# The Effect of Tortuosity on Permeability of Porous Scaffold

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## Abstract

**:**

^{−11}m

^{2}to 4.0 × 10

^{−10}m

^{2}. In addition, it was observed that the tortuosity parameter significantly affected the scaffold’s permeability and shear stress values. The tortuosity value of the NSP scaffold was in the range of 1.5–2.8. Therefore, tortuosity can be manipulated by changing the curvature of the surface scaffold radius to obtain a superior bone tissue engineering construction supporting cell migration and tissue regeneration. This parameter should be considered when making new scaffolds, such as our NSP. Such efforts will produce a scaffold architecturally and functionally close to the natural cancellous bone, as demonstrated in this study.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Parametric Design of Tortuous Microchannel in Scaffold

^{™}, Bruker micro-CT, Kontich, Belgium) with a voltage source of 100 kV, a current of 100 µA and a resolution of 17.20 µm. The tomographic reconstruction of these images using Materialize Mimics

^{®}software (Materialize, Wilfried, Leuven, Belgium) gives a three-dimension (3D) image volume dataset of 712 layers of two-dimension (2D) images of the cancellous bone structure. This dataset is 12 mm in height and 10 mm in diameter. The software extracted the desired region (ROI) with 60% porosity from the sample. Finally, three different scaffold structures (cubic, octahedron pillar (PO), and Schoen’s gyroid (SG)) and cancellous bone structures with a porosity (Φ) of 60% were developed and compared.

#### 2.2. Morphology Analysis

^{®}software features. 3D CAD models were exported to stereolithography (STL) format and imported into the slice software program Chitubox (CBD-Tech, Guangdong, China). The CAD model was sliced using a resolution of 17.20 µm. The remaining 244 slices and 500 × 500 pixels images were analyzed using Fiji (Image J, NIH). The trabecular thickness (Tb.Th) and separation (Tb.Sp) were calculated using the Fiji and BoneJ plugins. The image data set was then exported to MATLAB (MathWorks Corp., Natick, MA, USA) software to calculate the diffusion tortuosity. The open solver plugin Taufactor calculates the tortuosity factor based on the finite difference method (FDM) and directly uses image voxels as discretization meshes for simulation [22,23].

#### 2.3. Experimental Setup

#### 2.4. Fluid Properties and Boundary Conditions in CFD

^{®}software (COMSOL, Inc., Burlington, MA, USA). The fluid domain uses the Boolean subtraction method when using SolidWorks

^{®}software. This domain is exported in IGES format and imported into COMSOL Multiphysics

^{®}software. The boundary conditions are defined so that the volumetric flow rates on the inflow and outflow sides (zero outlet pressure) are 0.67 and 0.00 mL/min, respectively. In addition, the shape is symmetrical on the lateral side of the cube and non-slip on its interior surface [20], as shown in Figure 3a. This simulation used Simulated Body Fluid (SBF) liquid with a viscosity of 1 mPa.s and a density of 1 g/cm

^{3}at body temperature (37 °C) [27]. The boundary between the fluid and solid is characterized non-slip boundary during the Computational Fluid Dynamic (CFD) study where the fluid velocity at the boundary is equal to the velocity of the solid [28]. The outlet fluid pressure is set as zero. It is assumed that the fluid flow conditions are laminar and steady through the geometry of a 3D scaffold with tetrahedral elements (see Figure 3b). Convergence studies were carried out to obtain the optimal mesh size. The total number of elements for each model varies from 2,060,893 to 8,993,558. This simulation was carried out on a personal computer (Dell Precision 3630 Tower Workstation, TX, USA) with an Intel I7-8700 processor and 80 GB of RAM. This simulation uses the solver generalized minimal residual method (GMRES) as an iterative solver. First, the average pressure drop between the inflow and outflow is determined, and then the permeability is calculated according to Darcy’s law [28,29], which is shown in Equation (1),

^{2}), Q is the flow rate in (m

^{3}/s), $\mu $ is the fluid’s dynamic viscosity of the fluid in (Pa.s), L is the specimen length (m), A is the flow’s cross-sectional area in (m

^{2}), and $\Delta P$ is the pressure drop across the structure (Pa).

#### 2.5. Statistical Analysis

## 3. Results

#### 3.1. Morphology Indices

^{−1}, respectively.

