Table 2. Cell growth rate equations adjusted to account for different limitations in nutrient concentrations and cell contact inhibition.

Condition	Cell density equation
Base case	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_G}{K_G+C_G}\frac{C_{O,A}}{K_O+C_{O,A}}\frac{K_L}{K_L+C_L}\right)N\left(1-\frac{N}{N_{max}}\right)+D_N{\nabla }^{2}N$
Cell contact inhibition	$\frac{\partial N}{\partial t}=N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Oxygen	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_{O,A}}{K_O+C_{O,A}}\right)N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Glucose	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_G}{K_G+C_G}\right)N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Lactate	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{K_L}{K_L+C_L}\right)N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Oxygen and glucose	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_G}{K_G+C_G}\frac{C_{O,A}}{K_O+C_{O,A}}\right)N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Oxygen and lactate	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_{O,A}}{K_O+C_{O,A}}\frac{K_L}{K_L+C_L}\right)N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Glucose and lactate	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_G}{K_G+C_G}\frac{K_L}{K_L+C_L}\right)N(1-\frac{N}{N_{max}})+D_N{\nabla }^{2}N$
Oxygen, glucose and lactate only	$\frac{\partial N}{\partial t}=K_{max}\left(\frac{C_G}{K_G+C_G}\frac{C_{O,A}}{K_O+C_{O,A}}\frac{K_L}{K_L+C_L}\right)N+D_N{\nabla }^{2}N$