## 4 Numerical Experiments
### Simulation of Tissue Growth on Scaffold In 2D  
First, we simulate tissue growth in 2D space to validate the proposed mathematical model although the real scaffolds are 3D. We consider a relatively simple 2D scaffold in the domain \(\Omega=(0,1)\times(0,1)\). In all simulations below, the scaffold region is the region in which the value of a scalar field \(\psi\) is greater than zero. The simple 2D scaffold is written as follows:  
\[\psi(x,y)=\begin{cases}\;\;\;1&\text{if $0<x<0.2$ or $0.8<y<1$},\\ -1&\text{otherwise},\end{cases}\]  
which is shown in Figure 3a as the dark region. The surface is initially seeded with tissue layer \(\phi(x,y,0)\):  
\[\phi(x,y,0)=\begin{cases}\;\;\;1&\text{if $0<x<0.2+2h$ or $0.8-2h<y<1$},\\ -1&\text{otherwise},\end{cases}\]  
which is shown in Figure 3a as the green region. Note that for computational simplicity, we set \(\phi(x,y,0)\approx 1\), where \(\psi(x,y)>0\).  
In Figure 3b,c, we show simulation results of tissue growth on the scaffold in 2D at \(t=4000\Delta t\) and \(t=8000\Delta t\), respectively. Here, the space step \(h=1/50\), the time step size \(\Delta t=0.1h^{2}\), and \(\epsilon=\epsilon_{4}\) are used. The simulation results agree well with the previous solutions [9].  
Figure 3: (**a**–**c**) are snapshots of tissue growth on scaffold in the 2D plane at \(t=0\), \(4000\Delta t\), and \(8000\Delta t\), 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.  
Next, we perform a simulation on a 2D scaffold whose surface is given by a cosine curve which has concavity and convexity to verify the effect of curvature on tissue growth. The 2D scaffold in \(\Omega=(0,1)\times(-0.4,0.4)\) is given by  
\[\psi(x,y)=\begin{cases}\quad 1&\text{if $-0.4<y<0.3\cos(2\pi x)$},\\ -1&\text{otherwise}.\end{cases}\]  
The surface is initially seeded with \(\phi(x,y,0)\):  
\[\phi(x,y,0)=\begin{cases}\quad 1&\text{if $-0.4<y<0.3\cos(2\pi x)+3h$},\\ -1&\text{otherwise},\end{cases}\]  
and the initial cell layer is shown as a green region in Figure 4a. Here, \(h=1/200\), \(\Delta t=0.1h^{2}\), and \(\epsilon=\epsilon_{4}\) are used. As shown in Figure 4, the tissue grows where the curvature is positive.  
Furthermore, we present a simulation of tissue growth on a star-shaped scaffold which is more complex than the above cases. This simulation is carried out in the domain \(\Omega=(-0.5,0.5)\times(-0.5,0.5)\) with \(h=1/128\), \(\Delta t=0.1h^{2}\), and \(\epsilon=\epsilon_{4}\). An initial star-shaped scaffold is \(\psi>0\), where \(\psi(x,y)\) is given by  
\[\psi(x,y)=\tanh\left(\frac{\sqrt{x^{2}+y^{2}}-0.25-0.1\cos(6\theta)}{\sqrt{2} \epsilon}\right),\]  
\[\text{where}\quad\theta=\begin{cases}\quad\tan^{-1}\left(\frac{y}{x}\right), \quad\quad\quad\text{if $x>0$},\\ \pi+\tan^{-1}\left(\frac{y}{x}\right),\quad\quad\text{otherwise}.\end{cases}\]  
The initial cell layer is given by  
\[\phi(x,y,0)=\tanh\left(\frac{\sqrt{x^{2}+y^{2}}-0.25-0.1\cos(6\theta)+3h}{ \sqrt{2}\epsilon}\right),\]  
where \(\theta\) is defined as above. Figure 5 shows the temporal evolution of tissue on the star-shaped scaffold.  
Figure 4: (**a–c**) are snapshots of tissue growth simulation at \(t=0\), \(4000\Delta t\), and \(8000\Delta t\), 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.