# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 4 Results and Discussion  
The evolution of the volume of a healthy and a tumour cell over time was analysed in order to study the influence of oxygen and glucose on cell growth and, at the same time, the viability of the algorithm to simulate the twenty-five proposed combinations. As a reference, the duration of a cell cycle (i.e., from the beginning of the growth process until two cells are obtained), under normal conditions of oxygen and glucose, was defined as 24 h (0.857 h per iteration), for a healthy cell, and as 18 h (0.643 h per iteration), for a tumour cell [66; 67; 68]. Regarding the initial parameters that were used, the size of the domain was set to 1000 _per_ 1000 \(\upmu\)m, and the extracellular part was assumed to be homogenised with an elasticity modulus of 1 Pa [69] and a Poisson's coefficient of 0.45 [69; 70; 71]. The width of the cell was set to 100 \(\upmu\)m [72; 73], and the diffusion coefficients were defined as 2000 \(\upmu\)m/s (cell) and 3030 \(\upmu\)m/s (medium) for oxygen and as 2000 \(\upmu\)m/s (cell) and 3030 \(\upmu\)m/s (medium) for glucose [7; 23]. In addition, the same arrangement of the nodes and the same integration structure were used. A representation of these meshes can be seen in Figure 6. For all analyses, an integration scheme of 6 \(\times\) 6 integration points in each integration cell and influence domains with seven nodes were assumed. These parameters were optimised in a preliminary numerical study to increase the efficiency of the code (reduce the computational cost and maintain the accuracy and stability of the solution) and remained constant in all analyses. Figures 7-16 present the growth evolution of healthy and tumour cells over time and for each combination. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Extreme Hypoxia** & **Hypoxia** & **Normal Level** & **Hyperoxia** & **Extreme** \\ \hline \multirow{3}{*}{**Extreme**} & Constant Volume + & Constant Volume + & Constant Volume + & Constant Volume + & **Constant Volume +** \\  & Apoptosis & Apoptosis & Apoptosis & Apoptosis & Apoptosis & Apoptosis \\  & \(\text{t}_{\text{e}}=12\) & \(\text{t}_{\text{G}}=24\) & \(\text{t}_{\text{G}}=36\) & \(\text{t}_{\text{G}}=30\) & \(\text{t}_{\text{e}}=18\) \\  & Constant Volume + & Slower Growth & & Slower Growth & Constant Volume + \\  & Apoptosis & \(\alpha_{\text{t}_{\text{O}_{2}}m}\in[0.5,1]\) & \(\beta_{\text{t}_{\text{G}}m}\in[1,2]\) & \(\alpha_{\text{t}_{\text{O}_{2}}m}\in[0.5,1]\) & Apoptosis \\  & \(\text{t}_{\text{O}_{2}}=16\) & \(\beta_{\text{t}_{\text{G}}m}\in[0,1]\) & \(\beta_{\text{t}_{\text{O}}m}\in[0,1]\) & \(\text{t}_{\text{O}_{2}}=22\) \\  & Constant Volume + & & Slower Growth & Normal Growth & Slower Growth & Constant Volume + \\  & Apoptosis & \(\beta_{\text{t}_{\text{O}_{2}}m}\in[0,1]\) & \(\alpha_{\text{m}}=1\) & \(\beta_{\text{t}_{\text{O}_{2}}m}\in[0,1]\) & Apoptosis \\  & \(\text{t}_{\text{O}_{2}}=24\) & & \(\beta_{\text{t}_{\text{O}_{2}}m}\in[0,1]\) & \(\beta_{\text{t}_{\text{O}_{2}}m}=1\) & \(\text{t}_{\text{O}_{2}}=24\) \\  & Constant Volume + & & Slower Growth & Slower Growth & Constant Volume + \\  & Apoptosis & \(\alpha_{\text{t}_{\text{O}_{2}}m}\in[0,1]\) & \(\beta_{\text{t}_{\text{G}}m}\in[0,1]\) & \(\alpha_{\text{t}_{\text{O}_{2}}m}\in[0,1]\) & Apoptosis \\  & \(\text{t}_{\text{O}_{2}}=8\) & \(\beta_{\text{t}_{\text{G}}m}\in[0,1]\) & \(\beta_{\text{t}_{\text{G}}m}\in[0,1]\) & \(\text{t}_{\text{O}_{2}}=20\) \\  & Constant Volume + & & Constant Volume + & Constant Volume + & Constant Volume + & Constant Volume + \\  & Apoptosis & Apoptosis & Apoptosis & Apoptosis & Apoptosis \\  & \(\text{t}_{\text{e}}=4\) & \(\text{t}_{\text{G}}=18\) & \(\text{t}_{\text{G}}=36\) & \(\text{t}_{\text{G}}=24\) & \(\text{t}_{\text{e}}=16\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Schematic chart of the conditions implement in the algorithm for the twenty-five combinations, when a tumour cell is assumed.