Modeling and computational study of cancer treatment with radiotherapy using real data.

The numerical simulation of biological processes with non-integer ordering is attracting an increasing amount of interest from scientists and academics. Traditional biological systems can be presented in a fixed order, but fractional-order derivative systems are not considered stable orders. When the fractional derivative has a non-fixed order, it becomes more useful for simulating real-world problems. In this paper, we aim to study the dynamics of a novel technique that we propose, implement, and use in a radiation model for the treatment of cancer. We present some intriguing results for the cancer treatment fractal fractional model in the context of this innovative operator. Research has been done on the cancer model in both qualitative and quantitative manners. The first and second derivatives of the Lyapunov function are used to analyze the stability of the cancer fractal fractional model. Using the linear growth theory, the existence of a unique solution has been derived under the FFM. Lagrangian-piece-wise interpolation has been used to obtain numerical results for various fractal-fractional operators. The fractal fractional model was used to simulate the treatment process of three patients. Different values of fractional order [Formula: see text], fractal dimension [Formula: see text], and other parameter values have been used to show the graphs. Additionally, we looked at how radiation changed both healthy cells and malignant cells over time. The study confirmed the effectiveness of radiation medicine against populations as well as the occurrence of the memory effect during [Formula: see text] and [Formula: see text] transitions from 1. A biological process requires fractal-fractional processes which provide superior modeling capabilities compared to traditional fractional operators as well as classical operators. This research brings novel significance through its implementation of fractal-fractional operators as they provide a superior approach to model cancer treatment processes by better representing biological system complexities. Standard modeling systems cannot reproduce both important memory dynamics together with non-local communication patterns which play essential roles in cancer development and treatment analysis. The implementation of fractal-fractional derivatives enables our model to produce a more realistic representation of cancer cell and healthy cell radiotherapy responses throughout time. Our study has upgraded theoretical cancer dynamic analysis and developed optimized treatment methods for customization purposes. Wider understanding of cancer cell reactions to treatments enables healthcare providers to adopt personalized strategies that produce superior recovery outcomes for their patients. The model acquires stability strength through Lyapunov functions analysis to create a solid scientific foundation in oncology research.