Mathematical modeling of the immune-chemotherapeutic treatment of breast cancer under some control parameters

Ali Yousef,Thabet Abdeljawad,Fatma Bozkurt

doi:10.1186/s13662-020-03151-5

Abstract

We consider a system of fractional-order differential equations to analyze breast cancer growth in the immune-chemotherapeutic treatment process under some control parameters: ketogenic diet, immune booster, and anti-cancer drugs. The established model assumes the growth of the tumor density under chemotherapy treatment and the immune response during the interaction between the normal cells and tumor cells. For the local stability of the critical points (tumor-free critical point, dead critical point, and co-existing critical point), we used the Routh-Hurwitz criteria to show the necessary effect of the immune booster; moreover, we addressed the ketogenic rate in the treatment process. Our theoretical and numerical studies pointed out that on early detection of the tumor density (with weak Allee effect) the treatment should be supported by ketogenic nutrition. Several examples are shown to present our theoretical findings.

Highlights

According to the National Cancer Registry, cancer kills more people than tuberculosis, AIDS, and malaria combined
Various mathematical and analytical approaches have been developed to understand the interaction between the tumor cells and the immune response during the treatment process that was mainly established as integer-order differential equations (IDEs) [7,8,9]
7 Conclusions This study analyzed a mathematical model of breast cancer as a fractional-order system to analyze tumor growth under chemotherapeutic treatment and immune response

Summary

Introduction

According to the National Cancer Registry, cancer kills more people than tuberculosis, AIDS, and malaria combined. Various mathematical and analytical approaches have been developed to understand the interaction between the tumor cells and the immune response during the treatment process that was mainly established as integer-order differential equations (IDEs) [7,8,9]. Oke et al improved the model of Mudufza [28] in [3] They incorporated the control parameters such as ketogenic diet, immune booster, and anti-cancer drug to emphasize the point that there is an interaction between the cells due to the mutation in the tumor cell’s DNA. Φ1 is the tumor formation rate resulting from DNA mutation caused by the presence of excess estrogen, while (1 – k) represents the effectiveness of anti-cancer drugs. Μ2 represents the logistic rate of the tumor cell population.

The equilibrium points Consider the system

Conclusions