# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 3 Cellular Proliferation Algorithm Description
### Proliferation Algorithm
The region outside the cell maintains  
Figure 3: (**a**) Illustration of the domain used to simulate the evolution of the growth of healthy and tumour cells and of the periodic boundary condition imposed; (**b**) comparison between the proposed phenomenological law (Exp \(n=2\)) and the experimental curves available in the literature [54; 55] (Adapted with permission from Ref. [46]. Copyright 2022 ELSEVIER). the same pressure. This implies that the nodes inside the cell either cause it to shrink, grow, or maintain the same volume, as no pressure is imposed by the medium. The displacement field is obtained by calculating the internal pressure generated by the volume variation. As a result, a new nodal configuration can be established by adding the obtained displacements to the initial coordinates of the nodes. This new configuration is used in the subsequent iteration. This iterative process continues until the programme reaches a stopping condition. To obtain the volume variation, the fictitious volume that the cell should have in a particular iteration is calculated based on the concentrations of glucose and oxygen. If the mean concentrations of oxygen and glucose in the domain are within normal values (i.e., a concentration within the interval \([8.932\times 10^{-15};1.116\times 10^{-15}]\) mol/\(\upmu\)m\({}^{2}\) of oxygen [8; 15; 52; 56] or within \([1.944\times 10^{-14};3.500\times 10^{-14}]\) mol/\(\upmu\)m\({}^{2}\) of glucose [26; 53; 57], the cell grows according to the original proposed phenomenological law proposed in [46]. This law is given by the following expression:  
\[V_{f}=V_{i}\,\cdot\,\left[2\cdot\frac{e^{\frac{t_{i}}{t_{d}}}}{e}-\left(\frac {2}{e}-1\right)\cdot\left(1-\frac{t_{i}}{t_{d}}\right)\right]^{n}\cdot\frac{1 }{2^{n}}\cdot\frac{t_{i}}{t_{d}}+V_{i} \tag{44}\]  
where \(V_{f}\) is defined as the volume in an iteration \(i,V_{i}\) the initial volume, \(t_{d}\) the time that is necessary for the cell to divide into two new cells, \(t_{i}\) the time of each iteration, and \(n\) specifies the inclination of the curve. This phenomenological law was obtained by curve-fitting experimental data available in the literature (Figure 3b) [54; 55], and \(n=2\) was used for the simulations as it was the value that resulted in the smallest difference among the experimental and theoretical curves, as demonstrated in [46]. In the case of normal concentrations, the cell continues to grow until it reaches twice its initial volume, at which point it undergoes division into two new cells. When the cell reaches double its volume, the \"Division Block\" is initiated (Figure 4) in order to divide the cell (Figure 5a). To do that, the process of growth is blocked, and two points (\(X_{A}\) and \(X_{B}\)) are arbitrarily selected in opposite positions of the cell. After that, the points move to the midpoint of the cell until they are in the same position, having the same coordinates. In this case, the \(\Delta\)V inside the cell is 0, so it is possible to assume a hydrostatic stress state. This allows for the conversion of equivalent stress into equivalent forces, which can be used to determine the forces applied to the cell and enforce the displacement using the equation \(\mathbf{K}^{-1}\cdot\boldsymbol{f}=\boldsymbol{u}\). That way, the cell is compressed until the two points are positioned at the centre of the cell, and then the division into two new cells takes place. When the process of division is completed, the algorithm stops. Figure 4: Steps performed in the “Division Block” that is referred in Figure 2 (Adapted with permission from Ref. [46]. Copyright 2022 ELSEVIER). When abnormal concentrations are present in the domain, they can affect the growth process, leading to four main implications: slower or faster growth velocity, maintenance of the same cell volume, or initiation of the apoptosis process. These modifications in the normal proliferation process depend on the combination of oxygen and glucose concentration states within the specific iteration of the domain.