## 5 Discussion  
In this work, we developed a system of ordinary differential equations for the sustained release of virus and immature dendritic cells from a gel and the resulting interaction with a population of tumour cells and immune cells. Using our model, we recreated the in vitro and in vivo experiments of [PERSON] et al. [10] (Figures 2 and 3). Using the calibrated model, we have been able to provide insight into how the gel-release profile may be optimised to improve treatment. In the sequential fitting, attributes of the model were calibrated to the tumour response to combined Ad/IL12/GMCSF and immature DC therapy (Figure 3). To optimise the model to the Ad/IL12/GMCSF treatment of Lewis Lung Carcinoma (LLC) tumours by [PERSON] et al. [10], we fixed most parameters to values obtained from our previous optimisation of the Ad/IL12/GMCSF treatment of melanoma [36], fitting only the virus infection rate, \(\beta\), and killer T cell killing rate, \(\kappa\) (see Figure 3b). Both these parameters decreased for the treatment of LLC tumours compared to melanoma, suggesting that different tumour cells lines will ultimately result in different virus and immune characteristics. From the optimisation of the single immature DC injection (Figure 3c) and the combined DCs+Ad/IL12/GMCSF injection (Figure 3d), we found that immature APCs are stimulated more rapidly by infected as opposed to uninfected cells, parameters \(s_{AI}\) and \(s_{AUL}\), respectively. Additionally, we note that the killing rate, \(\kappa\), decreased for Ad/IL12/GMCSF alone, compared to DC+Ad/IL12/GMCSF. We hypothesise that the addition of immature DCs results in a heightened  
Figure 7: Genetic optimisation of the gel-release profile. Implementing a genetic algorithm we determined the optimal constant, linear and sigmoidal curves for the virus, \(u_{V}(t)\), and DCs, \(u_{DC}(t)\), see Table 5. The corresponding model dynamics are plotted for (**a**) the total number of immature DCs (\(D_{L}+D_{S}+A_{I}\)), (**b**) the virus (\(V\)) and (**c**) the total number of tumour cells (\(U+I\)). In panel (**c**), the mean and standard error from [PERSON] et al.’s tumour growth under DC+Ad/IL12/GMCSF loaded onto a gel is plotted in blue, and in grey is the model’s predicted tumour growth under DC+Ad/IL12/GMCSF in the absence of the gel. antitumour immune response, which results in a \"division of resources\" between the anticumour and antiviral immune response. [PERSON] et al. [10] reduced the tumour size by 50% on day 20 by using a GHPA hydrogel to release the DC+Ad/IL12/GMCSF treatment (Figure 3e). Using our calibrated model (Table 3), we numerically investigated alternative gel-release profiles to determine whether we could reduce the tumour burden further. Conserving the total amount of virus and DCs released from the gel, we first considered a constant gel-release profile. Simulating different release periods, \(t_{r}\), for the virus and DCs we found a minimum tumour size is attainable when the virus is released over a day, and the DCs are released for 13 days (Figure 4). This suggests that an initial burst of virus reduces the tumour volume through infection and stimulates an immune response that, under a prolonged constant release of DCs from the gel, results in a significantly reduced tumour volume. A natural extension was then to examine the tumour volume under different linear releases. If the release period, \(t_{r}\), is fixed for both the DCs and the virus, then release periods between \(t_{r}=5\) to 14 days reduce the tumour burden most significantly (Figure 5a,b). The global optimal of this regime occurs when \(a\approx 0\), i.e., a constant release profile, suggesting that the optimal protocol is constant administration if release periods for the virus and DCs cannot be distinct. For unique release times, the optimal release period is similar to the constant release profiles: virus released rapidly and the DCs released slowly over time (Figure 5c). Overall, the gel designed by [PERSON] et al.