Nisar, K. S., Farman, M., Zehra, A., & Hincal, E. (2026). Numerical and analytical study of fractional order tumor model through modeling with treatment of chemotherapy. 23International Journal of Modelling and Simulation, 46(1), 2–245.
In a recent article published in the International Journal of Modelling and Simulation, Near East University (NEU) researcher Prof. Dr. Evren Hıncal, together with Kottakkaran Sooppy Nisar (Prince Sattam bin Abdulaziz University, Saudi Arabia), Muhammad Farman (Near East University), and Anum Zehra (The Women University Multan, Pakistan), tackled a question that sits at the heart of modern oncology: how can mathematics help doctors design chemotherapy schedules that work better for individual patients?
The motivation is sobering. Cancer is the world’s second-biggest cause of death, responsible for around 10 million deaths in 2020, and that number is expected to reach 16 million by 2040. Chemotherapy remains one of the most important weapons against tumors, but choosing the right drug, the right dose, and the right timing is extraordinarily complex. The body’s response depends not only on what is happening right now, but also on everything that has happened before — previous doses, the immune system’s recent activity, how tumor cells have already adapted. Standard mathematical models, which only “see” the current moment, often miss this longer story.
To capture that longer story, the team built a mathematical model with four interacting populations: stem cells (which can regenerate healthy tissue), effector cells (the immune system’s tumor-fighting cells), tumor cells (the cancer itself), and the concentration of the chemotherapy drug. What sets the model apart is its use of fractional calculus — a branch of mathematics designed to describe systems that have memory. In simple terms, while ordinary equations only react to what is happening at the present instant, fractional equations “remember” the system’s past states and allow that history to shape its future. This is much closer to how a real patient’s body actually behaves under repeated treatment cycles.
Using a powerful mathematical tool called the Sumudu transform together with the Caputo fractal-fractional derivative, the authors showed that their model is mathematically well-behaved: it has a unique solution, it is stable in the rigorous sense required by Hilbert and Banach space theory, and it satisfies a strong stability property known as Ulam-Hyers-Rassias stability. In practical terms, this means the predictions the model makes are reliable and do not blow up or contradict themselves under small changes in the inputs — an essential property if such a model is ever to inform clinical decisions.
Numerical simulations, carried out in MATLAB at different “memory strengths” (in mathematical language, different fractional orders and fractal dimensions), produced clear and intuitive patterns. As the model is given more memory of past states, the populations of stem cells, immune cells, and tumor cells take longer to settle into a steady balance, and chemotherapy drugs appear to linger and act over a longer period. In other words, the fractional model captures something that classical models tend to miss: the slow, sustained, and sometimes unpredictable way that the body and the drug interact over time. The simulations also reveal complex, chaotic-looking patterns within a bounded, biologically realistic region — a hallmark of real-world tumor dynamics.
The clinical promise of this approach lies in personalization. By tuning the fractional order to match a particular patient’s biology, the model could in principle help oncologists choose dosing schedules that maximize tumor reduction while minimizing damage to healthy stem cells and the immune system. The authors see this as a step toward what is increasingly called precision oncology: treatments designed not only for the disease, but for the specific person living with it.
For collaboration and inquiries, Prof. Dr. Evren Hıncal can be contacted through the Department of Mathematics at Near East University, Nicosia.
About the researcher
Evren Hıncal was born in Nicosia in 1973. He completed his undergraduate studies in the Department of Mathematics at Hacettepe University and earned his master’s degree with distinction from the University of London, Imperial College. His Ph.D. was a collaborative program between the Department of Biology at Imperial College and the Department of Mathematics at Eastern Mediterranean University.
Since 2007, he has served as a full-time faculty member in the Department of Mathematics at Near East University, where he was appointed Professor in 2018. He currently holds multiple leadership roles, including Vice Dean of the Faculty of Arts and Sciences, Head of the Department of Mathematics, and President of the Mathematics Research Center, which he founded in 2018. In May 2023, Prof. Hıncal was awarded the “Dr. Suat İrfan Günsel Golden Medal of Honor” for his significant contributions to science.
His research focuses on mathematical modeling in areas such as cancer statistics, epidemiology, and infectious diseases, with numerous publications in respected international journals. He also serves as Advisor to the Board of Trustees of the Near East Foundation.
Disciplines: Mathematical Modeling, Cancer Statistics, Epidemiology, Infectious Diseases.
Abstract
Cancer is the world’s second-biggest cause of death, accounting for roughly 10 million deaths in 2020 and estimated to reach 16 million by 2040. In this study, we propose a novel technique for the treatment of tumor models with a power-law kernel with the Sumudu transform. The analysis was made for a generalized form of analytical solution that is unique and Picard K-stable by using Hilbert and Banach space results. The model investigates the Ulam-Hyers-Rassias stability, uniqueness of solutions, and impact of fractional derivatives with the power-law kernel. Reproductive number analysis with an equilibrium point shows the bounded solution in the feasible region. In the end, numerical simulations are drawn through figures at different fractional and fractal dimensions for the dynamics of treatment and the growth of normal cells. The analysis shows crucial criteria for system stability, ensuring the efficacy of therapeutic interventions through novel mathematical and biological insights.
