Treatment Optimization for Tumor Growth by Ordinary Differential Equations

Abstract

Cancer is the second leading cause of death worldwide and with the disease having over 200 variations, it has not been cured yet despite being the priority of the medical field for decades. Due to the difficulty of human subject research, animal studies, e.g., mouse and Chinese hamster V79 tumors have been widely used to test the modeling of tumor growth due to their dynamic nature and ability to grow to high volumes within short periods of time. Mathematical models, including ordinary differential equations (ODEs), have been utilized to model tumor growth and study treatment of cancer. With most current models being selected only for mathematical convenience, recent studies have been focusing on determining the optimal treatment schedule for the most popular existing treatments of chemotherapy and radiation therapy. In this paper, three of the most established ODE models: the Gompertz, Von Bertalanffy, and logistic models are utilized to analyze which model most accurately fits existing tumor growth data for the Chinese Hamster V79 fibroblast tumor, various forms of immunodeficient mice tumors, and glioblastoma based on the minimization of the normalized mean squared error (NMSE). Next, the ODEs themselves were modified to simulate the growth of the tumors when exposed to treatment and determined which treatment schedule produced the lowest final volume of the tumors. The results of this research identify the optimal treatment schedules based on data from all three ODE models and also determine the ODE models that produce curves that most precisely fit the datasets.