A Physical and Mathematical Study of Heat Distribution Induced by Radiotherapy in Biological Tissues Using the Bioheat Equation and the Implicit Euler Method

Abstract

This study aims to analyze the spatial and temporal distribution of temperature within biological tissues during radiotherapy by numerically solving the Bioheat Equation using the implicit Euler method. The results demonstrate that this approach provides high numerical stability and effectively suppresses non-physical oscillations, even with relatively large time steps, compared with explicit numerical methods. The model also highlights the significant influence of physiological tissue properties, particularly blood perfusion rate and thermal conductivity, on thermal behavior. Increased blood perfusion enhances heat dissipation and limits temperature elevation, whereas higher absorbed radiation energy leads to a noticeable increase in tissue temperature. The results show a gradual transition toward a quasi-steady thermal state, confirming the accuracy and reliability of the proposed method. The study concludes that the model is an effective tool for thermal dose planning in radiotherapy and for defining safe thermal limits, with the potential to be extended to three-dimensional modeling and to integrate patient-specific data from CT or MRI to support advanced clinical applications in thermal radiotherapy.