Competitive tumor growth modeling and optimal radiotherapy control via logistic equations

TL;DR

This manuscript develops and analyzes mathematical models that describe tumor response to radiotherapy by incorporating the Linear Quadratic model for cell survival, allowing for the comparison of constant versus optimized radiation strategies.

Abstract

The uncontrolled proliferation of cancer cells and their interaction with healthy tissue poses a major challenge in oncology. This manuscript develops and analyzes mathematical models that describe tumor response to radiotherapy by incorporating the Linear Quadratic model for cell survival. To improve therapeutic efficiency, the theory of optimal control is introduced on a system of coupled differential equations, allowing for the comparison of constant versus optimized radiation strategies. The analytical study of these models provides insights into the expected dynamics under different treatment scenarios, while numerical simulations validate the theoretical results and highlight the benefits of optimal control in reducing tumor burden with minimized collateral damage.

Figures (8)

• Figure 1: Numerical simulations of tumor growth dynamics in the absence of treatment, comparing the exponential, Gompertz, and Verhulst models as defined by \ref{['exp']}, \ref{['gom']}, and \ref{['verh']}, respectively.
• Figure 2: Simulation of a fractionated radiotherapy treatment consisting of 16 sessions, each of duration $0.2$ days and separated by $20$ days. The first session is administered at $t = 100$ days and the last one at $t = 400$ days. The sharp drops correspond to treatment pulses, while the blue and red curves represent the tumor and healthy cell populations respectively, following the piecewise dynamics described in this section.
• Figure 3: (a) Temporal evolution of healthy and cancer cell populations in the coexistence mechanisms. (b) Phase portrait of system \ref{['eq:coexistence_system']}, illustrating the vector field and the instability of the equilibrium points.
• Figure 4: (a) Temporal evolution of healthy $H(t)$ and cancer $C(t)$ cell populations under the competition model \ref{['eq:competition_system']}. (b) Phase portrait of the autonomous system showing the vector field. The labels indicate the transition of $(0, K)$ into a stable sink and $(K, 0)$ into an unstable saddle point, as derived in Table \ref{['tab:stability_competition']}.
• Figure 5: Sensitivity of the competition model to initial conditions using parameters from Table \ref{['tab:model_parameters']}. (a) Temporal progression of the tumor population $C(t)$ and (b) decline of the healthy population $H(t)$ for various $(H_0, C_0)$ pairs. (c) Combined phase portrait illustrating that all trajectories, regardless of the starting density, are globally attracted to the tumor-dominant equilibrium $(0, K)$.