Interaction and Development of Breast Cancer Cells with Immune Response Using First Order ODE

Breast cancer arises when cells develop uncontrollably in the breast to form tumour cells. The risk of having breast cancer rises as a woman continues to age. Thus, details about the early stages of cancer progression can help a woman make early treatment choices, preventing them from being diagnosed with these harmful cancers. It is believed that cytotoxic T lymphocytes (CTLs) act as effector cells to eradicate cancer cells. CTLs and tumor cells were discovered to be around a "predator-prey" relationship, with CTL acting as the predator along with tumor cells acting as the prey. In this paper, we examined steady-state solutions for two numerical differentiation using the Jacobian matrix. We will also examine the stability region of breast cancer cells in two different phases to describe its progress in different types of the human body at various phases. Also, compare tumour cells population development in the duration of interphase and metaphase in the presence and the absence of immune response, which is dependent on the CTL population, will be observed by applying Fourth Order Runge Kutta (RK4) method. We can see that the Runge-Kutta method is an important method for approximate solutions to ordinary systems with known initial conditions. We achieved populations value of tumour cells throughout interphase and mitosis as well as the population of the immune system using the method. Based on our observations, we draw the conclusion that persons with a higher immune system are more likely to be able to fight cancer for for a period of time than cancer sufferers with a poor immune system. We may observe that the cancer cells were reduced in a shorter period of time with a high immune response population compared to other lower CTL values