Table 1. The system of equations for the Single Glioma Cell Model.

$$\mathbf{\text{Glioma Cells}}:{\partial }_{t}G=\underset{\text{Passive diffusion of G cells}}{\underbrace{\delta \nabla ·(D\nabla G)}}-\underset{\text{Active transport of G cells}}{\underbrace{\eta \nabla ·(G\nabla B)}}+\underset{\text{Net production of G cells}}{\underbrace{MG}}-\underset{\text{Necrosis of G cells}}{\underbrace{\gamma FG}}$$ (1)

$$\mathbf{\text{Brain Cells}}:{\partial }_{t}B=\underset{\text{Necrosis of brain cells}}{\underbrace{-\gamma FB}}$$ (2)

$$\mathbf{\text{Necrotic Cells}}:{\partial }_{t}N=\underset{\text{Conversion of G and B to necrotic cells}}{\underbrace{\gamma F(B+G)}}$$ (3)

Assumptions regarding angiogenesis, hypoxia, mitosis, and necrosis: $$\mathbf{\text{Total cell concentration}}:C=G+B+N$$ (4)

$$\mathbf{\text{Mitotic Rate}}:M=\tau \left(\frac{tanh\left(100(C_{ltm}-C)\right)}{2}\right)$$ (5)

$$\mathbf{\text{Rate of necrosis}}:\gamma F=\gamma \left(\frac{1+G}{B+0.01}\right)\left(\frac{1-tanh\left(100(C_{ltm}-C)\right)}{2}\right)$$ (6)

$$\mathbf{\text{Necrotic threshold}}:C_{ltm}=\sigma \left[\text{log}(1+G)\right]+\Omega ,$$ (7) where σ = 1.0 to simulate angiogenesis, and σ = 0 to simulate anti-angiogenesis.

A description of the parameters and their values/units may be found in Table 2.