Mathematical modeling of cancer treatments with fractional derivatives: an overview

Abstract

This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment has become a significant research area that has attracted many mathematical models developed by mathematicians to represent cancer treatment processes such as hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives. The concept of fractional derivatives, which is still new to many mathematicians, is a generalized definition of a derivative whose order is a real number and has proved to be more effective and robust in modeling cancer treatments. Therefore, it is imperative to review fractional cancer treatment models to elucidate their significance and also predict future directions. The review was carried out by first presenting 22 various definitions of fractional derivatives. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the utilization of fractional derivative models was analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivatives for cancer treatment modeling. Moreover, the fractional derivative is a general definition for all derivatives. Furthermore, the review showed that the Caputo and its non-singular kernel versions are the most used in fractional derivative models. The current review concluded that the future direction of cancer treatment modeling lies in the adoption and effective use of fractional derivative models corroborated with accurate experimental or clinical data.