# Chemotherapy Regimen Optimization Using a Two-Archive Multi-Objective Squirrel Search Algorithm
## 3 Proposed Model and Method
### Mathematical Model of Tumor Dynamics
\[\frac{dP}{dt}=\{(a-b-c)(1-\beta_{A}*u(t-t_{a}-1)-\beta_{B}*u(t-t_{b}-1))\}P (t)+dQ(t)-(g_{A}(t)+g_{B}(t))P(t) \tag{1}\] \[\frac{dQ}{dt}=cP(t)-d(1-\beta_{A}*u(t-t_{a}-1))Q(t)-g_{A}(t)Q(t)\] (2) \[\frac{dP_{A}}{dt}=\{(a-b-c)(1-\beta_{B}*u(t-t_{b}-1))\}P_{A}(t)+dQ _{A}(t)+\beta_{A}*u(t-t_{a}-1)P(t)-g_{B}(t)P_{A}(t)\] (3) \[\frac{dQ_{A}}{dt}=cP_{A}(t)-dQ_{A}(t)+\beta_{A}*u(t-t_{a}-1)Q(t)\] (4) \[\frac{dP_{B}}{dt}=\{(a-b-c)(1-\beta_{A}*u(t-t_{a}-1))\}P_{B}(t)+dQ _{B}(t)+\beta_{B}*u(t-t_{b}-1)P(t)-g_{A}(t)P_{B}(t)\] (5) \[\frac{dQ_{B}}{dt}=cP_{B}(t)-d(1-\beta_{A}*u(t-t_{a}-1))Q_{B}(t)-g _{A}(t)Q_{B}\] (6) \[\frac{dP_{AB}}{dt}=(a-b-c)P_{AB}(t)+dQ_{AB}(t)+\beta_{A}*u(t-t_{a }-1)P_{B}(t)+\beta_{B}*u(t-t_{b}-1)P_{A}(t)\] (7) \[\frac{dQ_{AB}}{dt}=cP_{AB}(t)-dQ_{AB}(t)+\beta_{A}*u(t-t_{a}-1)Q_ {B}(t)\] (8) Figure 1:0  
where \(t\) denotes time, \(u(t)\) denotes the unit step signal used to indicate that the drugs all have their efficacy appear after the first administration, \(t_{a}\) and \(t_{b}\) denote the instant of administration of the two drugs, and \(g_{A}\) and \(g_{B}\) denote the killing effect of the two drugs, respectively. The model describes tumor cell growth linearly due to chemotherapy's efficacy in inhibiting growth near the carrying capacity. This linear kinetics simplifies calculations and offers transparency in interpreting underlying dynamics, particularly in the context of tumor growth dynamics. In the early stages of tumor growth, linear approximations provide a reasonable first-order approximation of the system's behavior. Additionally, the effectiveness of chemotherapeutic routines in inhibiting tumor growth close to its carrying capacity suggests that any nonlinearity may have a negligible impact within our study's scope. Furthermore, linear models often offer computational advantages, requiring fewer computational resources compared to nonlinear models, enabling faster simulations and facilitating exploratory analyses under various conditions. Our approach allows us to capture the fundamental dynamics of tumor growth without introducing unnecessary complexity. Unlike tumor cell growth, the bone marrow cell growth described in Equation (9) is formulated as a logistic equation, where \(\delta\) denotes the natural growth rate of the bone marrow cell population, \(K\) denotes the carrying capacity of the bone marrow cell population, and the term \((g_{A}(t)+g_{B}(t))M(t)\) denotes the amount of damage to the bone marrow cell population from exposure to the drug. The killing effect of drugs, denoted as \(g_{A}\) and \(g_{B}\), depend on the rate of change in drug concentration at the tumor site, and the pharmacokinetic equation used in this model is shown in the following equation:  
Figure 1:1  
Figure 1:2  
Figure 1:3  
Figure 1:4  
where \(U_{A}(t)\) and \(U_{B}(t)\) represent the drug doses injected into the patient's body at time \(t\), its unit is mg/mL per day. \(D_{A}\) and \(D_{B}\) represent the drug concentrations at the tumor site, \(\gamma_{a}\) and \(\gamma_{b}\) represent the decay rates of the two drugs, and \(\lambda_{1}\) and \(\lambda_{2}\) are constants representing the killing effect of the drug concentration on cells per time unit. The remaining tumor cells and the cumulative damaged bone marrow cell population are utilized as the two optimization objectives, and the solution of the TA-MOSSA represents the drug dosage regimen of the two drugs throughout the course of treatment. _Objective_1, representing the remained tumor cells, and _Objective_2, representing the cumulative damaged bone marrow cells, are calculated as shown in Equation (14) and Equation (15), respectively:  
Figure 1:5  
Figure 1:6  
Both objectives are to be minimized in this paper, representing the efficacy and the toxicity levels of chemotherapy regimens, respectively. Moreover, during the optimization process, constraints are imposed on the drug concentration within the patient's body and the population of bone marrow cells [22] to avoid overdose.