## 2 A Mathematical Model for the Treatment of CML with BCR-ABL1-Targeted and Immunomodulatory Drugs
### Scaling of Parameters  
We note that the dynamical system has various groups of symmetries that can be used to scale the variables and controls. Here we normalize all the \"\(C_{50}\)\" parameter values to 1 by rescaling the corresponding variables in terms of these quantities. This simply minimizes the number of parameters to be considered in the analysis of the system. For example, let \(Q_{\text{ref}}\) be a constant to be determined later, and define  
\[\widetilde{Q}=\frac{Q}{Q_{\text{ref}}},\qquad\widetilde{P}=\frac{P}{PC_{50}}, \qquad\widetilde{E}=\frac{E}{EC_{50}},\]  
and  
\[\widetilde{u}_{1}=\frac{u_{1}}{U1C_{50}},\qquad\widetilde{u}_{2}=\frac{u_{2}}{ U2C_{50}},\qquad\widetilde{u}_{3}=\frac{u_{3}}{U3C_{50}}.\]  
Then we have that  
\[\frac{E}{EC_{50}+E}=\frac{\widetilde{E}}{1+\widetilde{E}}\]  
and analogously for the other terms. \begin{table}
\begin{tabular}{c l c c} \hline \hline
**Symbol** & **Interpretation** & **Values Used** & **Values Used** \\  & **in Figure 2** & **in Figures 3 and 4** \\ \hline \multirow{2}{*}{\(u_{1}\)} & normalized concentration of a general BCR-ABL1 & & \\  & inhibitor (e.g., imatinib) & & \\ \hline \multirow{2}{*}{\(U1_{\text{max},1}\)} & maximum possible effect of \(u_{1}\) on slowing & 0.8 & 0.8 \\  & transfer of quiescent cells \(Q\) into \(P\) and & & \\  & inhibiting growth of proliferating cells \(P\) & & \\ \hline \multirow{2}{*}{\(U1_{\text{max},2}\)} & maximum possible effect of \(u_{1}\) on death of & 10 & 2 \\  & proliferating cells \(P\) & & \\ \hline \multirow{2}{*}{\(U1C_{50}\)} & concentration of \(u_{1}\) that gives half the maximum effect & 1 & 1 \\ \hline \multirow{2}{*}{\(u_{2}\)} & normalized concentration of a BCR-ABL1 inhibitor & & \\  & which also has immunomodulatory effects & & \\  & (e.g., dasatinib) & & \\ \hline \multirow{2}{*}{\(U2_{\text{max},1}\)} & maximum possible effect of \(u_{2}\) on death of leukemic & 2 & 0.01 \\  & cells (the same for \(P\) and \(Q\)) & & \\ \hline \multirow{2}{*}{\(U2_{\text{max},2}\)} & maximum possible effect of \(u_{2}\) slowing new \(P\) from & 0.6 & 0.03 \\  & \(Q\) and inhibiting growth of proliferating cells \(P\) & & \\ \hline \multirow{2}{*}{\(U2_{\text{max},3}\)} & maximum possible effect of \(u_{2}\) on & 10 & 0.01 \\  & death of proliferating cells \(P\) & & \\ \hline \multirow{2}{*}{\(U2_{\text{max},4}\)} & maximum possible effect of \(u_{2}\) on & 10 & 0.025 \\  & stimulating proliferation of effector T cells & & \\ \hline \multirow{2}{*}{\(U2_{\text{max},5}\)} & maximum possible effect of \(u_{2}\) on & 0.4 & 0.02 \\  & prevention of the death of effector T cells & & \\ \hline \multirow{2}{*}{\(U2C_{50}\)} & concentration of \(u_{2}\) that gives half the maximum effect & 0.6 & 0.8 \\ \hline \multirow{2}{*}{\(u_{3}\)} & normalized concentration of an immunomodulatory & & \\  & d agent (e.g., nivolumab) & & \\ \hline \multirow{2}{*}{\(U3_{\text{max},1}\)} & maximum possible effect of \(u_{3}\) on death of leukemic & 5 & 0.02 \\  & cells (the same for \(P\) and \(Q\)) & & \\ \hline \multirow{2}{*}{\(U3_{\text{max},2}\)} & maximum possible effect of \(u_{3}\) on & 5 & 0.05 \\  & stimulating proliferation of effector T cells & & \\ \hline \multirow{2}{*}{\(U3_{\text{max},3}\)} & maximum possible effect of \(u_{3}\) on & 0.7 & 0.07 \\  & prevention of the death of effector T cells & & \\ \hline \multirow{2}{*}{\(U3C_{50}\)} & concentration of \(u_{3}\) that gives half the maximum effect & 0.7 & 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Controls and pharmacodynamic parameters.