Fractional mathematical oncology: cancer-related dynamics under an interdisciplinary viewpoint - Valentim Junior, Carlos Alberto (2023)
Mathematical Oncology, an interdisciplinary field incorporating concepts from biology to materials science, employs mathematical models to gain a comprehensive understanding of cancer-related phenomena. Fractional calculus, a branch of mathematical analysis, offers tools to describe complex phenomena and enables models the potential to provide better insights into oncological characteristics. This thesis surveys and explores Fractional Mathematical Oncology, presenting new models and reviewing recent developments. The thesis demonstrates the advantages of using fractional models in tumor growth prediction, specifically in ODE-based population models. Analytical solutions for five such models are derived and compared against extant (still scarce) clinical data, highlighting their superior performance and potential for further exploration. Additionally, a multistep exponential model with a fractional variable-order is proposed to represent tumor evolution. Model parameters are fine-tuned based on variable fractional order profiles, and results demonstrate its superior ability to fit clinical time series data, offering new perspectives for modeling tumor growth. Moreover, the thesis introduces cellular-automata simulation strategies in the context of tumor growth and dynamic models. This agent-based computational model allows for monitoring independent single parameters that vary in time and space. The model captures both single-cell and cluster-cell behaviors, representing various complex tumor features through different parameter settings. The proposed stochastic cellular automaton model effectively simulates different scenarios of tumor growth, serving as a valuable in silico tool for mathematical oncology research, potentially facilitating improved diagnosis and personalized treatment options. By integrating fractional calculus, physics-based models and cellular-automata simulations, the thesis contributes to the advancement of mathematical oncology, exploring promising avenues for understanding cancer dynamics, suggesting prospective research and potentially aiding decision-making in areas of interest in clinical oncology.