Fractional calculus in mathematical oncology

Abstract

Even though, nowadays, cancer is one of the leading causes of death, too little is known about the behavior of this disease due to its unpredictability from one patient to another. Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, improvements are still necessary on these models. The primary objective of the present research is to prove the efficiency of fractional order calculus in mathematical oncology, more specifically in tumor growth modeling. For this, a generalization of the four most used differential equation models in tumor volume measurements fitting is realized, using the corresponding fractional order equivalent. Are established the fractional order Exponential, Logistic, Gompertz, General Bertalanffy-Pütter and Classical Bertalanffy-Pütter models for a treated and untreated dataset. The obtained results are compared by Mean Squared Error (MSE) with the integer order correspondent of each model. The results prove the superiority of the fractional order models. The MSE of fractional order models are reduced at least at half in comparison with the MSE of the integer order equivalent. It is demonstrated in this way that fractional order deterministic models can offer a good starting point in finding a proper mathematical model for tumor evolution prediction. Fractional calculus is a suitable method in this case due to its memory property, aspect that particularly characterizes biological processes.