## 5 Discussion
performed optimally and there are many increasing linear release profiles that would have performed significantly worse. Considering the more complex sigmoidal gel-release profile, we found that the tumour burden was reduced when the midpoint was below 0 and the curve was an increasing function (Figure 6a,b). In other words, when the gradient of the release rate was reducing and tending towards a constant release profile. Furthermore, explicitly increasing the steepness, \(k\), of the sigmoidal curve always increased the tumour size (Figure 6b,c). Unsurprisingly, the optimal release period for viruses and DCs was qualitatively similar to that of the constant and linear release periods (Figure 6d), implying that it is primarily the release period that optimises the treatment irrespective of the release curve. Implementing a genetic algorithm, we were able to determine the true optimal constant, linear and sigmoidal release curves (Table 5 and Figure 7). Unsurprisingly, the optimal release curves each had a short release time for the virus and an extended release time of 13 days for the DCs. Comparing the impact of the different optimal release curves (Figure 7c), we see that the difference in the tumour size was minimal. Although the optimised linear release curve reduced the tumour size the most, all release curves improved the treatment by a similar amount. This suggests that extending the complexity of the release mechanisms may not be necessary, as a constant release curve performs almost as well as a linear release curve, and the optimised sigmoidal curve is more constant than sigmoidal. Additionally, this also suggests that for the constraints considered on the release curves it is not possible to optimise the gel so that the tumour is completely eradicated. Previously, [PERSON] et al. [38] optimised the scheduling of discrete doses of oncolytic virus and DCs. They showed that higher doses of DCs were needed to result in a robust optimised therapy. The fact that our results suggest the need for sustained release of DCs complements their findings and demonstrates that therapy can be optimised when the timing of DC administration is optimal. Our results also align with the results of [PERSON] et al. [37], where they showed using a calibrated model that it is more effective to treat a tumour with immunostimulatory oncolytic viruses first and then follow up with a sequence of DCs than to alternate oncolytic virus and DC injections. A limiting assumption of this work is that the complex gel mechanics are modelled phenomenologically. As the goal was to investigate the effect of changing the function \(u_{DC}(t)\), we believe it is not necessary to model the gel-release kinetics explicitly and, as qualitatively the model formulation for the measurements with and without the gel is able to capture the model dynamics (see Figure 2), this was a suitable simplification. Future work, however, would look to develop a more appropriate realisation of the gel kinetics and investigate mechanistically the optimal release profile. Another simplification in the model is that the rate at which immune cells are stimulated is independent of the antigen type. In reality, immune cells are antigen specific [18] with either tumour-antigen- or virus-antigen-specific cells. Future extensions of this work could consider two populations of immune cells similar to the work of [PERSON] et al. [56] and investigate further why optimised gel-release profiles require an extended DC release. Using a system of ODEs, we were able to replicate the experimental conditions of DC+Ad/ IL12/GMCSF therapy and determine possible improvements to the gel-release profile. This model can be applied to other vivo-immunotherapies that aim to activate the adaptive antitumor immune response. As we only consider simple formulations for the gel-release rate, this model is easily translatable to other sustained-release delivery systems where the release functions \(u_{V}(t)\) and \(u_{DC}(t)\) may or may not be known.