# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 2 Materials and Methods
### Shape Function
\begin{bmatrix}\begin{array}{cccc}\mathbf{R}(r_{11})&\mathbf{R}(r_{21})&\ldots&\mathbf{ R}(r_{n1})\\ \mathbf{R}(r_{21})&\mathbf{R}(r_{22})&\ldots&\mathbf{R}(r_{n2})\\ \vdots&\vdots&&\vdots\\ R(r_{1n})&\mathbf{R}(r_{2n})&\ldots&\mathbf{R}(r_{m})\end{array}\end{bmatrix}  \tag{12}\]  
\[\mathbf{P}= \begin{bmatrix}\begin{array}{cccc}1&x_{1}&y_{1}&z_{1}\\ 1&x_{2}&y_{2}&z_{2}\\ \vdots&\vdots&\vdots\\ 1&x_{n}&y_{n}&z_{n}\end{array}\end{bmatrix}  \tag{13}\]  
Note that it is possible to obtain a unique solution for the non-constant coefficients \(\mathbf{a}\) and \(\mathbf{b}\) if the inverse of \(\mathbf{R}\) exists. In this case, these coefficients can be obtained through the next expression:  
\[ \begin{cases}\mathbf{a}\\ \mathbf{b}\end{cases} =\mathbf{G}^{-1} \begin{cases}\mathbf{u}\\ \mathbf{0}\end{cases}  \tag{14}\]  
Thus, combining this expression with Equation (2), it can be redefined as:  
\[\mathbf{u}(\mathbf{x}_{l})=\left\{\mathbf{R}^{\mathrm{T}}(\mathbf{x}_{l})\mathbf{p}^{\mathrm{T}}( \mathbf{x}_{l})\right\}\mathbf{G}^{-1} \begin{cases}\mathbf{u}\\ \mathbf{0}\end{cases} =\left\{\mathbf{\varphi}(\mathbf{x}_{l}),\mathbf{\Psi}(\mathbf{x}_{l}) \right\} \begin{cases}\mathbf{u}\\ \mathbf{0}\end{cases}  \tag{15}\]  
where \(\mathbf{\varphi}(\mathbf{x}_{l})\) is defined as the interpolation function vector given by:  
\[\mathbf{\varphi}(\mathbf{x}_{l})=\left\{\mathbf{\varphi}_{1}(\mathbf{x}_{l})\right.\ \ \mathbf{\varphi}_{2}(\mathbf{x}_{l})\ \ \ \ldots\ \ \ \mathbf{\varphi}_{\mathbf{n}}(\mathbf{x}_{l})\} \tag{16}\]  
and the residual vector that can be obtained by the expression below and it is given by (\(\mathbf{\Psi}(\mathbf{x}_{l})\))  
\[\mathbf{\Psi}(\mathbf{x}_{l})=\left\{\mathbf{\Psi}_{1}(\mathbf{x}_{l})\ \ \ \mathbf{\Psi}_{2}(\mathbf{x}_{l})\ \ \ \ldots\ \ \ \mathbf{\Psi}_{\mathbf{m}}(\mathbf{x}_{l})\right\} \tag{17}\]  
Bearing this in mind, Equation (15) may be expressed as:  
\[u(\mathbf{x}_{l})=\mathbf{\Phi}(\mathbf{x}_{l})^{T}\mathbf{u}=\left\{\mathbf{\varphi}(\mathbf{x}_{l}) ^{T},\mathbf{\Psi}(\mathbf{x}_{l})^{T}\right\} \begin{cases}\mathbf{u}\\ \mathbf{0}\end{cases}  \tag{18}\]  
and the partial derivatives of the interpolated field variable as:  
\[ \begin{cases}\frac{\partial u^{h}(\mathbf{x}_{l})}{\partial x}=\frac{\partial\bm {\Phi}(\mathbf{x}_{l})^{T}}{\partial x}\mathbf{u}=\left\{\frac{\partial\mathbf{\varphi}( \mathbf{x}_{l})^{T}}{\partial x};\ \frac{\partial\mathbf{\Psi}(\mathbf{x}_{l})^{T}}{\partial x} \right\}\begin{cases}\mathbf{u}\\ \mathbf{0}\end{cases}\\ \frac{\partial u^{h}(\mathbf{x}_{l})}{\partial y}=\frac{\partial\mathbf{\Phi}(\mathbf{x}_{ l})^{T}}{\partial y}=\left\{\frac{\partial\mathbf{\varphi}(\mathbf{x}_{l})^{T}}{ \partial y};\ \frac{\partial\mathbf{\Psi}(\mathbf{x}_{l})^{T}}{\partial y}\right\}\begin{cases} \mathbf{u}\\ \mathbf{0}\end{cases}\end{cases} \end{cases} \tag{19}\]  
Assuming that matrix \(\mathbf{G}\) is independent on \(\mathbf{x}_{l}\), it is possible to reorganise the terms above in order to be defined as:  
\[\left\{ \begin{array}{l}\left\{\frac{\partial\mathbf{\varphi}(\mathbf{x}_{l})^{T}}{ \partial x};\ \frac{\partial\mathbf{\Psi}(\mathbf{x}_{l})^{T}}{\partial x}\right\}=\frac{\partial \left(\left\{\mathbf{R}^{\mathrm{T}}(\mathbf{x}_{l})\mathbf{p}^{\mathrm{T}}(\mathbf{x}_{l}) \right\}\mathbf{G}^{-1}\right)}{\partial x}=\left\{\frac{\partial\mathbf{R}^{\mathrm{ T}}(\mathbf{x}_{l})}{\partial x};\ \frac{\partial\mathbf{p}^{\mathrm{T}}(\mathbf{x}_{l})}{\partial x} \right\}\mathbf{G}^{-1}\\ \frac{\partial\mathbf{\varphi}(\mathbf{x}_{l})^{T}}{\partial y},\ \frac{\partial\mathbf{\Psi}(\mathbf{x}_{l})^{T}}{ \partial y}\right\}=\frac{\partial\left(\left\{\mathbf{R}^{\mathrm{T}}(\mathbf{x}_{l}) \mathbf{p}^{\mathrm{T}}(\mathbf{x}_{l})\right\}\mathbf{G}^{-1}\right)}{\partial y}=\left\{ \frac{\partial\mathbf{R}^{\mathrm{T}}(\mathbf{x}_{l})}{\partial y};\ \frac{\partial\mathbf{p}^{\mathrm{T}}(\mathbf{x}_{l})}{\partial y} \right\}\mathbf{G}^{-1}\end{array} \right.