# Elucidating the Role of Matrix Porosity and Rigidity in Glioblastoma Type IV Progression
## 3 Computational Model
### Governing Equations  
The governing equations of the model used here are the mass balance equation for ECM, IF, TCs, nutrients, and the linear momentum balance equation for the mixture ECM+IF+TCs. The following equations are obtained from the general forms by introducing some simplifications and closure relationships (e.g., a Fickian type equation for diffusion of species, a generalized Darcy's equation for flow of the fluid phases, etc.). The primary variables of the model are differential pressures between phases \(t\), \(h\), \(l\), \(p^{hl}\) and \(p^{th}\), IF pressure \(p^{l}\), nutrient mass fraction, and displacement \(u^{s}\) of solid phase (ECM). It is recalled that \(p^{th}+p^{th}=p^{th}\). The saturation degree of a fluid phase \(\alpha\) is \(S^{\alpha}=\varepsilon^{\alpha}/\varepsilon\), with \(\alpha=t,h,l\) for TCs, HCs, and IF, respectively. In this paper the saturation degree of HCs is \(S^{th}=0\). From Equation (1) the constraint condition for saturation degrees for the system with just three phases follows  
\[S^{t}+S^{l}=1 \tag{2}\]  
The mass balance equation of the ECM is  
\[\frac{\partial\varepsilon}{\partial t}=\
abla\cdot\vec{\textbf{v}}^{\vec{ \textbf{v}}}+\frac{(1-\varepsilon)}{\rho^{s}}\frac{\partial\rho^{s}}{ \partial t}-\
abla\cdot\left(\varepsilon\vec{\textbf{v}}^{\vec{\textbf{v}}} \right)-\frac{\overset{l\rightarrow s}{M}}{\rho^{s}} \tag{3}\]  
The mass balance equation of TCs reads  
\[\begin{split}&\left[\frac{\varepsilon S^{t}}{K_{t}}+\frac{S^{t}(1- \varepsilon)}{K_{s}}\left(S^{t}+p^{th}\frac{\partial S^{t}}{\partial p^{th}} \right)+\varepsilon\frac{\partial S^{t}}{\partial p^{th}}\right]\frac{ \partial p^{th}}{\partial t}+\left[\frac{\varepsilon S^{t}}{K_{t}}+\frac{S  
The mass balance equation of IF reads  
\[\begin{array}{l}\frac{S^{l}(1-\varepsilon)}{K_{s}}\left(p^{th}\frac{\partial S ^{t}}{\partial p^{th}}+S^{t}\right)\frac{\partial p^{th}}{\partial t}+\left[ \frac{S^{l}(1-\varepsilon)}{K_{s}}\left(1-p^{th}\frac{\partial S^{l}}{ \partial p^{th}}-S^{l}\right)+\varepsilon\frac{\partial S^{l}}{\partial p^{th} }\right]\frac{\partial p^{th}}{\partial t}+\\ +\left(\frac{S^{l}(1-\varepsilon)}{K_{s}}+\frac{\varepsilon S^{l}}{K_{l}} \right)\frac{\partial p^{l}}{\partial t}=\
abla\cdot\left[\frac{k_{rd}^{l} \mathbf{k}}{\mu^{l}}\cdot\
abla p^{l}\right]-S^{l}\left(\mathbf{1}:\mathbf{d} \vec{\mathbf{\hat{g}}}\right)-\
abla S^{l}\cdot\left(\varepsilon\mathbf{v}^{ \vec{\mathbf{\hat{g}}}}\right)-\frac{1}{\rho^{l}}\left( \begin{array}{c}l \stackrel{{ l\to t}}{{M}}\\ \stackrel{{ l\to s}}{{lysi}}\end{array} \right)-\frac{1}{\rho^{l}} \frac{l\stackrel{{ l\to s}}{{LCM}}}{ \tag{5}\]  
where \(\mu^{\alpha}\) is the dynamic viscosity, \(k_{rd}^{\alpha}\) is relative permeability, \(\rho^{\alpha}\) is the density. Additionally, \(\mathbf{d}\vec{\mathbf{\hat{g}}}\) is the Eulerian rate of strain tensor, \(\mathbf{k}\) the intrinsic permeability tensor of the ECM, and \( \begin{array}{c}l\stackrel{{ l\to t}}{{M}}\\ \stackrel{{ growth}}{{lysi}}\end{array} \) is an interphase exchange of mass between the phases \(l\) and \(t\), and represents the mass of IF consumed by tumor cell growth. \(K_{i}\) is the compressibility of phase \(i\) (with \(i=s\), \(t\), \(h\) and \(l\)). \( \begin{array}{c}l\stackrel{{ l\to l}}{{M}}\\ \stackrel{{ lysi}}{{lysi}}\end{array} \) takes into account of mass exchange between the necrotic compartment of tumor cells and the IF phase and \( \begin{array}{c}l\stackrel{{ l\to s}}{{M}}\\ \stackrel{{ l\to s}}{{ECM}}\end{array} \) takes account of ECM deposition by the living tumor cells. The mass balance equation of the nutrient reads  
\[\varepsilon S^{l}\frac{\partial\omega^{nl}}{\partial t}-\
abla\cdot\left( \varepsilon S^{l}D_{eff}^{nl}\
abla\omega^{nl}\right)=\frac{1}{\rho^{l}} \left(\omega^{nl}\left( \begin{array}{c}l\stackrel{{ l\to t}}{{M}} \\ \stackrel{{ l\to t}}{{lysi}}\end{array} \right)+ \begin{array}{c}l \stackrel{{ l\to s}}{{M}}\\ \stackrel{{ l\to s}}{{ECM}}\end{array}