## 4 Numerical Experiments
### Simulation of Tissue Growth on Scaffold In 3D  
We consider tissue growth on a 3D scaffold in \(\Omega=(0,1)^{3}\). The numerical scheme is similar to the 2D case. The parameters used are \(h=1/50\), \(\Delta t=0.1h^{2}\), and \(\epsilon=\epsilon_{4}\). The solid scaffold structure is given as follows:  
\[\psi(x,y,z)=\left\{ \begin{aligned} & 1&\text{if }(x,y,z)\in(0,0.5)\times(0,1)\times(0,0.5)\cup\\ &\qquad(0,1)\times(0.5,1)\times(0,0.5)\cup(0,0.5)\times(0.5,1) \times(0,1),\\ -1&\text{otherwise},\end{aligned} \right.\]  
and is shown as a volume enclosed by the dark color surface in Figure 6a. The initial tissue layer is given by  
\[\phi(x,y,z,0)=\left\{ \begin{aligned} & 1&\text{if }(x,y,z)\in(0,0.5+2h)\times(0,1)\times(0,0.5+2h)\cup\\ &\qquad(0,1)\times(0.5-2h,1)\times(0,0.5+2h)\cup\\ &\qquad(0,0.5+2h)\times(0.5-2h,1)\times(0,1),\\ -1&\text{otherwise},\end{aligned} \right.\]  
and is shown as a volume enclosed by the green colored surface on the scaffold in Figure 6a. Figure 5: (**a–c**) are snapshots of tissue growth simulation with the star-shaped scaffold in 2D at \(t=0\), \(600\Delta t\), and \(1200\Delta t\), respectively. The green region is the tissue layer on the given scaffold which is represented by the dark region. For interpretation of the references to color in the figure, the reader is referred to the web version of the article. Figure 6: (**a–c**) are snapshots of tissue growth simulation in 3D on the scaffold at \(t=0\), \(3200\Delta t\), and \(6400\Delta t\), respectively. The green region is the tissue layer on the given scaffold which is represented by the dark region. For interpretation of the references to color in the figure, the reader is referred to the web version of the article. Snapshots of tissue growth simulation are shown in Figure 6b,c at \(t=3200\Delta t\) and \(6400\Delta t\), respectively. The tissue growth on the scaffold according to the curvature shows the similar simulation results from the previous test [9]. We consider tissue growth simulations on more complex surfaces such as triply periodic minimal surface (TPMS) which has interconnected pores and high porosity to ensure cell penetration, vascular ingrowth, nutrient diffusion, and waste product elimination [29]. TPMSs are widely used for the scaffold model structure, especially in tissue engineering applications. [PERSON] et al. [29] investigated the effect of the porous structure of TPMS scaffolds on the macroscopic permeability behavior using numerical and experimental methods. Simulations are conducted with Schwarz primitive (P), Schwartz diamond (D), and Schoen gyroid (G) surfaces, which are typical examples of TPMS. The approximated P, D, and G surfaces are given as follows, respectively:  
\[P(x,y,z) = \cos(2\pi x)+\cos(2\pi y)+\cos(2\pi z),\] \[D(x,y,z) = \cos(2\pi x)\cos(2\pi y)\cos(2\pi z)-\sin(2\pi x)\sin(2\pi y)\sin( 2\pi z),\] \[G(x,y,z) = \sin(2\pi x)\cos(2\pi y)+\sin(2\pi z)\cos(2\pi x)+\sin(2\pi y) \cos(2\pi z).\]  
The scaffold regions for the P, D, and G surfaces are defined by  
\[\psi(x,y,z) = \tanh\left(\frac{P(x,y,z)}{\sqrt{2}\epsilon}\right), \tag{7}\] \[\psi(x,y,z) = \tanh\left(\frac{D(x,y,z)}{\sqrt{2}\epsilon}\right),\] (8) \[\psi(x,y,z) = \tanh\left(\frac{G(x,y,z)}{\sqrt{2}\epsilon}\right). \tag{9}\]  
The initial cell layers for the P, D, and G surfaces are given as  
\[\phi(x,y,z,0) = \tanh\left(\frac{P(x,y,z)+4h}{\sqrt{2}\epsilon}\right), \tag{10}\] \[\phi(x,y,z,0) = \tanh\left(\frac{D(x,y,z)+4h}{\sqrt{2}\epsilon}\right),\] (11) Figure 6:0  
The simulation is performed in \(\Omega=(-0.5,0.5)^{3}\) with \(h=1/64\), \(\Delta t=0.1h^{2}\), and \(\epsilon=\epsilon_{4}\). Furthermore, \(N_{x}=N_{y}=N_{z}=64\) are used. We use the periodic boundary condition. Figure 7a-c show snapshots of the tissue growth simulation in 3D on the P, D, and G surface scaffolds (from top to bottom) at \(t=0\), \(500\Delta t\), and \(1000\Delta t\), respectively. The temporal evolutions of the volume and those of the surface area, \(V\) and \(S\), of cell layer over TPMSs are shown in Figure 8a,b, respectively.