# A Mathematical Model for the Treatment of Melanoma with the BRAF/MEK Inhibitor and Anti-PD-1
## 2 Medical Assumptions and Mathematical Equations
Under the above assumptions, the mathematical model includes nonlinear partial differential equations in the following form:  
**Equation for DCs (D)**  
\[\partial_{t}H-\Omega_{\delta_{H}}(H) = \lambda_{HN_{C}}\cdot N_{C}-d_{H}\cdot H \tag{1}\] \[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]  
Biological interpretation of the mathematical expressions is as follows: \(\lambda_{HN_{C}}\cdot N_{C}\), released from necrotic cancer cells; \(d_{H}\cdot H\), degradation; and \(\lambda_{N_{C}}\cdot C\), the life of cancer cells. **Dendritic cells are activated by HMGB-1**  
\[\partial_{t}D+\Pi_{\delta_{D}}(D) = D_{0}\cdot\Lambda_{DC}-d_{D}\cdot D \tag{3}\]  
The expression \(D_{0}\cdot\Lambda_{DC}\) describes the activation of dendritic cells by HMGB-1, and \(d_{D}\cdot D\) describes the death of dendritic cells. **Equation for \(CD4+T\) cells and \(CD8+T\)**  
\[\partial_{t}T_{CD4+T}+\Pi_{\delta_{T_{CD4+T}}}(T_{CD4+T}) = T_{CD4+T0}\cdot\Lambda_{T_{CD4+T}IL_{12}}\cdot Inh_{IL_{10}} \cdot Inh_{Treg}\cdot Inh_{PD_{1-L1}} \tag{4}\] \[+T_{CD4+T}\cdot\Lambda_{T_{CD4+T}IL_{2}}\cdot Inh_{PD_{1-L1}}-d_ {T_{CD4+T}T}\Upsilon_{CD4+T}\]  
\[\partial_{t}T_{CD8+T}+\Pi_{\delta_{T_{CD8+T}}}(T_{CD8+T}) = T_{CD8+T0}\cdot\Lambda_{T_{CD8+T}IL_{12}}\cdot Inh_{IL_{10}} \cdot Inh_{Treg}\cdot Inh_{PD_{1-L1}} \tag{5}\] \[+T_{CD8+T}\cdot\Lambda_{T_{CD8+T}IL_{2}}\cdot Inh_{PD_{1-L1}}-d_ {T_{CD8+T}}\cdot T_{CD8+T}\]The expressions in the RHS of these equations are the first terms describe the activation by IL-12, the inhibition by IL-10, and Treg. All these expressions are multiplied by the factor \(Inh_{PD_{1-1}}\), which describes inhibition by PD-1-PD-L1. The second term is IL-2-induced proliferation, and the last term is the death of \(CD4+T\) and \(CD8+T\) cells. **Equation for activated Tregs**  
\[\partial_{t}T_{reg}+\Pi_{\delta_{T_{reg}}}(T_{reg}) = T_{CD4+T0}\cdot\left(\Lambda_{T_{reg}T_{\beta}}+\Lambda_{T_{reg} Q}\right)-\
abla\cdot\left(\chi T_{reg}\
abla IL_{6}\right) \tag{6}\] \[-d_{T_{reg}}\cdot T_{reg}\]  
The term \(\Lambda_{T_{reg}T_{\beta}}\) is the \(TGF-\beta\)-induced proliferation, \(\
abla\cdot\left(\chi T_{reg}\
abla IL_{6}\right)\) is the activated Tregs recruited into the tumor by tumor-derived immunosuppressive cytokine IL-6, and the last term \(d_{T_{reg}}\cdot T_{reg}\) describes the death of Tregs. **Equation for activated MDSCs**  
\[\partial_{t}M+\Pi_{\delta_{M}}(M) = \lambda_{M}\cdot\left(M-M_{0}\right)\cdot\frac{IL_{6}}{K_{IL_{6} +IL_{6}}}-\
abla\cdot\left(\chi M\
abla IL_{6}\right)-d_{M}\cdot M \tag{7}\]  
The term \(\
abla\cdot\left(\chi M\
abla IL_{6}\right)\) describes the MDSCs that are chemotactically attracted to the tumor microenvironment by IL-6, and \(d_{M}\cdot M\) is the death of MDSC. **Equation for tumor cells**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]0 \[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]1  
The expression \(\lambda_{C}C(1-\frac{C}{C_{M}})\) is the proliferation of cancer cells, the term \(\frac{1}{1+\frac{B}{K_{BC}}}\) is the inhibition by BRAF/MEKi, the term \(\eta_{4}\cdot T_{CD4+T}\cdot C+\eta_{8}\cdot T_{CD8+T}\cdot C\) describes the killing by T cells, and the last term \(d_{C}\cdot C\) is the death of tumor cells. **Equation for IL-12**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]2  
The first term is the production by DCs, and the second term is the degradation of IL-12. **Equation for IL-2**  
\[\partial_{t}N_{C}+\Pi_{\delta_{N_{C}}}(N_{C}) = \lambda_{N_{C}}\cdot C-d_{N_{C}}\cdot N_{C} \tag{2}\]3  
The first term is the production by T1, and the second term is the degradation of IL-2.