**Figure 4.**Results of morphological indices of porous scaffolds: (

**a**) the relationship between BS/TV and porosity in the porous scaffold and cancellous bone (current study) and previous studies’ bone; (

**b**) the relationship between porosity and tortuosity; (

**c**) BS/TV versus porosity; and (

**d**) the relationship between BS/TV and tortuosity [33,34].

^{−1}, respectively.

#### 3.2. Mesh Convergence

#### 3.3. Fluid Flow Characterization of Scaffold

^{2}= 0.9. Furthermore, the permeability value of cubic scaffold was higher than that of other types of scaffolds, including natural cancellous bone. For the NSP scaffold, the permeability of the NSPr2 model was higher than that of the NSPr and NSPr1 models. An increase in porosity results in an increase in permeability. The permeability values for the NSPr, NSPr1, and NSPr2 models at 60% porosity were $2.1\times {10}^{-10}$ m

^{2}, $2.5\times {10}^{-10}$ m

^{2}, and $3.1\times {10}^{-10}$ m

^{2}, respectively. It indicates that the increase in porosity results in varying permeabilities. In addition, the permeability of the octahedron pillar model and natural cancellous bone is different in different planes despite having the same porosity.

^{2}= 0.9, each model also found a strong correlation between tortuosity and permeability. Figure 6b also shows that each structural model, such as cubic, SG, NSP, PO, and natural cancellous bone models, has the same porosity even though the tortuosity values differ. However, it was found that there is no correlation between tortuosity and permeability for the cubic model because the structures have similar tortuosity. Therefore, it is evident that porosity is not the only parameter that contributes to permeability. For example, the magnitude of the cancellous bone tortuosity in the X, Y, and Z planes is 1.57, 2.16, and 3.85, which indicates the permeability value is $7.61\times {10}^{-11}$ m

^{2}, $3.29\times {10}^{-11}$ m

^{2}, and $1.06\times {10}^{-11}$ m

^{2}, respectively.

^{2}= 0.9 for each model tested. The WSS value for the cubic model is lower than for other scaffolds. The WSS value of the NSPr model was higher than the NSPr1, NSPr2, and cubic models. At 60% porosity, the WSS values for the NSPr, NSPr1, and NSPr2 models were 0.084 Pa, 0.069 Pa, and 0.064 Pa, respectively. It shows that the same porosity produces different WSS.

^{−9}m in the Z direction, while the cancellous bone has a permeability of 10

^{−11}m with the same porosity. The polynomial relationship between tortuosity and permeability with different architectures at the same porosity contributes R

^{2}= 0.82 (see Figure 8c). The conclusion is that the change in the magnitude of the permeability of the scaffold is highly dependent on the porosity design with tortuous structures.

^{−10}m

^{2}to 1.5 × 10

^{−9}m

^{2}; PO 2.9 × 10

^{−10}m

^{2}to 5.6 × 10

^{−10}m

^{2}; SG 7.9 × 10

^{−10}m

^{2}; and NSP 2.0 × 10

^{−11}m

^{2}to 4.0 × 10

^{−10}m

^{2}and cancellous bone (present study) 1.1 × 10

^{−11}m

^{2}to 7.6 × 10

^{−11}m

^{2}. The permeability of cancellous bone from the literature varied from 2.5 × 10

^{−11}m

^{2}to 7.43 × 10

^{−8}m

^{2}. The cancellous bone samples were taken from the vertebral body of the calcaneus [24], the femoral bone, and the spine [35,36,37]. From Figure 10, the proposed structure is within the range of natural cancellous bone.

## 4. Discussion

^{2}= 0.8 (see Figure 8). Thus, tortuosity is a very important parameter in designing a superior tissue scaffold because the parameter is the most influential in affecting scaffold permeability, as in previous investigations [18,19]. In addition, there is also a polynomial correlation between the tortuosity and permeability of cancellous bone in the same structure. Structures with the same porosity and surface area have different tortuosities in different orientations. This finding is similar to previous studies’ results, where the permeability value of cancellous bone is different in the longitudinal and transverse directions [53]. According to Rabiatul et al. (2021) [53], this phenomenon occurs due to the orientation of the trabecular strut effect, which causes different bone marrow permeability. We assume that tortuosity is also the dominant parameter in controlling flow’s effective transport and direction to hard and soft tissues, regulating bone remodeling and cartilage regeneration. Differences in cancellous bone tortuosity related to cell migration and bone regeneration must be investigated in future studies.

