# In-Silico Prediction of Mechanical Behaviour of Uniform Gyroid Scaffolds Affected by Its Design Parameters for Bone Tissue Engineering Applications

^{1}

^{2}

^{3}

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*Computation*—Computational Engineering)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{3}. In the FE simulation, two plates (5.5 × 5.5 × 0.5 mm

^{3}) were kept above and below the scaffolds for uniform compression.

#### 2.1. Design of Scaffolds

#### 2.1.1. Implicit Description of TPMS

#### 2.1.2. Signed Distance Field

#### 2.1.3. Design of a Scaffold with an External Shape

_{c}) = Max (F(G), F(C))

_{c}) is the newly formed implicit gyroid surface in cuboid shape and F(G) and F(C) are the implicit functions to denote gyroid and cuboid surfaces, respectively.

#### 2.1.4. Design of Scaffolds Based on User-Desired Pore Size (PS) and Strut Size (SS)

- (i.)
- Calculate level constant (C) based on PS and SS:

^{5}+ 0.0162 (PSR)

^{4}− 0.1722 (PSR)

^{3}+ 0.9142 (PSR)

^{2}− 2.5329 (PSR) + 1.7889

- (ii.)
- Calculate pore size (P
_{2π}) and strut size (S_{2π}) for the period coefficient N_{o}= 2π:

_{2π}= −11.7311 C

^{5}− 0.1307 C

^{4}− 1.7987 C

^{3}+ 0.2070 C

^{2}− 186.9928 C + 433.0114

_{2π}= −11.7311C

^{5}− 0.0466 C

^{4}+ 1.7987 C

^{3}+ 0.0175 C

^{2}− 186.9928 C + 433.0937

- (iii.)
- Calculate the scale factor (SF) from either giving the desired pore or strut size;

- (iv.)
- Calculate suitable period coefficient (N).

_{o}= SF × 2π

- (i.)
- PS200 (Pore Size 200 µm and Strut Size 200 µm);
- (ii.)
- PS350 (Pore Size 350 µm and Strut Size 200 µm);
- (iii.)
- PS550 (Pore Size 550 µm and Strut Size 200 µm);
- (iv.)
- PS750 (Pore Size 750 µm and Strut Size 200 µm);
- (v.)
- PS1000 (Pore Size 1000 µm and Strut Size 200 µm).

#### 2.2. Creation of FE Volume Meshes

#### 2.2.1. Meshing

#### 2.2.2. Converting the Implicit Body (Gyroid Lattice) into the Surface Mesh

#### 2.2.3. Converting the Surface Mesh into a Volume Mesh

#### 2.2.4. Converting the Volume Mesh into a FE Volume Mesh

#### 2.3. Simulation

#### 2.3.1. FE Model

#### 2.3.2. Simulation Method and Static Analysis

## 3. Results and Discussion

#### 3.1. Design and Morphological Parameters

#### 3.2. FE Simulation—Von Mises Stress and Deformation Prediction

^{n}

#### 3.3. Discussion

#### 3.4. Limitations

## 4. Conclusions

- (i.)
- The advantage of having TPMS with the SDF method is that the end user can give the desired pore and strut sizes and porosity to achieve the required architecture of scaffolds for effective mechanical and degradation properties.
- (ii.)
- In the design of scaffolds, the level constant plays a vital role in tuning their interconnected architecture by deciding how many parts are to be solid (strut) or void (pores). This level constant influences the morphological parameters such as pore and strut sizes so that the pore–strut ratio decides the level constant variation, whereby a positive value results in more solid regions and a decrease in the level constant results in more solid regions and large pore sizes.
- (iii.)
- The porosity of scaffolds can be controlled by modifying the pore size of the scaffolds, keeping a constant strut size. Thus, these morphological properties affect the architecture of the lattice, which in turn alters the total mechanical properties.
- (iv.)
- The visual stress and deformation distributions are achieved using FE simulations, from which the values of mechanical responses are predicted.
- (v.)
- The maximum von Mises stress and the maximum deformation increase due to decreased volume fraction and increased porosity.
- (vi.)
- The effective elastic modulus of the scaffolds decreases with increased pore size and porosity. It was also predicted that the effective elastic moduli were in the 0.05 to 1.93 GPa range, matching that of trabecular bone.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B. Creation of a Gyroid Scaffold in nTopology Software

