## 3 Calibrating Parameters to In Vitro and In Vivo Time-Series Measurements
### Tumour, Immune and Viral Parameters
\begin{table}
\begin{tabular}{r c c c c c c c c} \hline \hline  & **Param** & **Units** & **Description** & **PBS\% Gel** & **Ad/I/G** & **DC** & **DC** + _Ad/I/G** & **DC** + _Ad/I/G + gel** \\ \hline  & \(d_{A1}\) & day\({}^{-1}\) & Immature DCs decay rate & & & 1.562 & 1.562 & 1.562 \\  & \(L\) & cells\(\times 10^{6}\) & carrying capacity & **13572** & 13572 & 13572 & 13572 & 135720 \\  & \(r\) & day\({}^{-1}\) & growth rate & **0.1045** & 0.1045 & 0.1045 & 0.1045 & 0.1045 \\  & \(U_{\rm D}\) & cells\(\times 10^{6}\) & initial tumour size & **20.35** & **85.90** & 85.90 & 85.90 & **47.45** \\  & \(\beta\) & day\({}^{-1}\) & infection rate & - & **0.7295** & - & 0.7295 & 0.7295 \\ Fit & \(\kappa\) & day\({}^{-1}\) & killing rate & - & **0.8225** & **0.218147** & **0.3626** & 0.3626 \\  & \(s_{A1}\) & day\({}^{-1}\) & APC activation rate by \(U\) & - & - & **0.001047** & 0.0022 & **0.05278** \\  & \(r_{A1}\) & day\({}^{-1}\) & recruitment rate of \(A_{1}\) & - & - & - & **0.0192** & 0.0192 \\  & \(s_{A1}\) & day\({}^{-1}\) & APC activation rate by \(I\) & - & - & - & **0.0011** & 0.0011 \\  & \(u_{V}\) & linear release slope (virus) & - & - & - & - & **0.33791** \\  & \(b_{V}\) & initial linear release (virus) & - & - & - & - & 2.35971 \\ \hline  & \(a\) & virus\(\times 10^{10}\) & viral burst size & - & 3500 & - & 3500 & 3500 \\  & \(d_{I}\) & day\({}^{-1}\) & burst rate & - & 1 & - & 1 & 1 \\  & \(d_{V}\) & day\({}^{-1}\) & viral decay rate & - & 2.3 & - & 2.3 & 2.3 \\  & \(d_{A}\) & day\({}^{-1}\) & decay of mature APCs & - & 0.23 & 0.23 & 0.23 & 0.23 \\
[FOOTNOTE:]Footnote : footnotemark: [ENDFOOTNOTE] & \(d_{A1}\) & day\({}^{-1}\) & decay of humature APCs & - & 1.562 & 1.562 & 1.562 \\
[FOOTNOTE:]Footnote : footnotemark: [ENDFOOTNOTE] & day\({}^{-1}\) & decay of helper T cells & - & 0.23 & 0.23 & 0.23 & 0.23 \\  & \(d_{K}\) & day\({}^{-1}\) & decay of killer T cells & - & 0.35 & 0.35 & 0.35 & 0.35 \\  & \(s_{A}\) & day\({}^{-1}\) & APC activation rate & - & 1.2 & - & 7.1 & 7.1 \\  & \(s_{K,A}\) & day\({}^{-1}\) & APC activation killer T cell & - & 7.1 & 7.1 & 7.1 \\  & \(s_{H}\) & day\({}^{-1}\) & helper T cell activation & - & 0.78 & 0.78 & 0.78 & 0.78 \\  & \(s_{KH}\) & day\({}^{-1}\) & helper T cell activate \(K\) & - & 1.6 & 1.6 & 1.6 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3. Parameter estimates from the sequential optimisation of the model following the algorithm in Table 2 to the experimental measurements in [10]. Certain parameters were fixed to their value in [36], and this is indicated in the table. Parameters that were fit for a particular experiment have their values in **bold**. Note that Ad/I.L12/GMCSF has been shortened to Ad/I/G. As a control, [PERSON] et al. [10] measured the tumour growth under a single injection of PBS and an empty gel. We assumed that the underlying tumour growth in both of these experiments should be identical and fit the growth rate, \(r\); carrying capacity, \(L\); and initial tumour size, \(U_{0}\), to both data sets, fixing \(I=V=A_{I}=A_{M}=H=K=0\) in Equations (1)-(8), see Figure 3a. Figure 3: Model optimisation of the measurements from [PERSON] et al. [10] following the algorithm in Table 2 for (**a**) control (PBS and gel) (**b**) Ad/IL12/GMCSF, (**c**) DC, (**d**) DC+Ad/IL12/GMCSF and (**e**) DC+Ad/IL12/GMCSF+gel. The individual mouse data are plotted as circles with the mean and standard error bar at each time point in blue. The model output for the experiment-optimised parameters is plotted as a solid black line. The parameter values are given in Table 3. In panel (**f**), the corresponding model populations are plotted for the DC+Ad/IL12/GMCSF+gel experiment in panel (**e**), note the use of both vertical axes. \(A_{I}\) was not plotted in panel (**f**) as the magnitude of the population was too small. Following the control experiments, [PERSON] et al. (2010) measured the tumour size under a single injection of Ad/IL12/GMCSF.