# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 3 Cellular Proliferation Algorithm Description
### Influence of Oxygen and Glucose Concentrations
\begin{array}{c}\beta_{a_{G}m}=2.5\times 10^{13}C_{Gm}-0.125\\ C_{Gm}\in[3.5\times 10^{-14},7.5\times 10^{-14}]\text{ mol}/\upmu\text{m}^{2}\end{array}  \tag{47}\]  
However, if this state is combined with hyperoxia or hypoxia, it is assumed that growth is negatively affected. Nevertheless, it is also assumed that the process is not as slow as when these states are combined with a normal glucose concentration or with hypoglycaemia. To achieve this behaviour, specific \(\alpha\) values were obtained for these two particular combinations (hyperoxia/hyperglycaemia and hypoxia/hyperglycaemia, respectively):  
\[ \begin{array}{c}\alpha_{a_{O_{2}}m}=-1.39949\times 10^{13}C_{O_{2}m}+1.125\\ C_{O_{2}m}\in\big{[}8.932\times 10^{-15},4.466\times 10^{-14}\big{]}\text{ mol}/\upmu\text{m}^{2}\\ \alpha_{b_{O_{2}}m}=4.56977\times 10^{14}C_{O_{2}m}-0.4898\\ C_{O_{2}m}\in\big{[}2.233\times 10^{-17},1.116\times 10^{-15}\big{]}\text{ mol}/\upmu\text{m}^{2}\end{array}  \tag{48}\]  
Regarding the extreme states (extreme hyperoxia (\(c_{O_{2}}\)), extreme hypoxia (\(d_{O_{2}}\)), extreme hyperglycemia (\(e_{G}\)) and extreme hypoglycaemia (\(f_{G}\))), cells are highly influenced by these changes in oxygen and glucose concentrations. Under these states, cells tend to cease growing and undergo cell death instead. Therefore, in the algorithm, cells were not allowed to grow for a specific period of time, and if this state was not reversed, the cell would initiate the process of apoptosis (Figure 5b). To define this last step, the parameters \(\alpha\) and \(\beta\) were defined as follows:  
\[ \begin{array}{c}if\quad t_{m}>t_{O_{2}}\\ \alpha_{c/d_{O_{2}}}=\frac{1}{t_{m}}\cdot\left(t_{O_{2}}-t_{m}\right)\\ if\quad t_{m}>t_{G}\\ \beta_{e/f_{G}}=\frac{1}{t_{m}}\cdot\left(t_{G}-t_{m}\right)\end{array}  \tag{49}\]  
where \(t_{m}\) represents the time of the iteration \(m\) in which the algorithm is, and \(t_{O_{2}}\) and \(t_{G}\) represent the durations for which the cell needs to remain in extreme oxygen or glucose conditions, respectively, in order to initiate the process of apoptosis. For extreme hyperoxia and extreme hypoxia, \(t_{O_{2}}\) was assumed to be 36 h if the concentration of glucose was within its normal range. In the case of extreme hyperglycaemia or extreme hypoglycaemia, the variable \(t_{G}\) was assumed to be 24 h when \(C_{O_{2}}\) is within its normal range. Note that these \(\alpha\) and \(\beta\) parameters only affect the growth process when the time periods mentioned above have passed (\(t_{m}\geq\ t_{O_{2}}\) or \(t_{m}\geq\ t_{G}\)), triggering the process of apoptosis, which leads to a 40% reduction in the initial cell volume and ultimately results in cell death [62; 63; 64; 65]. Until then (\(t_{m}<t_{O_{2}}\) or \(t_{m}<t_{G}\)), the cell volume remains constant if the extreme state in which the cell is located does not change. When extreme cases are combined with another abnormal state, the variables \(t_{O_{2}}\) and \(t_{G}\) are defined with lower values, depending on the aggressiveness of the other combined state. In cases where two extreme states are combined, the values of \(t_{O_{2}}\) and \(t_{G}\) are assumed to be equal. Thus, a generic \(t_{e}\) was considered, and a similar expression to the ones presented above was taken into account:  
\[ \begin{array}{c}if\quad\quad t_{m}>t_{e}\\ \alpha_{e}=\frac{1}{t_{m}}\cdot\left(t_{e}-t_{m}\right)\end{array}  \tag{50}\]  
Tables 2 and 3 summarise what occurred in each state for the healthy and tumour cells, respectively.