Ponente
Descripción
Cancer is a leading cause of death worldwide, accounting for nearly 10 million deaths in 2022, with lung, breast, and colorectal cancers being the most prevalent (Ferlay, J. et al., Global cancer observatory: Cancer today, International Agency for Research on Cancer, 2020). This epidemiological scenario highlights the need to develop rigorous therapeutic approaches. While conventional chemotherapy presents severe systemic toxicity, immunotherapy stimulates the host immune system to fight tumor cells in a targeted manner. The objective of this work is to investigate a mathematical model applied to oncology through local stability analysis, determining the equilibrium points and their biological stability conditions to understand the population dynamics and interaction between cells and therapeutic agents. The analyzed nonlinear system of coupled ordinary differential equations is based on (Rodrigues, D.S. et al., Math. Biosci. Eng. 10, 221-234, 2013) and (Wilson, S. & Levy, D., Bulletin of Mathematical Biology 74, 1485-1500, 2012). The system consists of four time-dependent variables that describe the populations of tumor cells ($T$), normal cells ($N$), chemotherapeutic agent dose ($Q$), and immunotherapeutic vaccination ($V$):
\begin{equation}
\left{
\begin{aligned}
\frac{dT}{dt} &= r_{1}T\left(1 - \frac{T}{k_{1}} - \alpha_{12}N\right) - \mu r_{1}T\left(1 - \frac{T}{k_{1}}\right)Q - r_{0}TV, \
\frac{dN}{dt} &= r_{2}N\left(1 - \frac{N}{k_{2}} - \alpha_{21}T\right) - \nu r_{2}N\left(1 - \frac{N}{k_{2}}\right)Q, \
\frac{dQ}{dt} &= q(t) - \lambda Q, \
\frac{dV}{dt} &= g(t) - \delta V.
\end{aligned}
\right.
\end{equation}
In this system, $r_1$ and $r_2$ are the intrinsic growth rates of $T$ and $N$, respectively, while $k_1$ and $k_2$ represent their carrying capacities; $\alpha_{12}$ and $\alpha_{21}$ are the competition coefficients between the cell lines. The parameters $\mu$ and $\nu$ denote the chemotherapy-induced mortality rates of $T$ and $N$, respectively, and $\lambda$ is the decay rate of the chemotherapeutic agent. Furthermore, $r_0$ represents the tumor mortality rate due to the action of $V$, and $\delta$ is the vaccine decay rate. Originally, the immunotherapy is modeled as a pulsed infusion of cytotoxic T lymphocytes described by the Dirac Delta function $g(t)=g_0\delta(t-3)$. However, to enable the qualitative analysis of local stability and complement previous numerical studies of the project, we consider the treatment functions as constant flows over time, where $q(t) = q > 0$ and $g(t) = g > 0$ represent the constant infusion rates. Under these conditions, four biologically feasible equilibrium points were determined: $E_1 = (0, 0, \frac{q}{\lambda}, \frac{g}{\delta})$, which represents the total extinction of both cell lines under excessive toxicity; $E_2 = (0, k_2, \frac{q}{\lambda}, \frac{g}{\delta})$, which reflects the ideal therapeutic success with tumor elimination and preservation of healthy tissue; $E_3 = (T^*, 0, \frac{q}{\lambda}, \frac{g}{\delta})$, which indicates severe protocol failure with tumor persistence and destruction of normal tissue; and $E_4 = (\bar{T}, \bar{N}, \frac{q}{\lambda}, \frac{g}{\delta})$, which represents coexistence in an endemic state. The results of this analysis prove that combining both treatments is more efficient in eliminating the tumor than using only one of them. We conclude that knowing these stability conditions offers an important theoretical basis, helping to understand the behavior of the system and to plan treatment protocols more effectively.