## 2 A Mathematical Model for the Treatment of CML with BCR-ABL1-Targeted and Immunomodulatory Drugs
### A Brief Review of the Mathematical Model
In the populations \(P\) and \(E\), replication rate constants are represented by \(r_{P}\) and \(s_{E}\), and carrying capacities (or steady states) by \(P_{\text{ss}}\) and \(E_{\text{ss}}\), respectively. The natural death rate constants of the respective populations are denoted by \(\delta_{Q}\), \(\delta_{P}\) and \(\delta_{E}\). The population \(Q\) consists of leukemic cells that are quiescent. Some or all of quiescent leukemic cells may be stem cells [14]. When quiescent cells divide, one copy is assumed to be the same kind as the original cell while the second copy may differentiate further into a proliferating type. For this reason, the transition term \(k_{P}Q\) is not subtracted from the quiescent cell population in (1). This term represents the rate at which quiescent cells produce differentiated proliferating cancer cells, with the population \(Q\) the source for the population \(P\). The control variables represent the concentrations of the respective drugs, and their effects (pharmacodynamics) are modeled by Michaelis-Menten terms with different maximum effectiveness on the various populations. In modeling the combined drug actions it is assumed that any two drugs act independently of each other. Thus the term  
\[\left(1-\frac{U1_{\text{max},1}u_{1}}{U1C_{50}+u_{1}}\right)\left(1-\frac{U2_ {\text{max},2}u_{2}}{U2C_{50}+u_{2}}\right)\left[k_{P}Q+r_{p}P\ln\left(\frac{P_ {\text{ss}}}{P}\right)\right]\]  
represents the effects that drugs 1 and 2 have on decreasing the proliferation of the population \(P\). A term of the type  
\[-\delta_{Q}\left(1+\frac{U2_{\text{max},1}u_{2}}{U2C_{50}+u_{2}}\right)\left(1 +\frac{U3_{\text{max},1}u_{3}}{U3C_{50}+u_{3}}\right)\frac{E_{\text{max},1}E}{ EC_{50}+E}Q\]  
represents the enhancement of the actions of the effector T cells \(E\) on the quiescent cells \(Q\) as a consequence of the activities of drugs 2 and 3. In each of the equations, the enhancement and inhibition effects of the drugs by means of the immune system are modeled additively. The \"\(C_{50}\)\" parameters \(U1C_{50}\), \(U2C_{50}\), and \(U3C_{50}\) represent the concentrations required to achieve half of the maximal effects of \(u_{1}\), \(u_{2}\), and \(u_{3}\), respectively. These and \(EC_{50}\) and \(PC_{50}\) are assumed to be fixed across effects being modeled. These represent \"potency\" levels depending intrinsically on the particular therapy or population, and not on the setting of the effect. The maximum possible effect size is allowed to depend on the setting. The equations above represent a semi-mechanistic, fit-for-purpose, minimal model. It is minimal in the sense that it only includes the levels of cell interactions needed to allow the controls to have their expected effects. Some of the terms are based on models validated with data, but other terms take forms that are more heuristic. For example, all of the control effect terms take a Michaelis-Menten or \"Emax\" form. This is because we wish to model very small effect at low levels of drug, as well as a limiting or asymptotic maximal effect at high levels of drugs. We chose the simplest among the models with this behavior that are typically used in drug development [15]. The states, controls, and related parameters are listed in Tables 1 and 2. Table 1 gives those parameters that are unrelated to the drug actions and make up the untreated, or uncontrolled, system; Table 2 lists the treatment-specific parameters in the model. In this paper, we do not fit or fix specific parameter values, and instead analyze the dynamic properties of the system (1)-(3) for large ranges of possible values. We include in the tables below two different sets of numerical values that we use to illustrate the dynamic properties of the system. These parameter values are purely for numerical illustration and do not reflect specific model fits or therapies. The focus of this paper is the mathematical analysis of the entire system rather than an analysis for particular parameter values.