Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm

Colon cancer is a complex disease with genetically unstable cell lines. In order to better understand the complexity of colon cancer cells and their metastatic mechanisms, we develop a mathematical model in this study. The model is based on a system of fractional-order differential equations and Fractional-Cancer-Informed Neural Networks (FCINN). The model captures a dynamic network of interactions between dendritic cells (DCs), cytotoxic T-cells (CD 8 + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8<^>+$$\end{document} ), helper T-cells (CD 4 + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4<^>+$$\end{document} ), and colon cancer cells through fractional differential equations. By varying the fractional order between 0 and 1, we can classify patients into different groups based on their immune patterns. The goal of this paper is to identify different immune patterns and cancer cell behaviors, as well as the parameters that play an important role in metastasis, control, or elimination of cancer cells in the model. However, several parameters in the model are difficult to estimate in a patient-specific manner. To address this challenge, we use FCINN as an effective deep-learning tool for parameter estimation and numerical simulation of the model. Our findings suggest that the most effective factors in controlling the progression and preventing metastasis of colon cancer are the initial number of cancer cells, the inhibiting rates of tumor cells by DCs, the source of DCs, and the activation of helper T-cells by DCs. These findings suggest that DCs can be used as an immunotherapy tool for the control and treatment of colon cancer.