## 4 Optimising the Gel Release Profile
### Implementing a Genetic Algorithm to Determine the True Optimal Release Curves  
In the previous sections, the gel-release dynamics have been qualitatively linked with the tumour burden on day 20. To determine the optimal parameter sets for the constant, linear and sigmoidal dosage functions, the tumour volume on day 20 was minimised under each release profile using Matlab's genetic algorithm function _ga_[53]. Genetic algorithms are heuristic global optimisation routines inspired by natural selection that are frequently employed to estimate parameters in computational biology models. They have previously been applied to study optimal dosing routines in immunology [54,55] and combined oncolytic virotherapy and immunotherapy [29].  
The tumour growth under the optimised constant, linear and sigmoidal release curves for virus and DCs are pictured in Figure 7. The parameter values from the genetic algorithm for each optimal release curve are given in Table 5. The constraints applied for the numerical simulations were also applied here for the different release curves. The optimal decreasing linear release curve has not been plotted as we know from the numerical simulations that it did not achieve a significant global minimum. Additionally, we found that the decreasing sigmoidal function's optimal parameters qualitatively matched that of the increasing sigmoidal function.  
Figure 7c shows that all three optimised release curves are able to reduce the tumour burden below the average size from [PERSON] et al.'s experiments [10]. The dynamics of the immature APC population (\(D_{L}+D_{S}+A_{I}\)) are similar for the constant and sigmoidal release curves, see Figure 7a, whereas the optimal linear release profile delays the initiation of the APC accumulation. As the virus dynamics are the same for the three release curves (Figure 7b), this suggests that the delay on the immature APC population expansion causes the decrease in tumour size under the optimised linear release therapy in Figure 7c.  
Overall, the parameters returned from the genetic algorithm matched the qualitative results of the numerical investigation for the constant and linear release functions. Both release curves required less than a day for the virus release and approximately 13 days for the DCs. The steepness of the linear curve \(a\) was much higher for the virus than the DCs, with the DC's value of \(a\) close to that plotted in Figure 5c. The sigmoidal release curve had the same dependence on the release curves for viruses and DCs as the constant and linear release period. Interestingly, the midpoint for both functions was significantly negative, meaning the curve would be similar to an increasing linear release profile. These results all suggest that the added complexity of the sigmoidal function with the time-varying gradient does not improve the efficacy of the gel-release mechanism.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Constant**} & \multicolumn{2}{c}{**Linear (Increasing)**} & \multicolumn{2}{c}{**Sigmoidal**} \\ \hline  & & \(t_{r}\) & \(t_{r}\) & \(a\) & \(t_{r}\) & \(k\) & \(x_{0}\) \\ \hline Ad/IL12/GMCSF & \(u_{V}(t)\) & 0.1261 & 0.1252 & 14.9893 & 0.05 & 9.2789 & \(-\)6.0499 \\ DCs & \(u_{DC}(t)\) & 13.7732 & 13.1629 & 2.8866 & 13.6311 & 9.1951 & \(-\)8.3333 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimised parameter values for the constant \(f_{c}\), linear \(f_{l}\) and sigmoidal \(f_{s}\) release curves (Equations (12)–(14)) for Ad/IL12/GMCSF and immature DCs from a gel. The virus and DCs were allowed to have individualised curves. Figure 7 is a plot of the tumour volume under the five optimised curves.