## 5. Conclusions

^{−11}m

^{2}and 4.0 × 10

^{−10}m

^{2}. The resultant permeability value suggests that it is comparable to cancellous bone permeability. Therefore, it is suggested that the NSP model’s tortuosity be considered when designing new scaffolds. In this study, an attempt was made to make a new scaffold similar in structure and function to natural cancellous bone.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic illustration of the scaffold design stage: (

**a**) process schematic diagram showing the principles of scaffold design based on the concept of meandering pore channels; (

**b**) NSP scaffolds with 60% porosity and different tortuosities can be controlled by adjusting the fluid pore size and radius of curvature; (

**c**) cubic, PO, SG and natural cancellous bone models with 60% porosity.

**Figure 3.**The steps used to characterize the fluid dynamics of the scaffold: (

**a**) The simulation’s boundary condition; (

**b**) a suitable convergent meshing model can be obtained.

**Figure 8.**Characterization of various scaffold structures: (

**a**) tortuosity of different structures with the same porosity (60%); (

**b**) permeability of different structures with the same porosity (60%); (

**c**) the relationship between permeability and tortuosity of different structures with the same porosity (60%).

**Figure 9.**Velocity streamlines, velocity contour, and pressure of porous structure with 60% porosity.

**Table 1.**Dimension parameter features of the NSP model CAD design (see also Figure 1a).

Dimensional Parametric Study | Value (mm) | ||
---|---|---|---|

Model | NSPr | NSPr1 | NSPr2 |

r | 0.4 | 0.45 | 0.49 |

Constant,$c$ | 0.745 | 0.745 | 0.745 |

y | $2c-r$ | $2c-r$ | $2c-r$ |

z | $c-\frac{r}{2}$ | $c-\frac{r}{2}$ | $c-\frac{r}{2}$ |

t | 2.1 | 2.1 | 2.1 |

$X$(Φ: 25%) | 0.450 | 0.421 | 0.41 |

$X$(Φ: 35%) | 0.540 | 0.515 | 0.51 |

$X$(Φ: 45%) | 0.620 | 0.603 | 0.600 |

$X$(Φ: 50%) | 0.660 | 0.646 | 0.645 |

$X$(Φ: 60%) | 0.740 | 0.731 | 0.735 |

$X$(Φ: 65%) | 0.780 | 0.775 | 0.778 |

**Table 2.**Permeability value of the NSPr model, as determined by both experimental and simulation results.

Porosity [%] | Permeability | |||
---|---|---|---|---|

Experimental | Simulation $\mathit{k}\times {10}^{-10}\left({\mathbf{m}}^{2}\right)$ | |||

Mean $\mathit{k}\times {10}^{-10}\left({\mathbf{m}}^{2}\right)$ | Standard Deviation $\mathit{\sigma}\times {10}^{-10}\left({\mathbf{m}}^{2}\right)$ | p-Value | ||

25 | 0.152 | 0.014 | 0.00287 | 0.148 |

45 | 1.014 | 0.694 | 0.00537 | 1.052 |

60 | 2.655 | 0.583 | 0.01571 | 2.537 |

65 | 3.143 | 0.976 | 0.03071 | 3.315 |

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## Share and Cite

**MDPI and ACS Style**

Prakoso, A.T.; Basri, H.; Adanta, D.; Yani, I.; Ammarullah, M.I.; Akbar, I.; Ghazali, F.A.; Syahrom, A.; Kamarul, T.
The Effect of Tortuosity on Permeability of Porous Scaffold. *Biomedicines* **2023**, *11*, 427.
https://doi.org/10.3390/biomedicines11020427

**AMA Style**

Prakoso AT, Basri H, Adanta D, Yani I, Ammarullah MI, Akbar I, Ghazali FA, Syahrom A, Kamarul T.
The Effect of Tortuosity on Permeability of Porous Scaffold. *Biomedicines*. 2023; 11(2):427.
https://doi.org/10.3390/biomedicines11020427

**Chicago/Turabian Style**

Prakoso, Akbar Teguh, Hasan Basri, Dendy Adanta, Irsyadi Yani, Muhammad Imam Ammarullah, Imam Akbar, Farah Amira Ghazali, Ardiyansyah Syahrom, and Tunku Kamarul.
2023. "The Effect of Tortuosity on Permeability of Porous Scaffold" *Biomedicines* 11, no. 2: 427.
https://doi.org/10.3390/biomedicines11020427