# Optimal Control Analysis in the Treatment of Solid Tumors using Combined Therapy

## DOI:

https://doi.org/10.5614/cbms.2024.7.1.7## Keywords:

Mathematical modeling, immune cells, cytokines, chemotherapy, optimal control## Abstract

In this paper, we develop and analyze a mathematical model that describes the dynamic interactions and competitions among tumor cells, normal cells, immune cells, and transforming growth factor-beta within the tumor microenvironment. We conducted qualitative analyses to examine the persistence or extinction of each cell population and analyzed the regions of stability and instability across various equilibria. Additionally, we formulated and solved an optimal control problem using the Pontryagin?s maximum principle, aiming to minimize tumor size and the concentration of transforming growth factor-beta while also reducing chemotherapy and siRNA drug-induced toxicity in patients. Numerical simulations are performed for the model with and without treatment. We demonstrate scenarios where neither individual treatment is capable of reducing both tumor and TGF-?, but their combination achieves a substantial reduction.

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*7*(1), 124-147. https://doi.org/10.5614/cbms.2024.7.1.7

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