The Pennes bioheat equation with Caputo fractional derivative applied to the thermal treatment of ductal breast cancer

Abstract

This article examines the Pennes bioheat equation in both its classical form and its extension using the Caputo fractional derivative to model tumor heating through magnetic hyperthermia with SPIONs. In the classical model (α = 1.0), simulations reach and maintain temperatures above 42 °C, consistent with the clinical and experimental results of Caizer et al., where nanoparticles raise and stabilize tissue within the therapeutic range. When incorporating the fractional derivative (α < 1.0), thermal memory effects emerge, allowing a more realistic description of tissue dynamics. Although the explicit L1 method exhibits numerical instability, the implicit L1 method provides stable and physically coherent solutions, showing slower and more localized heating for fractional orders, as expected in tissues with delayed diffusion. These fractional results computationally correspond to the three-dimensional simulations of Rahpeima & Lin, which report non-monotonic temperature patterns and diffusion dependent on SPION concentration. Overall, the implicit L1 method validates both the experimental behavior observed by Caizer and the numerical dynamics reported by Rahpeima & Lin, demonstrating that the fractional approach is promising for modeling tumor hyperthermia when stable numerical schemes are employed.