## 4 Optimising the Gel Release Profile
### Linear Release Profile  
Assuming both virus and DCs are released from the gel at a linear rate, \(f_{l}(t)\) in Equation (13), which is increasing, applying the constraint in Equation (15) gives  
\[u_{V}(t)=a\left(t-\frac{t_{r}}{2}\right)+\frac{V_{0}}{t_{r}},\quad u_{DC}(t)=a \left(t-\frac{t_{r}}{2}\right)+\frac{D_{0}}{t_{r}}. \tag{17}\]  
To investigate whether there was an optimal increasing linear release rate, we initially fixed \(t_{r}\) as common for virus and DCs and varied the gradient of the release curve \(a\), which was bounded by \(0<a<2A_{0}/t_{r}^{2}\) so as to be strictly positive and increasing. The resulting tumour size on day 20 is plotted in Figure 5a,b. As \(t_{r}\) increases from 1 day to 5 days, the overall tumour size decreases irrespective of the choice of \(a\). However, after a release period of \(t_{r}=6\) days, the minimum achievable tumour size at day 20 increases and so does the dependence of the tumour size at day 20 on the  
Figure 4: Effectiveness of a constant gel-release rate \(f_{s}(t)\) (Equation (16)) for DCs, \(u_{DC}(t)\), and virus, \(u_{V}(t)\). In panel (**a**), the tumour size at day 20 is given as a function of gel-release period \(t_{r}\), specific to the virus and DCs. The red points in panel (**a**) correspond to the simulated release profiles in panel (**b**), \(t_{r}=13\) days for DCs and \(t_{r}=1\) day for virus, and in panel (**c**), \(t_{r}=2.6\) days for DCs and \(t_{r}=9.8\) days for virus. The top row in panels (**b,c**) correspond to the total number of tumour cells \(U+I\), and the bottom corresponds to the release profile. Note the use of both vertical axes. steepness of the release rate \(a\). A minimum tumour size of 887 mm\({}^{3}\) was achieved for \(t_{r}=11\) days and \(t_{r}=13\) days when \(a=0\). In other words, if the release time is common for DCs and virus, the optimal linear release curve is approximately constant. Figure 5: Effectiveness of a linear gel-release rate \(f_{1}(t)\) (Equation (17)) for DCs, \(u_{DC}(t)\), and virus, \(u_{V}(t)\). In panels (**a**,**b**), the tumour size at day 20 is given as a function of the gradient of the release curve \(a\) for virus and DCs with planes corresponding to fixed values for the release time \(t_{r}\) which are noted on the plot. In panel (**c**), the tumour size at day 20 is given as a function of the release time for DCs and virus with planes corresponding to the labelled values of \(a\). In panel (**d**), the tumour size at day 20 for a decreasing linear release rate (Equation (18)) is plotted for varying initial release rate \(b\) for virus and DCs. The green point in panel (**c**) is the corresponding release profile from the results of [PERSON] et al. (2010) in Figure 3e. The red points in panels (**c**,**d**) correspond to the simulated release profiles in panel (**e**) where \(t_{r}=13\) for DCs and \(t_{r}=0.5\) for virus with \(a=3\), and in panel (**f**) where \(b=48.7\) for DCs and virus. The top row of figures in panels (**e**,**f**) corresponds to the total number of tumour cells \(U+I\) and the bottom row of figures is the corresponding release profile. Note the use of both vertical axes. We then numerically simulated the increasing linear release rate with \(t_{r}\) varying for virus and DCs and fixed \(a=0.001,3.3\) and \(7.4\), see Figure 5c. Qualitatively, the dependence on the value of \(t_{r}\) for \(u_{DC}(t)\) and \(u_{V}(t)\) is similar to that of the constant release profile in Figure 4a. A minimum tumour volume of 628.4 mm\({}^{3}\) was achieved for an interval of \(a\) between 3.3 and 4.6 for \(t_{r}=1\) for the virus and \(t_{r}\) between 11 and 13 for DCs. This suggests that it is possible to obtain an optimal efficacy with the linear release function, but only when the release time of the constituents are separated. Although the gel-release mechanisms measured by [PERSON] et al. [10] had an increasing gradient, it is worth considering how effective a treatment would be when released at a decreasing linear rate.