## 2 A Mathematical Model for the Treatment of CML with BCR-ABL1-Targeted and Immunomodulatory Drugs
### A Brief Review of the Mathematical Model  
Let \(Q\) be the concentration of quiescent leukemic cells, \(P\) the concentration of proliferating leukemic cells, and \(E\) the strength of immune system effects. We will consider \(E\) to represent effector T cell concentration levels, and will refer to \(E\) in the remainder as a concentration of effector T cells. The model contains three controls \(u_{1}\), \(u_{2}\) and \(u_{3}\) that all denote normalized levels of different therapies. The roles of the specific drugs are illustrated in Figure 1 taken from [12] with arrows indicating amplification of effects and vertical bars indicating inhibition. The control \(u_{1}\) represents the normalized concentration of a BCR-ABL1 inhibitor (such as imatinib) that mainly has an inhibitory effect on the highly-proliferating leukemic cells; \(u_{2}\) is a BCR-ABL1 inhibitor that inhibits BCR-ABL1 that also has immune effects (such as dasatinib); while \(u_{3}\) represents an immunomodulatory compound (such as nivolumab). Figure 1: Diagram of the dynamical system. The green circular areas represent the “populations” included in the model. Solid arrows extending from or to the populations represent changes in numbers, with inward-pointing arrows representing increases and outward-pointing arrows decreases. Dashed arrows indicate indirect effects on those increases or decreases. Bars represent inhibition of a production or an indirect effect, due to the represented treatment; arrows represent amplification of a rate or an indirect effect. The effects of the general BCR-ABL1 inhibitor \(u_{1}\) are shown using orange dashed bars and arrows, the effects of the BCR-ABL1 inhibitor \(u_{2}\) which also has immune effects are shown using wide red solid bars and arrows and the effects of the immunomodulatory compound \(u_{3}\) are shown using blue solid bars and arrows. Representing the pharmacodynamic effects of the drugs using Michaelis-Menten terms results in the following equations:  
\[\frac{dQ}{dt} =r_{\text{Q}}Q-\delta_{\text{Q}}\left[1+\left(1+\frac{U2_{\text{max },1}u_{2}}{U2C_{50}+u_{2}}\right)\left(1+\frac{U3_{\text{max},1}u_{3}}{U3C_{50}+ u_{3}}\right)\frac{E_{\text{max},1}E}{EC_{50}+E}\right]Q, \tag{1}\] \[\frac{dP}{dt} =\left(1-\frac{U1_{\text{max},1}u_{1}}{U1C_{50}+u_{1}}\right) \left(1-\frac{U2_{\text{max},2}u_{2}}{U2C_{50}+u_{2}}\right)\left[k_{p}Q+r_{p} P\ln\left(\frac{P_{\text{ss}}}{P}\right)\right]\] \[\qquad-\delta_{P}\left(1+\frac{U1_{\text{max},2}u_{1}}{U1C_{50}+ u_{1}}\right)\left(1+\frac{U2_{\text{max},3}u_{2}}{U2C_{50}+u_{2}}\right)P\] (2) \[\qquad-\delta_{P}\left(1+\frac{U2_{\text{max},1}u_{2}}{U2C_{50}+ u_{2}}\right)\left(1+\frac{U3_{\text{max},1}u_{3}}{U3C_{50}+u_{3}}\right)\frac{E_{ \text{max},2}E}{EC_{50}+E}P,\] \[\frac{dE}{dt} =s_{E}\left[1+\left(1+\frac{U2_{\text{max},4}u_{2}}{U2C_{50}+u_{ 2}}\right)\left(1+\frac{U3_{\text{max},2}u_{3}}{U3C_{50}+u_{3}}\right)\frac{P_ {\text{max},1}P}{PC_{50}+P}\right]E\ln\left(\frac{E_{\text{ss}}}{E}\right)\] \[\qquad-\delta_{E}\left[1+\left(1-\frac{U2_{\text{max},5}u_{2}}{U 2C_{50}+u_{2}}\right)\left(1-\frac{U3_{\text{max},3}u_{3}}{U3C_{50}+u_{3}} \right)\frac{P_{\text{max},2}P}{PC_{50}+P}\right]E. \tag{3}\]  
In this system, all parameters are non-negative. For \(P=0\) or \(E=0\), we extend the system by defining it using the limits as \(P\to 0\) or \(E\to 0\), respectively. The cell count numbers for \(Q\) are relatively small and are therefore modeled by an exponential function with growth coefficient \(r_{Q}\). For the proliferating cells \(P\) we model growth with a Gompertz function, as [PERSON] and [PERSON] state that this is best for describing CML growth [13]. The immune effect \(E\) (effector T cells) also has its rate of increase modeled by a Gompertz function, so as to have approximately exponential growth when numbers are very small, but still be bounded above.