ANALYSIS OF AN IMPULSIVE TUMOR-CHEMOTHERAPY MODEL WITH MICHAELIS-MENTEN NONLINEAR DRUG DEGRADATION

The $\text{Lambert } W$ function $ W(x) $ has two branch $W_{0}: [\exp\{-1\}, \infty]\to[-1, \infty]$ and $W_{-1}: [\exp\{-1\}, \infty]\to[\infty, -1]$.

Periodic solutions for tumor eradication under different initial conditions: (a) Time series of tumor cell concentration $ H(t) $: Both initial conditions $ (H_0, D_0)=(5, 0.8) $ and $ (H_0, D_0)=(2, 0.6) $ converge to 0, indicating tumor extinction. (b) Time series of drug concentration $ D(t) $: The drug variable converges to a stable positive periodic solution. (c) Phase portrait of $ H(t) $ vs. $ D(t) $: Tumor cells approach extinction while drug concentration converges to the periodic solution, indicating convergence of the full system to the tumor-free periodic solution. Other parameters $ r=0.72 $, $ \alpha=0.6 $, $ \eta=0.05 $, $ \omega=0.2 $, $ k=0.6 $, $ \beta=0.4 $, $ \tilde{\sigma}=0.2 $, $ K=1.8 $, $ \tau=3 $ [33].

Periodic solutions for tumor eradication under different initial conditions: (a) Time series of tumor cell concentration $ H(t) $: Both initial conditions $ (H_0, D_0)=(5, 0.8) $ and $ (H_0, D_0)=(2, 0.6) $ converge to 0, indicating tumor extinction. (b) Time series of drug concentration $ D(t) $: The drug variable converges to a stable positive periodic solution. (c) Phase portrait of $ H(t) $ vs. $ D(t) $: Tumor cells approach extinction while drug concentration converges to the periodic solution, indicating convergence of the full system to the tumor-free periodic solution. Other parameters $ r=0.72 $, $ \alpha=0.6 $, $ \eta=0.05 $, $ \omega=0.2 $, $ k=0.6 $, $ \beta=0.4 $, $ \tilde{\sigma}=0.2 $, $ K=1.8 $, $ \tau=3 $[33].

Dynamics under different injection doses. (a) Time series of tumor cell concentration $H(t)$: Red line ($\tilde{\sigma}=0.2$) converges to a stable periodic solution; green line ($\tilde{\sigma}=0.4$) and blue line ($\tilde{\sigma}=0.6$) converge to extinction. (b) Time series of drug concentration $D(t)$: Higher doses lead to higher residual concentrations. (c) Phase portrait of $H(t)$ vs. $D(t)$: Dose-dependent dynamical transitions, providing numerical evidence for dose optimization. Initial condition: $(H_0, D_0)=(2, 0.4)$. Other parameters are the same as that in Figure 2.

Sensitivity to parameter $r$ (intrinsic tumor growth rate). (a) Time series of tumor cell concentration $H(t)$: Red line ($r=0.72$) converges to extinction; green line ($r=0.685$) converges to a stable positive periodic solution. (b) Time series of drug concentration $D(t)$: Parameter $r$ has little effect on drug dynamics. (c) Phase portrait of $H(t)$ vs. $D(t)$: Small changes in $r$ induce a qualitative transition in system dynamics. Initial condition: $(H_0, D_0)=(5, 0.2)$. Other parameters are the same as that in Figure 2.

Comparison of linear and nonlinear drug elimination rates. (a) Time series of tumor cell concentration $H(t)$: Blue line (linear elimination, $D'(t)=-\beta D(t)$) shows simple decay; rad line (nonlinear elimination, Michaelis-Menten function) exhibits a two-phase effect (initial stimulation followed by inhibition). (b) Time series of drug concentration $D(t)$: Linear elimination (red) decays exponentially, while nonlinear elimination (blue) shows saturated decay. (c) Phase portrait of $H(t)$ vs. $D(t)$: Nonlinear elimination captures more complex dynamical behaviors consistent with clinical observations. Initial condition: $(H_0, D_0)=(5, 0.2)$. Other parameters are the same as that in Figure 2.