Investigating the wave profiles to the linear quadratic model in mathematical biology.

This study investigates the dynamic behavior of the linear quadratic model (LQM), a fundamental framework in radiation biology that describes cellular response to radiation, particularly in the context of DNA damage and cancer progression. The LQM was originally developed to quantify radiation-induced cell death and repair mechanisms, with a focus on double-stranded DNA breaks, the most critical type of radiation damage. Despite advances in tracking tumor cell dissemination, the mechanisms underlying cancer invasion remain poorly understood. Mathematical modeling, particularly through partial differential equations, has become an essential tool for simulating tumor growth and optimizing therapeutic strategies, bridging the gap between theoretical biology and clinical applications. In this work, we employ advanced analytical techniques, including the generalized Arnous method, modified F-expansion method, and generalized exponential rational function approaches to solve the model for the first time. By transforming the governing PDE into an ordinary differential equation using β-derivative and wave transformations, we derive exact solutions in the form of dark, bright, singular, mixed, complex, and combined soliton waves. These solutions, visualized through 2D and 3D plots, reveal the system's behavior under varying parameters, demonstrating the computational power and effectiveness of the applied methods. The results not only validate the proposed techniques but also enhance our understanding of the model's nonlinear dynamics. The novel findings presented here are expected to advance future research in radiation biology and cancer treatment optimization.