Abstract
This paper describes a mathematical model of tumor growth during chemotherapy, employing a fractional calculus approach to account for the interactions between cancer cells, normal cells, immune cells, and drugs. This method better reflects the role of biological memory, or the state of a system at a given point in time. Using the analysis of neural networks to solve the mathematical equations, it is shown how the treatment regimen impacts the development of these cell levels over time. These simulations test the following biologically relevant parameters: the strength of the immune response, the effectiveness of the drugs, the rates of growth of cancer cells versus other cells, and the spatial geometry of the tumor micro-environment. The results show how cancer cell decline, immune system stimulation, and healthy cell repopulation interact in time. It is shown that by adjusting the parameters of the chemotherapy regimen, it is possible to optimize tumor reduction while maintaining healthy cell levels. This study suggests that complex calculus models of biological systems like cancer development can provide the means for more effective chemotherapy.