# 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
Indeed, following the same argument presented above for \(\varepsilon\), a higher mechanical constriction results in lower cell proliferation. Notably, despite the huge variation in [PERSON]'s modulus, from 100 to 5000 Pa, the corresponding change in tumor size is quite moderate. At 8 days, the tumor size for \(E=100\) and 5000 Pa was, respectively, 6.07% higher and 6.74% lower as compared to the \(E=200\) Pa case (507.25 and 445.95 \(\upmu\)m as compared to 478.18 \(\upmu\)m, respectively). This evidently implies that tumor growth is more sensitive to variations in matrix porosity rather than rigidity. For the \(E=200\) Pa case there is a good agreement between computational results (solid line) and the experimental data (dots). As shown in Figure 9, tumors in COL tend to grow faster than in HA. Therefore, in the numerical simulations, matrix porosity was changed to reproduce this behavior. In Figure 10C, the dashed and dash-dot lines reproduce the cases for \(\varepsilon=0.5\) (low porosity) and \(\varepsilon=0.9\) (high porosity). As expected, lower porosities were associated with smaller tumor sizes over time. However, in both cases, the computational predictions were far from the experimental data. Thus, since COL are characterized by fibers along which tumor cells can more easily migrate, it was assumed that the matrix surrounding the initial spheroid could be characterized by alternating radial sectors of low (\(\varepsilon=0.5\) and 0.3) and high (\(\varepsilon=0.85\)) porosities. This configuration should favor tumor growth along the low porosity sectors as compared to the denser matrix regions. The dotted and dash-two dots lines in Figure 10C present data for these nonuniform porosity configurations. Within the first 3 days, both cases presented a similar behavior, with almost overlapping tumor growth curves. However, at longer time points, both cases predicted a faster growth as compared to the experimental data. Interestingly, at longer time points, the \(\varepsilon=0.5/0.85\) case showed a trend similar to the experimental data but shifted towards larger sizes. As detailed in Figure 11, the dual porosity configurations (\(\varepsilon=0.3/0.85\) and \(\varepsilon=0.3/0.5\)) predicted quite similar tumor growth behaviors, with the latter giving overall smaller tumor volumes because of the overall lower porosity of the matrix. Both simulations with dual configuration of porosity are not able to reproduce the experimental data (dots). More interestingly, the \(\varepsilon=0.3/0.85\) case resulted in tumor core volumes very similar to those estimated in the case of a uniformly porous matrix with \(\varepsilon=0.9\) (high porosity) (Figure 10C). These numerical results would imply that matrices with nonuniform porosities experience a more rapid tumor growth as compared to uniform and higher porosity matrices. However, none of the two general porosity configurations so far considered (uniform and nonuniform with radial sectors) accurately predict the tumor growth in COL. This is mostly because of the relatively low growth rate observed within the first 4 days of culturing. Therefore, the more realistic configuration for tumors growing in COL resulted in a combination of the above cited two cases: a rim with uniform porosity surrounding the initial tumor spheroid with \(\varepsilon=0.5\), which serves to modulate the initial growth rate; and alternating radial sectors of low (\(\varepsilon=0.5\)) and high (\(\varepsilon=0.85\)) porosities, characterizing the most exterior portion of the matrix. This case is identified as \(\varepsilon=0.5+0.5/0.85\) in the following. For this configuration, a good agreement (\(R^{2}=0.91\)) was achieved between the computational (solid line) and the experimental data (dots) for COL, as documented in Figure 10C.