# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 2 Materials and Methods
### Shape Function  
To obtain the interpolation shape functions, the RPIM resorts to the RPI technique, which is a combination of a polynomial basis function with a radial basis function [44; 47]. A field variable \(u(\mathbf{x})\) can be interpolated at an integration point \(\mathbf{x}_{l}\) with:  
\[u(\mathbf{x}_{l})=\sum_{l=1}^{n}R_{i}(\mathbf{x}_{l})\cdot a_{i}(\mathbf{x}_{l})+\sum_{j=1} ^{m}p_{j}(\mathbf{x}_{l})\cdot b_{j}(\mathbf{x}_{l})=\mathbf{R}^{T}(\mathbf{x}_{l})\mathbf{a}(\mathbf{x }_{l})+\mathbf{p}^{T}(\mathbf{x}_{l})\mathbf{b}(\mathbf{x}_{l}) \tag{1}\]  
Or as:  
\[u(\mathbf{x}_{l})=\left\{\mathbf{R}(\mathbf{x}_{l})^{\mathrm{T}},\mathbf{p}(\mathbf{x}_{l})^{T} \right\} \begin{Bmatrix}\mathbf{a}(\mathbf{x}_{l})\\ \mathbf{b}(\mathbf{x}_{l})\end{Bmatrix}  \tag{2}\]  
where \(n\) and \(m\) are the number of nodes of the influence domain on the point \(\mathbf{x}_{l}\), \(\mathbf{R}(\mathbf{x}_{l})\) the radial basis function, \(\mathbf{p}(\mathbf{x}_{l})\) the polynomial basis function, and \(a(\mathbf{x}_{l})\) and \(\mathbf{b}(\mathbf{x}_{l})\) the non-constant coefficients of each function, respectively. Moreover, considering \(\mathbf{R}(\mathbf{x}_{l})\), \(\mathbf{p}(\mathbf{x}_{l})\), \(\mathbf{a}(\mathbf{x}_{l})\) and \(\mathbf{b}(\mathbf{x}_{l})\), these matrices are obtained through the following expressions:  
\[\mathbf{R}(\mathbf{x}_{l})=\left[R_{1}(\mathbf{x}_{l})R_{2}(\mathbf{x}_{l})\dots R_{n}(\mathbf{x} _{l})\right]^{\mathrm{T}} \tag{3}\]  
\[\mathbf{p}(\mathbf{x}_{l})=\left[p_{1}(\mathbf{x}_{l})p_{2}(\mathbf{x}_{l})\dots p_{m}(\mathbf{x} _{l})\right]^{\mathrm{T}} \tag{4}\]  
\[\mathbf{a}^{\mathrm{T}}(\mathbf{x}_{l})=\left[a_{1}(\mathbf{x}_{l})a_{2}(\mathbf{x}_{l})\dots a _{n}(\mathbf{x}_{l})\right] \tag{5}\]  
\[\mathbf{b}^{\mathrm{T}}(\mathbf{x}_{l})=\left[b_{1}(\mathbf{x}_{l})\ \ b_{2}(\mathbf{x}_{l})\dots b _{m}(\mathbf{x}_{l})\right] \tag{6}\]  
In the case of the RPIM, \(\mathbf{R}_{\mathit{II}}(\mathbf{x}_{l})\) is a multiquadric radial basis function (MQ-RBF) [41; 44; 51], whose form is generically expressed using the following equation:  
\[\mathbf{R}_{\mathit{II}}(\mathbf{x}_{l})=\left(d_{\mathit{II}}^{2}+c^{2}\right)^{p} \tag{7}\]  
In this equation, \(c\) and \(p\) are defined as the shape parameters of the MQ-RBF. The efficiency of the method is dependent on them, but optimal values (\(c=0.0001\) and \(p=0.9999\)) are presented in the literature [47]. Regarding \(d_{\mathit{II}}\), also known as the Euclidean norm, it represents the distance between \(\mathbf{x}_{l}=\{x_{i},y_{i}\}^{T}\) and \(\mathbf{x}_{l}\). This can be defined as \(d_{\mathit{II}}=\sqrt{\left(x_{i}-x_{l}\right)^{2}+\left(y_{i}-y_{l}\right)^{2}}\). In the case of \(\mathbf{p}(\mathbf{x}_{l})\), different forms of this function can be used. Nevertheless, a standard form of this function can be given by the next equation:  
\[\mathbf{p}(\mathbf{x})^{T}=\left[1,x,y,x^{2},xy,y^{2},\dots\right] \tag{8}\]  
During this procedure, and in order to obtain only one solution, the equation below needs to be considered and included [41; 51]. \[\sum_{i=1}^{n}\mathbf{p}_{j}(\mathbf{x}_{i})a_{i}(\mathbf{x}_{i})=\mathbf{0} \tag{9}\]  
Having said this and taking into account all the previous steps, Equation (2) can be written as:  
\[ \begin{Bmatrix}\mathbf{u}_{\mathrm{s}}\\ \mathbf{0}\end{Bmatrix} = \begin{bmatrix}\mathbf{R}&\mathbf{P}\\ \mathbf{P}^{T}&\mathbf{0}\end{bmatrix}\begin{Bmatrix}\mathbf{a}\\ \mathbf{b}\end{Bmatrix} =\mathbf{G} \begin{Bmatrix}\mathbf{a}\\ \mathbf{b}\end{Bmatrix}  \tag{10}\]  
In this, \(\mathbf{u}_{\mathrm{s}}\) is the vector of the variable nodal values, and \(\mathbf{G},\mathbf{R}\), and \(\mathbf{P}\) are the moment matrix, radial moment, and polynomial moment matrices, respectively. The last three matrices can be defined, respectively, by the following expressions:  
\[\mathbf{u}_{\mathrm{s}}=\{u_{1},u_{2},\dots\ u_{n}\}^{T} \tag{11}\]\[\mathbf{R}=