This paper DOI: 10.5281/zenodo.4275629

Mathematical Modelling of the Dynamics of Tumor Growth and its Optimal Control

Jannatun Irana Ira1, Md. Shahidul Islam2, J C Misra3, and Md. Kamrujjaman 4*

1,2,4Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh

3Centre for Healthcare Science and Technology, Indian Institute of Engineering Science and

Technology, Shibpur, Howrah-711103, India

4Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada

*Corresponding author Email : kamrujjaman@du.ac.bd

Abstract: In the last few decades, the dynamics of tumor cells and their growths are presented viaclinical, experimental, and theoretical approaches, which leads to the development of the new idea of multiple cancer therapies to control and reduce the death rate for earlier detection. In this paper, we discussed the dynamics of tumor cell growth and its treatment process. We analyzed some simple mathematical models and generalized the study to understand the growth of tumor cells. The main proposed model is a system of ordinary dierential equations which combines interactions among natural killer cells, dendritic cells and cytotoxic CD8+ T cells. The model is solved numerically to explain how the tumor cells spread and become more dangerous as well as the treatment process of cancer. It is also studied that how the cell behaves in the presence of dierent therapy and drugs. The optimal control of chemotherapy has been discussed. It has also been explained how much the model is effective in reducing tumor cells over time. Finally, a couple of spatially distributed models are discussed for tumor cell growth.

Keywords: Mathematical models; tumor growth; chemotherapy; diffusion; optimal control.

AMS Subject Classi cation 2010: 35K61, 37N25, 49J15, 92D25.