# A Mathematical Model for the Treatment of Melanoma with the BRAF/MEK Inhibitor and Anti-PD-1
## 2 Medical Assumptions and Mathematical Equations
**Equation for TGF-\(\beta\)**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]4  
The term \(\lambda_{T_{\beta}C}\cdot C\) is the production by cancer cells, \(\lambda_{T_{\beta}T_{reg}}\cdot T_{reg}\) is the production by Tregs, \(\lambda_{T_{\beta}M}\cdot M\) is the production by MDSCs, and \(d_{T_{\beta}}T_{\beta}\) is the degradation of TGF-\(\beta\). **Equation for IL-6**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]5  
The first term \(\lambda_{IL_{6}C}\cdot C\) describes the production of IL-6 by cancer cells, and the second term \(d_{IL_{6}}\cdot IL_{6}\) is the degradation of IL-6. **Equation for IL-10**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]6The term \(\lambda_{IL_{10}C}\cdot C\) represents the production of IL-10 by cancer cells, \(\lambda_{IL_{10}M}\cdot M\) is the production of IL-10 by MDSCs, and is the \(d_{IL_{10}}\cdot IL_{10}\) degradation of IL-10. **Equation for PD-1, PD-L1 and PD-1-PD-L1**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]7  
where \(\Psi\equiv\rho_{P}\big{(}T_{CD4+T}+T_{CD8+T}+\epsilon_{T}\cdot T_{reg}\big{)}\). PD-1 is expressed on the surface of activated \(CD4+T\) cells, activated \(CD8+T\) cells, and Tregs. The mathematical relationship between the number of PD-1 and \(CD4+T\) and \(CD8+T\) is proportional to the factor \(\rho_{P}\) (it is constant when no anti-PD-1 drug is administered), and the number of PD-1 is smaller than Treg by the factor \(\epsilon_{T}=0.8\). In addition, PD-L1 is expressed on the surface of activated \(CD4+T\) cells, activated \(CD8+T\) cells, MDSCs, and tumor cells. The mathematical connection between the number of PD-L1 and \(CD4+T\), and \(CD8+T\) is proportional to the factor \(\rho_{L}\), and the number of PD-L1 is smaller than that of C cells by the factor \(\epsilon_{C}\) and smaller than that of the M cells by the factor \(\epsilon_{M}\); hence, \(L=\rho_{L}(T_{CD4+T}+T_{CD8+T}+\epsilon_{M}M+\epsilon_{C}C)\). **Equation for anti-PD-1**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]8  
where \(\mathcal{A}(t,q_{AntiPD1})\) is the treatment of anti PD-1, \(\mu_{PA}\cdot P\cdot A_{PD1}\) is the depletion through blocking PD-1, \(d_{A}\cdot A_{PD1}\) is the degradation of anti PD-1. **Equation for BRAF/MEK inhibitor**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]9  
where \(\mathcal{B}(t,q_{(\cdot)})\) is the treatment of BRAF/MEK, \(\mu_{BC}\cdot C\frac{BM}{K_{B}+BM}\) is the absorption of BRAF/MEKi by cancer cells, and \(d_{B}\cdot BM\) is the degradation of the BRAF/MEK inhibitor. **Equation for cells velocity**  
\[\partial_{t}D+\Pi_{\delta_{D}}(D) = D_{0}\cdot\Lambda_{DC}-d_{D}\cdot D \tag{3}\]0  
the treatment is biological, assuming all the cells have the same volume and surface area, the diffusion of all the cells are the same. **Equation for free boundary**  
\[\partial_{t}D+\Pi_{\delta_{D}}(D) = D_{0}\cdot\Lambda_{DC}-d_{D}\cdot D \tag{3}\]1  
we assumes that the free boundary moves with the velocity of cells.