Modeling and Analysis of Breast Cancer with Adverse Reactions of Chemotherapy Treatment through Fractional Derivative

Abstract

The abnormal growth of cells in the breast is called malignancy or breast cancer; it is a life-threatening and dangerous cancer in women around the world. In the treatment of cancer, the doctors apply different techniques to stop cancer cell development, remove cancer cells through surgery, or kill cancer cells. In chemotherapy treatment, powerful drugs are used to kill abnormal cells; however, it has adverse reactions on the patient heart which is called cardiotoxicity. In this paper, we formulate the dynamics of cancer in the breast with adverse reactions of chemotherapy treatment on the heart of a patient in the fractional framework to visualize its dynamical behaviour. We listed the fundamental results of the fractional calculus for the analysis of our model. The model is then analyzed for the basic properties, and the existence and uniqueness of the proposed breast cancer system are investigated through fixed point theory. Furthermore, the Adams-Bashforth numerical technique is presented for the solution of fractional-order system to illustrate the time series of breast cancer model. The dynamical behaviour of different stages of breast cancer is then highlighted numerically to show the effect of fractional-order $\vartheta$ and to visualize the role of input parameter on the dynamics of breast cancer.