Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

中文翻译：

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一维 Pennes 生物热方程的解析解

摘要 Pennes 生物热方程是研究暴露于射频能量的生物系统中传热最广泛使用的热模型。在他们的文章“表面冷却和血流对组织微波加热的影响”中，Foster 等人。发表了使用傅立叶变换获得的一维 (1-D) 问题的解析解。然而，他们的文章没有提供推导的任何细节。在这项工作中，我们重新审视了 1-D 问题，并提供了解析解的综合数学推导。我们的结果纠正了 Foster 解决方案中的一个错误，这可能是他们文章中的一个错字。与 Foster 等人不同，我们直接对偏微分方程进行积分。对于某些参数值，解的表达式有几个明显的奇异性，其中物理问题不是奇异的。我们证明所有这些奇点都是可移除的，并且我们推导出替代的非奇异公式。最后，我们扩展了我们的分析，以写出多电磁加热脉冲情况下的一维生物热方程的解析解。