## Appendix C. Creation of FE Volume Mesh in nTopology

## Appendix D. Static Structural Analysis

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**Figure 1.**(

**a**) A triply periodic minimal surface-based gyroid scaffold of pore size 1000 µm and strut size of 200 µm with a well-interconnected network of pores and struts helping the movement of oxygen, nutrients, and waste materials. (

**b**) Illustration of how relative density affects volume fraction and influences the morphology of a unit cell. (

**c**) Representative of a gyroid unit cell’s pore size (red) and strut size (black) [6].

**Figure 2.**Workflow of the given research project for the design of scaffolds and FEM-based compressive loading simulation. The nTop notebooks for design of scaffolds (Appendix A and Appendix D), creation of FE volume mesh (Appendix C) and static structural analysis (Appendix D) can be seen.

**Figure 4.**The Boolean intersection between the gyroid implicit surface and the required shape (cuboid) of 5 × 5 × 10 mm

^{3}to obtain a cuboid gyroid scaffold of 5 × 5 × 10 mm

^{3}.

**Figure 5.**Uniform Gyroid Scaffolds of different pore sizes (PS) ranging from 200 µm to 1000 µm with a constant strut size (SS) of 200 µm.

**Figure 8.**(

**a**) A volume mesh having tetrahedral elements, which are 3D solid volume elements, different from a surface mesh of 2D elements; (

**b**) A FE volume mesh having integration points of a geometric order added to a volume mesh.

**Figure 9.**Convergence plots of PS350 for (

**a**) displacement and (

**b**) Max. von Mises Stress (Remaining plots can be viewed in Supplementary Materials Figure S1).

**Figure 10.**Frontal and isometric views of an FE model, which is a combination of FE volume mesh with boundary conditions—a uniform force (Yellow) is applied on a top plate (movable), and a displacement restraint (red) is applied on a bottom plate (fixed).

**Figure 12.**Graph of Design and Morphological Properties: variations of (

**a**) surface area with poresize, (

**b**) level constant with pore / strut ratio, (

**c**) period coefficient with level constant, (

**d**) surface area to volume ration with pore size, (

**e**) porosity with pore size, (

**f**) level constant with pore size, (

**g**) volume fraction with level constant, and (

**h**) volume fraction with pore size.

**Figure 13.**FE models under compressive loading—the von Mises distribution of gyroid scaffold PS350. The stress values increment from violet (minimum value) to red colour (maximum value). The von Mises contours of other FE models can be seen in Supplementary Materials Table S2.

**Figure 14.**FE models under compressive loading. Displacement distribution of gyroid scaffold PS350. The displacement values increment from violet (minimum value) to red colour (maximum value). The displacement contours of other FE models can be seen in Supplementary Materials Table S3.

**Figure 15.**Graph of mechanical properties predicted from FE simulation: variations of effective elastic modulus with (

**a**) pore size, (

**b**) porosity, and (

**c**) volume fraction; variations of (

**d**) relative elastic modulus with volume fraction, (

**e**) maximum von Mises stress with maximum deformation and (

**f**) maximum von Mises stress with pore size.

$\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{scaffold}=1-\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{scaffold}}{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{same}\mathrm{sized}\mathrm{cuboid}}$ $\mathrm{Volume}\mathrm{Fraction}=1-\mathrm{Porosity}$ $\mathrm{Relative}\mathrm{Density}=\frac{\mathrm{Density}\mathrm{of}\mathrm{gyroid}\mathrm{lattice}}{\mathrm{Density}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{material}}$ |

Label | Element Count | Node Count | Edge Count | Vertex Count |
---|---|---|---|---|

PS200 | 7,052,136 | 11,570,625 | 9,867,119 | 1,703,506 |

PS350 | 8,386,979 | 12,702,213 | 10,945,522 | 1,756,691 |

PS550 | 2,576,201 | 4,442,317 | 3,753,745 | 688,572 |

PS750 | 1,794,654 | 3,107,398 | 2,623,529 | 483,869 |

PS1000 | 1,155,452 | 2,032,425 | 1,711,706 | 320,719 |

**Table 3.**Material Properties used in the simulation [48].

Material | Young’s Modulus (E) | Poisson Ratio (ν) | Yield Strength |
---|---|---|---|

