Fractional calculus in mathematical oncology

Area of Science:

• Mathematical Oncology
• Applied Mathematics
• Biophysics

Background:

• Cancer remains a leading cause of death, with unpredictable patient-specific behavior.
• Classical mathematical models are crucial for cancer treatment but require further refinement.
• Understanding tumor growth dynamics is essential for effective cancer management.

Purpose of the Study:

• To demonstrate the efficacy of fractional order calculus in mathematical oncology.
• To enhance tumor growth modeling using fractional calculus.
• To improve the prediction of tumor evolution.

Main Methods:

• Generalizing four established differential equation models (Exponential, Logistic, Gompertz, Bertalanffy-Pütter) into their fractional order equivalents.
• Applying these fractional models to both treated and untreated tumor datasets.
• Comparing the performance of fractional models against their integer-order counterparts using Mean Squared Error (MSE).

Main Results:

• Fractional order models exhibited superior performance in fitting tumor volume data.
• The Mean Squared Error for fractional models was reduced by at least 50% compared to integer order models.
• This indicates a significant improvement in predictive accuracy.

Conclusions:

• Fractional order deterministic models provide a robust framework for predicting tumor evolution.
• The 'memory property' inherent in fractional calculus makes it well-suited for modeling biological processes like tumor growth.
• Fractional calculus represents a promising advancement in mathematical oncology for personalized cancer treatment strategies.