# Structural design and study of the performance of a biomimetic scaffold of primitive minimal triple periodic surfaces Ti6Al4V

### Structure design method

Scaffold models were generated by rhino software (Robert McNeel & Assoc, USA), as shown in Fig. 1. The unit size was 1mm, the solid models were obtained by the wall offset of 0.05mm, and the surface structure was defined by the implicit function expression (1). Four groups of scaffold structures with different aspect ratios were designed, Fig. 1a–d, r1: r2 is 3:1, 2:1, 2:3 and 1:1 respectively. To facilitate the following description, the four groups of scaffolds have been named and simplified according to the characteristics of their shaft-to-diameter ratio. They were referred to as P1-3 scaffold, P1-2 scaffold, P2-3 scaffold and P1-1 scaffold respectively.

$${varphi }_{D} (x,y,z) ={lambda }_{1} {cos}left(xright)+{lambda }_{2} {cos}left (yright)+{lambda }_{3}{cos}left(zright)+mu$$

(1)

where (uplambda) and (upmu) are constants.

### Model preparation

Four groups of scaffold models were prepared by the SLM process. Ti6Al4V powder was purchased from Gaoke New Material Technology (Beijing) Co., LTD. The particle size is 12 to 50 μm. The 3D printing equipment used Shanghai Kewei Forming Technology Co., Ltd.’s SLM280, which is equipped with a 500W laser with a spot size of 60μm. The scanning layer thickness and scanning speeds are 15 μm and 400 mm/s respectively. Through the Boolean operation, the size of each sample was set to φ10 × 12. Four groups of representative scaffold models have been shown in Fig. 3.

### Mechanical properties test

The scaffolding compression test was carried out in accordance with the international standard for compression testing of metals (ISO13314:2011). Four groups of samples were compressed at the compression speed of 0.05 mm/min using a micro-controlled electronic universal testing machine. The stress-strain curves of each group of samples were plotted, and the modulus of elasticity and the yield stress of each group of samples were obtained to evaluate the mechanical properties of each group of samples.

### Computational fluid dynamics analysis

Permeability is one of the important characteristics of the structure of the bone scaffold, which affects the efficient transport of oxygen and nutrients in the scaffold19. It is not conducive to fluid flow in the scaffold with low permeability. The permeability is too large, it is easy to wash out cells and nutrients, which is also not conducive to tissue regeneration. Permeability was analyzed by computational fluid dynamics (CFD), the simulation process was completed in ANSYS 19.0 software (ANSYS, America, https://www.pcsoft.com.cn/soft/194402.html). Considering that the object of analysis was an incompressible fluid with constant density, the Navier-Stokes equation defined by Eq. (2) was used.

$$left{begin{array}{c}rho frac{partial{varvec{upsilon}}}{partial t}+rho left({varvec{upsilon)} bullet nabla right)upsilon +nabla P-mu {nabla }^{2}upsilon =F nabla bullet upsilon =0end{array}right.$$

(2)

where ρ is the density of the fluid (kg/m3); v is the fluid velocity (m/s); t is the time (s); is the operator; P is the pressure (Pa); μ is the dynamic viscosity coefficient of the fluid (Pa s); F is the acting force (N).

To simplify the calculation and analysis of the simulation, water has been selected as the fluid domain material. At normal body temperature, the density and viscosity of water is 1000 kg/m3 and 1.45e−9 MPa s respectively20.21. The scaffolds were meshed using tetrahedral elements with a maximum mesh size of 0.002 mm. The boundary condition of the fluid model was shown in Fig. 4, the whole light colored region was the fluid domain and the green part was the scaffold model. The entry speed applied to the scaffold was set at 1 mm/s. The outlet pressure is considered zero. It was assumed that the wall had no slip22.

The fluid flow in the biomimetic bone scaffold was simulated by ANSYS18.2 (ANSYS, America) and the law of the fluid flow in the scaffold was obtained. The pressure difference between the inlet and the outlet of the scaffold and the permeability of the scaffold were calculated by the equations. (3) and (4).

$$Delta P={P}_{Input}-{P}_{output}$$

(3)

$$K=frac{mu bullet nu bullet L}{Delta P}$$

(4)

where K is the permeability coefficient (mm2); L is the characteristic length (mm); ΔP is the pressure difference (MPa).

### Sedimentation and cell adhesion

The migration and settlement of cells in the scaffold can largely reflect the rationality of the scaffold design. In this article, cell movement and adhesion in the scaffold were simulated by Comsol software (Comsol, Sweden). The fluid model was split by the tetrahedral mesh method, as shown in Figure 5a). The top surface was evenly meshed and the size was 0.05mm. The cells were tuned to a spherical discrete phase with a diameter of 0.01 mm and a density of 1130 kg/m323. The meshes on the upper surface were evenly distributed, as shown in Figure 5b). It was assumed that the cell starts to settle from the top surface under the action of gravity and drag, the initial velocity of the cell was 0 and the fluid velocity was 1 mm/s. The drag satisfies the Stokes equation defined by Eqs. (5) and (6).

$${F}_{D}=frac{1}{{tau }_{P}}bullet mbullet (u-upsilon )$$

(5)

$${tau }_{p}=frac{rho bullet {d}^{2}}{18mu }$$

(6)

where m is the mass of the particles (kg); d is the particle diameter (m); (uprho) is the particle density (kg/mm3); μ is the dynamic viscosity coefficient of the fluid (Pa s).

Adhesion occurs when the cell movement touches the side wall and the inner wall of the scaffold, and when it moves towards the exit, it crosses the boundary. We used the software counter to count the number of adhesions in the process of cell migration to assess which group of scaffold structures is most conducive to cell adhesion.

### statistical analyzes

Analysis was performed using SPSS 20.0 software (SPSS Inc., Chicago, IL, USA). All data were expressed as mean ± standard deviation and analyzed with one-way ANOVA. In all cases, the results were considered statistically significant for p