# A Mathematical Model for the Treatment of Melanoma with the BRAF/MEK Inhibitor and Anti-PD-1
## 2 Medical Assumptions and Mathematical Equations  
In this section, we present the assumptions of a mathematical model that considers the correlation between PD-1 inhibitors and BRAF/MEK inhibitors in melanoma therapy [12]. The dynamic variables of the model are as follows: \(H\) is the HMGB-1 concentration, \(N_{C}\) is the density of necrotic cancer cells, \(D\) is the density of DCs, \(T_{CD4+T}\) is the density of activated CD4+T cells, \(T_{CD8+T}\) is the density of activated CD8+T cells, \(T_{reg}\) is the density of activated Treg cells, \(M\) is the density of activated MDSCs, \(C\) is the density of cancer cells, \(IL-12\) concentration, \(IL-2\) concentration, \(T_{\beta}\) is the TGF-\(\beta\) concentration, \(IL-6\) concentration, \(IL-10\) concentration, \(PD_{1}\) is the PD-1 concentration, \(PD_{L1}\) is the PD-L1 concentration, \(Q\) is the PD-1-PD-L1 concentration, \(A_{PD_{1}}\) is the anti-PD-1 concentration, and \(BM\) is the BRAF/MEK inhibitor concentration. The main medical assumptions of the model are as follows [13; 14; 15; 16; 17; 18; 19; 20]: the tumor is spherical, and growth depends on time, and its radius is denoted by \(R(t)\); the total density of cells within the tumor is constant in space and time, that is, \(C+D+T_{CD4+T}+T_{CD8+T}+T_{reg}+M=K_{0}\); and the densities of immature dendritic cells and naive CD4+T and CD8+T cells remain constant throughout the tumor tissue. All cells have the same velocity \(u\) dependent on space and time, and all cytokines and anti-tumor drugs diffuse within the tumor. The anti-PD-1 as well as the BRAF/MEK inhibitor are injected intradermally every three days for 30 days [21]. The cells involved in the interaction have approximately the same surface area and volume, and hence, the diffusion coefficients of all these cells are the same. All densities and concentrations are radially symmetric; hence, they are a function of the radial time space \((r,t)\). Let us define the following operators \(\Pi_{\delta_{\circ}}(\circ)\equiv\
abla\cdot(\vec{u}\circ)-\delta_{\circ}\
abla^{2}\circ\), where \(\circ\) is the dynamical variable of the system (for example, \(\Pi_{\delta_{\circ}}(P)=\
abla\cdot(\vec{u}P)-\delta_{\circ}\
abla^{2}P\), \(\Omega_{\delta_{\circ}}(\circ)\equiv\delta_{\circ}\
abla^{2}\circ\) such that \(\Omega_{\delta_{\vec{v}p}}(P)=\delta_{P}\
abla^{2}P\), and \(\Lambda_{\alpha_{\vec{v}\circ j}}=\lambda_{\vec{ij}}\frac{\circ_{j}}{K_{ \alpha_{j}}+\delta_{j}}\), where \(i,j\) are the dynamical variables of the system, for example \(\Lambda_{DC}=\lambda_{DC}\frac{c}{K_{C}+c}\). In addition, we define the inhibition by IL-10 as \(Inh_{IL_{10}}\equiv\frac{1}{1+\frac{T_{IL_{10}}}{KT_{IL_{10}}}}\), the inhibition by Tregs as \(Inh_{Treg}\equiv\frac{1}{1+\frac{T_{reg}}{KT_{Treg}}}\), and the inhibition by PD-1-PD-L1 as \(Inh_{PD_{1-L1}}\equiv\frac{1}{1+\frac{Q}{KT_{Q}}}\). The dynamic variables of the system (in units of g/cm\({}^{3}\)) are as follows:  
\(H:\)HMGB-1 concentration;  
\(N_{C}:\) density of necrotic cancer cells;  
\(D:\) Density of DCs;  
\(T_{CD4+T}:\) density of activated \(CD4+\) T cells;  
\(T_{CD8+T}:\) density of activated \(CD8+\) T cells;  
\(T_{reg}:\) density of activated Treg cells;  
\(M:\) density of activated MDSCs;  
\(C:\) density of cancer cells;  
\(IL_{12}:\) IL-12 concentration;  
\(IL_{2}:\) IL-2 concentration;  
\(T_{\beta}:\) TGF-\(\beta\) concentration;  
\(IL_{6}:\) IL-6 concentration;  
\(IL_{10}:\) IL-10 concentration;  
\(P:\) PD-1 concentration;  
\(A_{PD1}:\) anti-PD-1 concentration;  
\(BM:\) BRAF/MEK inhibitor concentration;  
\(R:\) tumor radius;  
\(u:\) cell velocity. The operators \(\Pi_{\delta_{\circ}}(\circ)\), \(\Omega_{\delta_{\circ}}(\circ)\) contain two mathematical terms. The first one, \(\
abla\cdot(\vec{u}\circ)\), expresses velocity, and the second one, \(\delta_{\circ}\
abla^{2}\circ\), expresses diffusion.