## 3 Calibrating Parameters to In Vitro and In Vivo Time-Series Measurements
### Tumour, Immune and Viral Parameters  
To obtain remaining parameter estimates for the model, we sequentially optimised the tumour time-series measurements of [PERSON] et al. (2010) for PBS, gel, Ad/IL12/GMCSF, DC+Ad/IL12/GMCSF and DC+Ad/IL12/GMCSF+gel. To reduce the degrees of freedom in the model, the parameters \(\alpha,d_{I},d_{V},d_{A},d_{H},d_{K},s_{A},s_{KA},s_{H}\) and \(s_{KH}\) were fixed to the values obtained from optimising tumour growth under Ad/IL12/GMCSF treatment in [PERSON] et al. (2016). The remaining parameters were obtained by sequentially fitting sub-models of Equations (1)-(9), where fitted parameter values were fixed for higher level models in accordance with the different treatments investigated. Table 2 gives a summary of the fitting algorithm, with parameter values in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Units** & **Description** & **Value** \\ \hline \(d_{S}\) & day\({}^{-1}\) & decay rate of short-term immature DCs & 1.562 \\ \(d_{L}\) & day\({}^{-1}\) & decay rate of long-term immature DCs & 0.318 \\ \(f\) & dimensionless & proportion of short-term injected/released DCs & 0.325 \\ \(a_{D}\) & DCs/day\({}^{2}\) & gradient of release rate & 9.725 \(\times 10^{4}\) \\ \(b_{D}\) & DCs/day\({}^{2}\) & constant release rate & 1.463 \(\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values obtained from fitting the release rate of the DCs and the short-term and long-term decay rates of the DCs to the in vitro viable DC count, see Figure 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Experiment**} \\ \hline
**PBS \& Gel** & **Ad/I/G** & **DC** & **DC+Ad/I/G** & **DC+Ad/I/G+Gel** \\ \hline \multirow{6}{*}{Relevant equations} & Equation (1) & Equation (1) & Equation (1) & Equation (1) \\  & Equation (2) & & Equation (2) & Equation (2) \\  & Equation (3) & Equation (3) & Equation (3) & \\  & & Equation (5) & Equation (5) & Equation (5) \\  & & Equation (4) & Equation (4) & Equation (4) \\  & & Equation (6) & Equation (6) \\  & Equation (7) & Equation (7) & Equation (7) & Equation (7) \\  & Equation (8) & Equation (8) & Equation (8) \\  & Equation (9) & Equation (9) & Equation (9) \\ \hline \multirow{3}{*}{Variables} & \(U\) & \(U,I,V\) & \(U\) & \(U,I,V\) & \(U,I,V\) \\  & & \(A,H\) & \(A_{I},A_{M}H\) & \(A_{I},A_{M},H\) & \(A_{I},A_{M},H\) \\  & & \(K\) & \(K\) & \(K\) \\ \hline \multirow{2}{*}{Parameters fit} & \(r,L,U_{0}\) & \(\beta,U_{0}\) & \(s_{AU},U_{0}\), & \(r_{AI},s_{AI},U_{0}\), & \(a_{V},b_{V},U_{0}\), \\  & & \(\kappa\) & \(\kappa\) & \(s_{AU}\) \\ \hline \multirow{2}{*}{Parameters fixed to Tables 1 and 3} & - & \(r,L\) & \(r,L\), & \(r,L,\beta\), & \(r,L,\beta\), \\  & & & \(d_{S},d_{L},d_{AI}\) & \(d_{S},d_{L}d_{AI},\kappa\) & \(d_{S},d_{L},a_{D},b_{D}\), \\  & & & & \(d_{AI},r_{AI},s_{AI},\kappa\) \\ \hline \multirow{2}{*}{Parameters fixed from [36]} & \(d_{V},\alpha,s_{H},d_{H}\) & \(s_{H},d_{H}\), & \(d_{V},\alpha,s_{H},d_{H}\), & \(d_{V},\alpha,s_{H},d_{H}\), \\  & & \(d_{I},s_{A},d_{A}\) & \(d_{A}\) & \(d_{I},s_{AU},d_{A}\), & \(d_{I},s_{AU},d_{A}\) \\  & & \(s_{KH},s_{KA},d_{K}\) & \(s_{KH},s_{KA},d_{K}\) & \(s_{KH},s_{KA},d_{K}\) & \(s_{KH},s_{KA},d_{K}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experiment-specific optimisation conditions for the measurements of [PERSON] et al. (2010). Equations used to optimise each experiment are listed along with the state variables considered and parameters fitted or fixed. Some parameters were fixed to previous optimisation work in [PERSON] et al. (2016).