized difference between the target (5 mPa) and the real average calculated WSS, and the design variables are the side length of the unit cell of the TPMS structure and the scaffold porosity. These variables are limited to the lower and upper bounds of 1 and 10 mm for the unit length and 60% and 80% for the porosity, respectively. These values were chosen to maintain pore sizes that promote cellular processes.

^{®}(Mathworks, Natick, MA, USA) script to calculate the objective function value of the designed scaffold and recorded it as the current solution (Z

_{c}). Subsequently, the algorithm defined an initial temperature (T

_{j}) equal to 1/5 of the current solution. The temperature was cooled after each iteration by multiplying it by the previously established cooling schedule variable (a).

_{n}). If Z

_{n}was lower than or equal to Z

_{c}, then the code always accepted the new value. Otherwise, the chance of accepting the new value was equal to ${\mathrm{e}}^{{\left(\right(\mathrm{Z}}_{\mathrm{c}}{-\mathrm{Z}}_{\mathrm{n}}{)/\mathrm{T}}_{\mathrm{j}})}$. The code then recorded the value of the parameters and objective function value and proceeded with the temperature cooling. Finally, when the algorithm ended, it returned the history of the optimization process.

## 3. Results

#### 3.1. Scaffold Design Parameters

#### 3.2. Surface Topology CFD Analysis

#### 3.3. Optimization Method

## 4. Discussion

_{j}of the algorithm was at its lowest).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Example meshes: (

**a**) original hexahedral scaffold mesh; (

**b**) smoothed hexahedral scaffold mesh.

**Figure 3.**Comparison between different values of the world unit parameter, resulting in tetrahedral meshes with varying element numbers, regarding the test scaffold’s: (

**a**) pressure drop; (

**b**) average WSS.

World Unit | 0.095 | 0.1 | 0.125 | 0.15 | 0.175 | 0.2 | 0.25 | 0.3 |

# Elements | 936246 | 840421 | 662816 | 503793 | 282630 | 203490 | 118248 | 78767 |

Pressure Drop (mPa) | 17.846 | 17.817 | 17.713 | 17.587 | 17.401 | 17.283 | 16.937 | 16.635 |

WSS (mPa) | 1.029 | 1.021 | 1.028 | 1.018 | 1.016 | 1.013 | 0.998 | 0.990 |

World Unit | 0.095 | 0.1 | 0.125 | 0.15 | 0.175 | 0.2 | 0.25 | 0.3 |

# Elements | 844429 | 777951 | 645069 | 499543 | 275998 | 190338 | 110911 | 72858 |

Pressure Drop (mPa) | 11.319 | 11.305 | 11.260 | 11.210 | 11.131 | 11.061 | 10.922 | 10.790 |

WSS (mPa) | 0.770 | 0.770 | 0.774 | 0.768 | 0.781 | 0.785 | 0.797 | 0.811 |

**Table 3.**Comparison of the pressure drop and average WSS between scaffolds with different length of the empty chamber before and after the scaffold.

SG70 Half Chamber | SG70 Full Chamber | SD70 Half Chamber | SD70 Full Chamber | |
---|---|---|---|---|

Pressure Drop (mPa) | 11.212 | 11.187 | 17.587 | 17.555 |

Relative Difference (%) | −0.224 | −0.172 | ||

Average WSS (mPa) | 0.768 | 0.768 | 1.018 | 1.021 |

Relative Difference (%) | −0.028 | 0.287 |

**Table 4.**Comparison of the average WSS between SG scaffolds with and without surface topology smoothing.

SG60 Original | SG60 Smoothed | SG70 Original | SG70 Smoothed | SG80 Original | SG80 Smoothed | |
---|---|---|---|---|---|---|

Average WSS (mPa) | 0.700 | 0.930 | 0.568 | 0.768 | 0.455 | 0.625 |

Relative Difference (%) | 32.843 | 35.221 | 37.320 |

**Table 5.**Comparison of the average WSS between SD scaffolds with and without surface topology smoothing.

SD60 Original | SD60 Smoothed | SD70 Original | SD70 Smoothed | SD80 Original | SD80 Smoothed | |
---|---|---|---|---|---|---|

Average WSS (mPa) | 0.955 | 1.278 | 0.749 | 1.018 | 0.607 | 0.821 |

Relative Difference (%) | 33.826 | 35.915 | 35.180 |

G60 | G70 | G80 | SD60 | SD70 | SD80 | |
---|---|---|---|---|---|---|

Original | 599.5 | 700.5 | 799.8 | 600.8 | 699.1 | 800.5 |

Smoothed | 599.4 | 701.4 | 801.6 | 599.1 | 699.2 | 801.71 |

Relative Difference (%) | −0.012 | 0.128 | 0.216 | −0.280 | 0.026 | 0.155 |

Optimization Iteration | Cubic Unit Length (mm) | Porosity (%) | Average WSS (mPa) |
---|---|---|---|

1 | 5.500 | 70 | 9.277 |

2 | 7.375 | 73.1 | 6.515 |

3 | 7.735 | 70.8 | 6.531 |

4 | 9.523 | 65.4 | 5.978 |

6 | 10.000 | 65.4 | 5.697 |

11 | 10.000 | 68.2 | 5.361 |

12 | 10.000 | 71.2 | 5.039 |

Optimization Iteration | Cubic Unit Length (mm) | Porosity (%) | Average WSS (mPa) |
---|---|---|---|

1 | 5.500 | 70 | 6.916 |

2 | 6.183 | 67.2 | 6.522 |

5 | 6.081 | 66.5 | 6.710 |

7 | 7.320 | 71.6 | 5.041 |

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**MDPI and ACS Style**

Pires, T.H.V.; Dunlop, J.W.C.; Castro, A.P.G.; Fernandes, P.R.
Wall Shear Stress Analysis and Optimization in Tissue Engineering TPMS Scaffolds. *Materials* **2022**, *15*, 7375.
https://doi.org/10.3390/ma15207375

**AMA Style**

Pires THV, Dunlop JWC, Castro APG, Fernandes PR.
Wall Shear Stress Analysis and Optimization in Tissue Engineering TPMS Scaffolds. *Materials*. 2022; 15(20):7375.
https://doi.org/10.3390/ma15207375

**Chicago/Turabian Style**

Pires, Tiago H. V., John W. C. Dunlop, André P. G. Castro, and Paulo R. Fernandes.
2022. "Wall Shear Stress Analysis and Optimization in Tissue Engineering TPMS Scaffolds" *Materials* 15, no. 20: 7375.
https://doi.org/10.3390/ma15207375