This study aims to develop a mathematical model to identify key factors influencing the anti-tumor response. We have been proposed a nonlinear fractional-order tumor dynamics model (LPIHE) using a novel piecewise approach. This model incorporates the effects of estrogen, providing a comprehensive understanding of tumor behavior and offering insights into using estrogen to control tumor growth. To validate the model, we establish the existence and uniqueness of solutions for the piecewise derivative system under Arzelà-Ascoli and Schauder conditions. To assess biological feasibility, we have been calculated the reproductive number \(R_0\) and conduct a sensitivity analysis. Key parameters \(\lambda _2, \alpha _2, \beta _3, \gamma _1\) are systematically varied to analyze their impact on \(R_0\) , providing insights into the model’s robustness and vulnerability. Newton’s polynomial approach is used to obtain numerical solutions with real data across various fractional orders. This model have been investigated the effects of classical and modified fractional calculus operators, with a particular focus on the classical Caputo piecewise operator. The interval \([0, m_2]\) , where \(m_2 \in \mathbb {R}\) , is divided into two subintervals: \([0, m_1]\) and \([m_1, m_2]\) . The classical derivative is applied within \([0, m_1]\) , while the modified operator is used in \([m_1, m_2]\) . Results indicate that higher estrogen levels reduce tumor growth rates, underscoring the importance of fractional operators in modeling tumor dynamics.
