## 4 Optimising the Gel Release Profile
### Sigmoidal Release Profile  
The final release curve considered was a sigmoidal function \(f_{\rm s}(t)\), given by Equation (14). Integrating and simplifying using the constraint in Equation (15) gives  
\[u_{V}(t)=\frac{kV_{0}}{\left(\ln\left(\frac{1+e^{-k(t-x_{0})}}{1+e^{x_{0}}} \right)+t,k\right)\left(1+e^{-k(t-x_{0})}\right)},\quad u_{DC}(t)=\frac{kD_{0} }{\left(\ln\left(\frac{1+e^{-k(t-x_{0})}}{1+e^{x_{0}}}\right)+t,k\right)\left( 1+e^{-k(t-x_{0})}\right)}. \tag{19}\]  
We imposed additional constraints on the sigmoidal function so that the function was always positive, i.e., when \(k>0\), the function within the \"In\" would always be positive, and when \(k<0\), the denominator was always negative.  
Assuming initially that the characteristics of the release curve for virus and DCs are equivalent, gives the tumour size at day 20 in Figure 6a for a release period of 5 days, 9 days and 13 days. The sign of \(k\) determines whether the sigmoidal function is increasing or decreasing. This explains the clear dynamical switch at \(k=0\). For increasing sigmoidal release curves (\(k>0\)), the tumour size is minimised when the mid-point \(x_{0}\) is larger. The overall minimum tumour size is achieved for increasing release curves when the midpoint is decreased below zero. The release period \(t_{r}\) influences the minimum achieved with \(t_{r}=9\) achieving a lower tumour volume than \(t_{r}=5\), and the planes associated with \(t_{r}=9\) and \(t_{r}=13\) approaching the same limit for \(x_{0}\to 0\).  
Investigating this further, the release time \(t_{r}\) was fixed to 9 days and the value of \(k\) varied for virus and DCs for \(x_{0}=-1,1\) and 3, see Figure 6b. Lower values of \(k\) for the DCs reduced the tumour volume when the midpoint \(x_{0}\) was positive. As \(x_{0}\) decreases, the variance in the plane reduces, and we see, overall, the minimum tumour burden is achieved irrespective of \(k\) for \(x_{0}=-1\).  
Simulating varying \(x_{0}>0\) for virus and DCs and fixing \(k=-1,1\) and \(3\), we see that the value of the midpoint needed to minimise the tumour burden depends most heavily on the midpoint for \(u_{DC}(t)\), Figure 6c. If \(k>0\), then a lower \(x_{0}\) is needed, whereas the opposite is exhibited if \(k<0\)  
Figure 6: Effectiveness of sigmoidal gel-release rate \(f_{s}(t)\) (Equation (14)) for DCs, \(u_{DC}(t)\), and virus, \(u_{V}(t)\). In panel (**a**), the tumour size at day 20 as a function of \(k\), \(x_{0}\) and \(t_{r}\) is plotted, where the parameters were common for virus and DC release functions. In panel (**b**), the tumour size at day 20 is plotted as a function of varying \(k\) for virus and DCs with \(t_{r}=9\) and planes representing \(x_{0}=-1,x_{0}=1\) and \(x_{0}=3\). In panel (**c**), the tumour size at day 20 is plotted as a function of varying \(x_{0}\) for virus and DCs with \(t_{r}=9\) and planes representing \(k=-1,1\) and \(k=3\). In panel (**d**), the tumour size at day 20 as a function of \(t_{r}\) for virus and DCs where \(x_{0}=1\) and \(k=1\). The red points in panels (**c,d**) correspond to the simulated release profiles in panels (**e,f**). In panel (**e**), \(k=1\), \(t_{r}=9\) and \(x_{0}=0.5\) for \(u_{V}\) and \(x_{0}=3\) for \(u_{DC}\). In panel (**f**), \(k=x_{0}=1\) and \(t_{r}=1.5\) for \(u_{V}\) and \(t_{r}=14\) for \(u_{DC}\). The top row of figures in panels (**e,f**) correspond to the number of tumour cells \(U+I\) and the bottom row of figures are the corresponding release profiles. Note the use of both vertical axes.  
Unsurprisingly, simulating the sigmoidal function for fixed values of \(k=1\) and \(x_{0}=1\) and varying \(t_{r}\) for DCs and virus, we have similar dynamics to the linear and constant release curves, see Figure 6d. For an increasing sigmoidal function, the optimal tumour size is obtained when the virus is released quickly and the DCs have an extended release. Two model simulations are provided in Figure 6e,f to demonstrate the different sigmoidal release functions considered.