# A Mathematical Model for the Treatment of Melanoma with the BRAF/MEK Inhibitor and Anti-PD-1
## Appendix A
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Description** & **Value and Units** \\ \hline \(\lambda_{T_{\text{reg}}T_{\beta}}\) & Activation rate of \(T_{\text{regs}}\) by \(TGF-\beta\) & 0.415 day\({}^{-1}\) \\ \hline \(\lambda_{DC}\) & Activation rate of DCs by tumor cells & 4 g/cm\({}^{3}\). day \\ \hline \(\lambda_{T_{\text{CD4}+TL-12}}\)Activation rate of \(CD4+T\) cells by \(IL-12\) & 18.64 day\({}^{-1}\) \\ \hline \(\lambda_{T_{\text{CD4}+TL-2}}\) Activation rate of \(CD4+T\) cells by \(IL-2\) & 0.25 day\({}^{-1}\) \\ \hline \(\lambda_{T_{\text{CBS}+TL-2}}\) Activation rate of \(CD4+T\) cells by \(IL-2\) & 0.25 day\({}^{-1}\) \\ \hline \(\lambda_{T_{\text{reg}}T_{0}}\) & Activation rate of \(T_{\text{regs}}\) by \(PD-1-PD-L1\) & 0.083 day\({}^{-1}\) \\ \hline \(\eta_{1}\) & Killing rate of tumor cells by \(CD4+T\) cells & 11.5 day\({}^{-1}\). cm\({}^{3}\)/g \\ \hline \(\eta_{8}\) & Killing rate of tumor cells by \(CD8+T\) cells & 46 day\({}^{-1}\). cm\({}^{3}\)/g \\ \hline \(\mu_{PA}\) & Blocking rate of \(PD-1\) by anti-\(PD-1\) & \(6.04\cdot 10^{6}\) cm\({}^{3}\)/g. day \\ \hline \(\mu_{DC}\) & Absorbtion rate of BRAF/MEKi by cancer cells & \(6.17\cdot 10^{-10}\) day\({}^{-1}\) \\ \hline \(\rho_{P}\) & Expression of \(PD-1\) in \(T\) cells & \(2.49\cdot 10^{-7}\) \\ \hline \(\rho_{P}\) & Expression of \(PD-L1\) in \(T\) cells & \(5.22\cdot 10^{-7}\) \\ \hline \(\epsilon_{C}\) & Expression of \(PD-L1\) in tumor cells & 0.01 \\ \hline \(\epsilon_{M}\) & Expression of \(PD-L1\) in MDSCs & 0.1 day\({}^{-1}\) \\ \hline \(d_{D}\) & Death rate of DCs & 0.1 day\({}^{-1}\) \\ \hline \(d_{T_{\text{CD4}+T}}\) & Death rate of \(CD4+T\) cells & 0.97 day\({}^{-1}\) \\ \hline \(d_{T_{\text{CBS}+T}}\) & Death rate of \(CD8+T\) cells & 0.18 day\({}^{-1}\) \\ \hline \(d_{T_{\text{reg}}}\) & Death rate of Tregs & 0.2 day\({}^{-1}\) \\ \hline \(d_{M}\) & Death rate of MDSCs & 0.03 day\({}^{-1}\) \\ \hline \(d_{C}\) & Death rate of tumor cells & 0.17 day\({}^{-1}\) \\ \hline \(d_{IL12}\) & Degradation rate of \(IL-12\) & 1.38 day\({}^{-1}\) \\ \hline \(d_{IL2}\) & Degradation rate of \(IL-2\) & 2.376 day\({}^{-1}\) \\ \hline \(d_{T_{\beta}}\) & Degradation rate of \(TGF-\beta\) & 499.066 day\({}^{-1}\) \\ \hline \(d_{IL_{6}}\) & Degradation rate of \(IL-6\) & 4.16 day\({}^{-1}\) \\ \hline \(d_{IL10}\) & Degradation rate of \(IL-10\) & 8.32 day\({}^{-1}\) \\ \hline \(d_{A}\) & Degradation rate of anti-\(PD-1\) & 0.046 day\({}^{-1}\) \\ \hline \(d_{B}\) & Degradation rate of BRAF/MEKi & 1.66 day\({}^{-1}\) \\ \hline \(D_{0}\) & Density of inactive DCs & \(2\cdot 10^{5}\) g/cm\({}^{3}\) \\ \hline \(T_{10}\) & Density of naive \(CD4+T\) cells in tumor & \(4\cdot 10^{-4}\) g/cm\({}^{3}\) \\ \hline \(T_{80}\) & Density of naive \(CD8+T\) cells in tumor & \(2\cdot 10^{-4}\) g/cm\({}^{3}\) \\ \hline CM & Carrying capacity of cancer cells & 0.8 g/cm\({}^{3}\) \\ \hline \(\hat{T}_{1}\) & Density of \(CD4+T\) cells from lymph node & \(4\cdot 10^{-3}\) g/cm\({}^{3}\) \\ \hline \(\hat{T}_{8}\) & Density of \(CD8+T\) cells from lymph node & \(2\cdot 10^{-3}\) g/cm\({}^{3}\) \\ \hline \(K_{T_{\text{CD4}+T}}\) & Half-saturation of \(CD4+T\) cells & \(2\cdot 10^{-3}\) g/cm\({}^{3}\) \\ \hline \(K_{T_{\text{CBS}+T}}\) & Half-saturation of \(CD8+T\) cells & \(1\cdot 10^{-3}\) g/cm\({}^{3}\) \\ \hline \(K_{T_{\text{reg}}}\) & Half-saturation of Tregs & \(5\cdot 10^{-4}\) g/cm\({}^{3}\) \\ \hline \hline \end{tabular}