ANALYSIS OF A FRACTIONAL ORDER MODEL FOR THE INTERACTION BETWEEN GROWTH AND CROWDING OF TUMOR STEM CELLS VIA CAPUTO FRACTIONAL DERIVATIVES

The dynamics of cancer, such as tumor growth and the existence of cancer stem cells, pose a persistent challenge to contemporary medicine. With consideration of cancer stem cells, we present a fractional-order model to explain an intricate relationship between cellular crowding and tumor growth and offer more accurate depictions of actual biological processes. Also, by using the fixed-point theorem, we examine the possibility of finding solutions to our model and their uniqueness. Further, we use the Laplace residual power series approach to obtain certain analytical solutions and arrive at explicit solutions that shed light on the dynamics of the given system and the development of tumors. Furthermore, we show the influence of fractional derivative dynamics and cancer stem cells on tumor growth, as the fractional derivative enriches our knowledge of the complex underlying mechanisms of tumor progression that, in turn, have consequences for the determination of targeted therapies. In addition, the main contribution of this work lies in expanding the literature on the mathematical modeling of tumors, as it emphasizes the critical importance of the impact of fractional derivative dynamics on the complex interactions in tumor microenvironments.