# Elucidating the Role of Matrix Porosity and Rigidity in Glioblastoma Type IV Progression
## 4 Results
### Predicting the Growth of U87-MG Spheroids in Different Scaffolds  
The multiphysics computational model described in Section 3, is used to predict the temporal development of GBM spheroids within different microenvironmental conditions. The role of two major mechanical parameters of the hydrogels was systematically assessed, namely the Young's modulus \(E\) and the porosity \(\varepsilon\), defined as 1--\(V_{s}\)/\(V_{tot}\) where \(V_{s}\) is the volume occupied by the solid porous ECM within the whole domain \(V_{tot}\). Therefore, the larger \(\varepsilon\) is and the smaller the volume fraction occupied by the solid ECM is, the higher is the porosity of the ECM. For HA, a [PERSON]'s modulus of 200 Pa was considered (see Figure 2) while the matrix porosity \(\varepsilon\) was varied between 0.8 and 0.3, and kept homogeneous across the whole domain surrounding the initial spheroid. Figure 10A shows the variation of the predicted tumor core size as a function of time for different values of \(\varepsilon\). For \(\varepsilon\) = 0.8, a good agreement (\(R^{2}\) = 0.78) was found between the computational (solid line) and the experimental data for HA (dots). On the other hand, as the porosity \(\varepsilon\) reduces, a significant decrease in tumor growth rate was observed. At 8 days, the tumor size for \(\varepsilon\) = 0.5 and 0.3 was, respectively, 18.52% and 39.48% smaller as compared to the \(\varepsilon\) = 0.8 case (389.62 and 289.35 um as compared to 478.18 um, respectively). Indeed, dense matrices (low \(\varepsilon\)) favor pressure build-up within the porous solid which eventually limits cell proliferation and induces cell death. Figure 9: Behavior of glioblastoma type IV spheroids in hydrogels. (**A,B**) Microscopy images—bright field (BF), fluorescent (GFP), and merged—of U87-MG GFP\({}^{+}\) spheroids documenting tumor growth over time COL and HA, respectively. The tumor mass appears as a dense (high fluorescence intensity) inner portion—core—surrounded by a lighter (low fluorescence intensity) outer portion—infiltrating region (scale bar = 50 μm). (**C,D**) Tumor spheroid progression over time, expressed in terms of overall radius of the tumor mass (grey bars) and radial dimension of the infiltrating region (dark bars) for COL and HA, respectively. Statistical analysis was done using \(t\)-test multiple comparison with a confidence level of 95% (\(p<0.05\)). GraphPad was used. Figure 10: Computational model predictions for tumor spheroid growth. (**A**) Tumor core radius variation over time for different ECM porosity conditions, ranging from high porosity (\(\varepsilon\) = 0.8) to low porosity (\(\varepsilon\) = 0.3). ECM properties are homogenous across the domain, externally to the central blue core. (**B**) Tumor core radius variation over time for different ECM Young’s moduli \(E\), ranging from a soft (\(E\) = 100 Pa) to a rigid (\(E\) = 5000 Pa) situation, for a constant porosity \(\varepsilon\) = 0.8. ECM properties are homogenous across the domain, externally to the central blue core. (**C**) Tumor core radius variation over time for different spatial organizations of the ECM porosity, namely uniform (left schematic), nonuniform with alternating radial sectors of different porosities (two middle schematics), and a combination of the previous two configurations (right schematic). The legend for the graph lines is given under the corresponding images. In the graph, continuous lines show numerical results, dots report the average values of the experimental tests (the error bars show the standard deviation from the average values). Then, starting from the conditions resembling the growth of the tumor core in HA, the effect of the Young's modulus \(E\) was investigated. Figure 10B reports on the tumor growth over time for \(E\) ranging between 100 and 5000 Pa, the latter being the [PERSON]'s modulus measured for COL (see Figure 4). As expected, an increase in matrix rigidity leads to a decrease in tumor growth.