Abstract
In this study, we investigate a new fractional-order mathematical model which considers population dynamics among tumor cells-macrophage cells-active macrophage cells, and host cells involving the Caputo fractional derivative. Firstly, the stability of the positive steady state of the model is studied. Subsequently, the conditions for existence and uniqueness of the solutions are examined. Then, the least squares curve fitting method (LSCFM) which is one of the prominent methods for parameter estimation is used to fit the parameters of the model. It is aimed to fit the relevant parameters with the help of the tumor tissue samples which were collected from the patient with non-small cell lung cancer who had chemotherapy-naive hospitalized at Kayseri Erciyes University hospital in Turkey. A total of 12 parameters in the model are estimated using the data of lung tumor cells of this patient for 14 days. Moreover, the numerical simulations are given by considering the different fractional orders and different parameters for the model. So, it is achieved how the change in alpha affects the dynamic behavior of the system. In the sequel, to point out the advantages of the fractional-order modeling, the memory trace and hereditary traits are taken into consideration. Finally, the interpretations in terms of biological science are provided in conclusion. We believe that this interdisciplinary study will open new doors for other similar studies and will shed light on the studies to be developed on the use of real data in the mathematical modeling of cancer.
Highlights
Cancer is the deadliest and complicated disease of our time
While SCLCs are malignant tumors that can be identified by neuroendocrine features, accounting for nearly 15 percent of lung cancers [9], non-small cell lung cancer (NSCLC) account for approximately 85 percent of all lung cancers [8,10]
A new fractional-order differential equation model for tumor–immune system interaction related to lung cancer has been studied by taking into account the models given by [20], and [59]
Summary
Cancer is the deadliest and complicated disease of our time. This illness is caused by the uncontrolled growth of abnormal or mutated cells in the body, and cancer cells are known for their capability to grow rapidly, divide and proliferate uncontrollably [1,2]. While SCLCs are malignant tumors that can be identified by neuroendocrine features, accounting for nearly 15 percent of lung cancers [9], NSCLCs account for approximately 85 percent of all lung cancers [8,10] This discrimination reflects the different clinical recognition, disease course, and therapeutic alternatives of the two subgroups [11]. A study on a mathematical model of immune response with cell mediated for tumor phenotype heterogeneity has been presented in [50]. In [55], the authors proposed two modeling approaches to predict lung tumor dynamics as an effect of radiotherapy. They used the real clinical information of non-small cell lung cancer (NSCLC) patients undergoing stereotactic body radiation therapy (SBRT) as the primary treatment method for numerical simulations.
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