# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 3 Cellular Proliferation Algorithm Description  
### Proliferation Algorithm  
The algorithm used in this study was developed to simulate the proliferation of a generic cell under normal and abnormal conditions. This process is dependent on the concentrations of oxygen (O\({}_{2}\)) and glucose (GL), as well as a growth phenomenological law that is based on documented observations that describe the volume of the cell over time. An illustration of the main steps performed by the algorithm is presented in Figure 2.  
To initiate the algorithm, specific data need to be input (referred to as the \"Input Data\" block in Figure 2). This includes the size of the domain, divisions along both the horizontal (\(Ox\)) and vertical (\(Oy\)) axes, cell size, maximum iteration count, and properties of the cell and the extracellular matrix (such as diffusion coefficients and concentrations of O\({}_{2}\) and GL). Once these initial parameters are defined, the algorithm creates a nodal mesh, assigns material properties to the cell and extracellular medium, and applies the boundary conditions. The essential boundary conditions (\(\Gamma_{u_{i}}\) and \(\Gamma_{u_{d}}\)) were assumed to be periodic (double mirror) for all the limits of the problem. The diffusion of the molecules was allowed to occur from \(\Gamma_{u}^{(+)}\) to \(\Gamma_{u}^{(-)}\), as well as in the reverse direction, as shown in Figure 3a. Figure 2: Illustration of the cell growth algorithm dependent on oxygen and glucose, where the oxygen concentration is represented by [O\({}_{2}\)], the glucose concentration by [GL], and the volume of the cell by V (Adapted with permission from Ref. [46]. Copyright 2022 ELSEVIER). Subsequently, the background integration mesh is constructed using square integration cells and a regular lattice. Each side of the square cells has a dimension equal to twice the average nodal distance. Within these integration cells, the integration points are placed using the Gauss-Legendre quadrature scheme. Additionally, different materials are defined in the domain to differentiate the cell from the medium based on their geometric positions. Using radial searches, the influence domains are created. For each integration point, a specific number of closest nodes are identified. This information is then used to construct the shape functions (i.e., the radial point interpolation functions), which allow for the interpolation of values within the influence domains. At this stage, an iterative loop begins, progressing through a fictitious time step. During the iterations, the concentrations of oxygen and glucose are enforced in the domain by solving the reaction-diffusion equations. This process enables the determination of the distribution of these concentrations at each iteration. Once these steps are completed and normal concentration values are assumed in the domain, the cell initiates the proliferation process by imposing a volume variation. Assuming the present iteration as the time step and considering the medium to be under hydrostatic strain (\(\varepsilon_{xx}\!=\!\varepsilon_{yy}\!=\!\varepsilon_{zz}\)), the corresponding strain required for this volume variation can be determined to be \(\Delta\mathcal{V}=\varepsilon_{xx}\varepsilon_{yy}\varepsilon_{zz}\) and \(\varepsilon_{xx}\!=\!\varepsilon_{yy}\!=\!\varepsilon_{zz}\!=\Delta\mathcal{V}/3\). On the other hand, assuming that \(\sigma=c\epsilon\), the stress field \(\sigma\) is determined, and it is possible to be imposed in the solid domain through an internal body force: \(\
abla\mathbf{\Lambda}+\mathbf{b}=0\Rightarrow\mathbf{b}=-\
abla\mathbf{\Lambda}\). The imposition of this pressure variation enforces growth in the cell nodes, which, in turn, causes an increase in the volume of the cell.