Mathematical modelling of breast cancer stages with drug resistance and cardiotoxicity progression: A modified Mittag-Leffler kernel approach

Abstract

Breast cancer $(\bc)$ remains a leading cause of mortality among women globally, with treatment strategies including surgery, chemotherapy, and immune modulation. While chemotherapy is effective in reducing tumor burden, it often leads to cardiotoxic side effects, complicating patient outcomes. Current models inadequately capture the multifaceted nature of breast cancer progression, treatment resistance, immune interactions, and treatment-related toxicities. In this work, we develop an advanced mathematical model for breast cancer patient dynamics, incorporating ten compartments to represent disease progression, drug resistance, cardiotoxicity, and immune response. The population is segmented into sub-populations for each cancer stage, including stage 1 $(\A)$, stage 2 $(\B)$, stage 3 without $(\C_1)$ and with $(\C_2)$ drug resistance, stage 4 without $(\D_1)$ and with $(\D_2)$ drug resistance, a disease-free compartment $(\F)$, and cardiotoxic stages $(\E_1, \E_2, \E_3)$ representing mild, moderate, and severe cardiotoxicity.  Stability of equilibrium points is examined using the Routh-Hurwitz criteria, and numerical simulations provide insight into the population distribution across compartments under varying initial conditions. A qualitative analysis of the model's solutions was conducted using fixed-point theory with $\m$ fractional operator. Numerical simulations, implemented with a predictor-corrector method, corroborated and enhanced our understanding of the $\bc$ dynamics.