Titanium Grade 5 (Ti-6Al-4V) | 114 GPa | 0.34 | 883 MPa |

$\mathrm{Compressive}\mathrm{Stress}=\frac{\mathrm{Reactive}\mathrm{force}\mathrm{on}\mathrm{the}\mathrm{fixed}\mathrm{lower}\mathrm{side}\mathrm{of}\mathrm{a}\mathrm{scaffold}}{\mathrm{Equivalent}\mathrm{area}\mathrm{of}\mathrm{a}\mathrm{scaffold}}$ $\mathrm{Strain}=\frac{\mathrm{Deformation}}{\mathrm{Height}\mathrm{of}\mathrm{a}\mathrm{scaffold}}$ $\mathrm{Effective}\left(\mathrm{or}\right)\mathrm{Compressive}\mathrm{Elastic}\mathrm{Modulus}=\frac{\mathrm{Compressive}\mathrm{Stress}}{\mathrm{Strain}}$ $\mathrm{Relative}\mathrm{Elastic}\mathrm{Modulus}=\frac{\mathrm{Effective}\mathrm{Elastic}\mathrm{Modulus}}{\mathrm{Elastic}\mathrm{Modulus}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{material}}$ |

Label | PS (µm) | SS (µm) | C | N | Surface Area of Scaffold (mm^{2}) | Volume of Scaffold (mm^{3}) | Porosity (%) | Volume Fraction (%) | SA:V (mm^{−1}) |
---|---|---|---|---|---|---|---|---|---|

PS200 | 200 | 200 | 0.01 | 13.52 | 1831.62 | 126.09 | 49.56 | 50.44 | 14.53 |

PS350 | 350 | 200 | −0.62 | 9.90 | 1191.99 | 74.18 | 70.33 | 29.67 | 16.07 |

PS550 | 550 | 200 | −1.01 | 7.27 | 720.39 | 41.50 | 83.40 | 16.60 | 17.36 |

PS750 | 750 | 200 | −1.18 | 5.72 | 469.93 | 26.75 | 89.30 | 10.70 | 17.57 |

PS1000 | 1000 | 200 | −1.30 | 4.54 | 295.11 | 15.56 | 93.78 | 6.22 | 18.97 |

Pore size ↓, Relative density of the lattice ↑, Volume Fraction ↑ Pore size ↑, Relative density of the lattice ↓, Volume Fraction ↓ |

Label | Reactive Force (×10 ^{−2} N) | Max. Deformation (µm) | Max. Von Mises Stress (GPa) | Strain (µm/m) | Stress (N/m ^{2}) | Effective Elastic Modulus (GPa) | Relative Elastic Modulus (×10 ^{−3}) |
---|---|---|---|---|---|---|---|

PS200 | 2.69 | 5.58 | 0.23 | 558.00 | 1075.40 | 1.93 | 16.92 |

PS350 | 3.03 | 10.90 | 0.35 | 1089.53 | 1212.00 | 1.11 | 9.74 |

PS550 | 4.35 | 63.35 | 1.34 | 6335.41 | 1740.00 | 0.27 | 2.37 |

PS750 | 7.91 | 186.34 | 2.21 | 18,633.80 | 3164.00 | 0.17 | 1.49 |

PS1000 | 9.40 | 725.57 | 6.82 | 72,557.00 | 3760.00 | 0.05 | 0.44 |

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## Share and Cite

**MDPI and ACS Style**

N. Musthafa, H.-S.; Walker, J.; Rahman, T.; Bjørkum, A.; Mustafa, K.; Velauthapillai, D.
In-Silico Prediction of Mechanical Behaviour of Uniform Gyroid Scaffolds Affected by Its Design Parameters for Bone Tissue Engineering Applications. *Computation* **2023**, *11*, 181.
https://doi.org/10.3390/computation11090181

**AMA Style**

N. Musthafa H-S, Walker J, Rahman T, Bjørkum A, Mustafa K, Velauthapillai D.
In-Silico Prediction of Mechanical Behaviour of Uniform Gyroid Scaffolds Affected by Its Design Parameters for Bone Tissue Engineering Applications. *Computation*. 2023; 11(9):181.
https://doi.org/10.3390/computation11090181

**Chicago/Turabian Style**

N. Musthafa, Haja-Sherief, Jason Walker, Talal Rahman, Alvhild Bjørkum, Kamal Mustafa, and Dhayalan Velauthapillai.
2023. "In-Silico Prediction of Mechanical Behaviour of Uniform Gyroid Scaffolds Affected by Its Design Parameters for Bone Tissue Engineering Applications" *Computation* 11, no. 9: 181.
https://doi.org/10.3390/computation11090181