# 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  
The different combinations of concentration states have a significant impact on the growth of the cell over time. When hyperoxia, hypoxia, and/or hypoglycaemia have been determined, the growth rate will be slower compared to when normal oxygen and/or glucose concentrations are present, since these three conditions tend to decrease the rate of cellular proliferation for both types of cells. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{O_{2}}\) **State** & \(\mathbf{[O_{2}](mol/\mu m^{2})}\) & **GL State** & \(\mathbf{[GL](mol/\mu m^{2})}\) \\ \hline Extreme Hyperoxia & \(>4.466\times 10^{-14}\) & Extreme Hyperglycaemia & \(>7.5\times 10^{-14}\) \\ Hyperoxia & \(\left]8.932\times 10^{-15},4.466\times 10^{-14}\right]\) & Hyperglycaemia & \(\left]3.5\times 10^{-14},7.5\times 10^{-14}\right]\) \\ Hypoxia & \(\left[2.233\times 10^{-17},1.116\times 10^{-15}\right[\) & Hypoglycaemia & \(\left[1.111\times 10^{-14},1.944\times 10^{-14}\right[\) \\ Extreme Hypoxia & \(<2.233\times 10^{-17}\) & Extreme Hypoglycaemia & \(<1.111\times 10^{-14}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Range of concentrations that defined each oxygen (\(\mathbf{O_{2}}\)) and glucose (GL) abnormal state used in the proposed algorithm. Figure 5: Colour maps of the evolution of the cell in terms of volume for the two generic possibilities: (**a**) cell proliferation and (**b**) apoptosis. Here, \(m\) represents the iteration and \(V\) the volume of the cell (\(\mu m^{3}\)) in a specific \(m\). Also, red represents the extracellular domain, blue represents one cell, and purple another one. Note that the presented maps do not correspond to sequential iterations, and the points A(450, 650), B(650, 650), C(650, 450), D(450, 450) delimit a section of the domain (Adapted with permission from Ref. [46]. Copyright 2022 EESEIVER). For hyperoxia (\(a_{O_{2}}\)) and hypoxia (\(b_{O_{2}}\)), a parameter \(\alpha\) was introduced to influence the growth of cell volume in a specific iteration while considering the specific concentration range for each case. The value of \(\alpha\) was determined through the following expressions:  
\[ \begin{array}{c}\alpha_{O_{2}m}=-2.79899\times 10^{13}C_{O_{2}m}+1.25\\ C_{O_{2}m}\in\big{[}8.932\times 10^{-15},4.466\times 10^{-14}\big{]}\text{mol}/ \upmu\text{m}^{2}\\ \alpha_{b_{O_{2}}m}=9.13954\times 10^{14}C_{O_{2}m}-2.04082\times 10^{-2}\\ C_{O_{2}m}\in\big{[}2.233\times 10^{-17},1.116\times 10^{-15}\big{]}\text{mol}/ \upmu\text{m}^{2}\end{array}  \tag{45}\]  
In these, \(C_{O_{2}}\) represents the mean oxygen concentration of the domain in a specific iteration \(m\). Regarding the healthy cell, a similar process was assumed for hyperglycaemia (\(a_{G}\)) and hypoglycaemia (\(b_{G}\)). For these conditions, in order to affect the growth of the cell, a parameter \(\beta\) was introduced to slow the process of growth. The parameter \(\beta\) is obtained for these cases as follows:  
\[ \begin{array}{c}\beta_{a_{G}}=-2.5\times 10^{13}C_{Gm}-1.875\\ C_{G}\in[3.5\times 10^{-14},7.5\times 10^{-14}]\text{ mol}/\upmu\text{m}^{2}\\ \beta_{b_{G}}=1.2\times 10^{14}C_{Gm}-1.333\\ C_{G}\in\big{[}1.111\times 10^{-14},1.944\times 10^{-14}\big{]}\text{mol}/ \upmu\text{m}^{2}\end{array}  \tag{46}\]  
Here, \(C_{G}\) is the mean glucose concentration of the domain in the iteration \(m\). It should be noted that when these four states are combined with each other or with normal concentrations of oxygen or glucose, the more aggressive state dominates. As a result, different combinations also result in variable durations of the proliferation process. For tumour cells under hypoglycemia, the procedure defined for healthy cells remains unchanged. However, in the case of hyperglyaemia (\(a_{G}\)), a new equation was introduced. Therefore, if the oxygen concentration is normal, the proliferation occurs at a faster rate compared to the normal glucose concentration. Hence, for hyperglyaemia, the parameter \(\beta\) is determined by the following equation:  
\[