## 1 Introduction  
In some fracture patients, the bone defects do not get well in the ordinary recuperation period or do not mend at all. In [1], the authors investigated the causes of impaired healing and proposed potential treatment strategies with the combined in silico-in vivo approach. In the field of tissue engineering, bioscaffolds are used as hosts for cell proliferation and extracellular matrix formation. There have been studies on the effects of materials of scaffold substrates [2]. Tissue growth is known to be controlled by substrate geometry such as roughness [3; 4] and curvature [5; 6; 7] of the substrate. Curvature affects cell density because of the contraction or expansion of the local surface area [8]. In previous studies, simulations of tissue growth on scaffold have been performed [9; 10; 11; 12] and applied in many fields [13; 14; 15; 16; 17]. Figure 1 shows new tissue formation on various scaffolds (triangular, square, and hexagonal shapes) [18].  
Figure 1: Formation of new tissue on triangular, square, and hexagonal shaped scaffolds. Reprinted from [PERSON] et al. [18], with permission from The Royal Society.  
The authors in [18] performed in vitro cell growth studies with osteogenic cells grown on biologically inductive substrate, hydroxyapatite. These cells, pre-osteoblastic mouse MC3T3-E1 cells, exhibit logarithmic growth when cultured in osteogenic conditions. In addition, the authors investigated the geometric features of tissue growth in aspects both the channel surface size and the shape.  
The main purpose of this study is to present a mathematical model and numerical simulation for tissue growth on a bioscaffold in two-dimensional (2D) and three-dimensional (3D) space. The mathematical model is based on the Allen-Cahn (AC) equation [19]:  
\[\frac{\partial\phi(\mathbf{x},t)}{\partial t} = -\mu(\mathbf{x},t)=-\left(\frac{F^{\prime}(\phi(\mathbf{x},t))}{ \epsilon^{2}}-\Delta\phi(\mathbf{x},t)\right),\ \ \mathbf{x}\in\Omega,\ t>0, \tag{1}\] \[\mathbf{n}\cdot\
abla\phi(\mathbf{x},t) = 0,\ \ \mathbf{x}\in\partial\Omega, \tag{2}\]  
where \(\Omega\subset\mathbb{R}^{d}\ \left(d=2,3\right)\) is a domain with an outer unit normal vector \(\mathbf{n}\) to \(\partial\Omega\). Note that Equation (2) can be replaced with other conditions such as the Dirichlet boundary condition or the periodic boundary condition. Here, \(\phi(\mathbf{x},t)\) is the difference between the concentrations of the tissue and the ambient medium, \(\mu(\mathbf{x},t)\) is the chemical potential, \(F(\phi)=0.25(\phi^{2}-1)^{2}\), and \(\epsilon\) is a positive constant. The AC equation was originally introduced as a mathematical model for antiphase domain coarsening in a binary alloy. The AC equation has been successfully used to model a class of problems such as crystal growth [20; 21], image processing [22; 23], mean curvature flows [24], and a mixture of two incompressible fluids [25]. We propose a simple mathematical model for tissue growth on scaffolds using features of the AC equation, that is, motion by mean curvature.  
This paper is organized as follows. In Section 2, we propose the mathematical model for tissue growth on scaffolds. In Section 3, we describe a numerical solution for solving the mathematical model. We present the numerical results for several examples in Section 4. In Section 5, the conclusions are drawn.