One-dimensional computational model of thermal state of the breast with an interstitial tumor

Akulova D.V., Sheremet M.A.
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The paper presents a computational model of the thermal state of the breast with an interstitial tumor. The model is based on the modified Pennes biothermal equation and describes a five-layered biological area including skin, subcutaneous fat, glandular and muscular tissues, as well as a neoplasm zone. Convective heat exchange with the environment is taken into account at the outer boundary, and body temperature is maintained at the internal boundary. In addition, the fabric surface is exposed to exponentially attenuating effects of spatial heating, such a heating scheme is actually based on the Bouguer – Lambert – Baer law. Tissue thermal conductivity and blood perfusion are modeled by linear functions of temperature, reflecting physiological thermoregulation. The boundary-value problem for the partial differential equation has been solved numerically using an explicit-implicit finite difference scheme; the system of algebraic equations getting after an approximation of the mentioned boundary-value problem is solved by the Thomas procedure. Numerical experiments have shown that even a small tumor increases the local temperature of tissues by half a degree due to increased metabolism and delayed blood perfusion. This anomaly is clearly manifested in tumors larger than ten millimeters. It was found that the depth of occurrence critically affects the thermal response: when the tumor is located closer to the surface, the maximum temperature shifts to the skin, whereas at a deeper position, a thermal peak forms inside the glandular tissue. The effectiveness of hyperthermic exposure was assessed by the integral criterion of thermal necrosis based on the Arrhenius law. At a radiation intensity that creates a surface thermal load of about five kilowatts per square meter and an attenuation factor of one hundred, tumor destruction begins after two to three minutes of exposure, while the surrounding healthy tissues remain within safe temperatures. Reducing the attenuation coefficient leads to the opposite effect: heat spreads deeper, and the glandular tissue is damaged first, which limits the therapeutic window. Additionally, maps of the distribution of temperature, time to necrosis, and the depth of thermal damage were constructed depending on the irradiation power, diameter, and position of the tumor.

Keywords: Pennes model, multilayer biological system, hyperthermia process, mathematical modeling
Citation in English: Akulova D.V., Sheremet M.A. One-dimensional computational model of thermal state of the breast with an interstitial tumor // Computer Research and Modeling, 2026, vol. 18, no. 1, pp. 169-184
Citation in English: Akulova D.V., Sheremet M.A. One-dimensional computational model of thermal state of the breast with an interstitial tumor // Computer Research and Modeling, 2026, vol. 18, no. 1, pp. 169-184
DOI: 10.20537/2076-7633-2026-18-1-169-184
URL: http://crm-en.ics.org.ru/journal/article/3694/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Akulova D.V., Sheremet M.A.

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