Mathematical Modeling of Pancreatic Cancer Cell Behavior under Radiotherapy in the Framework of Time Scale Theory

The objective of time scale theory is to bridge continuous and discrete models and to better capture hybrid dynamics arising in real-world applications. Within the framework of this theory, the dynamics of pancreatic cancer cells under radiotherapy will be investigated by incorporating both continuous and discrete mathematical formulations. In this study, the mathematical model describing pancreatic cancer cell behavior under radiotherapy will first be examined on an arbitrary time scale, allowing for a general representation of tumor growth and treatment effects. For this general case, explicit solutions of the proposed model will be derived, providing analytical insight into the underlying system dynamics. Subsequently, continuous and discrete models will be obtained as special cases of the general time scale formulation, corresponding to classical differential and difference equation models, respectively. A detailed comparison between these continuous and discrete models and their solutions will then be carried out in order to evaluate differences in tumor response and treatment effectiveness. This comparative analysis aims to identify the most appropriate time scale that leads to the fastest and most effective therapeutic response under radiotherapy. By capturing biological processes across different time scales in a unified manner, this approach offers a more realistic and flexible representation of pancreatic cancer dynamics. Ultimately, the proposed modeling framework contributes to a deeper understanding of cellular responses to radiotherapy and may support the development of improved treatment strategies.