# A Numerical Analysis of the Influence of Oxygen and Glucose in Healthy and Tumour Cells
## 2 Materials and Methods
### Elasticity Equations  
The elasticity theory can be used to simulate cell proliferation. Thus, considering the domain \(\Omega\), the equilibrium equations shown next can be used to describe a linear elastostatic problem:  
\[\left\{ \begin{array}{c}\
abla\cdot\mathbf{\Lambda}+\mathbf{b}=0\;in\;\Omega\\ \mathbf{\Lambda}\;\mathbf{n}=\overline{\mathbf{t}}\;on\;\Gamma_{t}\\ \mathbf{u}=\overline{\mathbf{u}}\;on\;\Gamma_{u}\end{array} \right. \tag{22}\]  
Here, \(\
abla\) represents the gradient operator and \(\mathbf{u}\) the displacement field. \(\mathbf{b}\) is the body force per unit volume, \(\Gamma\) the contour which presents the essential \(\Gamma_{\text{u}}\) and the natural boundaries \(\Gamma_{t}\), and \(\mathbf{\Lambda}\) the Cauchy stress tensor for a kinematically admissible \(\mathbf{u}\), \(\mathbf{b}\) the body force per unit volume and. In addition, \(\Gamma\in\Omega:\Gamma_{u}\cup\Gamma_{t}=\Gamma\;\bigwedge\;\Gamma_{u}\cap \Gamma_{t}=\varnothing\), \(\overline{\mathbf{u}}\) is the prescribed displacement on \(\Gamma_{u}\), \(\overline{\mathbf{t}}\) the traction on \(\Gamma_{t}\) and \(\mathbf{n}\) the unit outward normal to \(\Gamma\). Considering the [PERSON] formulation written using the Voigt notation, the weak form equation can be given by the following equation:  
\[\delta L=\int_{\Omega}\delta\varepsilon^{\text{T}}\sigma d\Omega-\int_{ \Omega}\delta\mathbf{u}^{\text{T}}\mathbf{b}d\Omega-\int_{\Gamma_{t}}\delta\mathbf{u}^{ \text{T}}\overline{\mathbf{t}}\;d\Gamma_{t}=0 \tag{23}\]  
where \(\varepsilon\) is the strain vector and \(\sigma\) the stress vector. In the case of \(\varepsilon\), this parameter can be obtained by:  
\[\varepsilon=\mathbf{L}\mathbf{u} \tag{24}\]  
where \(\mathbf{L}\) is the differential operator defined as:  
\[\mathbf{L}= \begin{bmatrix}\frac{\partial}{\partial x}&0&\frac{\partial}{ \partial y}\\ 0&\frac{\partial}{\partial y}&\frac{\partial}{\partial x}\end{bmatrix} ^{ \text{T}} \tag{25}\]  
If the Hooke Law is assumed, then \(\sigma\) can be obtained by resorting to the previous equation. Thus, \(\sigma\) can also be expressed as a function of \(\mathbf{u}\):  
\[\sigma=c\varepsilon=c\mathbf{L}\mathbf{u} \tag{26}\]  
where \(c\) represents the material constitutive matrix, which is represented as:  
\[\mathbf{c}= \begin{bmatrix}\frac{1-v^{2}}{E}&-\frac{v+v^{2}}{E}&0\\ -\frac{v+v^{2}}{E}&\frac{1-v^{2}}{E}&0\\ 0&0&\frac{1}{\varepsilon}\end{bmatrix} ^{-1} \tag{27}\]  
In this matrix, \(E\) is the elasticity modulus, \(v\) is the material Poisson coefficient, and \(G\) is the distortion modulus (\(G=E/(2+2v)\)). In the case of 2D problems, each node \(\mathbf{x}_{i}\) is defined by two degrees of freedom, which are defined at \(\mathbf{u}_{i}=\{u_{i},v_{i}\}\), the reference axis. To obtain the virtual displacement \(\delta\mathbf{u}(\mathbf{x}_{l})=\delta\mathbf{u}_{l}\) in an interesting point \(\mathbf{x}_{i}\), the interpolation function can be redefined as:\[\delta u_{I}= \begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{Bmatrix}\mathbf{\Phi}_{I}\\ \mathbf{\Phi}_{I}\end{Bmatrix} \delta u_{S}= \begin{bmatrix}\varphi_{1}(\mathbf{x}_{I})&0& \cdots&\varphi_{n}(\mathbf{x}_{I})&0\\ 0&\varphi_{1}(\mathbf{x}_{I})&\cdots&0&\varphi_{n}(\mathbf{x}_{I})\end{bmatrix} \delta u _{S} \tag{28}\]  
This way, and assuming the Hooke law and the strain/displacement relation, \(\int_{\Omega}\delta\varepsilon^{\mathsf{T}}\sigma d\Omega\) is equivalent to:  
\[\int_{\Omega}\delta\varepsilon^{\mathsf{T}}\sigma d\Omega=\int_{\Omega} \delta(\mathbf{L}\delta u)^{\mathsf{T}}\mathbf{c}(\mathbf{L}\mathbf{u})d\Omega=\int_{\Omega} \delta(\mathbf{L}\mathbf{H}_{I}\delta u_{S})^{\mathsf{T}}\mathbf{c}(\mathbf{L}\mathbf{H}_{I}\mathbf{u} _{S})d\Omega=\int_{\Omega}\delta\big{(}\mathbf{L}\mathbf{B}_{I}^{\mathsf{T}}\mathbf{c}\mathbf{ B}_{I}\mathbf{u}_{S}\big{)}^{\mathsf{T}}d\Omega \tag{29}\]  
In which \(\mathbf{B}_{I}\) is defined as the deformation matrix for the \(n\) nodes of the influence cell of \(\mathbf{x}_{I}\) and given by the next expression. \[\mathbf{B}_{I}=